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1. Liquidity Provision, Ambiguous Asset Returns and the Financial Crisis by Willem Spanjers (Kingston University and Rimini Centre for Economic Analysis) 15 th May 2013 University of Bologna at Rimini Rimini, Italy. Liquidity Provision, Ambiguous Asset Returns and the Crisis. 2. Content - PowerPoint PPT Presentation
Citation preview
Liquidity Provision Ambiguous Asset Returns and the Financial Crisis
by
Willem Spanjers
(Kingston University andRimini Centre for Economic Analysis)
15th May 2013
University of Bologna at Rimini
Rimini Italy
1
Content1 Introduction
2 Ambiguity
3 Updating
4 The Basic Model
5 Second Best Efficiency
6 Financial Sector
7 Regulation
8 The Impact of Ambiguity
9 Conclusions
Liquidity Provision Ambiguous Asset Returns and the Crisis 2
1 IntroductionBasic questionsbull Could the financial crisis be caused by a failure
to recognized the presence of incalculable riskbull Should financial regulation be tightened
Answersbull Updating ambiguous beliefs may lead to an
endogenous loss of confidence causing a crisisbull Crises should be dealt with if and when they
arisebull This policy should be public knowledge
preventing excessive caution by investors
Liquidity Provision Ambiguous Asset Returns and the Crisis 3
Related literaturebull Liquidity provision as in
- Jacklin and Bhattacharya (1988) and - Allen and Gale (JoF 1998)
bull Intuition of ambiguity as in - Knight (1921) and Keynes (1937)
bull Modelling of ambiguity- in the tradition of Schmeidler (19821989) - E(llsberg)-capacities as in Eichberger and Kelsey (1999)
Liquidity Provision Ambiguous Asset Returns and the Crisis 4
Keynes (1937) gives a description of what is meant by ambiguity
ldquoBy lsquouncertainrsquo knowledge let me explain I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject in this sense to uncertainty [] The sense in which I am using the term is that [] there is no scientific basis on which to form any calculable probability whatever We simply do not knowrdquo [pp 113-114]
5Liquidity Provision Ambiguous Asset Returns and the Crisis
To Keynes these implications are not without consequences for financial economics
ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]
6Liquidity Provision Ambiguous Asset Returns and the Crisis
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Content1 Introduction
2 Ambiguity
3 Updating
4 The Basic Model
5 Second Best Efficiency
6 Financial Sector
7 Regulation
8 The Impact of Ambiguity
9 Conclusions
Liquidity Provision Ambiguous Asset Returns and the Crisis 2
1 IntroductionBasic questionsbull Could the financial crisis be caused by a failure
to recognized the presence of incalculable riskbull Should financial regulation be tightened
Answersbull Updating ambiguous beliefs may lead to an
endogenous loss of confidence causing a crisisbull Crises should be dealt with if and when they
arisebull This policy should be public knowledge
preventing excessive caution by investors
Liquidity Provision Ambiguous Asset Returns and the Crisis 3
Related literaturebull Liquidity provision as in
- Jacklin and Bhattacharya (1988) and - Allen and Gale (JoF 1998)
bull Intuition of ambiguity as in - Knight (1921) and Keynes (1937)
bull Modelling of ambiguity- in the tradition of Schmeidler (19821989) - E(llsberg)-capacities as in Eichberger and Kelsey (1999)
Liquidity Provision Ambiguous Asset Returns and the Crisis 4
Keynes (1937) gives a description of what is meant by ambiguity
ldquoBy lsquouncertainrsquo knowledge let me explain I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject in this sense to uncertainty [] The sense in which I am using the term is that [] there is no scientific basis on which to form any calculable probability whatever We simply do not knowrdquo [pp 113-114]
5Liquidity Provision Ambiguous Asset Returns and the Crisis
To Keynes these implications are not without consequences for financial economics
ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]
6Liquidity Provision Ambiguous Asset Returns and the Crisis
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
1 IntroductionBasic questionsbull Could the financial crisis be caused by a failure
to recognized the presence of incalculable riskbull Should financial regulation be tightened
Answersbull Updating ambiguous beliefs may lead to an
endogenous loss of confidence causing a crisisbull Crises should be dealt with if and when they
arisebull This policy should be public knowledge
preventing excessive caution by investors
Liquidity Provision Ambiguous Asset Returns and the Crisis 3
Related literaturebull Liquidity provision as in
- Jacklin and Bhattacharya (1988) and - Allen and Gale (JoF 1998)
bull Intuition of ambiguity as in - Knight (1921) and Keynes (1937)
bull Modelling of ambiguity- in the tradition of Schmeidler (19821989) - E(llsberg)-capacities as in Eichberger and Kelsey (1999)
Liquidity Provision Ambiguous Asset Returns and the Crisis 4
Keynes (1937) gives a description of what is meant by ambiguity
ldquoBy lsquouncertainrsquo knowledge let me explain I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject in this sense to uncertainty [] The sense in which I am using the term is that [] there is no scientific basis on which to form any calculable probability whatever We simply do not knowrdquo [pp 113-114]
5Liquidity Provision Ambiguous Asset Returns and the Crisis
To Keynes these implications are not without consequences for financial economics
ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]
6Liquidity Provision Ambiguous Asset Returns and the Crisis
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Related literaturebull Liquidity provision as in
- Jacklin and Bhattacharya (1988) and - Allen and Gale (JoF 1998)
bull Intuition of ambiguity as in - Knight (1921) and Keynes (1937)
bull Modelling of ambiguity- in the tradition of Schmeidler (19821989) - E(llsberg)-capacities as in Eichberger and Kelsey (1999)
Liquidity Provision Ambiguous Asset Returns and the Crisis 4
Keynes (1937) gives a description of what is meant by ambiguity
ldquoBy lsquouncertainrsquo knowledge let me explain I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject in this sense to uncertainty [] The sense in which I am using the term is that [] there is no scientific basis on which to form any calculable probability whatever We simply do not knowrdquo [pp 113-114]
5Liquidity Provision Ambiguous Asset Returns and the Crisis
To Keynes these implications are not without consequences for financial economics
ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]
6Liquidity Provision Ambiguous Asset Returns and the Crisis
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Keynes (1937) gives a description of what is meant by ambiguity
ldquoBy lsquouncertainrsquo knowledge let me explain I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject in this sense to uncertainty [] The sense in which I am using the term is that [] there is no scientific basis on which to form any calculable probability whatever We simply do not knowrdquo [pp 113-114]
5Liquidity Provision Ambiguous Asset Returns and the Crisis
To Keynes these implications are not without consequences for financial economics
ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]
6Liquidity Provision Ambiguous Asset Returns and the Crisis
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
To Keynes these implications are not without consequences for financial economics
ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]
6Liquidity Provision Ambiguous Asset Returns and the Crisis
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
2 Ambiguity
We consider
bull Calculable vs incalculable risk
bull Sure Thing Principle
bull (Subjective) Expected Utility
bull Choquet Expected Utility
Liquidity Provision Ambiguous Asset Returns and the Crisis 7
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Calculable vs Incalculable Risk
bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity
bull Risk may fail to be calculable because - one cannot make a reasonable probability
estimate for the relevant states of natureandor - one does not know the outcome that is
obtained for the specific states of nature
Liquidity Provision Ambiguous Asset Returns and the Crisis 8
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)
Examples for situations with ambiguity arebull after the terrorist attacks of 911
(prob known outcomes unknown pessimism)bull BSE crisis
(prob unknown outcomes known pessimism)bull Dotcom bubble
(prob unknown outcomes known optimism)
Liquidity Provision Ambiguous Asset Returns and the Crisis 9
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Sure Thing Principlebull For some consumers the trade-off between the
amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have
bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible
bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state
Liquidity Provision Ambiguous Asset Returns and the Crisis 10
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with
U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)
where ndash (π
1 π
S) is a probability distribution
ndash u ℝ rarr ℝ is a von Neumann-Morgenstern
utility index
Liquidity Provision Ambiguous Asset Returns and the Crisis 11
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Choquet Expected Utilitybull Choquet Expected Utility extends the expected
utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty
estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the
ambiguity attitudebull An outcome (x(s))
sϵS is evaluated as
γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))
Liquidity Provision Ambiguous Asset Returns and the Crisis 12
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
E(llsberg) Capacitiesbull For some situations with ambiguity the
ambiguity is present in some parts of the state space S but not in all
bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns
bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures
Liquidity Provision Ambiguous Asset Returns and the Crisis 13
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]
- an additive partition E1E
n of the state
space S in additive components bull Let I(AE) be the indicator function with
I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise
bull The capacity v is now defined by v(E) = Σj=1
m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]
Liquidity Provision Ambiguous Asset Returns and the Crisis 14
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
bull For state contingent payouts x define m(Ej) as
the minimum of u(x(s)) over all s ϵ E
j
bull The Choquet Expected Utility for a pessimistic consumer now is obtained as
U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1
m π(Ej) m(Ej)
Liquidity Provision Ambiguous Asset Returns and the Crisis 15
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
3 Updating of Ambiguous Beliefs
For this we considerbull The multiple prior representation
bull Bayesian updating
bull Updating capacities
Liquidity Provision Ambiguous Asset Returns and the Crisis 16
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions
AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises
Liquidity Provision Ambiguous Asset Returns and the Crisis 17
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
The Multiple Prior Representationbull In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over the CEU approach
bull Advantages of Maxmin Expected Utility approach compared to CEU are
- it is more intuitive
- it allows for more general beliefs
bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way
Liquidity Provision Ambiguous Asset Returns and the Crisis 18
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
bullpmin
Π
Liquidity Provision Ambiguous Asset Returns and the Crisis 19
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set
C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)
v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if
the capacity is convex
Liquidity Provision Ambiguous Asset Returns and the Crisis 20
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
Core of v
p3 ge v3
p1 ge v1
p2 ge v2
p3 le 1 - v12p1 le 1 - v23
p2 le 1 - v13
Liquidity Provision Ambiguous Asset Returns and the Crisis 21
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
p3 ge γπ3
p1 ge γπ1 p2 ge γπ2
The core ofa simplecapacity
p3 le 1 - γπ12
p1 le 1 - γπ23
p2 le 1 - γπ13
Liquidity Provision Ambiguous Asset Returns and the Crisis 22
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
p3 le 1 - v12
p3 ge v3
p2 = 1 - v13
p1 le 1 - v23
p1 ge v1
Core of anE-capacity
The core ofan E-Capacity with E1=13and E2=2
Liquidity Provision Ambiguous Asset Returns and the Crisis 23
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Bayesian Updatingbull We consider updating of the probabilities of the
states of nature over a state space
bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA
bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as
PB|A = PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 24
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
bull
bull
Bayesian Updatingafter receiving the Information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 25
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Updating CapacitiesWhen updating capacities one would want todistinguish between
bull updating a capacity that describes the ambiguity experienced by a decision maker and
bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude
Liquidity Provision Ambiguous Asset Returns and the Crisis 26
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
The Dempster-Shafer Rulebull The main updating rule for capacities suggests to
restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur
bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule
bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by
Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers
Liquidity Provision Ambiguous Asset Returns and the Crisis 27
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as
vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]
bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA
= PBcapA PA
Liquidity Provision Ambiguous Asset Returns and the Crisis 28
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
Dempster-Shafer Updating of a convex capacity forthe information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 29
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
bull
bullbull
Dempster-Shafer Updating of an E-Capacity for theinformation 12
bull
Liquidity Provision Ambiguous Asset Returns and the Crisis 30
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
(001)
(100)
(010)
bull
bull
Dempster-Shafer Updating of an E-Capacity for the information 23
Liquidity Provision Ambiguous Asset Returns and the Crisis 31
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set
bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour
bull The updating of the E-capacity for the information 23 leads to a single point
bull Here updating made the ambiguity disappear
Liquidity Provision Ambiguous Asset Returns and the Crisis 32
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- ldquounknown outcomesrdquo and of
- ldquounknown probabilitiesrdquo
bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making
bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity
Liquidity Provision Ambiguous Asset Returns and the Crisis 33
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities
- zero interest money holdings
- illiquid assets which pay out
α1 lt 1 when liquidated prematurely
αh gt 1 when matured and successful
αℓ = 0 when matured and failure
34
Liquidity Provision Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Information regarding assets
Signal σ Return ρ Probability π σρ
b h δ b ℓ ε
g h 1 - (δ + ε)
g ℓ 0
Liquidity Provision Ambiguous Asset Returns and the Crisis 35
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
TimingPeriod 0 - investment decisions
Period 1 - individual liquidity preference t isinHL becomes privately known
- signal σ isinbg becomes publicly known
- interaction
- possibility for liquidation of assets - consumption
Period 2 - return ρ isinhℓ occurs
- consumption
36
Liquidity Provision Ambiguous Asset Returns and the Crisis
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Beliefsbull ex-ante (calculable) risk with respect to
- individual liquidity preference t isin HL
bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)
bull beliefs over HLtimesbgtimeshℓ consist of- an additive probability distribution P- an additive partition of the state space with components
FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)
FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P
Liquidity Provision Ambiguous Asset Returns and the Crisis 37
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Preferencesbull risk-neutral with vNM-utility index
u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)
where βH gt βL gt 1
bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times
πH times minimum utility over FH
plus πL times minimum utility over FL
Liquidity Provision Ambiguous Asset Returns and the Crisis 38
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence
- after a good signal σ = g the level of confidence becomes
γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ
- after a bad signal σ = b the level of confidence becomes
γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]
= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ
Liquidity Provision Ambiguous Asset Returns and the Crisis 39
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after
a bad signal
bull consequence little welfare enhancing ad interim redistribution to H-types
bull the money holdings are
μb(γ) = πH γb πhb αh (πL βL + πH γb πh
b αh)
Liquidity Provision Ambiguous Asset Returns and the Crisis 40
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
High reservesbull such that incentive constraint of L-types only holds after a
good signalbull consequence high amount of welfare enhancing ad interim
redistribution to H-typesbull the money holdings are
μg(γ) = πH γg πhg αh (πL βL + πH γg πh
g αh)
= πH γg middot αh (πL βL + πH γg αh)
Assumption on parametersbull the high reserves are second best efficient
Liquidity Provision Ambiguous Asset Returns and the Crisis 41
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
ICLb
ICLb γ = 1
γ lt 1 y2
y1 μE(γ) πH
α2(1-μE(γ)) πL
Second BestEfficiency
bull
VEff
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg
ICLg
VEff
μE(1) πH
α2(1-μE(1)) πL
42
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
5 The Financial Sectorbull Modelled as an unregulated competitive banking
sector offering deposit contracts bull Period 0 - deposit contracts are offered
- investment decisions Period 1 - withdrawal decisions
- possible liquidation of assets
bull A deposit contract specifies- promised repayments in Periods 1 and 2
- fraction of the deposits held as money reserves
- withdrawals in Period 1 have priority over withdrawals in Period 2
Liquidity Provision Ambiguous Asset Returns and the Crisis 43
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
bull Only ldquofundamentalrdquo bank runs are taken into account
bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are
liquidated after a bad signal
bull Low reserves - no bank run after either signal
- not second best efficient because reserves are too
low by assumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 44
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
ICLb(γ)
VEff VBank
y2
y1 μB(γ) πH
bull
bull
UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)
Liquidity Provision Ambiguous Asset Returns and the Crisis
ICLg(γ)
μE(γ) πH
α2(1-μB(γ)) πL
α2(1-μE(1)) πL
45
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
6 Regulationbull In the case of a bank run the regulator should
interfere by- making immediate withdrawal less attractive and- making waiting more attractive
bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting
bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious
Liquidity Provision Ambiguous Asset Returns and the Crisis 46
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
7 The Impact of Ambiguitybull If recognized ex-ante
- reduces efficient reserve holdings- no further impact
bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings
Liquidity Provision Ambiguous Asset Returns and the Crisis 47
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation
bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors
bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare
Liquidity Provision Ambiguous Asset Returns and the Crisis 48
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49
In the financial crisis bull on aggregate the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current and future income
bull this problem is similar to that faced by the aggregated unregulated banking sector
bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where
- the provision of liquidity is the counterpart of a tax on immediate consumption and
- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption
Liquidity Provision Ambiguous Asset Returns and the Crisis 49