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Prospect Theory
2000 FAME Research PrizeResearch Paper N° 16
FAME - International Center for Financial Asset Management and Engineering
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A Framework for Collateral Risk Control DeterminationHEC-University of Lausanne and FAMEDaniel AUNON-NERIN
HEC-University of Lausanne, FAME and IMDDidier COSSIN
European Central BankFernando GONZÁLEZ
Zhijiang HUANGHEC-University of Lausanne and FAME
Research Paper N° 61December 2002
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A FRAMEWORK FOR COLLATERAL RISK CONTROL DETERMINATION
Daniel AUNON-NERIN Didier COSSIN
Fernando GONZÁLEZ Zhijiang HUANG
December 2002
A Framework for Collateral Risk Control DeterminationDidier Cossin,
HEC, University of Lausanne, Fame and IMD∗
Zhijiang Huang,Fame and HEC, University of Lausanne∗∗
Daniel Aunon-Nerin,Fame and HEC, University of Lausanne∗∗∗
Fernando GonzálezEuropean Central Bank, Risk Management Division∗∗∗∗
December 2002
Contact address:∗HEC, University of Lausanne, CH-1015 Lausanne, Switzerland
Tel: 41 21 692 34 69Fax: 41 21 692 33 05E-mail: [email protected]
∗∗E-mail: [email protected]∗∗∗E-mail: [email protected]∗∗∗∗E-mail: [email protected]
JEL Classification: E50, E58, G21, G10Keywords: Collateral, Repurchase Transactions, Default Risk, Central Banks,
Monetary Policy OperationsAcknowledgements: We are grateful to Carlos Bernadell, Peng Cheng, Jean
Pierre Danthine, Michel Habib, Rene Stulz, Evangelos Tabakis, Elu Von Thad-den and other participants from the FAME workshop and from the EuropeanCentral Bank research seminar for their helpful comments and suggestions.
Financial support by the National Centre of Competence in Research ”Finan-cial Valuation and Risk Management” is gratefully acknowledged. The NationalCentres of Competence in Research are managed by the Swiss National ScienceFoundation on behalf of the Federal Authorities.
The views expressed in this paper are those of the authors and do not nec-essarily reflect those of the European Central Bank.
Abstract
This paper derives a general framework for collateral risk control determina-tion in repurchase transactions or repos. The objective is to treat consistentlyheterogeneous collateral so that the collateral taker has a similar risk exposurewhatever the collateral pledged. The framework measures the level of risk withthe probability of incurring a loss higher than a pre-specified level given twowell known parameters used to manage the intrinsic risk of collateral: markingto market and haircuts. It allows for the analysis in a self contained closedform of the way in which different relevant factors interact in the risk control ofcollateral (e.g. marking to market frequency, level of volatility of interest rates,time to capture and liquidity risk, probability of default of counterparty, etc.).The framework, which combines the recent theoretical literature on credit andinterest risk, provides an alternative quantifiable and objective approach to theexisting more ad-hoc rule-based methods used in haircut determination.
Executive Summary A very important type of credit transaction in financial markets is the so-called repurchase transaction or repo. In this type of operation, an agent buys or sells eligible assets (i.e. collateral) against cash under repurchase agreements. According to the 2002 ISMA European repo market survey, the estimated total value of the outstanding repo business in Europe is equivalent to EUR 3.3 trillion. The use of this type of credit transactions is not only made by private agents but also by central banks who allocate in this way significant funds to the banking system in their monetary policy operations. For instance, the European Central Bank has, since the start of Stage Three, allocated per year funds worth more than EUR 3 Trillion through this type of open market operations. The specific characteristics of repurchase transactions in a collateral taker context are described precisely for example in ECB (2002) ["The single monetary policy in stage three-general documentation on Eurosystem monetary policy instruments and procedures. April 2002]. The so-called collateral leg of the transaction plays a crucial role in this type of operations. The collateral leg is intended to mitigate the credit risk or default risk of the counterparty borrowing the cash. In case of default of the counterparty, the collateral taker can make good any loss by selling the collateral received. When the collateral considered is default-risk free (but not market-risk free, as with government bonds), collateralization is a way to transform credit risk (or risk of default) into market risk (more studied in the literature and easier to handle) as the default risky counterparty provides a default-risk free asset to guarantee its position. Even when the collateral itself can be defaultable (as in corporate bonds, OTC derivatives or loans), the credit risk is strongly mitigated. The goal of this paper is to establish a general framework that would help the treatment of heterogeneous collateral in a consistent manner in the context of repurchase transactions. Common practices have appeared to manage the intrinsic risk of collateral: marking to market (which helps reduce the level of loss by revaluing more or less frequently the collateral using market prices) and haircuts (which help reduce the risk of loss in case the counterparty defaults). In this paper, we consider both techniques in the establishment of the risk control determination framework with the aim of guaranteeing to the collateral taker a similar risk exposure and similar economic impact whatever the type of collateral pledged. We use the vast theoretical literature on credit risk and on interest rate risk to provide a solution to our problem of defining a consistent framework to collateral risk control policies. Two approaches to modeling credit risk currently dominate financial economics: structural form and reduced-form approaches. The structural form approach extends the Merton (1974) option-based framework and models credit risk as a short position in a put option (and in practice this relies on the microeconomics of the firm's solvency situation). Reduced-form approaches, as settled by Madan and Unal (98), Jarrow et al. (97) and Duffie and Singleton (99) consider an exogenous stochastic default, modeled traditionally by a jump process with stochastic intensity. Default is defined as the first jump of the default process (see Cossin and Pirotte (2000) for an overview of the credit risk literature and Cossin and Hricko (2000) for application to collateral policies of commercial banks). While the finance academic literature has focused on the issue of pricing credit and interest risk quite extensively, little research exist on the impact of collateral on credit risk exposures and collateral haircut determination models. Margrabe (1978) mentions the similarity between an exchange option and a margin account and provides the
pricing for a very simple marking-to-market. Stulz and Johnson (1985) study the impact of collateralization on the pricing of secured debt using contingent claim analysis. More recently, Jokivuolle and Peura (2000) present a model of collateral haircut determination for bank loans. Their model is geared to providing adequate loan-to-value ratios, which is similar to the concept of haircut, using an structural credit risk approach. In a related research, Cossin and Hricko (2000), present a methodology for haircut determination also using a structural form approach but with the final objective of pricing a credit risk instrument backed with collateral. Notwithstanding this research, current industry practice still relies on ad-hoc rule-based methods to establish the level of haircuts (see Cossin and Pirotte 2001 for a typical haircut schedule of a major bank). As Jokivuolle and Peura (2000) put it, "on the whole, we find the discussion on collateral haircuts and loan to value ratios up to date rather vague in many respects". For example, the precise degree of risk mitigation pursued by taking collateral is typically not defined and haircut levels used by market participants have been generally set without a consistent approach. The aim of this paper is to participate in clarifying this existing theoretical gap by providing a general framework that would incorporate the main relevant factors affecting the determination of haircuts in a quantifiable and objective manner using recent general results in the financial economics literature in the area of credit and interest risk modeling. In this paper the framework presented focuses on the level of risk taken by the collateral taker with the aim of making it homogenous across different collateral. Based on the collateral haircuts and predetermined loss level, we derive the probability of incurring a loss higher than allowed. Accordingly, if the collateral taker fixes its risk exposure, which is related to its risk appetite, the required haircut can be extracted. We rely on the reduced form literature as opposed to the approaches followed by Cossin and Hricko (2000) and Jokivuolle and Peura (2000) which are based on structural form methods. In our framework, default is determined exogenously while marking-to-market is also introduced and margin calls are derived endogenously by the evolution of the underlying used as collateral. This feature makes the problem easier to be generalized and to obtain a closed form solution for the case of single collateral when the interest rate is assumed to follow affine type processes. In addition to the marking-to-market policy, other realistic features are taken into account like time to capture and liquidity effects. We also study the case where there exists correlation between interest rate and the probabilities of default of the counterparty. The framework confirms many practical intuitions, such as the direct impact on the haircut level required to cover for riskier collateral and supports the intuition that the frequency of marking to market and collateral are substitutes. In this sense, the framework could be used as an objective basis to understand the relationship between a number of factors and the level of haircuts which is important not only for practitioners and academics but also for regulators.
1 Introduction
A very important type of credit transaction in financial markets is the so-calledrepurchase transaction or repo. In this type of operation, an agent buys orsells eligible assets (i.e. collateral) against cash under repurchase agreements.According to the 2002 ISMA European repo market survey, the estimated totalvalue of the outstanding repo business in Europe is equivalent to EUR 3.3trillion.
The so-called collateral leg of the transaction plays a crucial role in thistype of operations. The collateral leg is intended to mitigate the credit risk ordefault risk of the counterparty borrowing the cash. In case of default of thecounterparty, the collateral taker can make good any loss by selling the collateralreceived. When the collateral considered is default-risk free (but not market-riskfree, as with government bonds), collateralization is a way to transform creditrisk (or risk of default) into market risk (more studied in the literature andeasier to handle) as the default risky counterparty provides a default-risk freeasset to guarantee its position. Even when the collateral itself can be defaultable(as in corporate bonds, OTC derivatives or loans), the credit risk is stronglymitigated.
The goal of this paper is to establish a general framework that would help thetreatment of heterogenous collateral in a consistent manner from a risk manage-ment point of view in the context of repurchase transactions. Common practiceshave appeared to manage the intrinsic risk of collateral: marking to market1
(which helps reduce the level of loss by revaluing more or less frequently thecollateral using market prices) and haircuts2 (which help reduce the risk of lossin case the counterparty defaults). In this paper, we consider both techniques inthe establishment of the risk control determination framework with the aim ofguaranteeing to the collateral taker a similar risk exposure and similar economicimpact whatever the type of collateral pledged.
We use the vast theoretical literature on credit risk and on interest rate riskto provide a solution to our problem of defining a consistent framework to col-lateral risk control policies. Two approaches to modelling credit risk currentlydominate financial economics: structural form and reduced-form approaches.The structural form approach extends the Merton (1974) option-based frame-work and models credit risk as a short position in a put option (and in practicethis relies on the microeconomics of the firm’s solvency situation). Reduced-form approaches, as settled by Madan and Unal (98), Jarrow et al. (97) and
1Marking to market refers to the practice of a periodical monitoring of the value of thecollateral . If the difference between the collateral and the value of the notional amount (valueof the loan) is smaller than a determined trigger level, the counterparty will be required to addsome additional collateral. If the opposite happens, the amount of collateral can be decreased.
2 Increases in collateral required depend on collateral type. For example, equities maybe valued at 60% of their market value for collateral purposes, hence implying a haircut ontheir value of 40%. Haircuts would typically be smallest for short maturity governmentbonds, increase for longer maturity bonds, and be larger for corporate bonds and equities.The relationship that these haircuts should have from one to the other is what this paperinvestigates.
1
Duffie and Singleton (99) consider an exogenous stochastic default, modelledtraditionally by a jump process with stochastic intensity. Default is definedas the first jump of the default process (see Cossin and Pirotte (2000) for anoverview of the credit risk literature and Cossin and Hricko (2000) for applica-tion to collateral policies of commercial banks).
While the finance academic literature has focused on the issue of pricingcredit and interest risk quite extensively, little research exists on the impact ofcollateral on credit risk exposures and collateral haircut determination models.Margrabe (1978) mentions the similarity between an exchange option and amargin account and provides the pricing for a very simple marking-to-market.Stulz and Johnson (1985) study the impact of collateralisation on the pricingof secured debt using contingent claim analysis. More recently, Jokivuolle andPeura (2000) present a model of collateral haircut determination for bank loans.Their model is geared to providing adequate loan-to-value ratios, which is sim-ilar to the concept of haircut, using an structural credit risk approach. In arelated research, Cossin and Hricko (2000), present a methodology for haircutdetermination also using a structural approach but with the final objective ofpricing a credit risk instrument backed with collateral. Notwithstanding thisresearch, current industry practice still relies on ad-hoc rule-based methods toestablish the level of haircuts (see Cossin and Pirotte 2001 for a typical haircutschedule of a major bank). As Jokivuolle and Peura (2000) put it, ”on thewhole, we find the discussion on collateral haircuts and loan to value ratios upto date rather vague in many respects”. For example, the precise degree ofrisk mitigation pursued by taking collateral is typically not defined and haircutlevels used by market participants have been generally set without a consistentapproach. The aim of this paper is to participate in clarifying this existing the-oretical gap by providing a general framework that would incorporate the mainrelevant factors affecting the determination of haircuts in a quantifiable and ob-jective manner using recent general results in the financial economics literaturein the area of credit and interest risk modelling.
In this paper the framework presented focuses on the level of risk taken by thecollateral taker with the aim of making it homogenous across different collateral.Based on the collateral haircuts and predetermined loss level, we derive theprobability of incurring a loss higher than allowed. Accordingly, if the collateraltaker fixes its risk exposure, which is related to its risk appetite, the requiredhaircut can be extracted. We rely on the reduced form literature as opposedto the approaches followed by Cossin and Hricko (2000) and Jokivuolle andPeura (2000) which are based on structural form methods. In our framework,default is determined exogenously while marking-to-market is also introducedand margin calls are derived endogenously by the evolution of the underlyingused as collateral. This feature makes the problem easier to be generalisedand to obtain a closed form solution for the case of single collateral when theinterest rate is assumed to follow affine type processes. In addition to themarking-to-market policy, other realistic features are taken into account like
2
time to capture3 and liquidity effects. We also study the case where thereexists correlation between interest rate and the probabilities of default of thecounterparty. The framework confirms many practical intuitions, such as thedirect impact on the haircut level required to cover for riskier collateral andsupports the intuition that the frequency of marking to market and collateralare substitutes. In this sense, the framework could be used as an objectivebasis to understand the relationship between a number of factors and the levelof haircuts which is important not only for practioners and academics but alsofor regulators.
We proceed as follows: In section 2, we describe the basic set up, wheremarking to market policy is specified as well as the basic problem. In part 3the single collateral model is developed. Section 4 presents some extensionsfor realistic implementation: liquidity effects, time to capture, other types ofcollateral (equities, loans), stochastic default processes, positive trigger levelsand alternative interest rate models. In section 5 we conclude.
2 The Set-Up
We start with a simple set-up. There is a repurchase transaction contract be-tween the collateral taker and one counterparty. The counterparty borrowscash (the underlying) from the collateral taker. For simplicity of presentation,we assume that no interest is bearing on the cash borrowed. The extension toa constant interest rate bearing is straightforward.4 The collateral taker asksthe counterparty for αt units of a bond B(t, T ) as collateral. We assume thatthe time to maturity of the collateral is greater than the expiration date of thecontract between the collateral taker and the counterparty. For every collateralpledged, there exists a haircut h. We will assume a K periods contract, wheremargin calls can happenK times (for example, over a period of two weeks, therecould be 14 daily margin calls, or 10 if they can happen only on business days).We treat the general case with constant time intervals between possible mar-gin calls (no week ends) but provide in appendix the solution for non constanttime intervals, thus accounting for periods in which collaterals can not be called(such as during week ends). In this set up we will consider 0 as a trigger levelfor the margin call (i.e., as soon as the value of the collateral diverges from theunderlying value, there is a margin call in order to reestablish equivalency). Inthe extensions developed underneath, the trigger level will be set to a constant
3With time to capture we refer to the period between a default declaration and when thecollateral can be sold by the central bank. Some adjustments are needed due to the existenceof market risk in this period.
4 In practice, the underlying may not be constant indeed, and it would bear the interestwhich is set at the beginning and distributed evenly. Denote the interest rate for the operationis R, so the value of the underlying at t will be
Ut = U0eRt
Replacing the constant underlying value with the above in all equations underneath wouldprovide the result.
3
d.At the end of period k (k = 1, 2, ...,K) we face three situations:
1. The underlying U0 is smaller than the level of the adjusted collateral(taking the haircut in consideration), i.e. U0 < αk−1B(tk, T )(1 − h),where αk−1 is the amount of collateral at the beginning of period k. Inthis situation, the counterparty will receive an extra amount of collateralback such that U0 = αkB(tk, T )(1− h). The contract continues.
2. The underlying U0 is larger than the level of the adjusted collateral, i.e.U0 > αk−1B(tk, T )(1 − h). In this situation, a margin call happens andthe counterparty will be required to deposit more collateral such thatU0 = αkB(tk, T )(1 − h). On the other hand, there exists the exogenousprobability that the counterparty defaults in period k. If the counterpartydoes not default during period k, it will post the extra collateral and thecontract continues.
3. In the case where the margin call happens and the counterparty is requiredto deposit more collateral, if the counterparty defaults in this period, itwill not be able to fulfil the requirement, and the collateral taker will havea maximum loss of (U0 − αk−1B(tk, T )) . The contract will stop and enterinto a liquidation process.
The collateral taker is interested in the third situation when the loss is largerthan a predetermined level L = lU0, i.e. U0 − αk−1B(tk, T ) > lU0. In otherwords, the collateral taker is interested in the joint event X ∩ Y, where X isthat, at the end of period k, the collateral’s value is lower than (1− l)U0, andY is that the counterparty defaults in period k. Event X mainly reflects marketrisk, and event Y the credit risk of the counterparty. The total probabilityof X ∩ Y happening in the life of the contract is a measure of the risk thecollateral taker would take, which is determined by the collateral taker’s riskappetite. Meanwhile, the total probability is related to the haircut attached tothe collateral, so the collateral taker can adjust the haircut h to manage therisk.
In this set-up, we will first compute the haircut necessary for one singlecollateral, a default-free zero coupon bond B(t, T ), to protect for a given levelof risk. Once the haircuts are calculated, the set-up could be extrapolated totake into account collateral portfolios consisting of n default-free zero couponbonds. This paper, however, only deals with the single collateral model andleaves the collateral portfolio model for future research.
3 Single Collateral Model
To start with, we assume that the interest rate follows a Vasicek process as de-fined originally in Vasicek (1977). We examine the possibility of other processestruly leading interest rates later. The widely used Vasicek model has the ad-vantage over some alternatives (such as Cox-Ingersoll-Ross (1985), thereafter
4
CIR) to be able to represent any possible term structure. Its main limitationconsists in its possibly allowing for negative interest rates, something CIR doesnot. The Vasicek model is of course very extensively used in the theoretical lit-erature (most of the credit risk literature with stochastic interest rates uses thatmodel, see for example Shimko and alii (1993),Longstaff and Schwartz (1995))as well as in practice. We have as in Vasicek (1977):
drt = a(b− rt)dt+ σrdWt (1)
where the interest rate follows a mean reverting process. a is the speed ofmean reversion, b the long run or equilibrium value that the rate reverts to inthe long run with a speed of a, σr is the instantaneous volatility and dW is theincrement of a Wiener process,
Equivalently, the interest rate can be written:
rt = r0e−at + b(1− e−at) + e−atσr
∫t
0
eaudWu (2)
Denote Bt the price of the bond with maturity T at time t. When the shortrate rt follows Vasicek(1977), Bt is written as
Bt = emt−ntrt (3)
where both mt and nt are deterministic functions of time t :
nt =1− e−a(T−t)
a(4)
mt =(nt − T + t)(a2b−
σ2
r
2 )
a2−σ2rn
2t
4a(5)
Therefore, expressing the bond in its logarithmic form and substituting thevalue of rt we found previously, we obtain
lnBt = mt − nt[b+ (r0 − b)e−at + e−atσr
∫t
0
eaudWu] (6)
Therefore
xt = lnBt ∼ N(mt − nt[b+ (r0 − b)e−at], n2te−2atσ2
r
∫t
0
e2audu) (7)
Simplifying we obtain:
xt = lnBt ∼ N(mt − nt[b+ (r0 − b)e−at],n2tσ
2r
2a(1− e−2at)) (8)
The life of the contract is divided into K marking to market periods withinterval τ . At the beginning of period k(k = 1, 2, ...,K), the collateral is read-justed such that
5
U0 = (1− h)αk−1Bk−1 (9)
αk−1 =U0
(1− h)Bk−1(10)
where αk−1 is the quantity of bond collateral required at the end of period k−1. We use Bk−1 to indicate the value at the beginning of period k (immediatelyafter collateral readjustment at the end of period k−1) of a bond with maturityat T .
At the end of period k, before rebalancing, the collateral taker’s loss is de-termined by the difference between the underlying and the collateral
lossk = U0 − αk−1Bk = U0 −U0
1− h
Bk
Bk−1(11)
In order to calculate the corresponding haircut to apply, we need to computethe probability that the loss exceeds a predetermined level L. We also definethe loss level as a function of the underlying, or L = lU0. L or equivalentlyl defines the level of risk the collateral taker is willing to take. The modelwill give an equivalency between the level of risk taken, the probability thatthe losses exceed that level of risk and the collateral haircut chosen, so that thebank can choose any one variable and obtain the two others. Hence, looking attwo bonds, the collateral taker will be able to establish their respective haircutso that the risk exposure taken with either bond is the same. It will thus notfavor one bond versus another in terms of risk exposure as it may do with adhochaircut schedules.
To establish this framework, we calculate the probability that the actual lossexceeds the predetermined level L.
We thus start from L � lossk which can be rewritten:
L � U0 −U0
1− h
Bk
Bk−1⇐⇒
Bk
Bk−1� (1− l)(1− h) (12)
where we know, as shown above, that
Bk = emk−nkrk (13)
Bk−1 = emk−1−nk−1 rk−1
Therefore
ln
(Bk
Bk−1
)= (mk −mk−1)− (nkrk − nk−1rk−1) (14)
Or equivalently, expressing rk in terms of rk−1
6
ln
(Bk
Bk−1
)= (mk −mk−1)− nkb(1− e−aτ ) + (15)
(−nke−aτ + nk−1)rk−1 − nkσre
−akτ
∫ kτ
(k−1)τ
eaudWu
From equation (2), we know that rk−1 can be written as
rk−1 = b+ e−a(k−1)τ (r0 − b) + σr(1− e−2a(k−1)τ
2a)1
2 z1 (16)
where
z1 =
∫ (k−1)τ0 eaudWu
(∫ (k−1)τ0
e2audu)1
2
∼ N(0, 1) (17)
On the other hand, we also know that
−
∫ kτ
(k−1)τ
eaudWu = (
∫ kτ
(k−1)τ
e2audu)1
2 z2 where z2 ∼ N(0, 1) (18)
Therefore substituting into ln(
Bk
Bk−1
)we obtain
x = ln
(Bk
Bk−1
)= µk + σ1kz1 + σ2kz2 (19)
where
µk = (mk −mk−1) +1− e−aτ
a(be−a(T−kτ) + e−a(k−1)τ (r0 − b))
σ1k =1− e−aτ
aσr(
1− e−2a(k−1)τ
2a)1
2 (20)
σ2k = nkσr(1− e−2aτ
2a)1
2
Define a new random variable z:
z =σ1k
σkz1 +
σ2k
σkz2 (21)
where
σk =√σ21k + σ22k (22)
Since z1 and z2 are independent standard normal random variables, it canbe shown that z is also a standard normal random variable. Hence
x ∼ N(µk, σ2
k) (23)
7
Then, the probability that lossk is larger than the predetermined value L is:
Pr ob(lossk > L) = P (0 ≤Bk
Bk−1≤ (1− l)(1− h)) (24)
= P (−∞ ≤ lnBk
Bk−1≤ ln((1− l)(1− h))
= N(ln((1− l)(1− h))− µk
σk)
If Q represents the annualised probability of default, then τQ would repre-sent the probability of default in each period. As expressed earlier, this prob-ability of default is exogenous and therefore assumed independent of a margincall or a loss level. This assumption (a typical reduced form model assumption)appears better than the alternative, which would make the probability of defaultdependent on the margin call. The later assumption (typical of structural formmodels) would imply that the credit operation contract and the value of thecollateral alone would push the counterparty to default (and would assume thatthe position is large enough to force the counterparty to default). However, theposition taken in this approach is that the cash taker (a company or individual)might default for a wider variety of reasons which might be totally unrelatedto the specific contract considered (e.g. a company might experience liquidityproblems and file for bankruptcy). At the beginning of period k, the probabilitythat the counterparty has not defaulted is (1− τQ)k−1. The probability of thecounterparty defaulting and the collateral taker incurring a loss larger than L
in period k (i.e. the probability of a joint event) will be
P (k) = N(ln((1− l)(1− h))− µk
σk)τQ (25)
Taking into account that the counterparty can only default once, the prob-ability of incurring a loss larger than L in period k becomes
(1− τQ)k−1P (k) (26)
Hence, the total probability of incurring a loss larger than L withK markingto market periods will be:
P (loss > L) = P (1) + (1− τQ)P (2) + ...(1− τQ)K−1P (K) (27)
=K∑k=1
(1− τQ)k−1P (k)
If the collateral taker fixes L and Pr(loss > L),the required haircut can berecovered from the above formula.
For example, a collateral taker may decide, in a certain interest environment,to protect itself against a loss of more than 5% with counterparties that have
8
an average probability of default of one percent. It can use this frameworkto obtain the haircut that corresponds to this risk appetite for a collateral of agovernment bond of maturity 3 years. It can also calculate how much an increasein haircut should go with a collateral of maturity 10 years in order to obtain thesame risk exposure. It can thus propose a consistent haircut schedule so thatwhatever the maturity of the bond proposed as collateral, the risk exposure ofthe bank remains the same.
Proposition 1 For l big enough, given a fixed collateral haircut, the longer the
maturity of the bond used as collateral, the higher is the probability of incurring
a loss larger than L. To obtain the same probability of incurring a loss larger
than L, longer term bonds require higher haircuts.
Proof. See AppendixAs we expect, the model shows that, other things being equal, increasing the
haircut will decrease the probability of incurring a large loss, and that the longerthe maturity of the bond used as collateral, the higher the haircut required inorder to have the same probability of incurring the loss.
This result is well understood by practitioners. It is classical to have hair-cut schedules designed so that longer term bonds face higher haircuts thanshorter term bonds. Interestingly, the methodology here gives precisely theextra amount of haircut required for extra maturity in a rational way. It alsogives the sensitivity of the haircut level to other parameters, such as the interestrate environment (level and volatility of the interest rate), the frequency of themarking to markets (which reflect the control environment of the process) aswell as the probability of default (which reflects the quality of the counterpartiesconsidered).
The following section gives simple numerical examples of these sensitivities.
3.0.1 Example
Consider the following benchmark5 case for the values of the main parametersin our model
b a r0 T h Q σr0.05 0.25 0.04 10 0.01 0.01 0.04
With these values we compute what is the total probability of a loss of5% (l = 5%) for the collateral taker in a contract of 10 years with a singlebond pledged as a collateral. In other words, starting with a value of 100 forthe underlying (cash provided to the counterparty), what is the probability ofhaving a loss of 5, when the collateral pledged is a government bond of maturity1 year and of initial value of 100/99% = 101.01. We study the case of daily,
5We take as a benchmark typical parameters reported in the literature. These values shouldbe only interpreted as an example since they are not the result of a calibration procedure.Similar values can be found in Neftci (2000) for instance.
9
weekly and monthly possible marking to markets with possible margin calls.The results give low probabilities that will need to be modified (and increased)by the introduction of realistic features such as time to capture, liquidity effects,etc. These are introduced in the following sections. Nonetheless, the numbersare instructive to look at for a first understanding of what drives risk exposureswith collateral in place. Classical market risk factors (the interest rate variableshere), credit risk factors (the probability of default here, which is extended to astochastic default process in a later extension) and elements of collateral policysuch as the haircut schedule all drive probabilities of losses. We obtain thefollowing probabilities
Time ProbabilityDaily 3.26858×10−18
Weekly 1.01347×10−5
Monthly 6.1385×10−4
With these results as a benchmark, we analyze what is the effect of changingthe values of the different parameters of the models. We choose very extremecases to make differences more obvious.
Changes in T, maturity of the bond held as collateral
T Daily Weekly Monthly1.5 0. 0. 1.33392×10−9
20 7.14632×10−6 2.41159×10−5 7.9913×10−4
As expected, the longer the time to maturity of the bond, the higher theprobability of having losses. The usual practice of asking for higher haircuts forlonger maturity bonds is thus justified. The mismatch between the maturityof the bond and the maturity of the loan agreed has a strong impact on theprobability of losses.
Changes in h, haircut on bond considered for collateral
h Daily Weekly Monthly0.1 0. 2.59421×10−17 6.16681×10−7
0.001 2.75417×10−14 5.13204×10−11 9.25418×10−4
Also as expected, when the haircut is high , the probability of having a lossdecreases, while if the haircut is small, the probability increases. Haircuts are agood way to reduce risk exposure. We provide hereby the exact impact of thehaircuts considered.
Changes in Q, probability of default of the counterparty
Q=prob of default Daily Weekly Monthly0.01 3.16435×10−17 9.72023×10−5 5.89537×10−3
0.0001 3.27929×10−19 1.01774×10−6 6.16348×10−5
10
The quality of the counterparty considered plays of course a large role on thelevel of the haircuts that should be asked for. Most of the time, a collateral takerwill be required to establish a haircut schedule that is independent of the qualityof the counterparty. One approach will then be to set a haircut schedule linkedto the worst exposures, or to an average exposure. The methodology providesfor the level of exposure depending on the counterparties considered.
We next consider the impact of the interest rate environment. The volatilityof the interest rate process seems to have a particularly strong impact on thehaircut levels necessary (via the probability of losses).
Changes in r0, initial interest rate value
r0 Daily Weekly Monthly0.01 3.54892×10−18 1.10399×10−5 1.87388×10−3
0.08 2.93061×10−18 9.03382×10−6 2.04854×10−5
When the initial interest rate decreases, then the probability of losses in-creases. This can be explained by the fact that, if the initial interest rate isvery low (lower than the long term mean), there is a high probability that itwill increase. If the interest rate increases, the price of the bond will decrease,therefore, the probability of incurring losses is higher. As noted inmore detailsunderneath though, the impact of the difference with the long term mean is notanalyzed separately here while it has a distinct effect as can be seen in equation(25).
Changes in b, long term mean of interest rates
b Daily Weekly Monthly0.1 3.22775×10−18 9.95571×10−6 6.00103×10−4
0.01 3.30184×10−18 1.02807×10−5 6.25082×10−4
We can observe that when the long term mean of the interest rate increases,then the probability of losses decreases. Notice though that from the formula(25) above (as both interest rate level and long term mean affect only the driftµkand not the other components of the probability of losses), the long term
means has an impact via its absolute level (but a rather minimal impact as it ismultiplied by an exponential of low value, especially for long maturity bonds)and more importantly via the distance between the current interest rate andthe long term mean (as this is multiplied by a higher value). The two distincteffects are not separated well in our sensitivity analysis as the interest rate isfixed while the long term mean is modified.
Changes in a, speed of mean reversion
a Daily Weekly Monthly0.1 9.36419×10−9 3.48408×10−4 1.87388× 10−3
0.5 0. 7.1690910−11 2.04854×10−5
11
When the speed of convergence of the interest rate to its mean increases,then the probability of losses decreases. It is interesting to see that the speedof mean reversion has an important effect on the overall value of the probabilitiesof losses. This is, with the volatility of the interest rate process, one of the maindrivers, as far as the interest rate environment is concerned, of the bank riskexposure.
Changes in σr, volatility of the interest rate process
σr Daily Weekly Monthly0.015 0. 9.14667×10−19 1.613×10−5
0.05 5.0507×10−13 6.845×10−5 1.0111×10−3
Changes in the volatility of the interest rate is one of the most importantfactors affecting the probability of loss. When the volatility increases, then theprobability of losses also increases. In the benchmark we considered a quite highvolatility (σr = 0.04) in order to see more easily how the probabilities changewhen we modify the different parameters.
While these numerical examples give us a first sense of what affects collateralpolicies, much needs to be incorporated to make it more realistic. First andforemost may be the fact that counterparties typically post several collaterals(portfolios of collaterals) as pledged. This issue is considered next.
4 Extensions for Realistic Implementation
4.1 Introduction of time to capture and liquidity risk
When the counterparty defaults, the collateral taker has to wait some time forthe legal procedures to yield actual possession (and sale or liquidation) of thecollateral. We call this phenomenon the time to capture. Suppose the time tocapture is equal to δ marking to market intervals, i.e. δτ , and the counterpartydefaults during period k, so there is no rebalance at the end of period k. Theloss of the collateral taker will not depend on the value of the collateral at theend of the kτ period, but on the value of collateral at the end of period (k+δ)τ .
On the other hand, one of the main concerns for the collateral taker willbe, in the case of default, of how much cash can be obtained if the collateralposition is liquidated. Unfortunately, this value is exposed to risk itself since itis very likely that the difference between the initial market value and the valuerealized after liquidation is greater than 0. This is what is called liquidity risk.
We can distinguish between exogenous liquidity risk (normally associated tothe difference between bid and ask), which is due to the characteristics of themarket and it is unaffected by the actions of any participant, and endogenous
liquidity risk which refers to how individual actions can affect the price. Somecollateral takers write in their books the bid price of the collateral, hence the
12
exogenous liquidity risk disappears6 . For those cases, only endogenous liquidityrisk is taken into account. This risk consists in the possibility that the tradesmove the market in an adverse direction. In case of large positions, it is possiblethat the collateral taker sale would in itself affect the liquidity risk. The modelof endogenous liquidity risk is still an open question in the literature and itis far from the scope of this paper to try and solve it. We will simply takethe liquidity risk as a percentage loss when the collaterals are liquidated, i.e.liquidity loss is θ percent of the collateral value at the time of liquidation, whichcan be different for each collateral.
At the beginning of period k, t = (k − 1)τ , we have
U0 = (1− h)αk−1Bk−1 (28)
then
αk−1 =U0
(1− h)Bk−1
(29)
At t = (k + δ) τ the possible loss is equal to
Lossk = U0 − αk−1Bk+δ(1− θ) (30)
Or equivalently, by substituting αk−1
Lossk = U0 −
[U0
1− h
Bk+δ
Bk−1
](1− θ) (31)
From the single collateral case, we know that
ln
(Bk+δ
Bk−1
)= µk,δ + σk,δz (32)
where
z ˜ N(0, 1)
µk,δ = (mk+δ −mk−1) +1− e−a(δ+1)τ
a(be−a(T−(k+δ)τ) + e−a(k−1)τ (r0 − b)) (33)
σk,δ =√σ21k,δ + σ22k,δ
σ1k,δ =1− e−a(δ+1)τ
aσr(
1− e−2a(k−1)τ
2a)1
2 (34)
σ2k,δ = nk+δσr(1− e−2a(δ+1)τ
2a)1
2
Then, we have the following relationship between the predetermined maxi-mum loss, L = lU0, and z.
6We have modeled in the appendix the case where exogenous liquidity risk does not disap-
pear for the banks that do not follow the policy of keeping in their books the bid price of the
collateral.
13
lU0 ≤ lossk = U0 −
[U0
1− heµk,δ+σk,δz
](1− θ) (35)
=⇒ z ≤ln((1−l)(1−h)
1−θ
)− µk,δ
σk,δ(36)
The probability of incurring a loss larger than L will be
P (k) = P (lossk ≥ L)τQ (37)
= N
ln
((1−l)(1−h)
1−θ
)− µk,δ
σk,δ
τQ (38)
Therefore, the probability of incurring a loss larger than L in the life ofcontract will be
P (loss ≥ L) =K∑k=1
(1− τQ)k−1P (k) (39)
It is not difficult to find that the similarity between this extension with theoriginal case. Introduction of liquidity risk θ is simply to replace (1− l) (1− h)
with (1−l)(1−h)1−θ . In other words, the impact of liquidity risk θ is to adjust l or
h to corresponding smaller numbers l−θ1−θ or h−θ
1−θ , consequently to increase the
probability of incurring a loss larger than L.Introduction of time to capture δ is to replace µk and σk with µk,δ and σk,δ
accordingly. Since time to capture δ is always positive, it adds more uncertaintyand increase the probability of incurring a loss larger than L. When δ is large,the uncertainty during time to capture will be larger than the uncertainty duringone marking to market interval, so that the risk due to the time to capture willbe an important part (possibly dominant part) of the overall risk.
We thus have derived the first important extension of the model, integratingthe realistic features of a time to capture and liquidity effect.
4.2 Examples
In this section, we analyze the effect of introducing time to capture and liquidityeffects in a single collateral context.
Consider the following benchmark case for the values of the main parametersin our model
b a r0 T h Q σ loss Time to capture liquidity
0.05 0.25 0.04 10 0.01 0.01 0.015 0.05 1 month 0.03
14
These values imply that the collateral will not be captured before a monthafter the event of default. This may be due to legal constraints that will makeit impossible to liquidate the collateral before or special characteristic of thefinancial asset used as collateral which may require a long analysis period bythe prospective buyers (e.g. asset backed securities with special covenants).This problem is combined with the liquidity issue, in which a loss of 3% of valueof the bonds occurs at liquidation, in case of a thin market.
We first have a look to the effect of different marking to market intervals:
Marking to Market Daily Weekly Monthly2.10434×10−3 2.22007×10−3 2.66116×10−3
We can observe that with the introduction of the time to capture and theliquidity effect, the probabilities of loss have increased considerably. We furtheranalyze the effects of changing the time to capture interval and the liquiditylevel under a daily marking to market set up.
Time to capture
Time to capture Prob of loss2 weeks 1.35211×10−3
2 months 2.65833×10−3
The probability of loss increases when we increase the time to capture in-terval. This is due to the fact that we face more uncertainty about the price ofthe collateral at the moment of the liquidation.
Liquidity
Liquidity Prob of loss0 1.25153×10−3
0.02 9.91587×10−3
As expected, the probability of loss has increased when we increase the liq-uidity loss level. Clearly, some instruments (such as loans) will face strongliquidity problems. Our methodology thus allows to differentiate between in-struments for which liquidity is good and instruments for which liquidity is poor.The haircut can thus be set to compensate for that problem.
4.3 Further Extensions
4.3.1 Other instruments as collateral: Equity
In this section we introduce the use of equities as collateral.As classical since Black and Scholes(1973), we assume the stock price St
follows
dSt
St= µtdt+ σtdWt (40)
15
or equivalently
St = Sue∫t
u(µ
s−
1
2σ2s)ds+
∫t
uσsdWs , ∀ u < t (41)
where µt and σt are deterministic and bounded functions of time t. It canbe shown that
ln
(StSu
)∼ N(
∫ t
u
(µs −1
2σ2s)ds,
∫ t
u
σ2sds) (42)
Following the same methodology we have used until now, and assuming Kmarking to market periods with interval τ , at the end of period k − 1 (k =1, 2, ...,K),
U0 = (1− h)αk−1Sk−1 (43)
αk−1 =U0
(1− h)Sk−1(44)
where as before αk−1 is the quantity of equity collateral required at the endof period k−1. At the end of period k, before rebalancing, the loss is determinedas:
lossk = U0 − αk−1Sk = U0 − U0(1− h)
SkSk−1
(45)
Therefore, applying the same methodology we used for the bond collateralcase we obtain:
Pr ob(lossk > L) = P (0 ≤ SkSk−1
≤ (1− l)(1− h))
= P (−∞ ≤ lnSkSk−1
≤ ln((1− l)(1− h)) (46)
= N
ln((1− l)(1− h))− ∫ kτ
(k−1)τ(µs − 1
2σ2s)ds√∫ kτ
(k−1)τ σ2sds
The total probability of incurring a loss can be easily derived from the aboveformula using the methodology described in previous sections.
4.3.2 Poisson process for default of the counterparty
For simplicity, we have assumed for the time being a constant probability ofdefault for the counterparty considered. This is a simplification of realityas default probabilities are not constant, but rather stochastic. We use theclassical formalization of reduced form models (see Jarrow and Turnbull (1995),Madan and Unal (1998), Duffie and Singleton (1999)). The structure of the
16
problem remains the same as above. Difficulties arise when the default processis correlated to the interest rate.
Assume there exists a jump process J(t) with intensity process λ(t). Defaultoccurs at the first jump time of J(t). If default happens at τ , this implies bydefinition
τ := inf {t ≥ 0 | J(t) = 1} (47)
The probability of no default event occurring between time t and T is givenby
P [J(T )− J(t) = 0] = e−∫T
tλ(s)ds (48)
This implies that the distribution of the default time is given by
P [τ ≤ T ] = 1− e−∫T
0λ(s)ds (49)
and its density
f(t) = λ(t)e−∫t
0λ(s)ds (50)
Assume that the default intensity is determined as follows:
λt = λot + φrt (51)
where λot is a stochastic variable that follows a mean-reverting diffusionprocess:
dλot = aλ(bλ − λot )dt+ σλdWλt (52)
Recall that default-free interest rate rt follows
drt = a(b− rt)dt+ σrdWt (53)
Here Wλt and Wt are two independent Brownian motion, so {λot , 0 ≤ t ≤ T}
and {rt, 0 ≤ t ≤ T} are two independent stochastic processes. φ can be consid-ered as a constant. Notice that the empirical literature tends to show a negativecorrelation between interest rate and intensity of default (see for example Duf-fee (98)). Given this model, we can compute as done in the other set-ups theprobability of having a loss larger than lU0 in period k. The calculations areprovided in Appendix.
4.3.3 Introduction of non-zero trigger level
In practice, margin calls are not effected as soon as there is a difference in valuebetween the underlying and the adjusted collateral (after haircut). A certainlevel of difference is allowed (1% in the case of the EuroSystem) to avoid havingtoo frequent calls.
When the trigger level is zero, the collateral portfolio is revalued and ad-justed at each marking-to-market to keep the same weights of its components.
17
Meanwhile, the new value of the collateral portfolio, adjusted after haircuts, isequal to that of the underlying. In the case of non-zero trigger levels, the col-lateral portfolio is revalued but not always adjusted at each marking to marketperiod. When the value of the collateral portfolio, adjusted after haircuts, isinside a narrow range centered in the value of the underlying, no adjustment isrequired, and the same collateral portfolio will be kept to the next margin call.Otherwise, the collateral will be adjusted to keep its weights of components andits value, adjusted after haircuts, equal to that of the underlying.
It is very difficult to model the non-zero trigger level precisely because itstrongly depends on the specific trajectory. We study the approximation rep-resented by two boundaries, first in the case of a single collateral and then inthe case of a collateral portfolio. Denote d the trigger level. At the beginningof period k,
(1− d)U0 ≤ (1− h)αk−1Bk−1 ≤ (1 + d)U0
So,
(1− d)U0(1− h)Bk−1
≤ αk−1 ≤ (1 + d)U0(1− h)Bk−1
At the end of period k,
U0 − (1 + d)U01− h
Bk
Bk−1≤ Lossk = U0 − αk−1Bk ≤ U0 − (1− d)U0
1− h
Bk
Bk−1
Equivalently,
(1− l) (1− h)
1 + d≤ Bk
Bk−1≤ (1− l) (1− h)
1− d
Following the similar process, we will have
P (loss > L) =K∑k=1
(1− τQ)k−1P (k)
with
N(ln[(1−l)(1−h)
1+d
]− µk
σk)τQ ≤ P (k) ≤ N(
ln[(1−l)(1−h)
1−d
]− µk
σk)τQ
We thus obtain an approximation of haircuts required (or risk exposures)depending on the trigger level considered.
18
4.4 Alternative interest rate model
In order to check the assumption of interest rate following a Vasicek type ofprocess, we decide to try other interest rate model specifications. Concretely,we analyze the differences if we assume a CIR process for interest rate withour model. Unfortunately, we are not able to derive closed form solutions forthe case of CIR, therefore, simulation methods are used. While Monte Carlosimulations allow for more flexibility in the modelling (with the possibility ofusing other interest rate models and more complex features), they do not allowfor the use and speed of calculations of the analytical formulas derived above,nor do they allow for general results analysis and sensitivities as those providedabove.
CIR assumes the following process for the interest rate:
drt = a′(b′ − rt)dt+ σ′r√rtdWt (54)
At time t, the price of the bond can be found as:
Bt(T ) = At(T )e−n′
t(T )rt (55)
where
n′t(T ) =2(eγ(T−t) − 1)
(γ + a′)(eγ(T−t) − 1) + 2γ(56)
At(T ) =
[2γe(a
′+γ)(T−t)/2
(γ + a′)(eγ(T−t) − 1) + 2γ
]2a′b′/σ′2
r
(57)
with
γ =√a′2 + 2σ′2r (58)
In order to compare the results of Vasicek model with the CIR model, we dis-cretize the process followed by the interest rate and we simulate the randomnessintroduced by the Brownian Motion.
To check the error we are making with simulations, we also simulate theVasicek process and compare it with the real values as obtained from the an-alytical results given above. We obtain an acceptable accuracy7 (around 2%relative difference between theoretical Vasicek and simulated Vasicek results)with a time step of five times per day and a number of repetitions equal to1000. We use these parameters in order to perform our CIR simulations.
However, before carrying out our study, we must consider a case where theparameters of Vasicek and CIR are compatible. For that we use the parametersspecified in Chan et.al. (1992) where different interest rate models are estimated
7We measure the accuracy as the difference between the real value and the simulated value.
We repeat the simulation 10 times. In each simulation the time step is of five times per day
and the number of repetitions equals to 1000. Taking the average of the errors in every set of
simulations we found that this is around 2%.
19
for the same period (from June 1964 to December 1989). The parameters foundin the estimation (and the ones we use in our simulations) are:
Vasicek CIRa 0.1779 0.2339b 0.0867 0.080803σ 0.02 0.08544
(59)
Based on this parameters, we study the differences between Vasicek and CIRmodels for different values of the haircut level, the level of loss accepted and theinitial interest rate. The results follow in the table bellow:
• r0 = 0.08
Vasicek case
h = 0.05 h = 0.01 h = 0.005 h = 0.001l = 0.001 1.27159×10−5 2.53348×10−4 3.18439×10−4 3.74778×10−4
l = 0.01 4.95089×10−6 1.56547×10−4 2.07011×10−4 2.53348×10−4
l = 0.05 1.6815×10−6 4.95089×10−6 8.48165×10−6 1.27159×10−6
l = 0.1 2.87072×10−13 1.79085×10−9 4.43444×10−9 8.90452×10−9
CIR case
h = 0.05 h = 0.01 h = 0.005 h = 0.001l = 0.001 1.74088×10−5 2.34474×10−4 2.93948×10−4 3.56502×10−4
l = 0.01 1.04118×10−5 1.45685×10−4 1.88916×10−4 2.29979×10−4
l = 0.05 4.16424×10−7 1.0995×10−5 1.49098×10−5 1.84918×10−5
l = 0.1 0 0 8.32847×10−8 3.33194×10−7
The numbers indicate the probabilities values under Vasicek model and CIRmodel.
We observe that for the standard case ( l = 0.01 and h = 0.01) the differencesbetween both models are rather small. We can then conclude that the utiliza-tion of the extensively used Vasicek model is not a drastic restriction althoughinterest rate modeling may matter in some cases.
20
5 Conclusions
We have derived how a collateral taker should build collateral policies (andnotably haircut schedules) for the collateral policy to be consistent. We thusprovide for the framework a collateral taker should follow in order to treat differ-ent collaterals consistently. The exercise reveals itself to be quite complex whenone aims at taking into account in the modeling the muldimensional complex-ity of real use: the presence of time to capture, of liquidity effects, of differentclasses of collateral, etc. In deriving the structure of consistent collateral poli-cies, we also have obtained a number of results. The longer the maturity of theinstrument considered, the higher the haircuts that should be effected. Mark-ing to market intervals, interest rate dynamics, credit risk of the counterpartyall interact in the determination of haircut schedules. The framework could beused as an objective basis to understand the relationship between these fac-tors (liquidity effects, credit risk of counterparty, time to capture, trigger levels,stochastic default processes, alternative interest rate models) and the level ofhaircuts. We hope that the framework presented here could trigger renovatedinterest in an area that has been subject of little research to date but whoseimportance is key in today’s financial markets.
While we have derived how a consistent collateral policy should be con-structed, we do not provide for what is the level of collateral policy that shouldbe chosen. This will be decided by strategic and political factors as well asagency and other issues that can only be treated in a general equilibrium frame-work to provide full answers. In addition, the proposed framework for singlecollateral could be extrapolated to portfolio of collateral in future research toanalyse the possible risk reduction gains that could be obtained and determinethe corresponding haircut levels. Much research thus remains to be done to ob-tain what would be the optimal collateral policy for a collateral taker. Ideally,future research would combine the framework of collateral risk control deter-mination as obtained here with optimality issues linked to auctioning problemsand game theoretical agency and asymmetry of information issues to obtain arational collateral policy. To achieve this goal, researchers will have to combinegame theory with continuous time finance, a challenge that some researchersare currently aiming for.
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[24] Kiyotaki, N., and J. Moore, 1997, ”Credit Cycles”, Journal of PoliticalEconomy, 105, 211-248.
[25] Lotz, C., and L. Schlögl, 2000, “Default risk in a market model”, Journalof Banking & Finance, 24, 301-327.
[26] Madan, D. and H. Unal, 1998, ”Pricing the Risks of Default”, Review ofDerivatives Research, Vol 2, 121-160.
[27] Margrabe, W. 1978, ”The value of an option to exchange one asset foranother”, Journal of Finance, 33, March, 177-87.
[28] Merton, R.C., 1974, ”On the Pricing of Corporate Debt: The Risk Struc-ture of Interest Rates”, The Journal of Finance, Vol49, 449-470.
[29] Musiela, M., and M. Rutkowski, 1997, Martingale Methods in FinancialModelling. Springer-Verlag.
[30] Neftci, S., 2000, An Introduction to the Mathematics of Financial Deriva-tives. Academic Press.
[31] Nyborg, K. G., and I.A. Strebulaev, 2001, ”Collateral and Short Squeezingof Liquidity in Fixed Rate Tenders”, Forthcoming in Journal of Interna-tional Money and Finance.
[32] Pagès, H., 2001, “Can liquidity risk be subsumed in credit risk? A casestudy from Brady bond prices”, working paper, Bank for InternationalSettlements.
[33] Stiglitz, J. and A. Weiss, 1981, ”Credit Rationing in Markets with ImperfetInformation”, American Economic Review, 71, 393-410.
[34] Stulz, R. and H. Johnson, 1985, ”An Analysis of Secured Debt”, Journalof Financial Economic, 14 (1985), p. 501-521.
23
[35] Taylor, J.B., 1999, ”The Robustness and Efficiency of Monetary PolicyRules as Guidelines for Interest Rate Setting by the European collateraltaker”, Journal of Monetary Economics, 43, 655-679.
24
6 Appendices:
6.1 Appendix 1: Proof of proposition 1
We will proof in this section that bearing same risk, a collateral formed witha long term maturity bond requires a higher haircut than a collateral formedwith a short term maturity bond.
Let
P (lossk ≥ lU0) = N(ln((1− l)(1− h))− µk
σk) = f(T, h) (60)
where
µk = (mk −mk−1) +1− e−aτ
a(be−a(T−kτ) + e−a(k−1)τ (r0 − b))
σk =√σ21k + σ22k (61)
σ1k =1− e−aτ
aσr(
1− e−2a(k−1)τ
2a)1
2
σ2k = nkσr(1− e−2aτ
2a)1
2
Taking partial derivatives of function f with respect to h and T we obtain:
∂f
∂h= n(
ln((1− l)(1− h))− µkσk
)(−1)
1− h
1
σk(62)
∂f
∂T= n(
ln((1− l)(1− h))− µkσk
)−
∂µk
∂Tσk − (ln((1− l)(1− h))− µk)
∂σk∂T
σ2k(63)
where n(.) is the standard normal density function while N(.) is the standardnormal cumulative density function. We can already observe that ∂f
∂h< 0.
Continuing with the computations
∂µk∂T
=∂mk
∂T−
∂mk−1
∂T+
1− e−aτ
abe−a(T−kτ)(−a) (64)
Remembering the expressions for mk and nk
mk =(nk(T )− T + kτ)(a2b−
σ2r
2 )
a2−
σ2rn2k
4a
nk =1− e−a(T−kτ)
a
Computing the partial derivatives of nk and mk we obtain:
25
∂nk
∂T=
1
a(−e−a(T−kτ))(−a) = e−a(T−kτ) (65)
∂mk
∂T=
(a2b−σ2r
2 )
a2(e−a(T−kτ) − 1)−
σ2r4a
2nke−a(T−kτ) (66)
= −abnk +σ2r2n2k
Therefore,
∂µk∂T
= −abnk +σ2r2n2k + abnk−1 −
σ2r2n2k−1 (67)
−b(1− e−aτ )e−a(T−kτ)
Note that
nk(T )− nk−1(T ) =−e−a(T−kτ) + e−a(T−kτ+τ)
a(68)
=−e−a(T−kτ)
a(1− e−aτ )
n2k − n2k−1 = (nk + nk−1)(nk − nk−1) (69)
=−e−a(T−kτ)
a(1− e−aτ )
2− e−a(T−kτ) − e−a(T−kτ+τ)
a
Hence,
∂µk∂T
= be−a(T−kτ)(1− e−aτ ) +σ2r2(n2k − n2k−1) (70)
−b(1− e−aτ )e−a(T−kτ)
=σ2r2(n2k − n2k−1)
∂σk
∂T=
σ2k√σ21k + σ22k
∂nk
∂Tσr(
1− e−2aτ
2a)1
2 (71)
=σ2k
σke−a(T−kτ)σr(
1− e−2aτ
2a)1
2
Notice that
26
∂σk
∂T> 0 (72)
∂µk∂T
< 0
Define loss lU0 such that P (lossk ≥ lU0) is small (say, smaller than 5%)
such thatln((1−l)(1−h))−µ
k
σk< 0. Then, we have that
∂f
∂T= n(
ln((1− l)(1− h))− µkσk
)1
σk
[−(
∂µk∂T
+(ln((1− l)(1− h))− µk
σk)∂σk
∂T)
]∂f
∂T> 0 (73)
We can express the total variation in the probability of having a loss as:
dP =∂f
∂TdT +
∂f
∂hdh (74)
For the same probability (i.e., total variation is equal to 0), we have
dP =∂f
∂TdT +
∂f
∂hdh = 0 =⇒ (75)
dh
dT= −
∂f∂T∂f∂h
Since we have shown that ∂f∂T
> 0 and ∂f∂h
< 0, we have dhdT
> 0 which impliesthat h2 > h1 ⇔ T2 > T1 for a fixed level of probability of loss, i.e. for the samelevel of probability of loss, the bond with longer time to maturity requires ahigher haircut than the bond with shorter time to maturity.(Q.E.D.)
6.2 Appendix 2: Exogenous liquidity risk
Concerning the liquidity risk, we can distinguish between endogenous liquidity
risk, which refers how individual actions can affect the price, and exogenous
liquidity risk which is due to the characteristics of the market and it is unaffectedby the actions of any participant.
In the paper we have developed the concept of endogenous liquidity risk.This risk consists in the possibility that the trades move the market in anadverse direction. We consider this risk when the size of the position overcomesthe quote depth, where quote depth is defined as the volume of the asset availableat the market maker’s quoted price (bid or ask). In very liquid assets, the level ofquote depth is very high which implies that the majority of the transactions willnot pass this level and hence will not face this risk. In a very simple approach,we have modeled this risk as a constant that decreases the value of the bond.
27
On the other hand, exogenous liquidity risk is common to all the participantsin the market. It is normally related to the difference between bid and ask. Invery liquid securities, this difference will be very small, and therefore this riskis also small.
In this appendix we want to add the concept of exogenous liquidity risk sinceit is relevant for all kind of transactions, no matter their size. Moreover, thedata needed to quantify this exogenous liquidity risk is normally available.
To introduce the exogenous liquidity risk, we will follow basically the method-ology developed by Bangia et al. (98) adapting it to our model. The mentionedpaper assumes that the cost of liquidity is based on a certain average spreadplus a multiple of the spread volatility, to cover most of the spread situations.Concretely, the cost of exogenous liquidity is
1
2
[Bt(S + aσ̃)
](76)
where Bt is today’s mid-price for the bond pledge as collateral, S is theaverage relative spread (relative spread is defined as Ask−Bid
Mid), σ̃ is the volatility
of the relative spread and a is a scaling factor such that we achieve roughlythe desired probability coverage. All the parameters taking part to determinethe cost of exogenous liquidity are observable except a which depends on thedistribution function of the relative spread. Hence, it would be no difficult toincorporate this risk into our model.
By equation (11) we knew that the loss was determined as
lossk = U0 − αk−1Bk (T )
With the introduction of liquidity risk, this possible loss must increase asfollows:
lossk = U0 − αk−1Bk (T ) + αk−11
2
[Bk (T ) (S + aσ̃)
](77)
We again have to check whether the lossk is greater or equal to the prede-termined value L, i.e.
L � U0 −Bk (T )
Bk−1(T )
U0
(1− h)
[1−
1
2(S + aσ̃)
]⇐⇒ (78)
Bk (T )
Bk−1 (T )� (1− l)
(1− h)[1− 1
2(S + aσ̃)]
We can realize that the exogenous liquidity risk is introduced as a linearcombination of U0, which simplifies notably the computations. Let
(1− h)[1− 1
2(S + aσ̃)] = (1− h′) (79)
Then,
28
Bk (T )
Bk−1 (T )� (1− l)(1− h′) (80)
which is very similar to expression (12). Therefore, we can use the resultsderived earlier, this is, the probability of incurring in a big loss larger than L inperiod k will be now:
P (k) = N(ln((1− l)(1− h′))− µk
σk)τQ (81)
where
h′ = 1−(1− h)[
1− 12(S + aσ̃)
] (82)
In the same way we can extend the result to the case of a collateral formedwith a portfolio of bonds. At t = kτ the possible loss will be equal to:
Lossk = U0−
{αk−1Bk(T1)
[1−
1
2(S1 + aσ̃1)
]+ qαk−1Bk(T2)
[1−
1
2(S2 + aσ̃2)
]}(83)
or equivalently
Lossk = U0 −Bk(T1) + q′Bk(T2)
(1− h′1)Bk−1(T1) + (1− h′2)qBk−1(T2)U0 (84)
where
1− h′1 =1− h1[
1− 12(S1 + aσ̃1)
] (85)
1− h′2 =1− h2[
1− 12(S1 + aσ̃1)
]q′ = q
[1− 1
2(S2 + aσ̃2)][
1− 12(S1 + aσ̃1)
]and follow same methodology as we did.
6.3 Appendix 3: Non-Constant Marking to Market
In this part we will introduce the case where marking to market period is notconstant. The reason to think about this possibility comes from the observationin real world. For instance, we can observe a weekend effect, i.e. althoughthe marking to market is daily, during the weekend there are not margin calls,therefore, the marking to market period is not constant anymore. This new casedoes not affect our theoretical derivation, however, the effect on the probabilityof loss can be considerable.
29
Consider that the life of the contract can be divided in K periods tk (k =1, 2, ...K) where the interval τk is not fixed (τk = tk − tk−1 is not constant).Then, the possible loss at time tk can be written as:
lossk = U0 − αk−1Bk
(86)
or equivalently
lossk = U0 −U0
(1− h)
Bk
Bk−1
(87)
we know that
ln(Bk
Bk−1) = µk + σkz (88)
where
z ˜ N(0, 1)
µk = (mk −mk−1) +1− e−a(tk−tk−1)
a(be−a(T−tk) + e−atk−1(r0 − b))
σk =√σ21k + σ22k (89)
σ1k =1− e−a(tk−tk−1)
aσr(
1− e−2atk−1
2a)12
σ2k = nkσr(1− e−2a(tk−tk−1)
2a)12
Then, we have the following relationship between the determined maximumloss, L = lU0, and z.
lU0 ≤ lossk = U0 −U0
1− heµk+σkz
=⇒ z ≤ln((1− l)(1− h))− µk
σk(90)
Then,
P (lossk ≥ L) = N
(ln((1− l)(1− h))− µk
σk
)The result is very similar to the case with constant marking to market.
6. 4 App en di x 4: Def aul t Fol l owi ng a Poi s s on Pr o c es s
P (k) = Pr(Xk ∩ Yk) = Pr (Xk) · Pr(Yk |Xk) (92)
where
30
(91)
Xk = {Bk (T )
Bk−1 (T )≤ (1− l) (1− h)} (93)
Yk = {(k − 1)τ < ι ≤ kτ} (94)
where Pr (Xk) has been already computed before. From (48), we computethe conditional probability at time 0,
Pr(Yk |Xk) = Pr(k − 1)τ < ι ≤ kτ |Xk)
= Pr(τ > (k − 1)τ |Xk)− Pr(τ > kτ |Xk)
= E0(e−
∫ (k−1)τ0 λ(s)ds − e−
∫kτ
0λ(s)ds |Xk)
= E0(e−
∫ (k−1)τ0 λ(s)ds |Xk)−E0(e
−
∫kτ
0λ(s)ds |Xk)
Notice that, in the first term, e−∫ (k−1)τ0 λ(s)ds has nothing to do with Xk
because e−∫ (k−1)τ0 λ(s)ds and Xk happen at different times. So,
E0(e−
∫ (k−1)τ0 λ(s)ds |Xk) = E0(e
−
∫ (k−1)τ0 λ(s)ds)
= E0(e−
∫ (k−1)τ0 (λo
s+φrs)ds)
= E0(e−
∫ (k−1)τ0 λo
sds) ·E0(e
−
∫ (k−1)τ0 φrsds)
On the other hand, E0(e−
∫ (k−1)τ0 λo
sds) is similar to the pricing formula of
a zero coupon bond maturing at time tk−1. According to the Feynman-Kac
theorem, E0(e−
∫ (k−1)τ0 λo
sds) is the solution of a partial differential equation.
E0(e−
∫ (k−1)τ0 λo
sds) = emλ(0,(k−1)τ)−nλ(0,(k−1)τ)λ00 (95)
where
nλ(t, T ) =1− e−aλ(T−t)
aλ(96)
mλ(t, T ) =
(nλ(t, T )− T + t)(a2
λbλ−
σ2λ
2 )
a2λ
−σ2λn2λ(t, T )
4aλ
(97)
From (53), we have
d (φrt) = a′(b′ − φrt)dt+ σ′rdWt (98)
where
a′ = a (99)
b′ = bφ (100)
σ′r = φσr (101)
31
Similarly,
E0(e−
∫ (k−1)τ0 φrsds) = em
′(0,(k−1)τ)−n′(0,(k−1)τ)φr0 (102)
where
n′(t, T ) =1− e−a
′(T−t)
a′(103)
m′(t, T ) =(n′(t, T )− T + t)(a′2b′ −
σ′2r
2 )
a′2−σ′2rn′2(t, T )
4a′(104)
Therefore,
E0(e−
∫ (k−1)τ0 λ(s)ds |Xk) = emλ(0,(k−1)τ)−nλ(0,(k−1)τ)λ00 ·em
′(0,(k−1)τ)−n′(0,(k−1)τ)φr0
(105)Since λot is independent of default-free interest rate,
E0(e−
∫kτ
0λ(s)ds |Xk) = E0(e
−
∫kτ
0λosds) · E0(e
−
∫kτ
0φrsds |Xk)
Again we have,
E0(e−
∫kτ
0λosds) = emλ(0,kτ)−nλ(0,kτ)λ
00 (106)
Define a savings account
B(t) = e∫t
0rsds (107)
Under risk-neutral probability measure, the relative price B(t,T )B(t) is a mar-
tingale, and follows
d
(B(t, kτ)
B(t)
)=
B(t, kτ)
B(t)(−σrn(t, kτ))dWt (108)
So
B(kτ, kτ)
B(kτ)=B(0, kτ)
B(0)exp(
∫ kτ
0
−σrn(t, kτ)dWs −1
2
∫ kτ
0
(−σrn(t, kτ))2ds)
(109)Since B(0) = 1, B(tk, tk) = 1,
1
B(kτ)B(0, kτ)= exp(
∫ kτ
0
−σrn(t, kτ)dWs −1
2
∫ kτ
0
(−σrn(t, kτ))2ds) (110)
1
B(kτ)φB(0, kτ)φ= exp(
∫ kτ
0
−φσrn(s, kτ)dWs −1
2
∫ kτ
0
φ (−σrn(s, kτ))2 ds)
= ξ (kτ) exp(1
2
∫ kτ
0
φ(φ− 1)σ2rn2(s, kτ)ds)
32
where
ξ (kτ) = exp(
∫ kτ
0
−φσrn(s, kτ)dWs −1
2
∫ kτ
0
(−φσrn(s, kτ))2ds) (111)
Hence, we have
E0(e−
∫kτ
0φrsds |Xk) =
1
Pr(Xk)E0(e
−
∫kτ
0φrsds1Xk
)
=B(0, kτ)φ
Pr(Xk)E0(
1
B(kτ)φB(0, kτ)φ1Xk
)
=B(0, kτ)φ
Pr(Xk)E0(ξ (kτ)1Xk
) exp(1
2
∫ kτ
0
φ(φ− 1)σ2rn2(s, kτ)ds)
It can be shown that
B(0, kτ)φ exp(1
2
∫ kτ
0
φ(φ− 1)σ2rn2(s, kτ)ds) = em
′(0,kτ)−n′(0,kτ)φr0 (112)
m′ (0, tk) and n′ (0, tk) are defined in (103) and (104).Define a new probability measure P tk :
P kτ (A) = E(ξ (kτ)1Xk), ∀A ∈ Fkτ (113)
Under P tk ,
W kτt =Wt + φσr
∫ t
0
n(s, kτ)ds (114)
is a Brownian motion. The default-free interest rate under the new proba-bility measure follows:
drt = a(b−φσ2ra
n(t, kτ)− rt)dt+ σrdWkτt (115)
Similarly,
E0(ξ (kτ)1Xk) = Pkτ (Xk) = N(
ln(1− l)(1− h)/ (1− θ)− µ′kσ′k
)
where
33
(116)
µ′k = (m(kτ + δ, T )−mk−1(T )) +1− e−a(τk+δ)
a(be−a(T−kτ−δ) + e−a(k−1)τ (r0 − b))
−n (kτ + δ, T )(η (kτ + δ, tk + δ)− η ((k − 1)τ , kτ + δ) e−a(τk+δ
)
σ′k =
√(1− e−a(τk+δ)
aσr
)2(1− e−2a(k−1)τ
2a) + (n(kτ + δ, T )σr)
2(1− e−2a(τk+δ)
2a)
η (s, t) =φσ2ra2
[1
2e−a(t−s)(1− e−2as)− (1− e−as)
](117)
where δ is time to capture, and θ is the percentage representing liquidity.Hence,
E0(e−
∫kτ
0 λ(s)ds |Xk) = emλ(0,kτ)−nλ(0,kτ)λ00 · em
′(0,kτ)−n′(0,kτ)φr0 ·Pkτ (Xk)
Pr(Xk)(118)
Pr(Yk |Xk) = emλ(0,(k−1)τ)−nλ(0,(k−1)τ)λ00 · em
′(0,(k−1)τ)−n′(0,tk−1)φr0(119)
−emλ(0,kτ)−nλ(0,kτ)λ00 · em
′(0,kτ)−n′(0,kτ)φr0 ·Pkτ (Xk)
Pr(Xk)
When φ = 0, there is no correlation between the interest rate and the inten-sity of default, therefore we have
Pr(Yk |Xk) = emλ(0,(k−1)τ)−nλ(0,(k−1)τ)λ00 − emλ(0,kτ)−nλ(0,kτ)λ
00 (120)
The simplest case is that the intensity of the default process is a constant λ,in that case, Pr(Yk |Xk) can be written as:
Pr(Yk |Xk) = E0(e−
∫ (k−1)τ0 λ(s)ds |Xk)−E0(e
−
∫kτ
0λ(s)ds |Xk)
= e−λ(k−1)τ (1− e−λkτ ) (121)
Denote the probability that default will happen within one year,
Q = 1− e−λ (122)
Then
Pr(Yk |Xk) = (1−Q)ι1+ι2+...+ιk−1(1− (1−Q)kι)
≈k−1∏i=1
(1−Qιi) ·Qιk (123)
34
(123) is a very good approximation for high frequent marking to market.Finally we have that the total probability of having a loss larger than L is:
P =∑
P (k) (124)
where
P (k) = emλ(0,(k−1)τ)−nλ(0,(k−1)τ)λ00
·em′(0,(k−1)τ)−n′(0,(k−1)τ)φr0 · Pr(Xk) (125)
−emλ(0,kτ)−nλ(0,kτ)λ00 · em
′(0,kτ)−n′(0,kτ)φr0 · P kτ (Xk)
35
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N° 44: Nonparametric Tests for Positive Quadrant Dependence Michel DENUIT, Université Catholique de Louvain, Olivier SCAILLET, HEC Genève and FAME, University of Geneva; March 2002
N° 43: Valuation of Sovereign Debt with Strategic Defaulting and Rescheduling Michael WESTPHALEN, École des HEC, University of Lausanne and FAME; February 2002
N° 42: Liquidity and Credit Risk Jan ERICSSON, McGill University and Olivier RENAULT, London School of Economics; August 2001
N° 41: Testing for Concordance Ordering Ana C. CEBRIÁN, Universidad de Zaragoza, Michel DENUIT, Université Catholique de Louvain, Olivier SCAILLET, HEC Genève and FAME, University of Geneva; March 2002
N° 40: Immunization of Bond Portfolios: Some New Results Olivier de La GRANDVILLE, University of Geneva; February 2002
N° 39: Weak Convergence of Hedging Strategies of Contingent Claims Jean-Luc PRIGENT, Thema, Université de Cergy-Pontoise; Olivier SCAILLET, HEC Genève and FAME, University of Geneva; January 2002
N° 38: Indirect Estimation of the Parameters of Agent Based Models of Financial Markets Manfred GILLI, University of Geneva; Peter WINKER, International University in Germany; November 2001
N° 37: How To Diversify Internationally? A comparison of conditional and unconditional asset allocation methods. Laurent BARRAS, HEC-University of Geneva, International Center FAME; Dušan ISAKOV, HEC-University of Geneva, International Center FAME; November 2001
N° 36: Coping with Credit Risk Henri LOUBERGÉ, University of Geneva, Harris SCHLESINGER, University of Alabama; October 2001
N° 35: Country, Sector or Style: What matters most when constructing Global Equity Portfolios? An empirical investigation from 1990-2001. Foort HAMELINK, Lombard Odier & Cie and Vrije Universiteit; Hélène HARASTY, Lombard Odier & Cie; Pierre HILLION, Insead (Singapore), Academic Advisor to Lombard Odier & Cie; October 2001
N° 34: Variable Selection for Portfolio Choice Yacine AÏT-SAHALIA, Princeton University & NBER, and Michael W. BRANDT, Wharton School, University of Pennsylvania & NBER; February 2001 (Please note: The complete paper is available from the Journal of Finance 56, 1297-1351.)
N° 33: The Characteristics of Individual Analysts’ Forecast in Europe Guido BOLLIGER, University of Neuchâtel and FAME; July 2001
N° 32: Portfolio Diversification: Alive and Well in Euroland Kpaté ADJAOUTE, HSBC Republic Bank (Suisse), SA and Jean-Pierre DANTHINE, University of Lausanne, CEPR and FAME; July 2001
N° 31: EMU and Portfolio Diversification Opportunities Kpate ADJAOUTÉ, Morgan Stanley Capital International, Geneva and Jean-Pierre DANTHINE, University of Lausanne, CEPR and FAME; April 2000
N° 30: Serial and Parallel Krylov Methods for Implicit Finite Difference Schemes Arising in Multivariate Option Pricing Manfred GILLI, University of Geneva, Evis KËLLEZI, University of Geneva and FAME, Giorgio PAULETTO, University of Geneva; March 2001
N° 29: Liquidation Risk Alexandre ZIEGLER, HEC-University of Lausanne, Darrell DUFFIE, The Graduate School of Business, Stanford University; April 2001
N° 28: Defaultable Security Valuation and Model Risk Aydin AKGUN, University of Lausanne and FAME; March 2001
N° 27: On Swiss Timing and Selectivity: in the Quest of Alpha François-Serge LHABITANT, HEC-University of Lausanne and Thunderbird, The American Graduate School of International Management; March 2001
N° 26: Hedging Housing Risk Peter ENGLUND, Stockholm School of Economics, Min HWANG and John M. QUIGLEY, University of California, Berkeley; December 2000
N° 25: An Incentive Problem in the Dynamic Theory of Banking Ernst-Ludwig VON THADDEN, DEEP, University of Lausanne and CEPR; December 2000
N° 24: Assessing Market Risk for Hedge Funds and Hedge Funds Portfolios François-Serge LHABITANT, Union Bancaire Privée and Thunderbird, the American Graduate School of International Management; March 2001
N° 23: On the Informational Content of Changing Risk for Dynamic Asset Allocation Giovanni BARONE-ADESI, Patrick GAGLIARDINI and Fabio TROJANI, Università della Svizzera Italiana; March 2000
N° 22: The Long-run Performance of Seasoned Equity Offerings with Rights: Evidence From the Swiss Market Michel DUBOIS and Pierre JEANNERET, University of Neuchatel; January 2000
N° 21: Optimal International Diversification: Theory and Practice from a Swiss Investor's Perspective Foort HAMELINK, Tilburg University and Lombard Odier & Cie; December 2000
N° 20: A Heuristic Approach to Portfolio Optimization Evis KËLLEZI; University of Geneva and FAME, Manfred GILLI, University of Geneva; October 2000
N° 19: Banking, Commerce, and Antitrust Stefan Arping; University of Lausanne; August 2000
N° 18: Extreme Value Theory for Tail-Related Risk Measures Evis KËLLEZI; University of Geneva and FAME, Manfred GILLI, University of Geneva; October 2000
N° 17: International CAPM with Regime Switching GARCH Parameters Lorenzo CAPIELLO, The Graduate Institute of International Studies; Tom A. FEARNLEY, The Graduate Institute of International Studies and FAME; July 2000
N° 16: Prospect Theory and Asset Prices Nicholas BARBERIS, University of Chicago; Ming HUANG, Stanford University; Tano SANTOS, University of Chicago; September 2000
N° 15: Evolution of Market Uncertainty around Earnings Announcements Dušan ISAKOV, University of Geneva and FAME; Christophe PÉRIGNON, HEC-University of Geneva and FAME; June 2000
N° 14: Credit Spread Specification and the Pricing of Spread Options Nicolas MOUGEOT, IBFM-University of Lausanne and FAME; May 2000
N° 13: European Financial Markets After EMU: A First Assessment Jean-Pierre DANTHINE, Ernst-Ludwig VON THADDEN, DEEP, Université de Lausanne and CEPR and Francesco GIAVAZZI, Università Bocconi, Milan, and CEPR; March 2000
N° 12: Do fixed income securities also show asymmetric effects in conditional second moments? Lorenzo CAPIELLO, The Graduate Institute of International Studies; January 2000
N° 11: Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets George CHACKO, Harvard University; Luis VICEIRA, Harvard University; September 1999
N° 10: Assessing Asset Pricing Anomalies Michael J. BRENNAN, University of California, Los Angeles; Yihong XIA, University of Pennsylvania; July 1999
N° 9: Recovery Risk in Stock Returns Aydin AKGUN, University of Lausanne & FAME; Rajna GIBSON, University of Lausanne; July 1999
N° 8: Option pricing and replication with transaction costs and dividends Stylianos PERRAKIS, University of Ottawa; Jean LEFOLL, University of Geneva; July1999
N° 7: Optimal catastrophe insurance with multiple catastrophes Henri LOUBERGÉ, University of Geneva; Harris SCHLESINGER, University of Alabama; September 1999
N° 6: Systematic patterns before and after large price changes: Evidence from high frequency data from the Paris Bourse Foort HAMELINK, Tilburg University; May 1999
N° 5: Who Should Buy Long-Term Bonds? John CAMPBELL, Harvard University; Luis VICEIRA, Harvard Business School; October 1998
N° 4: Capital Asset Pricing Model and Changes in Volatility André Oliveira SANTOS, Graduate Institute of International Studies, Geneva; September 1998
N° 3: Real Options as a Tool in the Decision to Relocate: An Application to the Banking Industry Pascal BOTTERON, HEC-University of Lausanne; January 2000
N° 2: Application of simple technical trading rules to Swiss stock prices: Is it profitable? Dušan ISAKOV, HEC-Université de Genève; Marc HOLLISTEIN, Banque Cantonale de Genève; January 1999
N° 1:
Enhancing portfolio performance using option strategies: why beating the market is easy. François-Serge LHABITANT, HEC-University of Lausanne; December 1998
International Center FAME - Partner Institutions
The University of Geneva The University of Geneva, originally known as the Academy of Geneva, was founded in 1559 by Jean Calvin and Theodore de Beze. In 1873, The Academy of Geneva became the University of Geneva with the creation of a medical school. The Faculty of Economic and Social Sciences was created in 1915. The university is now composed of seven faculties of science; medicine; arts; law; economic and social sciences; psychology; education, and theology. It also includes a school of translation and interpretation; an institute of architecture; seven interdisciplinary centers and six associated institutes.
More than 13’000 students, the majority being foreigners, are enrolled in the various programs from the licence to high-level doctorates. A staff of more than 2’500 persons (professors, lecturers and assistants) is dedicated to the transmission and advancement of scientific knowledge through teaching as well as fundamental and applied research. The University of Geneva has been able to preserve the ancient European tradition of an academic community located in the heart of the city. This favors not only interaction between students, but also their integration in the population and in their participation of the particularly rich artistic and cultural life. http://www.unige.ch The University of Lausanne Founded as an academy in 1537, the University of Lausanne (UNIL) is a modern institution of higher education and advanced research. Together with the neighboring Federal Polytechnic Institute of Lausanne, it comprises vast facilities and extends its influence beyond the city and the canton into regional, national, and international spheres. Lausanne is a comprehensive university composed of seven Schools and Faculties: religious studies; law; arts; social and political sciences; business; science and medicine. With its 9’000 students, it is a medium-sized institution able to foster contact between students and professors as well as to encourage interdisciplinary work. The five humanities faculties and the science faculty are situated on the shores of Lake Leman in the Dorigny plains, a magnificent area of forest and fields that may have inspired the landscape depicted in Brueghel the Elder's masterpiece, the Harvesters. The institutes and various centers of the School of Medicine are grouped around the hospitals in the center of Lausanne. The Institute of Biochemistry is located in Epalinges, in the northern hills overlooking the city. http://www.unil.ch The Graduate Institute of International Studies The Graduate Institute of International Studies is a teaching and research institution devoted to the study of international relations at the graduate level. It was founded in 1927 by Professor William Rappard to contribute through scholarships to the experience of international co-operation which the establishment of the League of Nations in Geneva represented at that time. The Institute is a self-governing foundation closely connected with, but independent of, the University of Geneva. The Institute attempts to be both international and pluridisciplinary. The subjects in its curriculum, the composition of its teaching staff and the diversity of origin of its student body, confer upon it its international character. Professors teaching at the Institute come from all regions of the world, and the approximately 650 students arrive from some 60 different countries. Its international character is further emphasized by the use of both English and French as working languages. Its pluralistic approach - which draws upon the methods of economics, history, law, and political science -reflects its aim to provide a broad approach and in-depth understanding of international relations in general. http://heiwww.unige.ch
Prospect Theoryand Asset PricesNicholas BARBERISUniversity of Chicago
Ming HUANGStanford University
Tano SANTOSUniversity of Chicago
2000 FAME Research PrizeResearch Paper N° 16
FAME - International Center for Financial Asset Management and Engineering
THE GRADUATE INSTITUTE OFINTERNATIONAL STUDIES
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