Risk management for heavy tails, black swans and other …actuaries.org/FUND/Bogota/jose_garrido.pdf · 2018-05-22 · Risk management for heavy tails, black swans and other catastrophes

1. Classical risk management2. Risk management in L0

Risk management for heavy tails, black

swans and other catastrophes

Jose Garrido

Department of Mathematics and StatisticsConcordia University, Montreal, Canada

III Congreso Internacional de ActuarıaUniversidad de los AndesNovember 21–22, 2011

(joint work with Prof. A. Balbas, U. Carlos III of Madrid, Spain)

Research funded by the Natural Sciences and Engineering Research Council of Canada (NSERC)

Jose Garrido – Concordia University Risk management for heavy tails


[Source: The (CAS) Actuarial Review, May 2011]



Mathematics and the actuary



[Source: The (CAS) Actuarial Review, August 2011]



The perfect answer and the actuary



Abstract

Risk measures are commonly used now in actuarial andfinancial risk management alike, for problems such as pricing,reinsurance, capital allocations, portfolio management orcredit risk.

With the notable exception of Value at Risk (VaR), most wellaccepted measures apply only to risks with finite moments.Mathematically this restricts the set of risks random variablesto Lp, for some p ≥ 1, which excludes heavy tailed risks in L0.

Without getting into the subjective choice of what propertiesare reasonable for a risk measure, we revisit the riskmanagement problem for heavy tailed risks through a personalsurvey of ideas in functional analysis and convex optimization.



Abstract






Abstract






Overview:

1. Classical risk management1.1 Introduction1.2 Random variables in Lp

1.3 Risk measures in Lp

2. Risk management in L0

2.1 Heavy tail risks2.2 Random variables in L0

2.3 Risk measures in L0



1.1 Introduction1.2 Random variables in Lp


1. Classical risk management

1.1 Introduction

Does randomness really exist in nature or is it just aconvenient mathematical model?

For instance, the weather,genetic mutations, birth/death?

Oxford English Dictionary1 randomness: Having no definiteaim or purpose; not sent or guided in a particular direction;made, done, occurring, etc., without method or consciouschoice; haphazard.

1http://www.oed.com/viewdictionaryentry/Entry/270766#contentWrapper






1.1 Introduction

Does randomness really exist in nature or is it just aconvenient mathematical model? For instance, the weather,

genetic mutations, birth/death?








1.1 Introduction

Does randomness really exist in nature or is it just aconvenient mathematical model? For instance, the weather,genetic mutations,

birth/death?








1.1 Introduction

Does randomness really exist in nature or is it just aconvenient mathematical model? For instance, the weather,genetic mutations, birth/death?








1.1 Introduction

Does randomness really exist in nature or is it just aconvenient mathematical model? For instance, the weather,genetic mutations, birth/death?







What is a risk?

(Source: http://en.wikipedia.org/wiki/Risk)

. . . a life threatening hazard?





What is a risk?

(Source: http://en.wikipedia.org/wiki/Risk)

. . . a life threatening hazard?





What is a heavy tail risk?

The Actuarial Review

The Coming Storms

n a sunny day in the fall of 2008, the natural surroundings of my Atlanta home were so tranquil and beautiful that it made me forget for a moment all the financial market turmoil

that was going on.

Fast forward to early 2011. The U.S. economy now appears to be on a gradual recovery, especially if you look at the impressive run up of the stock markets. Suddenly on March 11, 2011, a devastating earthquake and tsunami hit Japan. The horrific scenes of natural disaster and human tragedy made my stomach wrench.

Now, looking ahead to the coming years, I sense an impending storm gathering. It is likely a perfect storm, combining natural disasters on a larger scale than the Japan earthquake and a second-dip market meltdown worse than the fall 2008 financial crisis.

The clouds for the coming storm are visible:

1) Four of the five costliest earthquakes and tsunamis of the last 30 years have occurred in the last 13 months. There is a geophysical linkage (via crustal plates) of Chile, New Zealand, Japan, and the Northwest U.S. as the Pacific “Ring of Fire.” Given the recent earthquakes in Haiti (January 12, 2010), Chile (February 27, 2010), New Zealand (September 4, 2010, and February 21, 2011), and Japan (March 11, 2011), the conditional probability of an earthquake within the next year in the northwest U.S. has increased significantly due to the changing pressures on the crustal plates.1,2

2) The world is on a brink of severe shortage of food and water, due to growing demand that supplies cannot keep up with. The world population has reached a new peak of nearly seven billion. The population growth rate is faster than exponential growth, given that the one billion mark was first reached around the year 1800. This population growth coincides with industrial consumptions for agricultural and water resources. Meanwhile the earth is facing the prospect of severe drought in the arable land areas (despite the floods in Australia).3 Severe drought and water scarcity will affect the food availability and cause

spikes in food prices.

3) There are other potential threats of natural and man-made disasters. NASA warns that solar flares from a “huge space storm” might cause devastation. Solar flares would be like “a bolt of lightning” and may cause disruptions to the communication and navigation systems. AR readers may have seen the profile of Nolan Asch’s nonactuarial pursuits in the February issue (“Saving the World from Asteroids”). These may be less likely than the ring of fire earthquake scenarios, but nevertheless represent possible

O

OPINIONSHAUN WANG

1

earth-core.html.2

3






What is a heavy tail risk?

The Actuarial Review

The Coming Storms

n a sunny day in the fall of 2008, the natural surroundings of my Atlanta home were so tranquil and beautiful that it made me forget for a moment all the financial market turmoil

that was going on.

Fast forward to early 2011. The U.S. economy now appears to be on a gradual recovery, especially if you look at the impressive run up of the stock markets. Suddenly on March 11, 2011, a devastating earthquake and tsunami hit Japan. The horrific scenes of natural disaster and human tragedy made my stomach wrench.

Now, looking ahead to the coming years, I sense an impending storm gathering. It is likely a perfect storm, combining natural disasters on a larger scale than the Japan earthquake and a second-dip market meltdown worse than the fall 2008 financial crisis.

The clouds for the coming storm are visible:

1) Four of the five costliest earthquakes and tsunamis of the last 30 years have occurred in the last 13 months. There is a geophysical linkage (via crustal plates) of Chile, New Zealand, Japan, and the Northwest U.S. as the Pacific “Ring of Fire.” Given the recent earthquakes in Haiti (January 12, 2010), Chile (February 27, 2010), New Zealand (September 4, 2010, and February 21, 2011), and Japan (March 11, 2011), the conditional probability of an earthquake within the next year in the northwest U.S. has increased significantly due to the changing pressures on the crustal plates.1,2

2) The world is on a brink of severe shortage of food and water, due to growing demand that supplies cannot keep up with. The world population has reached a new peak of nearly seven billion. The population growth rate is faster than exponential growth, given that the one billion mark was first reached around the year 1800. This population growth coincides with industrial consumptions for agricultural and water resources. Meanwhile the earth is facing the prospect of severe drought in the arable land areas (despite the floods in Australia).3 Severe drought and water scarcity will affect the food availability and cause

spikes in food prices.

3) There are other potential threats of natural and man-made disasters. NASA warns that solar flares from a “huge space storm” might cause devastation. Solar flares would be like “a bolt of lightning” and may cause disruptions to the communication and navigation systems. AR readers may have seen the profile of Nolan Asch’s nonactuarial pursuits in the February issue (“Saving the World from Asteroids”). These may be less likely than the ring of fire earthquake scenarios, but nevertheless represent possible

O

OPINIONSHAUN WANG

1

earth-core.html.2

3










Risk

Wikipedia1: Risk is the potential that a chosen action oractivity (including the choice of inaction) will lead to a loss(an undesirable outcome).

The notion implies that a choice having an influence on theoutcome exists (or existed). Potential losses themselves mayalso be called “risks”. Almost any human endeavour carriessome risk, but some are much more risky than others.

1http://en.wikipedia.org/wiki/RiskJose Garrido – Concordia University Risk management for heavy tails




Mathematical formalism

A financial risk is defined as a loss random variable.

A random variable X : Ω → R is a real–valued function,defined on a sample space Ω, that is F -measurable (aσ-algebra of events).

A probability measure P , on F , completes the definition of aprobability space (Ω,F ,P) and induces a probability measureP on the Borel sets σ(R):

PX ≤ x = Pω ∈ Ω; X (ω) ≤ x, x ∈ R

and a probability distribution FX of X on R as follows:

FX (x) = PX ≤ x, x ∈ R.





Examples:

A financial loss is, for instance, the difference

Xt = A− Pt

between the acquisition cost of a portfolio, A, and its currentmarket value Pt .

A pension fund manager needs to assess the risk of his/herportfolio. What losses are tolerable, which ones are not?

How can he/she measure the loss risk?





Examples:


Xt = A− Pt








Examples:


Xt = A− Pt








1.2 Random variables in Lp

Take p ∈ [1,∞) and q ∈ (1,∞) to be conjugate numbers

1

p+

1

q= 1

with q = ∞ if p = 1.

Lp (Lq) the usual Banach space of random variables such thatE(|X |p) < ∞ (E(|X |q) < ∞, or if p = 1, then X is essentiallybounded, as q = ∞).





1.2 Random variables in Lp

Take p ∈ [1,∞) and q ∈ (1,∞) to be conjugate numbers

1

p+

1

q= 1

with q = ∞ if p = 1.

Lp (Lq) the usual Banach space of random variables such thatE(|X |p) < ∞ (E(|X |q) < ∞, or if p = 1, then X is essentiallybounded, as q = ∞).





The dual

Note that these form a decreasing sequence of spaces

1 ≤ p1 ≤ p2 ≤ ∞ =⇒ Lp1 ⊃ Lp2 .

By Riesz’ representation theorem: Lq is the dual space of Lp

(Rudin, 1973, McG-H).





The dual

Note that these form a decreasing sequence of spaces

1 ≤ p1 ≤ p2 ≤ ∞ =⇒ Lp1 ⊃ Lp2 .

By Riesz’ representation theorem: Lq is the dual space of Lp

(Rudin, 1973, McG-H).Jose Garrido – Concordia University Risk management for heavy tails





Risk measures are commonly characterized through axioms:homogeneity, translation invariance, sub–additivity, etc. Thesebeing properties of the risk function.

In the mathematical finance literature, risk measures are alsorepresented through classical optimization arguments used infunctional analysis. The focus being on the space of randomvariables (risks), rather than on the properties of a particularrisk measure.

The two approaches are equivalent!





















Families of risk measures

A risk measure is a function ρ : Lp → R used to control therisk level of loss X ∈ Lp, say at the end of a period [0, T ].

What real–valued function ρ is a good risk measure?

To extend the discussion to insurance losses consider generalrisk functions used by traders/insurers to control the risk of afinal wealth X ∈ Lp at the end of a period [0, T ].

The set

∆ρ = Y ∈ Lq ; −E(XY ) ≤ ρ(X ), ∀X ∈ Lp,is convex.

With the representation theorem of risk measures ofRockafellar et al. (2006, Fin. & Stoch.) the following theoremcan be proved.









The set











The set











The set











The set







Theorem 1

The 2 following assertions are equivalent:

1. ∆ρ is convex and σ(Lq, Lp)-compact,

ρ(X ) = max−E(XY ) : Y ∈ ∆ρ

holds for every X ∈ Lp, there exists E ≥ 0 in R such that

∆ρ ⊂ Y ∈ Lq ; E(Y ) = E,

and the constant (0-variance) random variable Y = Ea.s. belongs to ∆ρ.





Theorem 1 (. . . continued)

2. ρ is continuous and satisfies:

(a) (translation invariance) ρ(X + k) = ρ(X )− E k, for everyX ∈ Lp and k ∈ R,

(b) (sub–aditivity) ρ(X1 + X2) ≤ ρ(X1) + ρ(X2), for everyX1, X2 ∈ Lp,

(c) (homogeneity) ρ(αX ) = αρ(X ), for every X ∈ Lp andα > 0,

(d) (mean dominated) ρ(X ) ≥ −E E(X ), for every X ∈ Lp.





Coherent risk measures

It is easily shown that if ρ satisfies (a)–(d) above with E = 1then it is coherent in the sense of Artzner et al. (1999,Math. Fin.).

That is ρ is decreasing if and only if

∆ρ ⊂ Lq+ = Y ∈ Lq ; P(Y ≥ 0) = 1.





Examples

Interesting examples of coherent continuous risk measuressatisfying (a)–(d) above with E = 1 are (among many others):

I Conditional Value at Risk (CVaR): defined in L1

CVaRα(X ) = −E[X |X < −VaRα(X )

],

where VaRα(X ) = − infx ; P(X ≤ x) > 1− α can beextended even on L0, but is not coherent,





More examples

I Weighted Value at Risk (WVaR): defined on L1

WVaRg (X ) =∫ 1

0VaRt(X )dg(t), for an appropriate

weighting function such that g(1)− g(0) = 1,

forexample the Dual Power Transform (DPTa(X )) is basedon WVaR for g(t) = 1− (1− t)a, a > 1, or the measureof Wang (2000, J.Risk & Ins.) for g(t) = Φ

[a + Φ−1(t)

],

a < 0, is defined on L2.

I Weighted Conditional Value at Risk (WCVaR):

WCVaRa(X ) =∫ 1

0CVaRt(X )dg(t), of Rockafeller et

al. (2006, Fin. & Stoch.) is also defined on L1.





More examples


WVaRg (X ) =∫ 1


weighting function such that g(1)− g(0) = 1, forexample the Dual Power Transform (DPTa(X )) is basedon WVaR for g(t) = 1− (1− t)a, a > 1,

or the measureof Wang (2000, J.Risk & Ins.) for g(t) = Φ

[a + Φ−1(t)

],



WCVaRa(X ) =∫ 1







More examples


WVaRg (X ) =∫ 1


weighting function such that g(1)− g(0) = 1, forexample the Dual Power Transform (DPTa(X )) is basedon WVaR for g(t) = 1− (1− t)a, a > 1, or the measureof Wang (2000, J.Risk & Ins.) for g(t) = Φ

[a + Φ−1(t)

],



WCVaRa(X ) =∫ 1







More examples


WVaRg (X ) =∫ 1


weighting function such that g(1)− g(0) = 1, forexample the Dual Power Transform (DPTa(X )) is basedon WVaR for g(t) = 1− (1− t)a, a > 1, or the measureof Wang (2000, J.Risk & Ins.) for g(t) = Φ

[a + Φ−1(t)

],



WCVaRa(X ) =∫ 1







Other representations

As properties (a)–(d) do not draw consensus, other families ofrisk measures have been proposed, again motivated by naturalproperties these must satisfy, and then justified throughrepresentation theorems like Theorem 1.

Particular examples are the expectation bounded and thedeviation risk measures of Rockafeller et al. (2006, Fin. &Stoch.), which are continuous risk measures satisfying (a)–(d)above with E = 0, such as the p−deviation, given by

ρ(X ) =[E

(∣∣E(X )− X∣∣p)]1/p

,





Other representations

As properties (a)–(d) do not draw consensus, other families ofrisk measures have been proposed, again motivated by naturalproperties these must satisfy, and then justified throughrepresentation theorems like Theorem 1.

Particular examples are the expectation bounded and thedeviation risk measures of Rockafeller et al. (2006, Fin. &Stoch.), which are continuous risk measures satisfying (a)–(d)above with E = 0, such as the p−deviation, given by

ρ(X ) =[E

(∣∣E(X )− X∣∣p)]1/p

,





Deviation measures

or the downside p−semi deviation:

ρ(X ) =[E

(∣∣ maxE(X )− X , 0∣∣p)]1/p

,

again, among many others.





Other applications of this representation

Such representations of risk measures based on convexoptimization have been successfully used in:

I reinsurance: Balbas et al. (2009, IME and 2011, EJOR)study the optimal (minimal risk) reinsurance design for abroad class of risk measures,

I credit risk: Okhrati et al. (2011, pre-print) define aρ–arbitrage for bond markets under general (finitevariation Levy) contingent claims. Among other things, itcan be used for credit rating of corporate bonds.





Other applications of this representation

Such representations of risk measures based on convexoptimization have been successfully used in:

I reinsurance: Balbas et al. (2009, IME and 2011, EJOR)study the optimal (minimal risk) reinsurance design for abroad class of risk measures,

I credit risk: Okhrati et al. (2011, pre-print) define aρ–arbitrage for bond markets under general (finitevariation Levy) contingent claims. Among other things, itcan be used for credit rating of corporate bonds.






All the above risk measures require at least a finite mean ofthe final wealth; X ∈ L1.

Actuarial risk can be heavy tailed (reinsurance costs in Japan).

Figure: Pareto distributions(Source: http://en.wikipedia.org/wiki/Pareto distribution)






All the above risk measures require at least a finite mean ofthe final wealth; X ∈ L1.

Actuarial risk can be heavy tailed (reinsurance costs in Japan).

Figure: Pareto distributions(Source: http://en.wikipedia.org/wiki/Pareto distribution)





2.1 Heavy tail risks

For X ∼Pareto(k , θ):

E(X ) =θ

k − 1, k > 1,

V(X ) = E(X )θ

(k − 2), k > 2,

VaRα(X ) = θ[(1− α)−1/k − 1

],

CVaRα(X ) = VaRα(X ) +θ(1− α)−1/k

k − 1, k > 1.





Financial heavy tail risks

Financial risks can exhibit heavier tails than the exponentialdecaying tails of normal distributions (credit risk crisis, blackswans).

Figure: Symmetric α-stable distributions(Source: http://en.wikipedia.org/wiki/Stable distributions)





Financial heavy tail risks

Financial risks can exhibit heavier tails than the exponentialdecaying tails of normal distributions (credit risk crisis, blackswans).

Figure: Symmetric α-stable distributions(Source: http://en.wikipedia.org/wiki/Stable distributions)





What is a heavy tail random variable?

In the actuarial literature a heavy tail distribution usuallymeans a subexponential distribution:

limx→∞

1− F ∗2(x)

1− F (x)= 2.

Subexponential distributions do not admit a momentgenerating function, which is used by some as a definition ofheavy tail.

Subexponential distributions form a wide class that includesslowly varying tail distributions:

1− F (x) ∼ x−γC (x), x →∞.







limx→∞

1− F ∗2(x)

1− F (x)= 2.



1− F (x) ∼ x−γC (x), x →∞.







limx→∞

1− F ∗2(x)

1− F (x)= 2.



1− F (x) ∼ x−γC (x), x →∞.





Other definitions

Some authors call light tailed the random variables that admita moment generating function only on a limited domain:

E(etX ) < ∞, t ∈ (−γ1, γ2),

such as the exponential/gamma or the inverse Gaussian onR+.

Others called them medium tailed.

In any case, the consensus is that distributions that admit amoment generating over all of R are light tail and have finitemoments E(|X |p) < ∞ of all orders (Poisson, normal).

Then in statistics and in applied probability they sometimescall heavy tailed any random variable that is not light tailed.





Other definitions


E(etX ) < ∞, t ∈ (−γ1, γ2),

such as the exponential/gamma or the inverse Gaussian onR+. Others called them medium tailed.







Other definitions


E(etX ) < ∞, t ∈ (−γ1, γ2),








Other definitions


E(etX ) < ∞, t ∈ (−γ1, γ2),








2.2 Random variables in L0

Is it realistic to use random variables with E(|X |) = ∞?

Or is it just a convenient model for heavy tailed losses withrandom variables in L0, the set of all random variables on(Ω,F ,P)?





2.2 Random variables in L0

Is it realistic to use random variables with E(|X |) = ∞?

Or is it just a convenient model for heavy tailed losses withrandom variables in L0, the set of all random variables on(Ω,F ,P)?





(a) Risk theory

(b) L0 risks





(a) Risk theory (b) L0 risks





(a) Risk theory (b) L0 risks





Dual of L0

Can Theorem 1 be extended beyond L1 to the larger L0?

In other words, can we define coherent risk measures ordeviation measures for risks with infinite expectation?

The answer to the first question is no: for 0 ≤ r < 1, Lr is ametric but not Banach space (see Rudin, 1973).

Its dual reduces to zero if P is atomless, so ∆ρ ⊂ 0 andTheorem 1 leads to trivial risk measures.





Dual of L0









Dual of L0









Dual of L0









Theorem 2

Let P be atomless and 0 ≤ r < 1. If ρ : Lr → R is continuousand satisfies (b) and (c) of Theorem 1, then ρ = 0.

The proof follows from Delbaen (2000). He proposesapproximations to substitute for the lack of characterization.

The only risk measures that can be used on L0 are those basedon percentiles, like VaR , as there are no continuous functionson Lr , hence no dual elements.





Theorem 2








Theorem 2








Naive solution

Statisticians routinely use variance reduction transforms, suchas ln X or

√X . Actuaries use similar tail reduction transforms

(Y = 1/X ).

These may lead to simple approximations when X /∈ L1:

E(X ) = E(1

Y) ≈ 1

E(Y )

[1 +

σ(Y )

3

],

where σ(y) = V(Y )/E(Y )2 is the coefficient of variation of Y .





Naive solution

Statisticians routinely use variance reduction transforms, suchas ln X or

√X . Actuaries use similar tail reduction transforms

(Y = 1/X ).

These may lead to simple approximations when X /∈ L1:

E(X ) = E(1

Y) ≈ 1

E(Y )

[1 +

σ(Y )

3

],

where σ(y) = V(Y )/E(Y )2 is the coefficient of variation of Y .






Neslehova, Chavez–Demoulin and Embrechts (2006, JOR):Define empirical risk measures for loss random variables thatinclude the Pareto(α, θ) case with bad exponent α < 1.

These random variables show up in operational risk studies.

If, for risk capital allocation, there is a need for concave (evencoherent) risk measures then there is no immediate solutionfor this problem.





















Other penguins

Delbaen (2009, MF):

“We show that when a real–valued risk measure is defined ona solid, rearrangement invariant space of random variables,then necessarily it satisfies a weak compactness, also calledcontinuity from below, property, and the space necessarilyconsists of integrable random variables . . . ”

“. . . As a result we see that a risk measure defined for, say,Cauchy–distributed random variable, must take infinite valuesfor some of the random variables.”

Uses conjugate Young functions to transform X ∈ L0 into anelement of an Orlicz space. The risk measure is E(X ) (linear,positive) when restricted to the Orlicz space.





Other penguins

Delbaen (2009, MF):








Other penguins

Delbaen (2009, MF):








Other penguins (. . . continued)

In Balbas & G (2011) we look for a representation, like that inTheorem 1, but in Lr , for 0 ≤ r < 1, that can overcomeTheorem 2.

We propose an extension ρ : Lp ⊕ (⊕kj=1Uj) → R, as described

in the following theorem.





Other penguins (. . . continued)

In Balbas & G (2011) we look for a representation, like that inTheorem 1, but in Lr , for 0 ≤ r < 1, that can overcomeTheorem 2.

We propose an extension ρ : Lp ⊕ (⊕kj=1Uj) → R, as described

in the following theorem.





Theorem 3

Consider 0 ≤ r < 1 ≤ p < ∞ and a family of linear spacesU1, . . . Uk , such that Uj ⊂ Lr , for j = 1, 2, . . . , k, andU1, . . . , Uk , L

p Frechet (locally convex, complete andmetric) satisfying:

uj ∈ Uj , for j = 1, . . . , k∑kj=1 uj = 0

⇒ u1 = · · · = uk = 0

for every (u1, . . . , uk) ∈ Lp.

Now suppose that ρj : Lp → R arecontinuous and satisfy the properties of (sub–aditivity)

ρj(v1 + v2) ≤ ρj(v1) + ρj(v2), v1, v2 ∈ Lp, (1)

and (homogeneity)

ρj(αv) = αρj(v), v ∈ Lp. (2)





Theorem 3

Consider 0 ≤ r < 1 ≤ p < ∞ and a family of linear spacesU1, . . . Uk , such that Uj ⊂ Lr , for j = 1, 2, . . . , k, andU1, . . . , Uk , L

p Frechet (locally convex, complete andmetric) satisfying:

uj ∈ Uj , for j = 1, . . . , k∑kj=1 uj = 0

⇒ u1 = · · · = uk = 0

for every (u1, . . . , uk) ∈ Lp. Now suppose that ρj : Lp → R arecontinuous and satisfy the properties of (sub–aditivity)

ρj(v1 + v2) ≤ ρj(v1) + ρj(v2), v1, v2 ∈ Lp, (1)

and (homogeneity)

ρj(αv) = αρj(v), v ∈ Lp. (2)






Then there exists ρ : Lp ⊕ (⊕kj=1Uj) → R such that:

a) ρ extends every ρj ,

b) ρ is continuous and also satisfies (1) and (2),

c) if θ : Lp ⊕ (⊕kj=1Uj) → R satisfies a) and b), above, and

θ ≤ ρ, then θ = ρ (that is ρ is minimal).

























Conclusion

With the construction in Theorem 3, extensions ofcoherent/expectation bounded risk measures are feasiblebeyond L1.

In particular, an extension of CVaR is possible which amountsto the CVaR for risks in L1 and VaR for heavy tailed risks,through a special parameter.





Conclusion

With the construction in Theorem 3, extensions ofcoherent/expectation bounded risk measures are feasiblebeyond L1.

In particular, an extension of CVaR is possible which amountsto the CVaR for risks in L1 and VaR for heavy tailed risks,through a special parameter.





Thank you for your attention!





Bibliography

Artzner, P., Delbaen, F., Eber, J-M. and D. Heath (1999)“Coherent measures of risk”, Mathematical Finance, 9, 3, 203–228.

Balbas, A., Balbas, B. and A. Heras (2009) “Optimal reinsurancewith general risk measures involving risk measures”, EuropeanJournal of Operational Research, 1–29,http://dx.doi.org/10.1016/j.ejor.2011.05.035

Balbas, A., Balbas, B. and A. Heras (2011) “Stable solutions foroptimal reinsurance problems”, Insurance: Mathematics andEconomics, 44, 374–384.

Balbas, A. and J. Garrido (2011) “Heavy tails and risk measures”,preprint.

Delbaen, F. (2000) “Coherent risk measures”, preprint.





Bibliography (. . . continued)

Delbaen, F. (2009) “Risk measures for non–integrable randomvariables”, Mathematical Finance, 19, 329–333.

Neslehova, J., Chavez–Demoulin, V. and Embrechts, P. (2006)“Infinite-mean models and the LDA for operational risk”, Journalof Operational Risk, 1, 1, 3–25.

Okhrati, R., Balbas, A. and J. Garrido (2011) “Defaultable claimsunder finite variation Levy processes”, preprint, submitted toStochatic Processes and their Applications.

Rockafellar, R.T., Uryasev, S. and M. Zabarankin (2006)“Generalized deviations in risk analysis”, Finance and Stochastics,10, 51-74.

Rudin, W. (1973) Functional Analysis, McGraw–Hill.





Bibliography (. . . end)

Wang, S.S. (2000) “A class of distortion operators for financialand insurance risks”, Journal of Risk Insurance, 6, 7, 15-36.


Documents

Risk management for heavy tails, black swans and other …actuaries.org/FUND/Bogota/jose_garrido.pdf · 2018-05-22 · Risk management for heavy tails, black swans and other catastrophes