risk, coherency and cooperative game · harry markowitz’s mpt (developed in 1950-1970, received...

Risk Assessment Coherent Risks: An Axiomatic Approach Relation with Cooperative Game Concluding Remarks

Risk, Coherency and Cooperative Game

Haijun Li

[email protected] of MathematicsWashington State University

Tokyo, June 2015

Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 1 / 30


Outline

1 Risk AssessmentRisk = Volatility?Modern Portfolio Theory (MPT)Risk = Volatility + Loss

2 Coherent Risks: An Axiomatic ApproachCoherent Risk and Its Dual RepresentationExpected ShortfallsRelation Between CVaR and VaR

3 Relation with Cooperative GameConvex GameDistortion

4 Concluding Remarks



Risk Factors and Univariate Risk measures

Fix a probability space (Ω,F ,P) generated by a stochasticsystem (e.g., financial portfolio).

Let L denote a set of random risk factors defined on(Ω,F ,P); e.g., profit/loss variables or returns X in afinancial portfolio.Assume that L is a convex cone.A univariate risk measure % : L 7→ R is a functionalsatisfying some operational properties.Multivariate (or set-valued) risk measures have also beenstudied in the literature.



Although risk measures are closely related to survivabilitymeasures in reliability modeling in engineering and survivalanalysis used in medical fields, we will stick with thecontext of financial applications (due to widely availablepublic data in finance).

Why should we care about good risk measures?

1 Risk assessment %(X) is used to set up capital reserves atbanks (Basel II International Banking Accords, 2004).

2 Risk assessment is widely used in capital allocations indiversification of risky investment.

3 Risk measure % can be used to price financial derivativesand calculate insurance premiums.



Although risk measures are closely related to survivabilitymeasures in reliability modeling in engineering and survivalanalysis used in medical fields, we will stick with thecontext of financial applications (due to widely availablepublic data in finance).Why should we care about good risk measures?

1 Risk assessment %(X) is used to set up capital reserves atbanks (Basel II International Banking Accords, 2004).

2 Risk assessment is widely used in capital allocations indiversification of risky investment.

3 Risk measure % can be used to price financial derivativesand calculate insurance premiums.



Risk = Volatility?

One method for risk assessment is to use some volatilitymeasures, such as the variance (and co-variance).

DefinitionThe variance of a risk factor X on (Ω,F ,P) is defined as

Var(X) :=

∫Ω

(X(ω)− µ)2P(dω) = ||X − µ||L2 = E(X − µ)2,

where µ =∫

X(ω)P(dω) denotes the mean value of X.

The risk measure Var(X) is appropriate for risk factors thathave Gaussian (normal) distributions, because a Gaussiandistribution is completely determined by its mean andvariance.The standard deviation σ =

√Var(X) is also used as a risk

measure.



Risk = Volatility?

One method for risk assessment is to use some volatilitymeasures, such as the variance (and co-variance).

DefinitionThe variance of a risk factor X on (Ω,F ,P) is defined as

Var(X) :=

∫Ω

(X(ω)− µ)2P(dω) = ||X − µ||L2 = E(X − µ)2,

where µ =∫

X(ω)P(dω) denotes the mean value of X.

The risk measure Var(X) is appropriate for risk factors thathave Gaussian (normal) distributions, because a Gaussiandistribution is completely determined by its mean andvariance.The standard deviation σ =

√Var(X) is also used as a risk

measure.Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 5 / 30


Modern Portfolio Theory (MPT)

Harry Markowitz’s MPT (developed in 1950-1970, received theNobel Memorial Prize in 1990) is a mathematical formulation ofdiversification in investing. The basic ingredients are as follows.

Investors are assumed to be rational and financial marketsare assumed to be efficient.Asset’s returns are assumed to be jointly normallydistributed, and so the portfolio return has a Gaussiandistribution.Risk is measured by the standard deviation σ of returns.MPT aims to minimize the variance of the portfolio for agiven level of expected return, by carefully choosing theproportions (weights) of its assets.Investing is a tradeoff between risk and expected return.MPT explains how to find the best possible diversificationstrategy for specific definitions of risk and return.




Mean-Variance Efficient Frontier (mathworks.com)portfolio_fig1_wl.jpg (JPEG Image, 750x643 pixels) http://www.mathworks.com/company/newsletters/news_notes/oct06/imag...

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Does MPT really model financial markets?

The main criticisms:Market participants are not rational (e.g., herding), andmarkets are not efficient (such as information asymmetryand statistical arbitrage). The assumptions of MPT arestrongly challenged by behavioral economists (e.g., RobertShiller).Asset returns are not (jointly) normally distributed.The variance (or standard deviation) is a scale parameterand it cannot be used to measure extreme risk at the tailsof loss distributions.The L2-norm based risk measures, such as the variance,overlook clustering of extreme values that frequentlyappears in financial data.




Financial Log-Return Series (Xn, n ≥ 1)

Stylized FactsExtreme returns appear in clusters.Returns are heavy-tailed (i.e., P(Xn > x) ∼ x−β as x→∞).Return series are not iid although they show little temporalcorrelation. Series of absolute or squared returns showprofound temporal correlation (higher order long-rangedependence).

These facts seem to apply to the majority of risk-factorchanges, such as log-returns on equities, indexes,exchange rates, and commodity prices.These facts hold in multiple time scales: intra-daily, daily,weekly and monthly returns.




Returns are Heavy-Tailed

Histogram of Monthly U.S. SP500 Financials

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−30 −20 −10 0 10 20

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Mean Excess Plot

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Exploratory QQ Plot

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ThresholdFigure: S&P 500 Financials 1990-2010 (Erin Lu, WSU Department ofFinance)




Autocorrelations of U.K. FTSE 100: 1998-2007

showImage.php (PNG Image, 500x500 pixels) http://fedc.wiwi.hu-berlin.de/quantnet/graphics/showImage.php?i=ed360...

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showImage.php (PNG Image, 500x500 pixels) http://fedc.wiwi.hu-berlin.de/quantnet/graphics/showImage.php?i=ed360...

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Figure: Left = returns, Middle = returns2, Right = |returns|




Government National Mortgage Association Data(JPEG Image, 480x480 pixels) https://mail.google.com/mail/?ui=2&ik=2a21981ea2&view=att&th=12f5...

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Figure: Ginnie Mae MBS, 1990-2010 (Erin Lu, WSU Dept of Finance)Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 12 / 30


Risk = Volatility + Loss

Value-at-Risk (VaR) of a Loss Variable X

The VaR with confidence level 1− α (0 ≤ α ≤ 1) is defined as

VaRα(X) := supx : P(X > x) ≥ α.For example, VaRα(X) = σΦ−1(1− α) + µ for a normallydistributed loss X with mean µ and standard deviation σ, whereΦ is the CDF of the standard Gaussian.

var.jpg (JPEG Image, 468x394 pixels) http://www.ima.umn.edu/2009-2010/MM8.2-11.10/var.jpg

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VaR is a widely used risk measure of the risk of loss.VaR was developed by a division at J. P. Morgan in 1994,which was later spun off into an independent for-profitbusiness, called RiskMetrics Group.The Basel II recommends VaR as the preferred measure ofmarket risk.The Actuarial exams test regularly VaRs of various lossdistributions.

VaR enjoys some nice operational properties:1 ∀ X1,X2 ∈ L such that X1 ≤ X2 almost surely,

VaRα(X1) ≤ VaRα(X2).2 ∀ X ∈ L, ∀ λ > 0, VaRα(λX) = λVaRα(X).3 ∀ X ∈ L, ∀ l ∈ R, VaRα(X + l) = VaRα(X) + l.




Drawbacks of VaR

VaR focuses on the manageable risks near the center ofprofit/loss distributions and underestimate tail risk, leadingto excessive risk-taking and over-leverage at financialinstitutions.VaR is not subadditive; that is,

VaRα(X1 + X2) > VaRα(X1) + VaRα(X2), ∃ X1,X2 ∈ L.

That is, the diversification can lead to increase of risk, ifrisk is measured by VaR.VaR is subadditve within the class of loss variables thathave elliptical distributions (including Gaussian,generalized hyperbolic distributions, ...).



Coherent Risk and Its Dual Representation

Univariate Coherent Risk Measures

Let L be the convex cone consisting of all the variables X whichmay represent losses of portfolios at the end of a given period.

Artzner, Delbaen, Eber and Heath (1999)A functional % : L 7→ R is called a coherent risk measure if %satisfies the four coherent axioms:

(monotonicity) For X1,X2 ∈ L with X1 ≤ X2 almost surely,%(X1) ≤ %(X2).(subadditivity) For all X1,X2 ∈ L, %(X1 + X2) ≤ %(X1) +%(X2).(positive homogeneity) For all X ∈ L and every λ > 0,%(λX) = λ%(X).(translation invariance) For all X ∈ L and every l ∈ R,%(X + l) = %(X) + l.




Dual Representation (Delbaen, 2000)Under some regularity conditions (Fatou property, ...), acoherent risk measure %(X) arises as the supremum ofexpected values of loss X under various scenarios:

%(X) = supQ∈S

EQ(X)

where S is a convex set of probability measures on physicalstates, that are absolutely continuous with respect to theunderlying measure P.

Q P (absolute continuity): P(A) = 0 implies thatQ(A) = 0, A ∈ F .Q P implies that Q(A) =

∫A f (ω)P(dω), where f = dQ/dP

is known as the Radon-Nikodym derivative.




Interpretation in Finance

The measure P describes what random events might occur.Any measure Q ∈ S describes likelihoods of a generalized“scenario” in the future that takes some uncertainty (e.g.,interest rate hike, ...) into account. A scenario measure Q,which can be just finitely additive, is also called a distortionmeasure.Any coherent risk measure is the worst expected lossunder a collection of generalized scenarios that couldhappen in the real world.The SPAN margin system (Chicago Mercantile Exchange,1995) considered 16 scenarios (14 “regular scenarios” + 2“extreme scenarios”) to measure risk for marginrequirements.




Another Interpretation ...

Let C := X : %(X) ≤ 0 is clearly a convex cone.C is a set of positions with acceptable risk.

... From Regulator/Supervisor’s Point of View

For any || · ||L∞-closed convex cone C ⊂ L∞(Ω,F ,P) such thatL−∞(Ω,F ,P) ⊂ C,

%(X) = infl : l− X ∈ C

is a coherent risk measure.

The risk %(X) for loss X corresponds to the amount of extracapital requirement that has to be invested in some secureinstrument so that the resulting position %(X)− X is acceptableto regulator/supervisor.




A general theory of coherent risk measures for arbitrary,univariate loss variables was developed in Delbaen (2000).Set-valued coherent risk was studied in Jouini, Meddeband Touzi (2004).The asymptotic properties of set-valued coherent risk werestudied in Joe and Li (2010) using the intensity measuresof multivariate extremes.One can also consider a wider class of the convex riskmeasures that combine sub-additivity and positivehomogeneity into the convexity property. The convex riskmeasures was extended to stochastic processes inCheridito, Delbaen and Klüppelberg (2004), and to ageneral space that may include deterministic, stochastic,single or multi-period cash-stream structures (Föllmer andSchield, 2002).



Expected Shortfalls

Worst Conditional Expectation

Consider the following class of scenario probability measures:

Sα := P(· | A) : P(A) > α,A ∈ F= Q : ||dQ/dP||L∞ ≤ 1/α, 0 < α < 1.

The corresponding coherent risk is called the worst conditionalexpectation and given by,

WCEα(X) = supE(X|A) : P(A) > α,A ∈ F.

Since α is usually small, Sα contains all the scenarioprobability measures conditioning on events with occurringprobability at least α, including rare events.WCEα(X) is the worst expected loss which could beincurred from various random events with occurringprobability at least α.



Expected Shortfalls

Conditional VaR = Tail Conditional Expectation

If the profit/loss variable X is continuous, then

WCEα(X) = E(X|X > VaRα(X)) =: CVaRα(X)

which is called the tail conditional expectation (or conditionalVaR or expected shortfall) with confidence level 1− α.

CVaR has been adopted by the finance and insuranceregulators both in Canada and US.Let t = VaRα(X), then

0 ≤ E(X − t|X > t) = CVaRα(X)− VaRα(X)

is known as the mean residual life widely used in reliabilitymodeling and in survival analysis long before coherent riskwas introduced in 1999.



Relation Between CVaR and VaR

CVaRα(X) = 1α

∫ α0 VaRξ(X)dξ.

For any light-tailed loss variable X (i.e., its distribution tailsdecay exponentially), CVaRα(X) ≈ VaRα(X), as α is small.For a heavy-tailed loss variable X (i.e., P(X > x) ∼ x−β asx→∞), if tail index β > 1, then

CVaRα(X) ≈ β

β − 1VaRα(X), when α is small.

var.jpg (JPEG Image, 468x394 pixels) http://www.ima.umn.edu/2009-2010/MM8.2-11.10/var.jpg

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Relation Between CVaR and VaR

CVaR is Better, But Is CVaR the “Best One”?

Question: VaR is not coherent, but can we find the smallestcoherent risk measure, say %, that dominates VaR (i.e.,%(X) ≥ VaRα(X), ∀X ∈ L)?

Theorem (Delbaen, 2000)Assume that some regularity conditions (such as the Fatouproperty) hold, and 0 < α < 1 and X ∈ L are fixed.

1 VaRα(X) = inf%(X) : % is coherent with % ≥ VaRα.2 If X is continuous and % is coherent, then % ≥ VaRα implies

that % ≥ CVaRα.

Answer: Yes, CVaR is the best one in the sense that (1) it iscoherent and (2) it dominates VaR.



Convex Game

Consider a set function ν : F 7→ R+ with ν(∅) = 0. Intuitively,ν(A) = payoff gained by forming the coalition A ∈ F .

Convex Game (Shapley, 1971)

A set function ν is called a convex game defined on (Ω,F ,P) if1 (supermodular) ν(A∩B)+ν(A∪B) ≥ ν(A)+ν(B), ∀A,B ∈ F .2 (absolutely continuous) for coalitions A,B ∈ F with

P(A = B) = 1, ν(A) = ν(B).

The supermodularity is equivalent to

ν(A ∪ ω)− ν(A) ≤ ν(B ∪ ω)− ν(B), ∀ A ⊆ B, ∀ω ∈ Ω.

That is, one should join a larger coalition because thepayoff is larger.The absolute continuity is equivalent to that P(A) = 0implies that ν(A) = 0.



Convex Game

The Core

Define the core of a convex game ν:

C(ν) := µ : µ is a finitely additive, non-negative measure,µ(Ω) = ν(Ω), µ(A) ≥ ν(A),∀A ∈ F.

Intuitively, the core contains all the profit distribution rules thatmake every member in the game happy (no incentive to leavethe grand coalition).

The core of a convex game is non-empty.The Shapley value (the unique fair profit distribution rule)of a convex game is the center of gravity of its core.P(A) = 0 implies that µ(A) = 0, ∀µ ∈ C(ν) (i.e., µ P).



Distortion

For simplicity, consider only bounded, non-negative lossvariables X defined on (Ω,F ,P). Define

%(X) := supµ∈C(ν)

Eµ(X).

Rewrite %(X) in terms of Choquet integrals, and one has

%(X) =

∫ ∞0

ν(ω : X(ω) > x)dx.

If ν(Ω) = 1, then %(X) is a coherent risk measure.

Example: Let f : [0, 1] 7→ [0, 1] be increasing and concave suchthat f (0) = 0 and f (1) = 1. The the set function ν(A) := f (P(A)),A ∈ F , defines a convex game. The coherent risk measure:

%(X) =

∫ ∞0

f (P(X > x))dx

is known as the distortion risk measure in actuarial science(Denneberg, 1989).



Distortion

For simplicity, consider only bounded, non-negative lossvariables X defined on (Ω,F ,P). Define

%(X) := supµ∈C(ν)

Eµ(X).

Rewrite %(X) in terms of Choquet integrals, and one has

%(X) =

∫ ∞0

ν(ω : X(ω) > x)dx.

If ν(Ω) = 1, then %(X) is a coherent risk measure.Example: Let f : [0, 1] 7→ [0, 1] be increasing and concave suchthat f (0) = 0 and f (1) = 1. The the set function ν(A) := f (P(A)),A ∈ F , defines a convex game. The coherent risk measure:

%(X) =

∫ ∞0

f (P(X > x))dx

is known as the distortion risk measure in actuarial science(Denneberg, 1989).



Distortion

A coherent risk measure % is said to be comonotone if%(X + Y) = %(X) + %(Y) for all loss variables of the followingform:

X = g(Z),Y = h(Z)

where g, h : R 7→ R+ are increasing.

Coherent Risks Arising from Convex Games

A coherent risk measure % arises from a convex game

%(X) = supµ∈C(ν)

Eµ(X)

if and only if % is comonotone.

CVaR is comonotone, and thus can be (easily) written as adistortion risk.



Distortion

Coherent Risk Measures Arising from Convex Games

Coherent Risk Measures

Distortion Risk measures



Concluding Remarks

The theory of univariate coherent risk measures is fairlycomplete.But multivariate (or set-valued) coherent risk measures arestill an open field. For example:

1 How to establish the asymptotic expressions of coherentrisks for extreme events? How to utilize limitinghomogeneous structures to estimate these expressionsbased on observed extreme values, which are usuallyrare?

2 How would the dependence of loss variables amongdifferent portfolios affect systemic risk assessment, and inparticular, the systemic risk arising from a cooperativegame?


risk, coherency and cooperative game · harry markowitz’s mpt (developed in 1950-1970, received...

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