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VaR vs CVaR inRisk Management and OptimizationStan UryasevJoint presentation withSergey Sarykalin, Gaia Serraino and Konstantin KalinchenkoRisk Management and Financial Engineering Lab, University of FloridaandAmerican Optimal Decisions1

Agenda Compare Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) definitions of VaR and CVaR basic properties of VaR and CVaR axiomatic definition of Risk and Deviation Measures reasons affecting the choice between VaR and CVaR risk management/optimization case studies conducted withPortfolio Safeguard package by AORDA.com2

Risk Management Risk Management is a procedure for shaping a loss distributionValue-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) arepopular function for measuring riskThe choice between VaR and CVaR is affected by:differences in mathematical properties,stability of statistical estimation,simplicity of optimization procedures,acceptance by regulators Conclusions from these properties are contradictive 3

Risk Management Key observations: CVaR has superior mathematical properties versus VaR Risk management with CVaR functions can be done very efficiently VaR does not control scenarios exceeding VaR CVaR accounts for losses exceeding VaR Deviation and Risk are different risk management concepts CVaR Deviation is a strong competitor to the Standard Deviation4

VaR and CVaR Representation5

VaR, CVaR, CVaR and CVaR-RiskCVaR CVaRCVaRVaRx6

Value-at-RiskX a loss random variableVaRα ( X ) min{z FX ( z ) α } for α ]0,1[VaRα (X ) is non convex and discontinuous function of theconfidence level α for discrete distributionsVaRα (X ) is non-sub-additivedifficult to control/optimize for non-normal distributions:VaR has many extremums for discrete distributions7

Conditional Value-at-Risk Rockafellar and Uryasev, “Optimization of Conditional Value-at-Risk”,Journal of Risk, 2000 introduced the term Conditional Value-atRiskFor α ]0,1[ CVaRα ( X ) αzdF X ( z) wherewhen z VaRα ( X ) 0 αFX ( z ) FX ( z ) αwhen z VaRα ( X ) 1 α8

Conditional Value-at-RiskCVaR (Upper CVaR): expected value of X strictly exceeding VaR(also called Mean Excess Loss and Expected Shortfall) CVaRα ( X ) E[ X X VaRα ( X )] CVaR-(Lower CVaR): expected value of X weakly exceeding VaR(also called Tail VaR)CVaRα ( X ) E[ X X VaRα ( X )]Property: C V a Rα ( X ) is weighted average of C V a R α ( X ) and V a R α ( X ) λα ( X ) VaRα ( X ) (1 λα ( X )) CVaRα ( X ) if FX (VaRα ( X )) 1CVaRα ( X ) if FX (VaRα ( X )) 1 VaRα ( X )λα ( X ) FX (VaRα ( X )) α1 αzero for continuous distributions!!!9

Conditional Value-at-Risk Definition on previous page is a major innovation CVaRα ( X ) and VaRα ( X ) for general loss distributions arediscontinuous functions CVaR is continuous with respect to α CVaR is convex in X VaR, CVaR- ,CVaR may be non-convex VaR CVaR- CVaR CVaR 10

VaR, CVaR, CVaR and CVaR-RiskCVaR CVaRCVaRVaRx11

CVaR: Discrete Distributions α does not “split” atoms:VaR CVaR- CVaR CVaR ,λ (Ψ- α)/(1- α) 0Six scenarios,p1 p 2 L p 6 16 , α CVaR C VaR 12f5 12f1Loss f6CVaR 12

CVaR: Discrete Distributions α “splits” the atom:VaR CVaR- CVaR CVaR ,λ (Ψ- α)/(1- α) 0Six scenarios,p1 p 2 L p 6 16 , α CVaR 15 VaR 54 CVaR 15f4 f1f5 aRCVaR--1216f6f5 12 f6 CVaR 13

CVaR: Discrete Distributions α “splits” the last atom:VaR CVaR- CVaR ,CVaR is not defined, λ (Ψ - α)/(1- α) 014

CVaR: Equivalent Definitions Pflug defines CVaR via an optimization problem, as in Rockafellarand Uryasev (2000) Acerbi showed that CVaR is equivalent to Expected Shortfalldefined byPflug, G.C., “Some Remarks on the Value-at-Risk and on the Conditional Value-at-Risk”, ProbabilisticConstrained Optimization: Methodology and Applications, (Uryasev ed), Kluwer, 2000Acerbi, C., “Spectral Measures of Risk: a coherent representation of subjective risk aversion”, JBF, 200215

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TWO CONCEPTS OF RISK Risk as a possible lossMinimum amount of cash to be added to make a portfolio (orproject) sufficiently safeExample 1. MaxLoss- Three equally probable outcomes, { -4, 2, 5 };MaxLoss -4; Risk 4- Three equally probable outcomes, { 0, 6, 9 };MaxLoss 0; Risk 0Risk as an uncertainty in outcomesSome measure of deviation in outcomesExample 2. Standard Deviation- Three equally probable outcomes, { 0, 6, 9 }; StandardDeviation 0

Risk Measures: axiomatic definition Afunctionalextended sense if:R1:R2:R3:R4:is a coherent risk measure in thefor all constant Cfor λ ]0,1[(convexity)when X X ' (monotonicity)whenwith(closedness) Afunctionalis a coherent risk measure in the basicsense if it satisfies axioms R1, R2, R3, R4 and R5:R5:for λ 0 (positive homogeneity)18

Risk Measures: axiomatic definition A functionalis an averse risk measure in theextended sense if it satisfies axioms R1, R2, R4 and R6:R6:for all nonconstant X (aversity) A functionalis an averse risk measure in the basicsense if it satisfies axioms R1, R2, R4, R6 and R5 Aversity has the interpretation that the risk of loss in anonconstant random variable X cannot be acceptable unless EX 0 R2 R5(subadditivity)19

Risk Measures: axiomatic definition Examples of coherent risk measures: A ZExamples of risk measures not coherent: , λ 0, violates R3 (monotonicity) violates subadditivity is a coherent measure of riskin the basic sense and it is an averse measure of risk !!! Averse measure of risk might not be coherent, a coherentmeasure might not be averse20

Deviation Measures: axiomatic definitionA functionalis called a deviation measure in the extendedsense if it satisfies:D1:for constant C, butfor nonconstant XD2:for λ ]0,1[ (convexity)D3:whenwith(closedness)A functionalis called a deviation measure in the basicsense if it satisfies axioms D1,D2, D3 and D4:D4:(positive homogeneity)A deviation measure in extended or basic sense is also coherent if itadditionally satisfies D5:D5:(upper range boundedness)21

Deviation Measures: axiomatic definition Examples of deviation measures in the basic sense: Standard Deviation Standard Semideviations Mean Absolute Deviationα-Value-at-Risk Deviation measure:α-VaR Dev does not satisfy convexity axiom D2deviation measureα-Conditional Value-at-Risk Deviation measure:it is not aCoherent deviation measure in basic sense !!!22

Deviation Measures: axiomatic definition CVaR Deviation Measure is a coherent deviation measure in thebasic sense23

Risk vs Deviation Measures Rockafellar et al. (2006) showed the existence of a one-to-onecorrespondence between deviation measures in the extended senseand averse risk measures in the extended sense:Rockafellar, R.T., Uryasev, S., Zabarankin, M., “Optimality conditions in portfolio analysiswith general deviation measures”, Mathematical Programming, 200624

Risk vs Deviation MeasuresDeviationMeasureCounterpart RiskMeasurewhere25

Chance and VaR Constraints Let f i ( x, ω ), i 1,.m be some random loss function. By definition: VaRα ( x) min{ε : Pr{ f ( x, ω ) ε } α } Then the following holds:Pr{ f ( x, ω ) ε } α VaR α ( X ) ε In general VaRα (x) is nonconvex w.r.t. x, (e.g., discretedistributions)VaR α ( X ) ε and Pr{ f ( x, ω ) ε } α may be nonconvex constraints26

VaR vs CVaR in optimization VaR is difficult to optimize numerically when losses are notnormally distributedPSG package allows VaR optimizationIn optimization modeling, CVaR is superior to VaR: For elliptical distribution minimizing VaR, CVaR or Variance isequivalent CVaR can be expressed as a minimization formula (Rockafellarand Uryasev, 2000) CVaR preserve convexity27

CVaR optimizationFα ( x, ζ ) ζ 1E{[ f ( x, ξ ) ζ ] }1 αTheorem 1:1.2.Fα ( x, ζ )is convex w.r.t. αα -VaR is a minimizer of F with respect to ζ :VaR α ( f ( x, ξ ) ) ζ α ( f ( x, ξ ) ) arg min Fα ( x, ζ )ζ3.α - CVaR equals minimal value (w.r.t. ζ ) of function F :CVaR α ( f ( x, ξ ) ) min Fα ( x, ζ )ζ28

CVaR optimization Preservation of convexity: if f(x,ξ) is convex in x then CVaRα ( X )is convex in xIf f(x,ξ) is convex in x then Fα ( x, ζ ) is convex in x and ζ If f(x*,ζ*) minimizesover X ℜ thenis equivalent to29

CVaR optimization In the case of discrete distributions:Fα ( x, ζ ) ζ (1 α ) 1N k 1pk [ f ( x, ξ k ) ζ ] z max { z, 0} The constraint Fα (x, ζ ) ω can be replaced by a system ofinequalities introducing additional variables ηk :η k 0, f ( x, yk ) ζ η k 0, k 1,., N1ζ 1 αN pk 1kηk ω30

Generalized Regression Problem Approximate random variable Y by random variables X 1 , X 2 ,., X n . Error measure satisfies axiomsRockafellar, R. T., Uryasev, S. and M. Zabarankin:“Risk Tuning with Generalized Linear Regression”, accepted for publication in Mathematics ofOperations Research, 200831

Error, Deviation, Statistic For an error measure : the corresponding deviation measure is the corresponding statistic is32

Theorem: Separation Principle General regression problemis equivalent to33

Percentile Regression and CVaR DeviationKoenker, R., Bassett, G.Regression quantiles.Econometrica 46, 33–50(1978)min ( E[ X C ] (α 1 1) E[ X C ] ) CVaRα ( X EX )C R34

Stability of Estimation VaR and CVaR with same confidence level measure “differentparts” of the distributionFor a specific distribution the confidence levels α1 and α2 forcomparison of VaR and CVaR should be found from the equationVaRα1 ( X ) CVaRα 2 ( X ) Yamai and Yoshiba (2002), for the same confidence level: VaR estimators are more stable than CVaR estimators The difference is more prominent for fat-tailed distributions Larger sample sizes increase accuracy of CVaR estimation More research needed to compare stability of estimators for thesame part of the distribution.35

Decomposition According to Risk FactorsContributions For continuous distributions the following decompositions ofVaR and CVaR hold:VaR α ( X ) n i 1CVaR α ( X ) VaR α ( X )z i E [ X i X VaR α ( X )] z i z in i 1 CVaR α ( X )z i E [ X i X VaR α ( X )] z i zi When a distribution is modeled by scenarios it is easier toestimate E[ X i X VaRα ( X )] than E[ X i X VaRα ( X )] Estimators ofof VaR α ( X ) zi CVaR α ( X ) ziare more stable than estimators36

Generalized Master Fund Theorem and CAPM Assumptions:Several groups of investors each withU j ( ER j , Dj ( R j ))utility functionUtility functions are concave w.r.t. mean and deviationincreasing w.r.t. meandecreasing w.r.t. deviationInvestors maximize utility functions s.t. budget constraint.Rockafellar, R.T., Uryasev, S., Zabarankin, M. “Master Funds in Portfolio Analysis withGeneral Deviation Measures”, JBF, 2005Rockafellar, R.T., Uryasev, S., Zabarankin, M. “Equilibrium with Investors using a Diversity ofDeviation Measures”, JBF, 200737

Generalized Master Fund Theorem and CAPM Equilibrium exists w.r.t.Each investor has its own master fund and invests in its own masterfund and in the risk-free assetGeneralized CAPM holds:Covar(G j , rij ) E[(G j EG j )(rij Erij )]is expected return of asset i in group jis risk-free rateis expected return of market fund for investor group jis the risk identifier for the market fund j40

Generalized Master Fund Theorem and CAPMWhenthenWhenthenWhenthen41

Classical CAPM Discounting FormulaAll investors have the same risk preferences:standard deviationDiscounting by risk-free rate with adjustment foruncertainties (derived in PhD dissertation ofSarykalin)Eζ iπi 1 r0 β i (rM r0 )1(Eζ i β i (rM r0 ) )πi 1 r0βi cov(ri , rM )σ M2

Generalized CAPM There are different groups of investors k 1, ,K Risk attitude of each group of investors can be expressedthrough its deviation measure Dk Consequently:Each group of investors invests its own Master Fund M

Generalized CAPM()cov ri , QMDβi D(rM )Eζ iπi 1 r0 β i (rM r0 )1(Eζ i β i (rM r0 ) )πi 1 r0D deviation measureQMD risk identifier for D corresponding to M