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Var Vs Cvar in Risk Management and Optimization Stan Uryasev VaR vs CVaR in Risk Management and Optimization Stan Uryasev Joint presentation with Sergey Sarykalin, Gaia Serraino and Konstantin Kalinchenko Risk Management and Financial Engineering Lab, University of Florida and American Optimal Decisions 1 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 with Portfolio Safeguard package by AORDA.com 2 Risk Management ` Risk Management is a procedure for shaping a loss distribution ` Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are popular function for measuring risk ` The 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 Deviation 4 VaR and CVaR Representation 5 VaR, CVaR, CVaR+ and CVaR- CVaR+ Risk CVaR CVaR- VaR x 6 Value-at-Risk X a loss random variable α ∈ [1,0] VaR( )α X= min{ zX | F≥ (α ) zfor } ` VaRα()X is non convex and discontinuous function of the confidence level α for discrete distributions ` VaRα()X is non-sub-additive ` difficult to control/optimize for non-normal distributions: VaR has many extremums for discrete distributions 7 Conditional Value-at-Risk ` “ ,ellar and RockafUrvyasealue-at-RiskV Optimization of Conditional ”, alue-at-V oduced the term Conditional 2000 intrournal of Risk,J Risk +∞ ` For ]α ∈ 0 ,CVaR 1 [ ()() X= α zdF z α ∫ X − ∞ ewher 0⎧ whez< VaR n α X ( ) α ⎪ FX() z= ⎨F() z−α X whenz ≥ VaR() X ⎩⎪ 1−α α 8 Conditional Value-at-Risk ` CVaR+ (Upper CVaR): expected value of X strictly exceeding VaR (also called Mean Excess Loss and Expected Shortfall) + CVaR( Xα )= E [ X | > Xα ( VaR )] X `CVaR- (Lower CVaR): expected value of X weakly exceeding VaR (also called Tail VaR) − CVaRXEXX(α )= [ | ≥VaRα X ()] + Property: CVaR α () X is weighted average of CVaR α () X and VaRα () X + ⎪⎧λλαα()XVaRX ()(1())+ −< α X CVaRX α () if FX ( VaRXα ())1 CVaRα () X = ⎨ ⎩⎪VaRα ( X ) if FX ( VaRα ( X ))= 1 F( VaR ( ))−α X λ ()X = X α α 1−α zero for continuous distributions!!! 9 Conditional Value-at-Risk ` Definition on previous page is a major innovation + ` CVaRα X )( and VaR α X )( for general loss distributions are discontinuous 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- CVaR+ Risk CVaR CVaR- VaR x 11 CVaR: Discrete Distributions ` α does not “split” atoms: VaR < CVaR- < CVaR = CVaR+, λ = (Ψ- α)/(1- α) = 0 124 Six scenarios, pp12====L p 6636, α == + 11 CVaR=+ CVaR = 22ff56 Probability CVaR 1 1 1 1 1 1 6 6 6 6 6 6 f f f 1 2 f3 4 f5 f6 Loss CVaR-- CVaR + VaR 12 CVaR: Discrete Distributions • α “splits” the atom: VaR < CVaR- < CVaR < CVaR+, λ = (Ψ- α)/(1- α) > 0 1 7 Six scenarios, pp12= ==L p 6 =61, α =2 14+ 122 CVaR=+55 VaR CVaR = 555fff456 ++ Probability CVaR 1 1 1 1 1 1 1 6 6 6 6 12 6 6 f1 f2 f3 f4 f5 f6 Loss -- 11 + VaR CVaR 22ff56+=CVaR 13 CVaR: Discrete Distributions • α “splits” the last atom: VaR = CVaR- = CVaR , CVaR+ is not defined, λ = (Ψ - α)/(1- α) > 0 14 CVaR: Equivalent Definitions ` Pflug defines CVaR via an optimization problem, as in Rockafellar and Uryasev (2000) ` Acerbi showed that CVaR is equivalent to Expected Shortfall defined by Pflug, G.C., “Some Remarks on the Value-at-Risk and on the Conditional Value-at-Risk”, Probabilistic Constrained Optimization: Methodology and Applications, (Uryasev ed), Kluwer, 2000 Acerbi, C., “Spectral Measures of Risk: a coherent representation of subjective risk aversion”, JBF, 2002 15 RISK MANAGEMENT: INSURANCE Payment Payment Deductible Premium Accident lost Accident lost PDF PDF ∞ ∞ ∞ Deductible Payment Premium Payment TWO CONCEPTS OF RISK Risk as a possible loss Minimum amount of cash to be added to make a portfolio (or project) sufficiently safe Example 1. MaxLoss -Three equally probable outcomes, { -4, 2, 5 }; MaxLoss = -4; Risk = 4 -Three equally probable outcomes, { 0, 6, 9 }; MaxLoss = 0; Risk = 0 Risk as an uncertainty in outcomes Some measure of deviation in outcomes Example 2. Standard Deviation -Three equally probable outcomes, { 0, 6, 9 }; Standard Deviation > 0 Risk Measures: axiomatic definition ` A functional is a coherent risk measure in the extended sense if: R1: for all constant C R2: for λ ∈ ]0,1[ (convexity) R3: when ≤ XX '(monotonicity) R4: when with (closedness) ` A functional is a coherent risk measure in the basic sense if it satisfies axioms R1, R2, R3, R4 and R5: R5: for λ > 0 (positive homogeneity) 18 Risk Measures: axiomatic definition ` A functional is an averse risk measure in the extended sense if it satisfies axioms R1, R2, R4 and R6: R6: for all nonconstant X (aversity) ` A functional is an averse risk measure in the basic sense if it satisfies axioms R1, R2, R4, R6 and R5 ` Aversity has the interpretation that the risk of loss in a nonconstant random variable X cannot be acceptable unless EX<0 ` R2 + R5 (subadditivity) 19 Risk Measures: axiomatic definition ` Examples of coherent risk measures: ` A ` Z ` Examples of risk measures not coherent: ` , λ>0, violates R3 (monotonicity) ` violates subadditivity ` is a coherent measure of risk in the basic sense and it is an averse measure of risk !!! ` Averse measure of risk might not be coherent, a coherent measure might not be averse 20 Deviation Measures: axiomatic definition A functional is called a deviation measure in the extended sense if it satisfies: D1: for constant C, but for nonconstant X D2: for λ ∈ [1,0] (convexity) D3: when with (closedness) A functional is called a deviation measure in the basic sense if it satisfies axioms D1,D2, D3 and D4: D4: (positive homogeneity) A deviation measure in extended or basic sense is also coherent if it additionally 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 Deviation ` α-Value-at-Risk Deviation measure: ` α-VaR Dev does not satisfy convexity axiom D2 it is not a deviation measure ` α-Conditional Value-at-Risk Deviation measure: Coherent deviation measure in basic sense !!! 22 Deviation Measures: axiomatic definition ` CVaR Deviation Measure is a coherent deviation measure in the basic sense 23 Risk vs Deviation Measures ` Rockafellar et al. (2006) showed the existence of a one-to-one correspondence between deviation measures in the extended sense and averse risk measures in the extended sense: Rockafellar, R.T., Uryasev, S., Zabarankin, M., “Optimality conditions in portfolio analysis with general deviation measures”, Mathematical Programming, 2006 24 Risk vs Deviation Measures Deviation Counterpart Risk Measure Measure where 25 Chance and VaR Constraints ` Let(f , xi ), ω i = 1 m ,.. be some random loss function. ` By definition:( )VaR min{α = x ε :f Pr{ xω ≤ (ε ,≥ α ) } } ` wing holds:olloThen the f Pr{ ( ,f ) xω ≤ }ε ≥ α ↔ α VaRX ≤ ε ( ) ` In general VaR α() is x vexnoncon w. x.r,t. ete discr(e.g., distributions) VaR` ( )α X ≤ εandf xPr{ω ≤ ε ≥ (α ,y be ) mavexnoncon } aintsconstr 26 VaR vs CVaR in optimization ` VaR is difficult to optimize numerically when losses are not normally distributed ` PSG package allows VaR optimization ` In optimization modeling, CVaR is superior to VaR: ` For elliptical distribution minimizing VaR, CVaR or Variance is equivalent ` CVaR can be expressed as a minimization formula (Rockafellar and Uryasev, 2000) ` CVaR preserve convexity 27 CVaR optimization 1 F(,)ζ x = ζ + {[E ( fξ , x − ) ζ+ ] } α 1−α Theorem 1: 1. F(,)α xζ is convex w.r.t. α 2. α -VaR is a minimizer of F with respect to ζ : VaRαα()f (xfxFx ,ξζ )==( ( ,ξ )) arg minα ( ,ζ ) ζ 3. α -CVaRequals minimal value (w.r.t. ζ ) of function F : CVaRαα()fx ( ,ξζ )= min F ( x , ) ζ 28 CVaR optimization ` Preservation of convexity: if f(x,ξ) is convex in x thenCVaR α() X is convex in x ` If f(x,ξ) is convex in x then F (,)α xζ is convex in x and ζ ` ` If f(x*,ζ*) minimizes over X × ℜ then ` is equivalent to 29 CVaR optimization ` In the case of discrete distributions: N −1 k + Fxα (,ζζ )=+− (1αξ )∑ pk [ fx (, )−ζ ] k =1 zz+ = max { , 0} ` The constraint F (,)α x ζ ≤ ω can be replaced by a system of inequalities introducing additional variables ηk : η0k ≥ , f x(k y ,−ζ0− ) η ,k ≤k= 1,.., N 1 N ζ + ∑ pηk k ≤ ω 1−α k =1 30 Generalized Regression Problem ` Approximate random variable Y by random variables X12,XX ,...,n . ` Error measure satisfies axioms Rockafellar, R. T., Uryasev, S. and M. Zabarankin: “Risk Tuning with Generalized Linear Regression”, accepted for publication in Mathematics of Operations Research, 2008 31 Error, Deviation, Statistic ` For an error measure : ` the corresponding deviation measure is ` the corresponding statistic is 32 Theorem: Separation Principle ` General regression problem is equivalent to 33 Percentile Regression and CVaR Deviation Koenker, R., Bassett, G. Regression quantiles. Econometrica 46, 33–50 (1978) −1 minE [XC−+ ]+− (α − 1) EXC [ − ] = CVaRXEXα ( − ) CR∈ ( ) 34 Stability of Estimation ` VaR and CVaR with same confidence level measure “different parts” of the distribution ` For a specific distribution the confidence levels α1 and α2 for comparison of VaR and CVaR should be found from the equation = XCVaRXVaR )()( α1 α2 ` 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 the same part of the distribution.
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