CVaR and Examples of Deviation Risk Measures

CVaR and Examples of Deviation Risk Measures Jakub Černý Department of Probability and Mathematical Statistics Stochastic Modelling in Economics and Finance November 10, 2014 1 / 25

Contents CVaR - Dual Representation 1 CVaR - Dual Representation 2 3 2 / 25

Contents CVaR - Dual Representation 1 CVaR - Dual Representation 2 3 3 / 25

General Function Let P be a probability measure on (Ω, F) and consider spaces X := L 1 (Ω, F, P) and Y := L (Ω, F, P). For constants ε 1 > 0 and ε 2 > 0, consider the function ρ(x ) = E P [X ] + φ(x ), where φ(x ) := inf = inf z R Ω z R { ε1 [z X (ω)] + + ε 2 [X (ω) z] + } dp(ω) { ε1 [z x] + + ε 2 [x z] + } dg(x), and G(x) := P({ω : X (ω) x}) is cumulative distribution function (cdf) of X (ω) with respect to the probability measure P. 4 / 25

α-quantile CVaR - Dual Representation The infimum at the right-hand side of previous equation is attained at any z such that P [X z] α and P [X z] 1 α, where α := ε 2 ε 1 + ε 2 = 1 ε 1 ε 1 + ε 2 ; the point z is called a α-quantile of the cdf G(x). Therefore and ε 2 = αε 1 1 α ε 1 [z X ] + + ε 2 [X z] + = = ε 1 { z X + [X z]+ } + ε2 [X z] + = ε 1 { z X + [X z]+ } + αε 1 = ε 1 { z X + 1 1 α [X z] + 1 α [X z] + }. 5 / 25

CVaR Risk Function Consequently, ρ(x ) = E P (X ) + inf z R = (1 ε 1 )E P (X ) + ε 1 inf { ε 1 z x + 1 } 1 α [x z] + dg(x) z R = (1 ε 1 )E P (X ) + ε 1 CVaR α (X ) { z + 1 } dg(x) 1 α [x z] + } {{ } := CVaR α(x ) So in case that ε 1 = 1 we obtain well-known conditional value at risk. ρ(x ) = CVaR α (X ) 6 / 25

Dual Representation The respective conjugate function is the indicator function of the set U := dom(ρ ) and ρ can be represented in the dual form where ρ(x ) = sup h, X h U U := {h Y : 1 ε 1 h(ω) 1 + ε 2, a.e. ω Ω, E P [h] = 1} The set U is a set of probability measures if ε 1 1. This shows that for ε 1 (0, 1] and ε 2 > 0, the function ρ is a risk function. In particular for ε 1 = 1 the dual representation of CVaR holds with the set U := { h Y : 0 h(ω) (1 α) 1, a.e. ω Ω, E P [h] = 1 } 7 / 25

Subdifferential Because ρ is convex, positively homogeneous, and continuous, we have that for any X X, the subdifferential ρ(x ) is nonempty and is given by formula ρ(x ) = arg max h, X. h Y So the risk function ρ is subdifferentiable at every X X and ρ(x ) = arg max hx dp : γ 1 h(ω) γ 2, a.e. ω Ω, E h Y P [h] = 1, where γ 1 = 1 ε 1 and γ 2 = 1 + ε 2. Ω 8 / 25

Argmax Problem Let us consider maximization problem on the right-hand side of previous equation. The Lagrangian of that problem is L(h, λ) = h, X + λ (1 E P [h]) = h, X λ + λ We can write it in the max-min form max inf (X λ)hdp + λ γ 1 h( ) γ 2 λ R. And its dual is the problem min sup (X λ)hdp + λ. λ R γ 1 h( ) γ 2 Ω Ω 9 / 25

Dual Problem Because 0 < γ 1 < γ 2 we have that sup h, X λ = max [γ 1 (X λ), γ 2 (X λ)] dp. γ 1 h( ) γ 2 Ω Therefore the dual problem is equivalent to min max [γ 1 (X λ), γ 2 (X λ)] dp + λ λ R. Ω In case that γ 1 = 0, i.e. CVaR α (X ), we obtain { min (1 α) 1 E P [X λ] + + λ }. λ R 10 / 25

Solution CVaR - Dual Representation Let λ be an optimal solution of the dual problem. Considering leftand right-side derivatives, at λ, of the objective function in the dual problem, we obtain 1 γ 1 P(X < λ) γ 2 P(X λ) 0 1 γ 1 P(X λ) γ 2 P(X > λ) This can be rewritten as follows: ε 1 P(X < λ) ε 2 P(X λ) 0 ε 1 P(X λ) ε 2 P(X > λ). Recalling α = ε 2 / (ε 1 + ε 2 ), we conclude that the set of optimal solutions is the set of α-quantiles of the cdf G, i.e. P(X λ) ε 1 ε 1 + ε 2, P(X λ) ε 2 ε 1 + ε 2. 11 / 25

Solution cont. Recall that the α- quantile of the cdf G is called Value-at-Risk and is denoted by VaR α (X ). For simplicity suppose that the set of α- quantiles of G is a singleton, i.e. α-quantile or VaR α (X ) is defined uniquely. Then arg max set at the beginning is given by such h(ω) that h(ω) = γ 2 if X (ω) > VaR α (X ), ρ(x ) = h : E P [h] = 1, h(ω) = γ 1 if X (ω) < VaR α (X ), h(ω) (γ 1, γ 2 ) if X (ω) = VaR α (X ), and h(ω) for CVaR α (X ) h(ω) = (1 α) 1 if X (ω) > VaR α (X ), h(ω) = 0 if X (ω) < VaR α (X ), h(ω) (0, (1 α) 1 ) if X (ω) = VaR α (X ), 12 / 25

Contents CVaR - Dual Representation 1 CVaR - Dual Representation 2 3 13 / 25

Definition CVaR - Dual Representation One of well-known examples of deviation measures is CVaR-deviation defined as CVaR α (X ) = CVaR α (X E[X ]) for α [0, 1). For α = 1 follows CVaR1 (X ) = E[X ] E[X ] = 0 is not a deviation measure, since it is zero for all random variables (not only for constants). 14 / 25

Risk Envelope Deviation measure D(X ) have dual characterization in terms of risk envelopes Q L 2 (Ω) defined by properties: 1 Q is nonempty, closed and convex, 2 for every nonconstant X there is some Q Q such that E[XQ] < E[X ], 3 E[Q] = 1 for all Q Q. 4 Q 0 for all Q Q. One-to-one correspondence between deviation measures and risk envelopes was shown: D(X ) = E[X ] inf Q Q E[XQ] Q = {Q L 2 (Ω), D(X ) E[X ] E[XQ] for all X } 15 / 25

Risk Envelopes cont. Risk envelope can ve viewed as a set of probability measures providing alternatives for the given probability measure P. In this case, the corresponding deviation measure D(X ) = E[X ] inf Q Q E[XQ] = E P[X ] inf Q Q E Q[X ] estimates the difference of what the agent can expect under P and under the worst probability distribution. 16 / 25

Risk Envelope CVaR-deviation The CVaR-deviation corresponding risk envelope is given by D(X ) = CVaR α (X ) Q = { Q, E[Q] = 1, 0 Q (1 α) 1} For example, if X is discretely distributed with P(X = x k ) = p k, k = 1,..., n then with the risk envelope representation, the CVaR-deviation are simply restated into the linear programming form CVaR α (X ) = = E[X ] min q k { n q k p k x k, 0 q k (1 α) 1, k=1 } n q k p k = 1. k=1 17 / 25

Contents CVaR - Dual Representation 1 CVaR - Dual Representation 2 3 18 / 25

General Function Let P be a probability measure on (Ω, F) and consider the space X := L p (Ω, F, P) for some p [1, ) and Y := L q (Ω, F, P) where 1/p + 1/q = 1. Define ρ(x ) := E P [X ] + cψ p (X ), where c 0 is a constant and ψ p (X ) := X E P [X ] p = Ω 1/p X (ω) E P [X ] p dp(ω). Note that for p = 2, the function ρ( ) corresponds to the classical mean-variance model of Markowitz, but with the standard deviation instead of variance. 19 / 25

Dual Representation The function ρ is convex continuous and positively homogeneous. Also (E P [ X p ]) 1/p = X p = sup h q 1 h, X, and hence (E [ X E P [X ] p ]) 1/p = sup h, X E P [X ] h q 1 = sup h E P [h], X h q 1 It follows that dual representation holds with the set A given by A = { h Y : h = 1 + h E P [h], h q c } 20 / 25

Coherence Conditions, p = 1 Now we know that ρ satisfies convexity, positive homogenity and translation invariance. But the monotonicity condition is more involved. Suppose that p = 1. Then q = +, i.e. the dual norm h is given by the essential maximum of h(ω), ω Ω, and hence for any h A and a.e. ω Ω we have h (ω) = 1 + h(ω) E P [h] 1 h(ω) E P [h] 1 2c for a.e. ω Ω. It follows that if c [0, 1/2], then h A are almost everywhere nonnegative values, and hence A is a set of probability measures. Therefore all coherent conditions are satisfied. 21 / 25

Coherence Conditions, p = 1, cont. Conversely, take h := c ( ) 1 A + 1 Ω/A. for some A F, and h = 1 + h E P [h]. Then h = c, Ω h dp = c[1 2P(A)], and hence 1 + h(ω) h dp = 1 2c + 2cP(A). Ω It follows that if c > 1/2, then h (ω) < 0 for all ω A, provided that P(A) is small enough. We obtain that for c > 1/2 the monotnonicity property does not hold if the following condition is satisfied: For any ε > 0 there exists A F such that ε > P(A) > 0. ( ) 22 / 25

Coherence Conditions, p > 1 Suppose now that p > 1. For a set A F and α > 0 let us take h := α1 A and h = 1 + h E P [h]. Then h q = αp(a) 1/q and h (ω) = 1 α + αp(a) ω Ω If p > 1, then for any c > 0 the mean deviation measure does not satisfy monotonicity condition if ( ) holds. Because and αp(a) 1/q c 1 α + αp(a) 1 cp(a) 1/q + cp(a)p(a) 1/q = 1 c 1 P(A) P(A) 1/q. 23 / 25

CVaR - Dual Representation P.Krokhmal, M.Zabarankin, S.Uryasev: Modeling and Optimization of Risk, Surveys in Operations Research and Management Science 16, 49-66, 2011. A. Ruszczynski, A. Shapiro: Optimization of Convex Risk Functions Mathematics of Operations Research 31(3),pp. 433-452, 2006. A. Shapiro, D. Dentcheva, A. Ruszczynski: Lectures on Stochastic Programming Modeling and Theory Mathematical modeling society, 2009. 24 / 25

Thank you for your attention. 25 / 25