CVaR and Examples of Deviation Risk Measures

Transcription

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

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

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

4 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

5 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 α [X z] + 1 α [X z] + }. 5 / 25

6 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

7 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

8 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

9 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

10 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

11 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 / 25

12 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

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

14 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

15 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

16 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

17 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

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

19 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

20 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

21 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

22 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

23 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

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

25 Thank you for your attention. 25 / 25