Stochastic Optimization with Risk Measures

Transcription

1 Stochastic Optimization with Risk Measures IMA New Directions Short Course on Mathematical Optimization Jim Luedtke Department of Industrial and Systems Engineering University of Wisconsin-Madison August 8, 2016 Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 1 / 25

2 Motivating Application: Portfolio Optimization Let s make some money! Given a set of assets I, what fraction of my portfolio should I invest each asset to maximize return? Return on asset i I is random: R i 0 Let x i = fraction of portfolio invested in asset i Constraints: Invest all, and no short-selling i I x i = 1, x 0 What is optimal solution if we want to maximize expected value? [ ] max E i I R ix i = i I E[R i]x i Invest everything in one asset with highest expected return! Anybody see a problem with that solution? Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 2 / 25

3 Mean-Variance Model Markowitz key observation: Variance of return is also important For given expected return, want portfolio with smallest variance (or vice versa) How to deal with two objectives? Constrain expected return above L, minimize variance (vary L) Constraint variance below L, maximize expected return (vary L) Maximize expected return less λ*variance (vary λ) Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 3 / 25

4 Mean-Variance Model Random returns R i, i I: Expected return r i, i I Variance Σ ii, Covariance Σ ij What is variance of return if x i is invested in asset i? Recall V(Z) = E [ (Z µ Z ) 2], Cov(Z, Y ) = E[(Z µ Z )(Y µ Y )] [ ( E R i x i i i ) ] 2 [ ] r i x i =E (R i r i )(R j r j )x i x j R i R j = i,j i j Cov(R i, R j )x i x j = x Σx where Σ is the variance-covariance matrix Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 4 / 25

5 Mean-Variance Model Mean-variance model: max { }{{} x r λ } x {{ Σx } : e x = 1, x 0 } E[f(x,ξ)] V[f(x,ξ)] Σ 0 Convex optimization model This is one example of a mean-risk optimization model Possible drawback: Variance is symmetric Penalizes high returns as well as low returns If returns are joint normally distributed, portfolio return is normal Fine But is normal a reasonable distribution of return? Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 5 / 25

6 Value at Risk Another popular quantification of risk used in finance α Value at Risk (VaR): VaR α (Z) = inf{t : P[Z t] α} = inf{t : F Z (t) α} Constraint on Value at Risk is equivalent to a chance constraint: VaR α (f(x, ξ)) 0 P[f(x, ξ) 0] α. NB: In portfolio problem, evaluate VaR of loss: f(x, ξ) = R x Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 6 / 25

7 Risk Measures More Generally Classic stochastic programming model: min{e[f(x, ξ)] : x X} Replace expected value with a risk measure: where ρ is a risk measure: min{ρ[f(x, ξ)] : x X} Maps a random variable to a number E.g., ρ(z) = E[Z], ρ(z) = E[Z] + λv(z), ρ(z) = E[Z] + λvar(z), ρ(z) = E[u(Z)], where u is a disutility function Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 7 / 25

8 Average Value at Risk Another popular risk measure, often called Conditional Value at Risk Minimizing, so wish to measure/limit risk of large values Definition α Average value at risk (AVaR) AVaR α (Z) def { 1 = min y + y R 1 α E[Z y] } + Whaa?!? Theorem (Intuitive Definition) If, P(Z = VaR α (Z)) = 0, then AVaR α (Z) = E[Z Z VaR α (Z)] Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 8 / 25

9 Proving Equivalent Definition Lemma Assume P(Z = VaR α (Z)) = 0. Then E[Z Z VaR α (Z)] = VaR α (Z) α E[Z VaR α(z)] +. AVaR α (Z) def { 1 = min y + y R 1 α E[Z y] } + Using Lemma, AVaR α (Z) = E[Z Z VaR α (Z)] follows if we can show v := VaR α (Z) is a minimizer of (*). (*) Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 9 / 25

10 Optimizing Average Value at Risk Assume finite support distribution: P(ξ = ξ s ) = p s, s = 1,..., S min{avar α (f(x, ξ)) : x X} x { 1 = min y + x,y 1 α E[f(x, ξ) y] + : x X, y R } S = min x,y y α s=1 p s w s s.t. w s f(x, ξ s ) y, s = 1,..., S w 0, y R, x X Convex program when X is convex and f(x, ξ s ) is convex for each s Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 10 / 25

11 Properties of Average Value at Risk Assume (only for simplicity) probability space is finite: Ω = S. Theorem Rndom variable Z vector z R S. AVaR α : Function from R S R. ρ( ) = AVaR α ( ) satisfies the following properties: (A1) Convexity. β (0, 1), Z 1 and Z 2 rvs ρ(βz 1 + (1 β)z 2 ) βρ(z 1 ) + (1 β)ρ(z 2 ) (A2) Monotonicity. Z 1 Z 2 a.s. ρ(z 1 ) ρ(z 2 ) (A3) Translation invariance. If Z is a r.v. and a R, then ρ(z + a) = ρ(z) + a. (A4) Positive homogeneity. ρ(tz) = tρ(z) t > 0. Reference: Rockafellar and Uryasev [2000] Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 11 / 25

12 Coherent Risk Measures AVaR α is an example of a coherent risk measure Definition A risk measure ρ( ) is called coherent if it satisfies the following properties: (A1) Convexity. β (0, 1), Z 1 and Z 2 rvs ρ(βz 1 + (1 β)z 2 ) βρ(z 1 ) + (1 β)ρ(z 2 ) (A2) Monotonicity. Z 1 Z 2 a.s. ρ(z 1 ) ρ(z 2 ) (A3) Translation invariance. If Z is a r.v. and a R, then ρ(z + a) = ρ(z) + a. (A4) Positive homogeneity. ρ(tz) = tρ(z) t > 0. References: Artzner et al. [1999], Ruszczynski and Shapiro [2006] Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 12 / 25

13 Distributional Robustness Recall assumption: Ω = S Distribution P vector µ in R S + with s µ s = 1 Random variable Z vector z R S In practice, distribution µ of random variable is not known with certainty Assume instead µ A, where A is a collection of distributions Find minimum cost over worst case distribution Distributionally Robust Stochastic Optimization min max E µ[f(x, ξ)] x X µ A Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 13 / 25

14 Distributional Robustness Coherent Risk Measures Suppose we are given a family of distributions A Let ρ A (Z) := max µ A E µ [Z] (= max µ A s µ sz s ) Then DRO problem is equivalent to: Theorem min ρ A[f(x, ξ)] x X Distributional robustness provides a means to define a risk measure NB: Evaluating ρ A (Z) is tractable A is a tractable convex set Suppose e µ = 1 and µ 0 for all µ A. Then ρ A is a coherent risk measure. Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 14 / 25

15 Coherent Risk Measures Distributional Robustness Theorem Suppose ρ is a coherent risk measure. Then there exists a convex set A such that for any random variable Z ρ[z] = max µ A E µ[z] where for each µ A, s µ s = 1 and µ 0. Example: Consider random variable with P(Z = z s ) = p s, where p R S + and s p s = 1 AVaR α (Z) = max µ A α E µ [Z] where A α = {µ R S + : s µ s = 1, µ s p s /(1 α)} Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 15 / 25

16 Proof Background Given a function ρ : R S R, its conjugate ρ : R S R is: ρ (µ) = max z R S{µ z ρ(z)} Let A := {µ R S : ρ (µ) < + } ρ is a convex function and A is a convex set Theorem If ρ is a convex function, then ρ(z) = max µ A {µ z ρ (µ)}. Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 16 / 25

17 Another Example: Mean α-semi-deviation Let α 1, and for λ 0 define: [ ] 1/α ρ λ [Z] = E[Z] + λe (Z E[Z]) α + For discrete distribution with P(Z = z s ) = p s, s = 1,..., S ( ρ λ [Z] = p z + λ p s (z s s i p i z i ) α + ) 1/α Theorem If λ [0, 1] then ρ λ ( ) is a coherent risk measure. The dispersion part by itself is NOT a coherent risk measure (it violates monotonicity). Proof of monotonicity of ρ λ ( ) is nontrivial. Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 17 / 25

18 Properties of Other Risk Measures Mean Variance: ρ MV [Z] = E[Z] + λv[z] Convex? Yes Monotone? No(!) Translation invariant? Yes Positively homogeneous? No Mean Standard-deviation: ρ MS [Z] = E[Z] + λv(z) 1/2 Convex? Yes Monotone? No(!) Translation invariant? Yes Positively homogeneous? Yes Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 18 / 25

19 Nonmonotonicity Example Suppose there are two outcomes: P(1) = 0.95, P(2) = 0.05 ω Z 1 Z E V V 1/ Z 1 Z 2 with probability 1, but mean variance and mean standard-deviation measures would prefer Z 1 for modest values of λ Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 19 / 25

20 Properties of Risk Measures VaR α (Z): Convex? No Monotone? Yes Translation invariant? Yes Positively homogeneous? Yes, if P(Z 0) = 1 Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 20 / 25

21 Nonconvexity of Value at Risk Suppose there are three equally likely outcomes ω Z 1 Z Z Z VaR E VaR 0.6(Z 1 ) VaR 0.6(Z 1 ) < VaR 0.6 ( 1 2 Z Z2 ) Diversified portfolio looks worse, despite having same expected value Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 21 / 25

22 Disutility Based Risk Measure Let g : R R be a convex increasing function. Let ρ g [Z] = E[g(Z)] Properties: Convex? Yes Montone? Yes Translation invariant? No. (Unless g(z) z.) Positively homogeneous? Yes, if g is positively homogeneous. Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 22 / 25

23 Modified Disutility Based Risk Measure Let g : R R be a convex increasing function. Let Properties: Convex? Yes Monotone? Yes Translation invariant? Yes! ρ g [Z] def = inf E[Z + g(z t)] t Positively homgeneous? Yes, if g is positively homgeneous. Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 23 / 25

24 Optimizing Risk Measures Now consider the problem: Theorem min{ρ[f(x, ξ)] : x X} Assume ρ is a convex and monotone risk function and f(x, ξ s ) is convex for s = 1,..., S. Then φ(x) := ρ[f(x, ξ)] is a convex function of x. Let h : R n R S, be defined by h s (z) = f(x, ξ s ). h is a convex function. ρ is a convex and increasing function. φ(x) = ρ(h(x)) is convex. Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 24 / 25

25 Optimizing Risk Measures Now consider the problem: min{φ(x) = ρ[f(x, ξ)] : x X} Let ρ be a coherent risk measure with representation: ρ(z) = max µ A E µ[z] Assume f(x, ξ s ) is convex for each s = 1,..., S. Subgradients Let x X and µ argmax µ A { s µ sf( x, ξ s )}, and let d s be a subgradient of f(x, ξ s ) at x for s Ω. Then is a subgradient of φ(x) at x. µ s d s s Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 25 / 25

26 Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath. Coherent measures of risk. Mathematical finance, 9(3): , R.T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk. Journal of Risk, 2:21 41, A. Ruszczynski and A. Shapiro. Optimization of convex risk functions. Mathematics of operations research, 31(3): , Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 25 / 25