Inverse Stochastic Dominance Constraints Duality and Methods

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1 Duality and Methods Darinka Dentcheva 1 Andrzej Ruszczyński 2 1 Stevens Institute of Technology Hoboken, New Jersey, USA 2 Rutgers University Piscataway, New Jersey, USA Research supported by NSF awards DMS and DMS-646

2 Outline 1 Stochastic Dominance Definition Characterization of Stochastic Dominance by Lorenz Functions 2 Dominance Constrained Optimization 3 Conjugate Function Method for Equal Probabilities 4 Quantile Cutting Plane Methods for General Probability Spaces The Scaled Methods The Unscaled Method 5 Numerical Experience Performance Comparison Implied Utility Functions Implied Risk Measures

3 Stochastic Dominance Distribution Functions F 1 (X; η) = F k (X; η) = η η P X (dt) = P{X η} for all η R F k 1 (X; t) dt for all η R, k = 2, 3,... k th order Stochastic Dominance X (k) Y F k (X, η) F k (Y, η) for all η R

4 Second-Order Stochastic Dominance F 2 (X, η) = η F 1 (X, t) dt = E(η X) + for all η R F 2 (Y, η)f 2 (X, η) η

5 The Lorenz Function (O. Lorenz, 195) Quantile function F ( 1) (X; p) = inf{η : F 1 (X; η) p} Absolute Lorenz function F ( 2) (X; p) = p F ( 1) (X; t) dt for < p 1, F ( 2) (X; ) = and F ( 2) (X; p) = + for p [, 1]. µ x F ( 2) (X; p) 1 p

6 Integrated Distribution Function and the Lorenz function F η µ 2 (X; η) x µ x η µ x F ( 2) (X; p) 1 p Fenchel conjugate function F (p) = sup{pu F (u)}. u Ogryczak - Ruszczyński (22) F ( 2) (X; ) = [F 2 (X; )] and F 2 (X; ) = [F ( 2) (X; )]

7 Characterization of Stochastic Dominance by Lorenz Functions µ x µ y F ( 2) (X; p) F ( 2) (Y ; p) 1 p X (2) Y F ( 2) (X; p) F ( 2) (Y ; p) for all p 1.

8 Dominance Constrained Optimization Given Y L 1 (Ω, F, P) - benchmark random outcome max f (z) (P 2 ) s.t. F ( 2) (G(z); p) F ( 2) (Y ; p), p [α, β], z Z Z is a closed subset of a vector space Z, [α, β] (, 1) G : Z L 1 (Ω, F, P) and f : Z R are continuous. Equivalent formulation max f (z) z,η( ) s.t. η(p) 1 p E max(, η(p) G(z)) 1 p F ( 2)(Y ; p) p [α, β] η( ) nondecreasing, z Z with η(p) a p-quantile of G(z).

9 Discrete Equiprobable Probability Space Assumption Ω = {ω 1,..., ω N } with probabilities p i = P{ω i } = 1/N, i = 1,..., N. The Lorenz curves F ( 2) (G(z); ) and F ( 2) (Y ; ) are convex and piecewise linear with break points at π i = i/n. For η = (η 1,..., η N ), and g m (z) = [G(z)](ω m ), set ϱ i = 1 π i F ( 2) (Y ; π i ) = 1 i i y [k]. k=1 Equivalent formulation max z,η f (z) subject to η i 1 i z Z N max(, η i g m (z)) ϱ i m=1 i = 1,..., N

10 Conjugate Function Method Step : Set k =. Step 1: Solve the problem and obtain (ẑ k, ˆη k ): max z,η f (z) s.t. η ij 1 i j z Z m A j (η ij g m (z)) ϱ ij j = 1,..., k { Step 2: Let δ k = max ϱ i ˆη i k 1 i N + 1 i N m=1 If δ k, stop; otherwise, continue. } max(, ˆη i k g m (ẑ k )). Step 3: Let i k be the index at which the maximum in Step 2 is achieved, and let A k = {m : ˆη k i k > g m (ẑ k )}. Step 4: Increase k by one, and go to Step 1.

11 New Dominance Characterization for General Probability Spaces Theorem Suppose X, Y L 1 (Ω, F, P). Then X (2) Y if and only if E[X A] 1 P(A) F ( 2)(Y ; P(A)) A F : P(A) >. Corollary X (2) Y if and only if E[X X t] 1 F(X; t) F ( 2)(Y ; F (X; t)) t R, F(X; t) >. Corollary Suppose X has a discrete distribution with N realizations, and Y L 1 (Ω, F, P). Then X (2) Y if and only if j π k x [k] F ( 2) (Y ; k=1 j π k ), j = 1,..., N. k=1

12 Quantile Cutting Plane Methods: The Scaled Method Step : Set k =. Step 1: Solve the problem to obtain z k and X k = G(z k ): max f (z) s.t. E[G(z) S j ] 1 P(S j ) F ( 2)(Y ; P(S j )) z Z j = 1,..., k Step 2: Consider the sets A k t δ k = sup t = {X k t} and let { 1 P(A k t )F ( 2)(Y ; P(A k t )) E[X k A k t ] : P(A k t ) > If δ k, stop; otherwise, continue. Step 3: Find t k such that P(X k t k ) > and E[X k A k t k ] 1 P(A k t k ) F ( 2)(Y ; P(A k t k )) δ k 2. Step 4: Set S k+1 = A k t k, increase k by one, and go to Step 1. }.

13 Convergence of the Scaled Method Theorem. Assume that Z R n is compact, f ( ) is continuous, and the operator G( ) : R n L (Ω, F, P) is Lipschitz continuous on Z. If problem (P 2 ) has a nonempty feasible set, then either the scaled method stops at a solution of it, or every accumulation point of the sequence {z k } generated by the method is a solution of (P 2 ). In finite probability space Ω, the scaled method converges in finitely many iterations. Modification If the operator G( ) : R n L 1 (Ω, F, P) is Lipschitz continuous on Z, then we can modify the method to ensure convergence.

14 Quantile Cutting Plane Methods: The Modified Scaled Method Assumption The operator G( ) : R n L 1 (Ω, F, P) is Lipschitz continuous on Z. We modify the method to ensure convergence by changing the following steps Step 2a: Consider the sets A k t = {X k t} and let { 1 } δ k = sup t P(A k t )F ( 2)(Y ; P(A k t )) E[X k A k t ] : P(A k t ) ε k. Step 2b: If δ k, replace ε k by ε k /2 and go to Step 2a; otherwise, continue. Step 4: If δ k < ε k then set ε k+1 = min{δ k, ε k }/2; otherwise set ε k+1 = ε k. Set S k+1 = A k t k, increase k by one, and go to Step 1. Observe that it is possible for the method to cycle between Steps 2a and 2b, without increasing the iteration index k.

15 Quantile Cutting Plane Methods: The Unscaled Method The method uses cutting planes in a different form. Step 2: Consider the sets A k t = {X k = G(z k ) t} and let { } δ k = sup t F ( 2) (Y ; P(A k t )) F ( 2) (X k ; P(A k t )) : P(A k t ) >. If δ k, stop; otherwise, continue. Step 3: Find t k such that P(X k t k ) > as well as Difference: no normalization by P(A k t ). F ( 2) (X k ; P(A k t k )) F ( 2) (Y ; P(A k t k )) δ k 2. Theorem. Assume that Z R n is compact, f ( ) is continuous, and the operator G( ) : R n L 1 (Ω, F, P) is Lipschitz continuous on Z. If problem (P 2 ) has a nonempty feasible set, then either the unscaled method stops at a solution of (P 2 ), or every accumulation point of the sequence {z k } is a solution of (P 2 ).

16 Portfolio Optimization Assets j = 1,..., n with random return rates R j Reference return rate Y (e.g. index, existing portfolio, etc.) Decision variables z j, j = 1,..., n, Z -polyhedral set Portfolio return rate R(z) = n j=1 z jr j max f (z) n s.t. z j R j Y j=1 z Z f (x) = E [ R(x) ] or f (x) = ϱ [ R(x) ] : measure of risk.

17 Linear Programming Formulation maximize s.t. N i=1 p i n r ik z k k=1 n r ik z k + s ij y j, i = 1,..., N, j = 1,..., N, k=1 N p i s ij F 2 (Y ; y j ), j = 1,..., N, i=1 s, z Z.

18 Numerical Experience Table: Dimensions of the three formulations. n = 5 and Y is the return rate of the S&P 5 index. Events Direct LP Conjugate Formulation Quantile Formulations N Variables Constraints Variables Constraints Variables Constraints Table: Performance on problems with equal probabilities of elementary events. Direct Linear Conjugate Function Quantile Cutting Plane Methods Events Programming Method Scaled Unscaled N CPU Iter. CPU Cuts Iter. CPU Cuts Iter. CPU Cuts Iter

19 Numerical Experience Table: Performance on problems with unequal probabilities of elementary events. Direct Linear Quantile Cutting Plane Methods Events Programming Scaled Unscaled N CPU Iter. CPU Cuts Iter. CPU Cuts Iter

20 Orders and Utility Functions in Economics For any two random variables X, Y L 1 (Ω, F, P) Risk-Averse Consistency via Expected Utility X (2) Y E u(x) E u(y ) nondecreasing concave u : R R. Hadar and Russell (1969) Risk-Averse Consistency via Rank Dependent Utility X (2) Y holds true if and only if for all nondecreasing concave functions w : [, 1] R that are subdifferentiable at 1 F ( 1) (X; p) dw(p) 1 F ( 1) (Y ; p) dw(p). Dentcheva and Ruszczyński (26)

21 The Implied Rank Dependent Utility Function Uniform inverse dominance condition (UIDC) for (P 2 ) { } z Z such that F ( 2) (G( z); p) F ( 2) (Y ; p) >. inf p [α,β] Lagrangian-like functional 1 1 Φ(z, w) = f (z) + F ( 1) (G(z); p) dw(p) F ( 1) (Y ; p) dw(p) W ([α, β]) contains all concave and nondecreasing functions w : [, 1] R such that w(p) = for all p [β, 1] and w(p) = w(α) + c(p α) with some c >, for all p [, α]. The dual functional Ψ(w) = sup Φ(z, w). z Z The dual problem (D 2 ) min w W ([α,β]) Ψ(w).

22 The Implied Rank Dependent Utility Function Theorem Under the UIDC if problem (P 2 ) has an optimal solution, then problem (D 2 ) has an optimal solution and the optimal values of both problems coincide. The optimal solutions of the dual problem (D 2 ) are the rank dependent utility functions ŵ W ([α, β]) satisfying the dominance constraint and 1 F ( 1) (G(ẑ); p) dŵ(p) = 1 F ( 1) (Y ; p) dŵ(p) for an optimal solution ẑ of the problem max Φ(z, ŵ). z Z

23 The Implied Rank Dependent Utility Function Utility 1 Utility 2 Utility 3-1.4

24 Measures of Risk A measure of risk ϱ assigns to an uncertain outcome X L p (Ω, F, P) a real value ϱ(x) on the extended real line R = R {+ } { }. A coherent measure of risk is a functional ϱ : L 1 (Ω, F, P) R satisfying the axioms: Convexity: ϱ(αx + (1 α)y ) αϱ(x) + (1 α)ϱ(y )for all X, Y L 1 (Ω, F, P) and α [, 1]. Monotonicity: If X, Y L 1 (Ω, F, P) and Y (ω) X(ω) ω Ω, then ϱ(y ) ϱ(x). Translation Equivariance : If a R and X L 1 (Ω, F, P), then ϱ(x + a) = ϱ(x) a. Positive homogeneity: If t > and X L 1 (Ω, F, P), then ϱ(tx) = tϱ(x). Artzner, Delbaen, Eber and Heath; Ruszczynski, Shapiro

25 Kusuoka Representation of Measures of Risk Definition Average Value at Risk of X at level p is defined as Theorem AVaR p (X) = 1 p F ( 2)(X; p) = 1 p p VaR t (X) dt. For every law invariant, finite-valued coherent measure of risk on L (Ω, F, P) on atomless space Ω, a convex set M P((, 1]) exists such that for all X. ϱ(x) = sup µ M 1 AVaR p (X) µ(dp). Definition A measure of risk ϱ is called spectral if M is a singleton.

26 Mean-Risk Models as a Lagrangian Relaxation Theorem (necessary optimality conditions): Under the UIDC, if ẑ is an optimal solution of (P 2 ), then a spectral risk measure ˆϱ and a constant κ exist such that G(ẑ) is also an optimal solution of the problem max z Z { f (z) κˆϱ(g(z)) } and (1) κˆϱ(g(ẑ)) = κˆϱ(y ). (2) If the dominance constraint is active, then condition (2) takes on the form ˆϱ(G(ẑ)) = ˆϱ(Y ). The support of the measure µ in the spectral representation of ˆϱ( ) is included in [α, β].

27 The Implied Measure of Risk.6.5 measure 1 measure 2 measure ϱ 1 (X) =.169AVaR.1772 (X) +.14AVaR.311 (X) +.274AVaR.3636 (X) +.577AVaR.493 (X) +.373AVaR.4594 (X) AVaR.4967 (X) +.177AVaR.581 (X) +.576AVaR.557 (X) +.213AVaR.5647 (X) +.66AVaR.575 (X)