Portfolio optimization with stochastic dominance constraints

Transcription

1 Charles University in Prague Faculty of Mathematics and Physics Portfolio optimization with stochastic dominance constraints December 16, 2014

2 Contents Motivation 1 Motivation

3 Contents Motivation 1 Motivation

4 Different approaches to the portfolio problem I Let R 1,...,R N be random returns of assets 1,...,N. Let X be a random return of a portfolio consisting of assets 1,...,N. The aim is to invest our capital in these assets in order to obtain some desirable characteristics of the total return rate on the investment. The main difficulty in formulating the optimization problem is the definition of the preference structure among feasible portfolios. Different approaches to the problem the mean-risk portfolio optimization problem max E[X] λρ(x), X X here λ 0 represents desirable compromise between the mean and the risk. select a certain utility function u : R R and formulate the following optimization problem max X X E[u(X)]. incorporate risk aversion into the model via VaR or CVaR constraints formulate stochastic dominance constraints

5 Different approaches to the portfolio problem II In some applications a reference outcome Y in L k 1 (Ω, F,P) is available. The intension is to find the new outcome X preferable over Y. We have the following optimization problem maximize subject to f(x) X (k) Y X C, where C L k 1 (Ω, F,P) is convex and closed, and f : C R is a concave continuous functional. The dominance constraint quarantees that for any decision maker, the solution X of the problem will satisfy the relation E[u(X)] E[u(Y)], thus the relation holds for arbitrary u U (k). In further application we consider the second order stochastic dominance.

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7 Definition of stochastic dominance relation I Consider a random variable X L k 1 (Ω, F,P) having distribution function F(X;η) = P[X η] for η R. Define recursively the function η F j (X;η) = F j 1 (X;α) dα for η R, j = 2,...,k, where F 1 (X; ) = F(X; ) Definition 1 (Dentcheva and Ruszczyński (2003)) A random variable X L k 1 (Ω, F,P) dominates in the k-th order another random variable Y L k 1 (Ω, F,P), denoted as X (k) Y, if F k (X;η) F k (Y;η) for all η R.

8 Definition of stochastic dominance relation II first order stochastic dominance condition: F(X;η) F(Y;η) for all η R second order stochastic dominance condition: η F(X;α) dα η F(Y;α) dα for all η R first order stochastic dominance implies second order stochastic dominance distribution function investment F investment G investment H Figure: Cumulative distribution functions of rate of return of three different investments.

9 Some theorems about SSD I Theorem 1 (Dentcheva and Ruszczyński (2003)) An equivalent representation of the second order stochastic dominance relation X (2) Y is E[(η X) +] E[(η Y) +] for all η R. Proof. Changing the order of integration we get F 2 (X;η) = = = = = η F(X; α)dα η α df(x; y) dα η η dα df(x;α) y η (η y)df(x; y) (η y) +df(x; y)

10 Some theorems about SSD II Theorem 2 (Dentcheva and Ruszczyński (2003)) For each X, Y, L 1 (Ω, F,P) the relation X (2) Y is equivalent to E[u(X)] E[u(Y)] for all u U (2), where U (2) is the set of concave nondecreasing functions u : R R. Proof. If E[u(X)] E[u(Y)] is satisfied for all u U (2), then it is particularly satisfied for u(x) = (η x) +, which implies that X (2) Y. Conversely, assume X (2) Y. For every function of the form u n(x) = c n n α i (η i x) +, i=1 where α i 0, we have E[u n(x)] E[u n(y)]. Let u U (2). We can construct a sequence of functions {u n} such that u n( ) is a piecewise linear approximation of u( ), Then E[u n(x)] n E[u(X)] for all X L 1 (Ω, F,P).

11 Some theorems about SSD III Theorem 3 (Dentcheva and Ruszczyński (2006)) The second order stochastic dominance relation X (2) Y is equivalent to the continuum of CVaR constraints CVaR α( X) CVaR α( Y) for all α [0, 1]. Proof. Consider a real random variable V L 1 (Ω, F,P) and define Then according to Theorem 1 F ( 2) (p; V) = sup η R {ηp E[(η V) +]}. F ( 2) (p; V) = sup η R {ηp F 2 (V;η)}. The second order dominance relation is equivalent to F 2 (X;η) F 2 (Y;η) for all η R, which implies F ( 2) (p; X) F ( 2) (p; Y) for all p [0, 1]. Since F 2 (V;η) is continuous, the oposite implication stems from the Fenchel duality. The statement of the theorem follows then from the following definition of CVaR { CVaR α( V) = inf η R η + 1 E[( V η)+] 1 α } = F ( 2)(1 α; V). 1 α

12 Some theorems about SSD for discrete random variables I Theorem 4 (Dentcheva and Ruszczyński (2003)) Assume that Y has a discrete distribution with realizations y i, i = 1,...,m, where a y i b for all i. Then the following inequalities are equivalent E[(η X) +] E[(η Y) +] for all η [a, b], E[(y i X) +] E[(y i Y) +] for i = 1,...,m. Theorem 5 (Dupačová and Kopa (2012)) Let X and Y be discrete random variables. Let qi x = i j=1 px j for i = 1,...,m and q0 x = 0. Then X (2) Y if and only if CVaR α( X) CVaR α( Y) for all α = qi x, i = 0,...,m 1. Let X and Y be discrete random variables with equal probabilities. Then X (2) Y if { } and only if CVaR α( X) CVaR α( Y) for all α 0, 1 m, 2 m 1,...,. m m

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14 The optimization problem I The original optimization problem maximize subject to f(x) X (2) Y X C, where Y is a reference random variable in L 1 (Ω, F,P), the set C L 1 (Ω, F,P) is a convex and closed, and f : C R is a concave continuous functional. Remark.The set of X defined as A (2) (Y) = { X L 1 (Ω, F,P) : X (2) Y } is also convex and closed. Consider a relaxed version of the previous problem (the dominance relation is enforced on an interval [a, b]) maximize f(x) subject to E[(η X) +] E[(η Y) +] for all η [a, b] X C.

15 The optimization problem II We introduce a decision vector S : [a, b] Ω R to represent the shortfall. We obtain the following problem maximize f(x) subject to X(ω)+S(η,ω) η for a.a. (η,ω) [a, b] Ω (1) E[S(η,ω)] E[(η Y) +] for all η [a, b] (2) S(η,ω) 0 for a.a. (η,ω) [a, b] Ω (3) X C. Suppose Ω = {ω 1,...ω m}. Denote p k = P[{ω k }], v i = E[(y i Y) +], x k = X(ω k ), and s ik = S(y i,ω k ). Then constraints (1),(2) and (3) can be rewritten as x k + s ik y i i = 1,...,m, k = 1,...,m m p k s ik v i i = 1,...,m k=1 s ik 0 i = 1,...,m, k = 1,...,m

16 The optimization problem III Constraints (1),(2) and (3) can be equally expressed as CVaR α( X) CVaR α( Y) for all α [0, 1]. If we consider Ω = {ω 1,...ω m} then the previous condition simplyfies to CVaR α( X) CVaR α( Y) for all α = q x i, i = 0,...,m 1, where q x i = i k=1 p k = i k=1 P[{ω k}].

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18 The portfolio optimization problem Let R 1...,R N L 1 (Ω, F,P) be random returns of assets 1,...,N. The aim is to invest our capital in these assets in order to maximize the expected return on condition that our portfolio is preferable over reference portfolio Y. Suppose R n, n = 1,...,N, has discrete distribution with equiprobable scenarios and denote r nk the return of asset n in year k, k = 1,...,m. Thus P(R n = r nk ) = 1/m for k = 1,...,m. year asset 1 asset 2 asset Table: Asset returns (in %).

19 Optimization problem with stoch. dom. constraint We arrive at the following linear programming problem maximize subject to 1 m m N r nk z n k=1 n=1 N r nk z n + s ik y i i = 1,...,m, k = 1,...,m n=1 1 m s ik v i i = 1,...,m m k=1 s ik 0 i = 1,...,m, k = 1,...,m N z n = 1 n=1 z n 0, n = 1,...,N.

20 Optimization problem with CVaR constraint The optimization problem with CVaR constraint (where index i is fixed) maximize subject to 1 m m N r nk z n k=1 n=1 N r nk z n + s k + a 0 n=1 a+ 1 (1 α i )m s k 0 N z n = 1 n=1 z n 0, a R. m k=1 k = 1,...,m n = 1,...,N, k = 1,...,m s k y i + v i 1 α i

21 Comparison of both approaches return of the optimal portfolio risk of the optimal portfolio (CVaR) SD constraint CVaR constraint Figure: Return and risk of the optimal portfolios.

22 Contents Motivation 1 Motivation

23 Motivation we provided a brief introduction to stochastic domanance we introduced a general approach to optimization with stochastic dominance constraints which was futher reformulated employing some statements about stochastic dominance relation we considered the problem of constructing a portfolio of finitely many assets whose return rates were described by a discrete distribution in the computational part we compared two optimization approaches, one with the stochastic dominance constraint and the other with CVaR constraint

24 Thank you for your attention

25 Motivation D. Dentcheva and A. Ruszczyński. Optimization with stochastic dominance constraints. SIAM J. Optimization, 14: , D. Dentcheva and A. Ruszczyński. Portfolio optimization under stochastic dominance constraints. Journal of Banking and Finance, 30: , J. Dupačová and M. Kopa. Robustness in stochastic programs with risk constraints. Annals of Operations Research, 200:55 74, 2012.