Consistency with stochastic dominance

45 the appropriate utility function that represents well our preferences and whose application leads to non-trivial and meaningful solutions of 3.13.

3.4 Consistency with stochastic dominance

The concept of stochastic dominance is related to an axiomatic model of risk-averse preferences Fishburn [52]. It originated from the theory of majorization Hardy, Litltewood and Polya [70], Marshall and Olkin [109] for the discrete case and was later extended to general distributions Quirk and Saposnik [146], Hadar and Russell [66], Hanoch and Levy [68], Rothschild and Stiglitz [155]. It is nowadays widely used in economics and finance Bawa[7], Levy [99]. In the stochastic dominance approach, random returns are compared by a point-wise comparison of some performance functions constructed from their distribution functions. Figure 3.3. Mean–risk analysis. Portfolio x is better than portfolio y in the mean–risk sense, but none of them is efficient. For a real random variable V, its first performance function is defined as the rightcontinuous cumulative distribution function of V : R P ∈ ≤ = η η η for V F V }, { . A random return V is said Lehmann[94], Quirk and Saposnik [144] to stochastically dominate another random return S to the first order, denoted S V FSD f , if R ∈ = η η η all for F F S V , . The second performance function 2 F is given by areas below the distribution function F, ∫ ∞ − = η α α η d V F V F ; ; 2 for R ∈ η 3.14 46 and defines the weak relation of the second-order stochastic dominance SSD. That is, random return V stochastically dominates S to the second order, denoted S V SSD f , if ; ; 2 2 η η S F V F ≤ for R ∈ η . 3.15 see Hadar and Russell [66], Hanoch and Levy [99]. The corresponding strict dominance relations FSD f and SSD p are defined in the usual way S V FSD f ⇔ S V SSD p . and S V SSD f 3.16 For portfolios, the random variables in question are the returns defined by 3.11. To void placing the decision vector, x, in a subscript expression, we shall simply write It will not lead to any confusion, we believe. Thus, we say that portfolio x dominates portfolio y under the FSD rules, if ; ; y F x F η η = for all R ∈ η , where at least one strict inequality holds. Similarly, we say that x dominates y under the SSD rules y R x R SSD f , if ; ; 2 2 y F x F η η = for all R ∈ η , with at least one inequality strict. ; η η x F x F R = and ; 2 2 η η x F x F R = . Figure 3.4. The expected shortfall function. Stochastic dominance relations are of crucial importance for decision theory. It is known that y R x R FSD f if and only if ] [ ] [ y R U x R U E E ≥ for any nondecreasing function U· for which these expected values are finite. Also, y R x R SSD f if and only if ] [ ] [ y R U x R U E E ≥ for every nondecreasing and concave U· for which these expected values are finite see, e.g., Levy [97]. 47 For a set P of portfolios, a portfolio P ∈ x is called SSD-efficient or FSD-efficient in P if there is no P ∈ y such that x R y R SSD f or x R y R FSD f . We shall focus our attention on the SSD relation, because of its consistency with risk- averse preferences: if y R x R SSD f , then portfolio x is preferred to y by all risk- averse decision makers. By changing the order of integration we can express the function ; 2 x F ⋅ as the expected shortfall Orgyczak ang Rusczynski [129]: for each target value η we have ] , [max ; 2 x R E x F − = η η . 3.17 The function ; 2 x F ⋅ is continuous, convex, nonnegative and nondecreasing. Its graph is illustrated in Figure 3.4. Following [42, 43], we introduce the following definition. Definition 2.1 Ruszczy´nski and Vanderbei [156] The mean-risk model , ρ μ is consistent with SSD with coefficient α , if the following relation is true y y x x y R x R SSD λρ μ λρ μ − ≥ − ⇒ f for all α λ ≤ ≤ . for all α λ ≤ ≤ In fact, as we shall see in the proof below, it is sufficient to have the above inequality satisfied for α ; its validity for all α λ ≤ ≤ follows from that. The concept of consistency turns out to be fruitful. In [129] we have proved the following result. Theorem 3.1. The mean–risk model in which the risk is defined as the absolute semideviation, } , {max x R x x − = μ δ E , 3.18 is consistent with the second-order stochastic dominance relation with coefficient 1. We provide an easy alternative proof here. Proof. First, it is clear from 3.17 that the line x μ η − is the asymptote of ; 2 x F ⋅ for ∞ → η . Therefore y R x R SSD f implies that y x μ μ ≥ . 3.19 48 Secondly, setting x μ η = in 4 we obtain } , {max y R x x − ≤ μ δ E . Since ≥ − y x μ μ , we have , max , max , max y R x y x y R y y x y R x − + − ≤ − + − = − μ μ μ μ μ μ μ Taking the expected value of both sides and combining with the preceding inequality we get y y x x δ μ μ δ + − = , which can be rewritten as y y x x δ μ δ μ + ≥ − . 3.20 Combining inequalities 3.19 and 3.20 with coefficients λ − 1 and λ , where ] 1 , [ ∈ λ , we obtain the required result. An identical result under the condition of finite second moments has been obtained in Ogryczak and Ruszczynski [128] for the standard semideviation, and further extended in Ogryczak and Ruszczynski [129] to central semideviations of higher orders and stochastic dominance relations of higher orders see also Gotoh and Konno [64]. Elementary calculations show that for any distribution 2 1 x x δ δ = , where x δ is the mean absolute deviation from the mean: x x R x μ δ − = E . 3.21 Thus, x δ is a consistent risk measure with the coefficient 2 1 = α . The mean–absolute deviation model has been introduced as a convenient linear programming mean–risk model by Konno and Yamazaki [86]. Another useful class of risk measures can be obtained by using quantiles of the distribution of the return Rx. Let x q p denote the p-th quantile In the financial literature, the quantity x q p W, where W is the initial investment, is sometimes called the Value at Risk of the distribution of the return Rx, i.e., 49 x δ . We may define the risk measure ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − − = z x R x R z p p x p , 1 max E ρ . 3.22 In the special case of 2 1 = p the measure above represents the mean absolute deviation from the median. For small p, deviations to the left of the p-th quantile are penalized in a much more severe way than deviations to the right. Although the p-th quantile x q p might not be uniquely defined, the risk measure x p ρ is a well defined quantity. Indeed, it is the optimal value of a certain optimization problem: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − − = z x R x R z p p x p , 1 max min E X ρ . 3.23 It is well known that the optimizing z will be one of the p-th quantiles of Rx see, e.g., Bloomfield [17]. In Ogryczak and Ruszczynski [127] we have proved the following result. Theorem 2. The mean–risk model with the risk defined as x p ρ is consistent with the second-order stochastic dominance relation with coefficient 1, for all 1 , ∈ p . Again, we provide here an alternative proof. Proof. Let us consider the composite objective in our mean–risk model scaled by p: ; x p x p x p G p ρ μ − = . 3.24 If follows from 3.23 that we can represent it as an optimal value: [ ] [ ] , 1 max sup ; z x R p x R z p p x p G − − − − = E μ X . 3.25 Clearly, we have the identity z x R p x R z z x R p x R z p − + − = − − − , max , 1 max . Using this in 3.25 we obtain [ ] , sup[ ; 2 x z F pz x p G − = X . 3.26 50 Figure 3.6 The absolute Lorenz curve Therefore, the function ; x G ⋅ is the Fenchel conjugate of ; 2 x F ⋅ see Fenchel [50] and Rockafellar[148]. Consequently, the second-order dominance y R x R SSD f implies that ; ; y p G x p G ≥ for all ] 1 , [ ∈ p . Recalling 13 we conclude that y y x x p p ρ μ ρ μ − ≥ − . Since we also have 8, Definition 1 is satisfied with all ] 1 , [ ∈ λ . ■ Interestingly, the function ; x G ⋅ can also be expressed as the integral: ∫ = η α α , d x q x p G 3.27 non-uniqueness of the quantile does not matter here. Indeed, it follows from 3.26 that the quantile x q p , which is the maximizer in 3.26, is a subgradient of ; x G ⋅ at p see Fenchel [50] and Rockafellar[150]. The integral in 3.27 is called the absolute Lorenz curve and is frequently used for nonnegative variables and in a normalized form in income inequality studies see Arnold [2], Gastwirth [59], Lorenz [106], Hardy, Litlewood and Polya [70] and the references therein. It is illustrated in Figure 3.6. 51 C HAPTER 4 THE DOMINANCE-CONSTRAINED PORTFOLIO PROBLEM

4.1 Introduction