IR / July. On Stochastic Dominance and Mean-Semideviation Models

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1 IIASA International Institute for Alied Systems Analysis A-2361 Laenburg Austria Tel: Fa: Web: INTERIM REPORT IR / July On Stochastic Dominance and Mean-Semideviation Models Włodzimierz Ogryczak Wlodzimierz.Ogryczak@mimuw.edu.l) Andrzej Ruszczyński rusz@iiasa.ac.at) Aroved by Gordon J. MacDonald macdon@iiasa.ac.at) Director, IIASA Interim Reorts on work of the International Institute for Alied Systems Analysis receive only limited review. Views or oinions eressed herein do not necessarily reresent those of the Institute, its National Member Organizations, or other organizations suorting the work.

2 ii Abstract We analyse relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean risk aroaches. The concet of α-consistency of these aroaches is defined as the consistency within a bounded range of mean risk trade-offs. We show that mean risk models using central semideviations as risk measures are α-consistent with stochastic dominance relations of the corresonding degree if the trade-off coefficient for the semideviation is bounded by one. Key Words: Decisions under risk, Stochastic dominance, Mean risk models, Portfolio otimization.

3 iii Contents 1 Introduction 1 2 Stochastic dominance 2 3 Necessary conditions of stochastic dominance 4 4 Mean semideviation models 8 5 Stochastic efficiency in a set 10 6 Conclusions 11

4 1 On Stochastic Dominance and Mean-Semideviation Models W lodzimierz Ogryczak Wlodzimierz.Ogryczak@mimuw.edu.l) Andrzej Ruszczyński rusz@iiasa.ac.at) 1 Introduction Uncertainty is the key ingredient in many decision roblems. Financial lanning, cancer screening and airline scheduling are just few eamles of areas in which ignoring uncertainty may lead to inferior or simly wrong decisions. There are many ways to model uncertainty; one that roved articularly fruitful is to use robabilistic models. We consider decisions with real valued outcomes, such as return, net rofit or number of lives saved. Although we sometimes discuss imlications of our analysis in the ortfolio selection contet, we do not assume any secificity related to this or any other alication. Whatever the alication, the fundamental question is how to comare uncertain outcomes. This has been the concern of many authors and will remain our concern in this aer. The general assumtion that we make is that larger outcomes are referred over smaller outcomes. Two methods are frequently used for modeling the choice among uncertain rosects: stochastic dominance Whitmore and Findlay, 1978; Levy, 1992) and mean risk aroaches e.g., Markowitz, 1987). The first one is based on an aiomatic model of risk averse references, but does not rovide a simle comutational recie. It is, in fact, a multile criteria model with a continuum of criteria. The second aroach quantifies the roblem in a lucid form of two criteria: the mean, that is the eected outcome, and the risk a scalar measure of the variability of outcomes. The mean risk model is aealing to decision makers and allows a simle trade off analysis, analytical or geometrical. On the other hand, the mean risk aroach is unable to model the entire richness of risk-averse references. Moreover, for tyical disersion statistics used as risk measures, the mean risk aroach may lead to obviously inferior solutions. The seminal ortfolio otimization model of Markowitz 1952) uses the variance as the risk measure. It is, in general, not consistent with the stochastic dominance rules; the use of the semivariance rather than variance was already recommmended by Markowitz 1959) himself. Porter 1974) showed that a fied target semivariance as the risk measure makes the mean risk model consistent with the stochastic dominance. This aroach was etended by Fishburn 1977) to more general risk measures associated with outcomes below some fied target. Warsaw University, Deartment of Mathematics & Comuter Science, Warsaw, Poland

5 2 Our aim is to develo relations between the stochastic dominance and mean risk aroaches that use more natural measures of risk, associated with all underachievemets below the mean. Therefore, we focus our analysis on the central semideviations: µ 1/k = µ ξ) k P dξ)), k =1,2,..., 1) where P denotes the robability measure induced by the random variable on the real line, and µ = E{} = ξp dξ). In articular, 1) for k = 1 reresents the absolute semideviation 1) = µ = ξ) P dξ) = µ 1 ξ µ P dξ), 2) 2 and for k =2thestandard semideviation: 2) = σ = µ 1/2 µ ξ) 2 P dξ)). 3) We shall show that mean risk models using semideviations as risk measures are consistent with the stochastic dominance order, if the mean risk trade-off is bounded by one. In Section 2 we recall the notion of stochastic dominance and establish its basic roerties. Section 3 develos new necessary conditions of stochastic dominance. In Section 4 we use these conditions to establish relations between stochastic dominance and mean risk models, and in Section 5 we resent simle sufficient conditions of stochastic efficiency. Finally, we have a conclusions section. 2 Stochastic dominance Stochastic dominance is based on an aiomatic model of risk averse references Fishburn, 1964). It originated from the majorization theory for the discrete case Hardy, Littlewood and Polya, 1934; Marshall and Olkin, 1979) and was later etended to general distributions Hanoch and Levy, 1969; Rothschild and Stiglitz, 1969). Since that time it has been widely used in economics and finance see Bawa, 1982; Levy, 1992 for numerous references). In the stochastic dominance aroach random variables are comared by the ointwise comarison of their distribution functions F. For a real random variable the first function F 1) is the right continuous cumulative distribution function The kth function F F 1) η) =F η)= η for k =2,3,...) is defined as F η) = η P dξ)=p{ η} for η R. 4) F k 1) ξ) dξ for η R. 5) The relation of the kth degree stochastic dominance ksd) is understood in the following way: y F η) F y η) for all η R. 6) The corresonding strict dominance relation is defined by the standard rule y y and y. 7)

6 3 Thus, we say that dominates y by the ksd rules η R, with at least one strict inequality. Clearly, k 1) y imlies y and k 1) y imlies y, rovided that the y), if F η) F y η) for all kth degree function F is well defined. We shall emloy a sightly more general aroach to the toic. Let Ω, F, P) bean abstract robability sace, and let E = ω) Pdω) denote the eected value of a random variable. The sace of real random variables such that E{ k } < is denoted, as usual, L k Ω, F, P) we frequently write simly L k ). The norm in L k is defined as 1/k. k = E{ }) k The distribution functions 5) are closely related to the norms in the saces L k. Proosition 1 Let k 1 and L k. Then for all η R η F k+1) η)= 1 η ξ) k P dξ)= 1 k! k! ma0,η ) k k. Proof. If k = 0, the left equation follows directly from 4) with the convention 0! = 1). Assuming that it holds for k 1 we shall show it for k. Wehave F k+1) ζ) = = 1 ζ k 1)! 1 k 1)! ζ η ζ ) η ξ) k 1 P dξ) dη ) η ξ) k 1 dη P dξ), where the order of integration could be changed by Fubini s theorem see, e.g., Billingsley, 1995). Evaluation of the integral with resect to η gives the result for k. 2 ξ and the corre- Remark 1 Proosition 1 allows to define the distribution functions F κ) sonding dominance relations for arbitrary real κ>0: F κ+1) η)= 1 Γκ +1) ma0,η ) κ κ, where Γ ) denotes the Euler s gamma function. In the sequel, however, we shall consider only integer κ. Equation 1) and Proosition 1 imly the following observation. 1/k. Corollary 1 Let k 1 and L k.then = k!f k+1) µ )) It is also clear that the functions F are nondecreasing and conve, but the conveity roerty can be strengthened substantially. Proosition 2 Let k 1 and L k. Then for all a, b R and all t [0, 1] one has F k+1) 1 t)a + tb) 1 t) F k+1) a) ) 1/k + t F k+1) b)) 1/k ) k. 8)

7 4 Proof. Let t [0, 1]. Consider the random variables A =ma0,a ), B =ma0,b ), and U =ma0,1 t)a + tb ). By the conveity of the function z ma0,z ), with robability one 0 U 1 t)a + tb. Therefore, U k 1 t)a + tb k 1 t) A k + t B k, where we used the triangle inequality for k. By Proosition 1, k!f k+1) a) = A k k. Similarly, k!f k+1) b)= B k k,and k!f k+1) 1 t)a + tb) = U k k. Substitution into the last inequality yields the required result. 2 In a similar way we can rove the following roerties. Proosition 3 Let k 1 and, y L k. Then for all η R and all t [0, 1] one has F 1 k+1) 1 t)+ty η) t) F k+1) ) 1/k ) 1/k k η) + t F y η)) k+1). 9) Proof. We define the random variables A =ma0,η ), B =ma0,η y), and U =ma0,η 1 t) ty), and roceed eactly as in the roof of Proosition 2. 2 Proosition 4 Let k 1 and, y L k. Then for all t [0, 1] one has 1 t)+tyη) 1 t) + t y. 10) Proof. Define A =ma0,µ ), B =ma0,µ y y), and U =ma0,1 t)µ + tµ y 1 t) ty), and roceed as in the roof of Proosition Necessary conditions of stochastic dominance The simlest necessary condition of the kth degree stochastic dominance establishes the corresonding inequality for the eected values Fishburn, 1980). Proosition 5 Let k 1 and, y L k.if k+1) y,thenµ µ y. Our objective is to develo necessary conditions that involve central semideviations. At first we establish some technical results. Lemma 1 Let k 1 and L k.then ) 1/i 1/k ) 1/i 1/k i!f i+1) η) k!f η)) k+1) P{<η} for i =1,...,k. Proof. We have i!f i+1) η)=e { ma0,η )) i} = E { ma0,η )) i 1l <η }, where 1l <η denotes the indicator function of the event { <η}. Define A = ma0,η )) i, B =1l <η, = k/i and q = k/k i). From Hölder s inequality E{AB} A B q see, e.g., Billingsley, 1995) we obtain i!f i+1) η) ma0,η )) i i/k k i)/k 1l <η k/i k/k i) ) k i)/k. = ma0,η ) k i P{<η} Raising both sides to the ower 1/i we obtain the result. 2

8 5 Lemma 2 Let k 1,, y L k and let k+1) y.then ) 1/i ) 1/i 1/k i) i!f i+1) µ y ) y P{<µy } for all i =1,...,k; ) 1/i ii) if y > 0, then i!f i+1) µ y ) < y for all i =1,...,k 1. Proof. By Lemma 1 and the dominance, i!f i+1) µ y )) 1/i k!f k+1) 1/k µ y )) P{<µy } 1/k k!f y k+1) µ y )) P{<µy } ) 1/i 1/k ) 1/i 1/k = y ) 1/i 1/k, P{<µy } 11) for i =1,...,k, which comletes the roof of i). To rove ii), note that Proosition 5 imlies that P{<µ y } P{<µ }<1. 2 We are now ready to state the main result of this section. Theorem 1 Let k 1 and, y L k.if k+1) y then µ µ y and µ µ y where the last inequality is strict whenever µ >µ y. Proof. By 5) and 6), F k+1) µ )=F k+1) µ y )+ µ F µ y y, µ ξ)dξ F y k+1) µ y )+ µ y F ξ)dξ. 12) Let k>1. Owing to Proosition 5, µ µ y, and the assertion needs to be roved only inthecaseof Proosition 2: µ µ y F ξ) dξ = µ µ y ) > y. The integral on the right hand side of 12) can be estimated by µ µ y ) F 1 t)µ y + tµ ) dt ) 1/k 1) ) 1/k 1) k 1 1 t) F µ ) + t F µ y )) dt. Using Lemmas 1 and 2 with i = k 1), and integrating we obtain: µ F ξ) dξ µ µ 1 y 1 t) µ y k 1)! P{<µ }) 1/kk 1) 0 µ µ y k 1)! = µ µ y k! t y P{<µ y}) 1/kk 1)) k 1 dt 1 t) ) k y y ) k 1 + t y dt ) k. 13) Substitution into 12) and simlification with the use of Corollary 1 yield y µ µ y, 14)

9 6 which was set out to rove. We shall now rove that 14) is strict, if µ >µ y. Suose that Lemma 2ii), inequality 13) is strict, which makes 14) strict, too. If µ >µ y and =0,wemusthaveP{<µ y }=0,so y P{<µ } 1/k µ µ y ) <µ µ y. y > 0. By virtue of and 14) is strict again. If k = 1 the integral on the right hand side of 12) can be simly bounded by µ µ y, and we get 14) in this case, too. Moreover, F ξ) < 1forξ<µ, and the inequality is strict whenever µ y <µ. 2 Since the dominance k+1) y imlies m) y for all m k +1such thatf m) well-defined, we obtain the following corollary. is Corollary 2 If k+1) y for k 1, thenµ µ y and µ m k such that E{ m } <. m) µ y m) y for all In the secial case of the second degree stochastic dominance our results have a useful grahical interretation. For a random outcome having a bounded variance we consider the grah of the function F 2) : the Outcome-Risk O-R) diagram Figure 1). By Corollary 1, the first two semimoments are easily identified in the O-R diagram: the absolute semideviation 1) = is the value F 2) µ ), and the semivariance σ 2 = 2) ) 2 is the doubled area below the grah from to µ.wealsohaveamanifestationofthelyaunov inequality σ Lemma 1 with η = µ, k =2andi= 1), because the shaded area contains the triangle with the vertices µ, 0), µ, )andµ,0). 6 2) F η) η µ 1 2 σ2 µ - η Figure 1: The O-R diagram and semimoments Now consider two random variables and y in a common O-R diagram. If 2) y 1) 1) then Theorem 1 imlies that µ µ y y. This is obvious from Figure 2, because the sloe of F 2) does not eceed one. But we also have a more sohisticated relation illustrated in Figure 3. The area below F 2) to the left of µ,equalto 1 2 σ2 2), is not larger than the area below F y to the left of µ y, increased by the area of the traezoid with the vertices: µ y, 0), µ y, y ), µ, 0), and µ, ). Emloying the Lyaunov inequalities σ and y σ y, we obtain a grahical

10 7 6 y +η µ y F 2) y η) F 2) η) y +µ µ y y µ y µ - η Figure 2: First necessary condition: 2) y y + µ µ y 6 F 2) y η) F 2) η) 1 2 σ2 y y 1 2 σ2 µ y µ - η Figure 3: Second necessary condition: 2) y 1 2 σ2 1 2 σ2 y µ µ y ) + y )

11 8 roof of Corollary 2 for k =1andm= 2. For more details on the roerties of the O-R diagram the reader is referred to Ogryczak and Ruszczyński, 1997). A careful analysis of the roof of Theorem 1 reveals that its assertion can be slightly strengthened. Indeed, estimating in 13) the quantities P{ <µ y }and P{ <µ }from above by some constant ρ, P{<µ } ρ 1, we obtain the necessary condition ρ 1/k µ ρ 1/k µ y Unfortunately, it does not ossess the searability roerties of the assertion of Theorem 1, because the right hand side contains a factor deendent on. In the secial case of symmetric distributions, however, we can use ρ = 1/2. We can also use central deviations y. 1/k δ = µ ξ k P dξ)) =2 1/k, and obtain a stronger necessary condition. Corollary 3 If and y are symmetric random variables and k+1) y for k 1, then µ µ y and µ δ m) µ y δ y m) for all m k such that E{ m } <. 4 Mean semideviation models Mean risk aroaches are based on comaring two scalar characteristics summary statistics) of each outcome: the eected value µ and some measure of risk r. The weak relation of mean risk dominance is defined as follows µ/r y µ µ y and r r y. The corresonding strict dominance relation µ/r is defined in the standard way 7). Thus we say that dominates y by the µ/r rules µ/r y), if µ µ y and r r y where at least one inequality is strict. An imortant advantage of mean risk aroaches is the ossibility to erform a ictorial trade-off analysis. Having assumed a trade-off coefficient λ 0 one may directly comare real values of µ λr and µ y λr y. This aroach is consistent with the mean risk dominance in the sense that µ/r y µ λr µ y λr y for all λ 0. 15) Therefore, an outcome that is inferior in terms of µ λr for some λ 0 cannot be suerior by the mean risk dominance relation. A mean risk model is said to be consistent with the stochastic dominance relation of degree k if y µ/r y. 16) Such a consistency would be highly desirable, because it would allow us to search for stochastically non-dominated solutions by the rule: µ/r y y,

12 9 or by the imlied rule involving the scalarized objective: µ λr >µ y λr y for some λ 0 y. We would then know that using simlified aggregate measures of form µ λr would not lead to solutions that are inferior in terms of stochastic dominance. A natural question arises: can mean risk models be consistent with the stochastic dominance relation? The most commonly used risk measure is the variance see Markowitz, 1987). Unfortunately, the resulting mean risk model is not, in general, consistent with stochastic dominance. The use of fied-target risk measures is a ossible remedy, because stochastic dominance relations are based on norms of fied-target underachievents Proosition 1). We shall try to address the question in a different way. We modify the concet of consistency to accomodate scalarizations, and we use central semideviations as risk measures. Definition 1 We say that a mean risk model is α-consistent with the kth degree stochastic dominance relation if y µ µ y and µ αr µ y αr y, 17) where α is a non-negative constant. By virtue of 15), the consistency in the sense of 16) imlies α-consistency for all α 0. Moreover, µ µ y and µ αr µ y αr y µ λr µ y λr y for all 0 λ α 18) combine the inequalities at the left side with the weights 1 λ/α and λ/α). Thus α- consistency imlies λ-consistency for all λ [0,α]. It still guarantees that the mean risk analysis leads us to non-dominated results in the sense that µ λr >µ y λr y for some 0 λ α y. With these general definitions we can return to our main question: how can the risk measure be defined to maintain α-consistency with the stochastic dominance order? The answer follows immediately from Theorem 1. Theorem 2 In the sace L k Ω, F,P) the mean risk model with r = is 1-consistent with all stochastic dominance relations of degrees 1,...,k+1. Proof. By Theorem 1, µ >µ y y y k+1). The imlication y i+1) y i), i = k,...,1, comletes the roof. 2 In the secial case of k = 1 we conclude that the mean absolute deviation model of Konno and Yamazaki 1991) is 1 2-consistent with the first and second degree stochastic dominance. Indeed, the absolute deviation δ 1) =2 1), and Theorem 2 imlies the result. For k = 2 we see that the use of the central semideviation as the risk measure instead of the variance in the Markowitz model) guarantees 1-consistency with stochastic dominance relations of degrees one, two and three. The constant α = 1 in Theorem 2 cannot be increased for general distributions, as the following eamle shows: P{ =0}=1+ε) k,p{=1}=1 1 + ε) k,andy=0.

13 10 Obviously k+1) y, but for each α>1 we can find ε>0forwhichµ α <0= µ y α y. For symmetric distributions we can use Corollary 3 to get a wider range of tradeoffs for semideviations, which allows us to relace semideviations with the corresonding deviations. Corollary 4 In the class of symmetric random variables in L k Ω, F,P) the mean risk model with r = δ is 1-consistent with all stochastic dominance relations of degrees 1,...,k+1. 5 Stochastic efficiency in a set Comarison of random variables is usually related to the roblem of choice among risky alternatives in a given feasible attainable) set Q. For eamle, in the simlest roblem of ortfolio selection Markowitz, 1987) the feasible set of random variables is defined as all conve combinations of a given collection of investment oortunities securities). A feasible random variable Q is called efficient by the relation if there is no y Q such that y. Consistency 17) leads to the following result. Proosition 6 If the mean risk model is α-consistent with the kth degree stochastic dominance relation with α > 0, then, ecet for random variables with identical µ and r, every random variable that is maimal by µ λr with 0 <λ<αis efficient by the kth degree stochastic dominance rules. Proof. Let 0 <λ<αand Q be maimal by µ λr. Suose that there eists z Q such that z. Then from 17) we obtain and µ z µ, µ z αr z µ αr. Due to the maimality of, µ z λr z µ λr. All these relations may be true only if they are satisfied as equations; otherwise, combining the first two with weights 1 λ/α and λ/α we obtain a contradiction with the third one. Consequently, µ z = µ and r z = r. 2 It follows from Proosition 6 that for mean risk models satisfying 17), an otimal solution of roblem ma{µ λr : Q} with 0 <λ<α, is efficient by the ksd rules, rovided that it has a unique air µ,r ) among all otimal solutions. In the case of nonunique airs, however, we only know that the otimal set contains a solution which is efficient by the ksd rules. Combining Proosition 6 and Theorem 2 we obtain the following sufficient condition of stochastic efficiency. Theorem 3 If is the unique solution of the roblem ma{µ λ : Q} 19) for some λ 0, 1] and k 1, then it is efficient by the rules of stochastic dominance of degrees 1,...,k+1.

14 11 Proof. It remains to consider the case λ = 1. Suose that there eists z Q such that z. Then from Theorem 2, µ z µ and µ z λ z µ λ. Since is the unique maimal solution of 19), z =. 2 Theorem 3 can be etended on risk measures defined as conve combinations of semideviations of various degrees. By alying Theorem 2 for i = k,...,mthe following sufficient condition of stochastic efficiency can be obtained. Corollary 5 If is the unique solution of the roblem ma{µ m i=k λ ii) : Q} 20) for m k 1 such that E{ m } <, and some λ i 0 satisfying 0 < m i=k λ i 1, then it is efficient by the rules of stochastic dominance of degrees 1,...,k+1. Owing to Corollaries 3 and 4, for symmetric distributions we can use the central deviation δ i) as the risk measures in 20). It is ractically imortant that the simlified objective functionals in 19) and 20) are concave see Proosition 4). 6 Conclusions The stochastic dominance relation k+1) y is rather strong and difficult to verify: it is an inequality of two distribution functions, F k+1) F y k+1). The necessary conditions of Section 3 establish useful relations: µ λ µ y λ y, for all λ [0, 1], that follow from the dominance µ and denote the eectation and the kth central semideviation of ). This allows us to relate stochastic dominance to mean risk models with the risk reresented by the kth central semideviation. The key observation is that maimizing simlified objective functionals of the form µ λ, where λ 0, 1), we obtain solutions which are efficient in terms of stochastic dominance. This may hel to quickly identify rosective candidates in comle decision roblems under uncertainty. Still, sufficient conditions of stochastic dominance in some secific classes of decision roblems under uncertainty require more attention. We hoe to address these issues in the near future.

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