From stochastic dominance to mean-risk models: Semideviations as risk measures 1

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1 European Journal of Operational Research 116 (1999) 33±50 Theory and Methodoloy From stochastic dominance to mean-risk models: Semideviations as risk measures 1 Wøodzimierz Oryczak a, *, Andrzej Ruszczynski b a Department of Mathematics and Computer Science, Warsaw University, Banacha 2, Warsaw, Poland b Department of Manaement Science and Information Systems, Ruters University, 94 Rockafeller Rd, Piscataway, NJ 08854, USA Received 1 November 1997; accepted 1 April 1998 Abstract Two methods are frequently used for modelin the choice amon uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quanti es the problem in a lucid form of two criteria with possible tradeo analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resultin mean±variance (Markowitz) model is, in eneral, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second deree stochastic dominance, provided that the trade-o coe cient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome±Risk (O±R) diaram, which appears to be particularly useful for comparin uncertain outcomes. Ó 1999 Elsevier Science B.V. All rihts reserved. Keywords: Decisions under risk; Portfolio optimization; Stochastic dominance; Mean-risk model 1. Introduction * Correspondin author. [email protected] 1 This work was partially supported by the International Institute for Applied Systems Analysis, Laxenbur, Austria, and distributed on-line as the IIASA Report IR (June 1997). Comparin uncertain outcomes is one of fundamental interests of decision theory. Our objective is to analyze relations between the existin approaches and to provide some tools to facilitate the analysis. We consider decisions with real-valued outcomes, such as return, net pro t or number of lives saved. A leadin example, oriinatin from - nance, is the problem of choice amon investment opportunities or portfolios havin uncertain returns. Althouh we discuss the consequences of our analysis in the portfolio selection context, we do not assume any speci city related to this or /99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rihts reserved. PII: S ( 9 8 )

2 34 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 another application. We consider the eneral problem of comparin real-valued random variables (distributions), assumin that larer outcomes are preferred. We describe a random variable X by the probability measure P X induced by it on the real line R. It is a eneral framework: the random variables considered may be discrete, continuous, or mixed (Pratt et al., 1995). Owin to that, our analysis covers a variety of problems of choosin amon uncertain prospects that occur in economics and manaement. Two methods are frequently used for modelin choice amon uncertain prospects: stochastic dominance (Whitmore and Findlay, 1978; Levy, 1992), and mean-risk analysis (Markowitz, 1987). The former is based on an axiomatic model of riskaverse preferences: it leads to conclusions which are consistent with the axioms. Unfortunately, the stochastic dominance approach does not provide us with a simple computational recipe ± it is, in fact, a multiple criteria model with a continuum of criteria. The mean-risk approach quanti es the problem in a lucid form of only two criteria: the mean, representin the expected outcome, and the risk: a scalar measure of the variability of outcomes. The mean-risk model is appealin to decision makers and allows a simple trade-o analysis, analytical or eometrical. On the other hand, mean-risk approaches are not capable of modelin the entire amut of risk-averse preferences. Moreover, for typical dispersion statistics used as risk measures, the mean-risk approach may lead to inferior conclusions. The seminal Markowitz (1952) portfolio optimization model uses the variance as the risk measure in the mean-risk analysis. Since then many authors have pointed out that the mean± variance model is, in eneral, not consistent with stochastic dominance rules. The use of the semivariance rather than variance as the risk measure was already suested by Markowitz (1959) himself. Porter (1974) showed that the mean-risk model usin a xed-taret semivariance as the risk measure is consistent with stochastic dominance. This approach was extended by Fishburn (1977) to more eneral risk measures associated with outcomes below some xed taret. There are many aruments for the use of xed tarets. On the other hand, when one of performance measures is the expected return, the risk measure should take into account all possible outcomes below the mean. Therefore, we focus our analysis on central semimoments which measure the expected value of deviations below the mean. To be more precise, we consider the absolute semideviation (from the mean) d X ˆ Z l X ˆ 1 2 l X n P X dn jn l X j P X dn and the standard semideviation 0 r X Z l X 1 l X n 2 P X dn A 1=2 1 ; 2 where l X ˆ EfX. We show that mean-risk models usin standard or absolute semideviations as risk measures are consistent with the stochastic dominance, if a bounded set of mean-risk tradeo s is considered. In the portfolio selection context these models correspond to the Markowitz (1959, 1987) mean±semivariance model and the Konno and Yamazaki (1991) MAD model with absolute deviation. The paper is oranized as follows. In Section 3 we recall the basics of the stochastic dominance and mean-risk approaches. We also specify what we mean by consistency of these approaches. In Section 3 we introduce a convenient raphical tool for the stochastic dominance methodoloy: the Outcome±Risk (O±R) diaram, and we examine various risk measures within the diaram. We further use the O±R diaram to establish consistency of mean-risk models usin the absolute semideviation (Section 4) and the standard semideviation (Section 5), respectively, with the second deree stochastic dominance rules. In a similar way we reexamine the standard deviation as a possible risk measure in Section 6. Owin to the use of the O±R diaram all proofs are easy, nevertheless, riorous.

3 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33± Stochastic dominance and mean-risk models Stochastic dominance is based on an axiomatic model of risk-averse preferences (Fishburn, 1964). It oriinated in the majorization theory (Hardy et al., 1934) for the discrete case and was later extended to eneral distributions (Hanoch and Levy, 1969; Rothschild and Stilitz, 1970). Since that time it has been widely used in economics and - nance (see Bawa (1982) and Levy (1992) for numerous references). In the stochastic dominance approach random variables are compared by pointwise comparison of some performance functions constructed from their distribution functions. Let X be a random variable with the probability is de ned as the riht-continuous cumulative distribution function itself: measure P X. The rst performance function F 1 X F 1 X ˆ F X ˆ PfX 6 for 2 R: The weak relation of the rst deree stochastic dominance (FSD) is de ned as follows: X FSD Y () F X 6 F Y for all 2 R: 3 The second performance function X is iven by areas below the distribution function F X : X ˆ F X n dn for 2 R; and de nes the weak relation of the second deree stochastic dominance (SSD): X SSD Y () X 6 Y for all 2 R: 4 The correspondin strict dominance relations FSD and SSD are de ned by the standard rule X Y () X Y and Y ² X : 5 Thus, we say that X dominates Y under the FSD rules (X FSD Y ), if F X 6 F Y for all 2 R, where at least one strict inequality holds. Similarly, we say that X dominates Y under the SSD rules (X SSD Y ), if X 6 Y for all 2 R, with at least one inequality strict. Certainly, X FSD Y implies X SSD Y and X FSD Y implies X SSD Y. Note that F X expresses the probability of underachievement for a iven taret value. Thus the rst deree stochastic dominance is based on the multidimensional (continuum-dimensional) objective de ned by the probabilities of underachievement for all taret values. The FSD is the most eneral relation. If X FSD Y, then X is preferred to Y within all models preferrin larer outcomes, no matter how risk-averse or riskseekin they are. For decision makin under risk most important is the second deree stochastic dominance relation,. If X SSD Y, then X is preferred to Y within all risk-averse preference models that prefer larer outcomes. It is therefore a matter of primary importance that an approach to the comparison of random outcomes be consistent with the second deree stochastic dominance relation. Our paper focuses on the consistency of mean-risk approaches with SSD. Mean-risk approaches are based on comparin two scalar characteristics (summary statistics), the rst of which ± denoted l ± represents the expected outcome (reward), and the second ± denoted r ± is some measure of risk. The weak relation of meanrisk dominance is de ned as follows: associated with the function X X l=r Y () l X P l Y and r X 6 r Y : The correspondin strict dominance relation l=r is de ned in the standard way, as in Eq. (5). We say that X dominates Y under the l=r rules (X l=r Y ), if l X P l Y and r X 6 r Y, and at least one of these inequalities is strict. Note that random variables X and Y such that l X ˆ l Y and r X ˆ r Y are indi erent under the l=r rules (X l=r Y and Y l=r X ). We say then that X and Y enerate a tie in the l=r model. An important advantae of mean-risk approaches is the possibility of a pictorial trade-o analysis. Havin assumed a trade-o coe cient k between the risk and the mean, one may directly compare real values of l X kr X and l Y kr Y. Indeed, the followin implication holds: X l=r Y ) l X kr X P l Y kr Y for all k > 0: We say that the trade-o approach is consistent with the mean-risk dominance.

4 36 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Suppose now that the mean-risk model is consistent with the SSD model by the implication X SSD Y ) X l=r Y : Then mean-risk and trade-o approaches lead to uaranteed results: X l=r Y ) Y ² SSD X ; l X kr X > l Y kr Y for some k > 0 ) Y ² SSD X : In other words, they cannot strictly prefer an inferior decision. In this paper we show that some mean-risk models are consistent with the SSD model in the followin sense: there exists a positive constant a such that for all X and Y X SSD Y ) l X P l Y and l X a r X P l Y a r Y : 6 In particular, for the risk measure r de ned as the absolute semideviation (1) or standard semideviation (2), the constant a turns out to be equal to 1. Yitzhaki (1982) showed a similar result for the risk measure de ned as the RGini's R mean (absolute) di erence r X ˆ C X ˆ 1 2 jn j PX dn P X d. Relation (6) directly expresses the consistency with SSD of the model usin only two criteria: l and l a r. Both, however, are de ned by l and r, and we have l X P l Y and l X a r X P l Y a r Y ) l X kr X P l Y kr Y for 0 < k 6 a: Consequently, Eq. (6) may be interpreted as the consistency with SSD of the mean-risk model, provided that the trade-o coe cient is bounded from above by a. Namely, Eq. (6) uarantees that l X kr X > l Y kr Y ) Y ² SSD X : for some 0 < k 6 a Comparison of random variables is usually related to the problem of choice amon risky alternatives in a iven feasible set Q. For instance, in the simplest problem of portfolio selection (Markowitz, 1987) the feasible set of random variables is de ned as all convex combinations (weihted averaes with nonneative weihts totalin 1) of a iven number of investment opportunities (securities). A feasible random variable X 2 Q is called e cient under the relation if there is no Y 2 Q such that Y X. Consistency (6) leads to the followin result. Proposition 1. If the mean-risk model satis es (6), then except for random variables with identical l and r, every random variable that is maximal by l kr with 0 < k < a is e cient under the SSD rules. Proof. Let 0 < k < a and X 2 Q be maximal by l kr. This means that l X kr X P l Y kr Y for all Y 2 Q. Suppose that there exists Z 2 Q such that Z SSD X. Then, from Eq. (6), l Z P l X and l Z ar Z P l X ar X : 7 Addin these inequalities multiplied by 1 k=a and k=a, respectively, we obtain 1 k=a l Z k=a l Z ar Z P 1 k=a l X k=a l X ar X ; which after simpli cation reads: l Z kr Z P l X kr X : But X is maximal, so we must have l Z kr Z ˆ l X kr X ; 8 that is, equality in Eq. (8) holds. This combined with Eq. (7) implies l Z ˆ l X and r Z ˆ r X : Proposition 1 justi es the results of the meanrisk trade-o analysis for 0 < k < a. This can be extended to k ˆ a provided that the inequality l X a r X P l Y a r Y turns into equality only in the case of l X ˆ l Y. Corollary 1. If the mean-risk model satis es (6) as well as X SSD Y and l X > l Y ) l X a r X > l Y a r Y 9

5 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 37 then, except for random variables with identical l and r, every random variable that is maximal by l kr with 0 < k 6 a is e cient under the SSD rules. Proof. Due to Proposition 1, we only need to prove the case of k ˆ a. Let X 2 Q be maximal by l ar. Suppose that there exists Z 2 Q such that Z SSD X. Hence, by Eq. (6), l Z P l X. If l Z > l X, then Eq. (9) yields l X ar X < l Z ar Z, which contradicts the maximality of X. Thus, l Z ˆ l X and, by Eq. (6) and the maximality of X, one has l X ar X ˆ l Z ar Z. Hence, l Z ˆ l X and r Z ˆ r X : It follows from Proposition 1 that for mean-risk models satisfyin Eq. (6) the optimal solution of the problem maxfl X k r X : X 2 Q 10 with 0 < k < a, if it is unique, is e cient under the SSD rules. However, in the case of nonunique optimal solutions, we only know that the optimal set of Problem (10) contains a solution which is e cient under the SSD rules. The optimal set may contain, however, also some SSD-dominated solutions. Exactly, due to Proposition 1, an optimal solution X 2 Q can be SSD dominated only by another optimal solution Y 2 Q which enerates a l=r tie with X (i.e., l Y ˆ l X and r Y ˆ r X ). A question arises whether it is possible to additionally reularize (re ne) problem (10) in order to select those optimal solutions that are e cient under the SSD rules. We resolve this question durin the analysis of speci c risk measures. In many applications, especially in the portfolio selection problem, the mean-risk model is analyzed with the so-called critical line alorithm (Markowitz, 1987). This is a technique for identifyin the l=r e cient frontier by parametric optimization (10) for varyin k > 0. Proposition 1 uarantees that the part of the e cient frontier (in the l=r imae space) correspondin to trade-o coe cients 0 < k < a is also e cient under the SSD rules. 3. The O±R diaram The second deree stochastic dominance is based on the pointwise dominance of functions. Therefore, properties of the function are important for the analysis of relations between the second deree stochastic dominance and the meanrisk models. The followin proposition summarizes the basic properties which we use in our analysis. Proposition 2. If EfjX j < 1, then the function X is well de ned for all 2 R and has the followin properties: P1. X is continuous, convex, nonneative and nondecreasin. P2. If F X 0 > 0, then X is strictly increasin for all P 0. P3. X ˆ R n df X n ˆ R n P X dn ˆ PfX 6 Ef X jx 6. P4. lim! X ˆ 0. P5. X l X ˆ R 1 n df X n ˆ R 1 n P X dn ˆ PfX P EfX jx P. P6. X l X is a continuous, convex, nonneative and nonincreasin function of. P7. lim!1 X l X Š ˆ 0. P8. For any iven 0 2 R X P X 0 0 supff X n j n < 0 P X 0 0 for all < 0 ; X 6 X 0 0 supff X n j n < 6 X 0 0 for all > 0 : Properties P1±P4 are rather commonly known but frequently not expressed in such a riorous form for eneral random variables. Properties P5± P8 seem to be less known or at least not widely used in the stochastic dominance literature. In Appendix A we ive a formal proof of Proposition 2. From now on, we assume that all random variables under consideration are interable in the sense that EfjX j < 1. Therefore, we are allowed us to use all the properties P1±P8 in our analysis.

6 38 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Note that, due to property P3, X ˆ PfX 6 Ef X j X 6 thus expressin the expected shortae for each taret outcome. So, in addition to bein the most eneral dominance relation for all risk-averse preferences, SSD is a rather intuitive multidimensional (continuum-dimensional) risk measure. Therefore, we will refer to the raph of X as to the O±R diaram for the random variable X (Fi. 1). The raph of the function X has two asymptotes which intersect at the point l X ; 0. Speci cally, the -axis is the left asymptote (property P4) and the line l X is the riht asymptote (property P7). In the case of a deterministic outcome (X ˆ l X ), the raph of X coincides with the asymptotes, whereas any uncertain outcome with the same expected value l X yields a raph above (precisely, not below) the asymptotes. Hence, the space between the curve ; X, 2 R, and its asymptotes represents the dispersion (and thereby the riskiness) of X in comparison to the deterministic outcome of l X. We shall call it the dispersion space. Both size and shape of the dispersion space are important for complete description of the riskiness. Nevertheless, it is quite natural to consider some size parameters as summary characteristics of riskiness. As the simplest size parameter one may consider the maximal vertical diameter. By properties P1 and P6, it is equal to X l X. Moreover, property P3 yields the followin corollary. Corollary 2. If EfjX j < 1, then X l X ˆ d X. The absolute semideviation d X turns out to be a linear measure of the dispersion space. There are many aruments (see, e.., Markowitz, 1959) that only the dispersion related to underachievements should be considered as a measure of riskiness. In such a case we should rather focus on the downside dispersion space, that is, to the left of l X. Note that d X is the larest vertical diameter for both the entire dispersion space and the downside dispersion space. Thus d X seems to be a quite reasonable linear measure of the risk related to the representation of a random variable X by its expected value l X. Moreover, the absolute deviation d X ˆ jn l X j P X dn 11 is symmetric in the sense that d X ˆ 2 d X for any (possible nonsymmetric) random variable X. Thus absolute mean d also can be considered a linear measure of riskiness. A better measure of the dispersion space should be iven by its area. To evaluate it one needs to calculate the correspondin interals. The followin proposition ives these results. Proposition 3. If EfX 2 < 1, then X f df ˆ 1 2 n 2 P X dn ˆ 1 2 PfX 6 Ef X 2 jx 6 ; 12 Fi. 1. The O±R diaram.

7 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 39 X f f l X Š df ˆ 1 2 n 2 P X dn ˆ 1 2 PfX P Ef X 2 jx P : 13 Formula (12) was shown by Porter (1974) for continuous random variables. The second formula seems to be new in the SSD literature. In the Appendix A we ive a formal proof of both formulas for eneral random variables. Corollary 3. If EfX 2 < 1, then Z lx r 2 X ˆ 2 X f df; 14 Z lx r 2 X ˆ 2 X f df 2 X f f l X Š df: l X 15 Hereafter, whenever considerin variance r 2 or semivariance r 2 (standard deviation r or standard semideviation r) we will assume that EfX 2 < 1. Therefore, we are eliible to use formulas (14) and (15) in our analysis. By Corollary 3, the variance r 2 X represents the doubled area of the dispersion space of the random variable X, whereas the semivariance r 2 X is the doubled area of the downside dispersion space. Thus the semimoments d and r 2, as well as the absolute moments d and r 2, can be rearded as some risk characteristics and they are well depicted in the O±R diaram (Fis. 2 and 3). In further sections we will use the O±R diaram to prove that the mean-risk model usin the semideviations d and r is consistent with the second deree stochastic dominance. Geometrical relations in the O±R diaram make the proofs easy. However, as the eometrical relations are the consequences of Propositions 2 and 3, the proofs are riorous. To conclude this section we derive some additional consequences of Propositions 2 and 3. Let us observe that in the O±R diaram the diaonal line X 0 0 is parallel to the riht as- Fi. 2. The O±R diaram and the semimoments. Fi. 3. The O±R diaram and the absolute moments.

8 40 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 ymptote l X and intersects the raph of X at the point 0 ; X 0. Therefore, property P8 can be interpreted as follows. If a diaonal line (parallel to the riht asymptote) intersects the raph of X at ˆ 0, then for < 0, X is bounded from below by the line, and for > 0, X is bounded from above by the line. Moreover, the boundin is strict except in the case of supff X n j n < 0 ˆ 1 or F X 0 ˆ 1, respectively. Settin 0 ˆ l X we obtain the followin proposition (Fi. 4). Proposition 4. If EfX 2 < 1, then r X P d X and this inequality is strict except in the case r X ˆ d X ˆ 0. Proof. From P8 in Proposition 2, X > X l X l X for all < l X ; since supff X n j n < l X < 1. Hence, in the case of X l X > 0, one has 1 2 r2 X > 1 d 2 2 X and r X > d X. Otherwise r X ˆ d X ˆ 0: Recall that, due to the Lyapunov inequality for absolute moments (Kendall and Stuart, 1958), the standard deviation and the absolute deviation satisfy the followin inequality: r X P d X : 16 Proposition 4 is its analoue for absolute and standard semideviations. While considerin two random variables X and Y in the common O±R diaram one may easily notice that, if l X < l Y, then the riht asymptote of X (the diaonal line l X ) must intersect the raph of Y at some 0. By property P8, X P l X P Y for P 0 : Moreover, since l Y is the riht asymptote of (property P7), there exists 1 > 0 such that Y X > Y for P 1 : Thus, from the O±R diaram one can easily derive the followin, commonly known, necessary condition for the second deree stochastic dominance (Fishburn, 1980; Levy, 1992). Proposition 5. If X SSD Y, then l X P l Y. While considerin in the common O±R diaram two random variables X and Y with equal expected values l X ˆ l Y, one may easily notice that the functions X and Y have the same asymptotes. It leads us to the followin commonly known result (Fishburn, 1980; Levy, 1992). Proposition 6. For random variables X and Y with equal means l X ˆ l Y ; X SSD Y ) r 2 X 6 r2 Y ; 17 X SSD Y ) r 2 X < r2 Y : Absolute deviation as risk measure In this section we analyze the mean-risk model with the risk de ned by the absolute semideviation d iven by Eq. (1). Recall that dx ˆ X l X (Corollary 2) and it represents the larest vertical diameter of the (downside) dispersion space. Fi r2 X P 1 2 d 2 X.

9 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 41 Hence, d is a well de ned eometrical characteristic in the O±R diaram. Consider two random variables X and Y in the common O±R diaram (Fi. 5). If X SSD Y, then, is bounded from above by Y, and, by Proposition 5, l X P l Y. For P l Y, Y is bounded from above by d Y l Y (second inequality of P8 in Proposition 2). Hence, by the de nition of SSD, X d X ˆ X l X 6 Y l X 6 d Y l X l Y : This simple analysis of the O±R diaram allows us to derive the followin necessary condition for the second deree stochastic dominance. Proposition 7. If X SSD Y, then l X P l Y and l X d X P l Y d Y, where the second inequality is strict whenever l X > l Y. Proof. Due to the considerations precedin the proposition, we only need to prove that l X d X > l Y d Y whenever X SSD Y and l X > l Y. Note that from the second inequality of P8 ( ˆ l X, 0 ˆ l Y ), in such a case, d X ˆ X l X 6 X l Y l X l Y supff X n j n < l X < d Y l X l Y : Proposition 7 says that the l= d mean-risk model is consistent with the second deree stochastic dominance by the rule (6) with a ˆ 1. Therefore, a l= d comparison leads to uaranteed results in the sense that l X k d X > l Y k d Y for some 0 < k 6 1 ) Y ² SSD X : For problems of choice amon risky alternatives in a iven feasible set, due to Corollary 1, the followin observation can be made. Corollary 4. Except for random variables with identical mean and absolute semideviation, every random variable X 2 Q that is maximal by l X k d X with 0 < k 6 1 is e cient under the SSD rules. The upper bound on the trade-o coe cient k in Corollary 4 cannot be increased for eneral distributions. For any e > 0 there exist random variables X SSD Y such that l X > l Y and l X 1 e d X ˆ l Y 1 e d Y : As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 1= 1 e, PfX ˆ 1 ˆ e= 1 e ; and Y de ned as PfY ˆ 0 ˆ 1. Konno and Yamazaki (1991) introduced the portfolio selection model based on the l=d meanrisk model. The model is very attractive computationally, since (for discrete random variables) it leads to linear prorammin problems. Therefore, it is recently applied to various nance problems (e.., Zenios and Kan, 1993). Note that the absolute deviation d is a symmetric measure and the absolute semideviation d is its half. Hence, Proposition 7 is also valid (with factor 1/2) for the l=d mean-risk model. Thus, for the l=d model there exists a bound on the trade-o s such Fi. 5. SSD and the absolute semideviations: X SSD Y ) d X 6 d Y l X l Y.

10 42 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 that for smaller trade-o s the model is consistent with the SSD rules. Speci cally, due to Corollary 4, the followin observation can be made. Corollary 5. Except for random variables with identical mean and absolute deviation, every random variable X 2 Q that is maximal by l X kd X with 0 < k 6 1=2 is e cient under the SSD rules. The upper bound on the trade-o coe cient k in Corollary 5 can be substantially increased for symmetric distributions. Proposition 8. For symmetric random variables X and Y, X SSD Y ) l X P l Y and l X d X P l Y d Y : Proof. If X SSD Y then, due to Proposition 5, l X P l Y. From the second inequality of P8 in Proposition 2, 1 2 d X ˆ X l X 6 X l Y l X l Y supff X n j n < l X d Y 1 2 l X l Y ; since for symmetric random variables supff X n j n < l X 6 1=2: Hence, l X d X P l Y d Y : For problems of choice amon risky alternatives in a iven feasible set, Propositions 1 and 8 imply the followin result. Corollary 6. Within the class of symmetric random variables, except for random variables with identical mean and absolute deviation, every random variable X 2 Q that is maximal by l X kd X with 0 < k < 1, is e cient under the SSD rules. The bound on the trade-o coe cient k in Corollary 6 cannot be increased. There exist symmetric random variables X SSD Y such that l X > l Y and l X d X ˆ l Y d Y. As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 0:5, PfX ˆ 4 ˆ 0:5; and Y de ned as PfY ˆ 0 ˆ 0:5, PfY ˆ 2 ˆ 0:5. It follows from Corollary 4 that the optimal solution of the problem maxfl X k d X : X 2 Q; 0 < k 6 1; 19 is e cient under the SSD rules, if it is unique. In the case of multiple optimal solutions, the optimal set of problem (19) contains a solution which is e cient under SSD rules but it may contain also some SSD dominated solutions. Exactly, due to Corollary 4, an optimal solution X 2 Q can be SSD dominated only by another optimal solution Y 2 Q which enerates a l= d tie with X (i.e., l Y ˆ l X and d Y ˆ d X ). A question arises how di erent can the random variables be that enerate a tie (are indi erent) in the l= d mean-risk model. Absolute semideviation is a linear measure of the dispersion space and therefore many di erent distributions may tie in the l= d comparison. Note that two random variables X and Y with the same expected value l X ˆ l Y are l= d indi erent if X l X ˆ Y l X, independently of values of X and Y for all other 6ˆ l X. Consider two nite random variables: X de ned as PfX ˆ 20 ˆ 0:5, PfX ˆ 20 ˆ 0:5; and Y de ned as PfY ˆ 000 ˆ 0:01, PfY ˆ 0 ˆ 0:98, PfY ˆ 1000 ˆ 0:01. They are l= d indi erent, because l X ˆ l Y ˆ 0 and d X ˆ d Y ˆ 10. Nevertheless, X SSD Y and X < Y for all 0 < jj < As an extreme one may consider the case when X < Y for all 2 R except for ˆ l X ˆ l Y, and despite that X and Y are l= d indi erent (Fi. 6). Therefore, the l= d model, althouh consistent with the second deree stochastic dominance for bounded trade-o s, dramatically needs some additional reularization to resolve ties in comparisons. Ties in the l= d model can be resolved with additional comparisons of standard deviations or variances. By its de nition, a tie in the l= d model may occur only in the case of equal means, and this is exactly the case when Proposition 6 applies. We can simply select from X and Y the one that has a smaller standard deviation. It can be for-

11 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 43 malized as the followin lexicoraphic comparison: l X k d X ; r X P lex l Y k d Y ; r Y () l X k d X > l Y k d Y or l X k d X ˆ l Y k d Y and r X P r Y : The lexicoraphic relation de nes a linear order. Hence, for problems of choice amon risky alternatives in a iven feasible set, the lexicoraphic maximization of l k d; r is well de ned. It has two phases: the maximization of l k d within the feasible set, and the selection of the optimal solution that has the smallest standard deviation r. Owin to Eq. (18), such a selection results in SSD e cient solutions (even in the case of multiple optimal solutions). Corollary 7. Every random variable X 2 Q that is lexicoraphically maximal by l X k d X ; r X with 0 < k 6 1 is e cient under the SSD rules. For the l=d portfolio selection model (Konno and Yamazaki, 1991) the results of our analysis can be summarized as follows. While identifyin the l=d e cient frontier by parametric optimization maxfl X k d X : X 2 Q 20 for trade-o k varyin in the interval 0; 0:5Š the correspondin imae in the l=d space represents SSD e cient solutions. Thus it can be used as the mean-risk map to seek a satisfactory l=d compromise. It does not mean, however, that the solutions enerated durin the parametric optimization (20) are SSD e cient. Therefore, havin decided on some values of l and d one should apply the reularization technique (minimization of standard deviation) to select a speci c portfolio which is SSD e cient. 5. Standard semideviation as risk measure In this section we analyze the mean-risk model with the risk de ned by the standard semideviation r iven by Eq. (2). Recall that the standard semideviation is the square root of the semivariance which equals to the doubled area of the downside dispersion space (Corollary 3). Hence, r is a well de ned eometrical characteristic in the O±R diaram. Consider two random variables X and Y in the common O±R diaram (Fi. 7). If X SSD Y, then, is bounded from above by Y, and, by Proposition 5, l X P l Y. Due to the convexity of X, the downside dispersion space of X is no reater than the downside dispersion space of Y plus the area of the trapezoid with the vertices: l Y ; 0, l X ; 0, l X ; X l X and l Y ; Y l Y. Formally, by the de nition of SSD, X 1 2 r2 X r2 Y 1 2 l X l Y d X d Y : 21 This inequality allows us to derive new necessary conditions for the consistency with SSD of the bicriteria mean-risk model usin standard semideviation as the risk measure. Fi. 7. SSD and the semivariances: X SSD Y ) 1 2 r2 X r2 Y 1 2 l X l Y d X d Y.

12 44 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Proposition 9. If X SSD Y, then l X P l Y and l X r X P l Y r Y, where the second inequality is strict whenever l X > l Y. Proof. If X SSD Y then, due to Proposition 5, l X P l Y. Moreover, inequality (21) is valid. From Proposition 4 we have r X P d X and r Y P d Y. Usin these inequalities in (21) we et r 2 X r2 Y 6 l X l Y r X r Y : Hence, r X r Y 6 l X l Y, and nally l X r X P l Y r Y. Moreover, from Proposition 4, r X ˆ d X and r Y ˆ d Y can occur only if r X ˆ r Y ˆ 0. Hence, X SSD Y and l X > l Y ) l X r X > l Y r Y ; which completes the proof. The messae of Proposition 9 is that the l=r mean-risk model is consistent with the second deree stochastic dominance by the rule (6) with a ˆ 1. Therefore, l=r comparisons lead to uaranteed results in the sense that l X kr X > l Y kr Y for some 0 < k 6 1 ) Y ² SSD X : For problems of choice amon risky alternatives in a iven feasible set, Corollary 1 results in the followin observation. Corollary 8. Except for random variables with identical mean and standard semideviation, every random variable X 2 Q that is maximal by l X kr X with 0 < k 6 1 is e cient under the SSD rules. The upper bound on the trade-o coe cient k in Corollary 8 cannot be increased for eneral distributions. For any e > 0 there exist random variables X SSD Y such that l X > l Y and l X 1 e r X ˆ l Y 1 e r Y. As an example one may consider two nite random variables: X de- ned as PfX ˆ 0 ˆ 1 e 2, PfX ˆ 1 ˆ 1 1 e 2 ; and Y ˆ 0. It follows from Corollary 8 that the optimal solution of the problem maxfl X k r X : X 2 Q; 0 < k is e cient under the SSD rules, if it is unique. In the case of nonunique optimal solutions, however, we only know that the optimal set of problem (22) contains a solution which is e cient under SSD rules. Thus, similar to the l= d model, the l=r model may enerate ties (Fi. 8) and the optimal set of problem (22) may contain also some SSD dominated solutions. However, two random variables that enerate a tie (are indi erent) in the l=r mean-risk model cannot be so much di erent as in the l= d model. Standard semideviation r X is an area measure of the downside dispersion space and therefore it takes into account all values of X for 6 l X. Note that, if two random variables X and Y enerate a tie in the l=r model, then l X ˆ l Y and Z l X Z l Y X f df ˆ Y f df: Fi. 8. A tie in the l=r model: l X ˆ l Y, r X ˆ r Y and X SSD Y.

13 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 45 Functions are continuous and nonneative. Hence, if X SSD Y enerate a l=r tie, then X ˆ Y for all 6 l X. Thus a tie in the l=r model may happen for X SSD Y but the second deree stochastic dominance X over Y is then related to overperformances rather than the underperformances. Summin up, the l=r model needs some additional reularization to resolve ties in comparisons, but it is not such a dramatic need as in the l= d model. Similar to the l= d model, ties in the l=r model can be resolved by additional comparisons of standard deviations or variances. In the case when l X kr X ˆ l Y kr Y, one may select from X and Y the one that has a smaller standard deviation. It can be formalized as the followin lexicoraphic comparison: l X kr X ; r X P lex l Y kr Y ; r Y () l X kr X > l Y kr Y or l X kr X ˆ l Y kr Y and r X P r Y : For problems of choice amon risky alternatives in a iven feasible set, the lexicoraphic maximization of l kr; r has two phases aain: maximization of l kr on the feasible set, and selection of the optimal solution that has the smallest standard deviation r, if many optimal solutions occur. Due to Eq. (18), such a selection results in SSD e cient solutions (even in the case of multiple optimal solutions). Corollary 9. Every random variable X 2 Q that is lexicoraphically maximal by l X kr X ; r X with 0 < k 6 1 is e cient under the SSD rules. The mean±semivariance optimization approach was proposed by Markowitz (1959). It is quite an intuitive modi cation of the mean±variance model, since an investor worries about underperformance rather than overperformance. Nevertheless, it is less used in portfolio optimization. One reason is that it is more di cult to compute the mean± semivariance e cient frontier that for the mean± variance model. Still, Markowitz et al. (1993) have developed a critical line alorithm for the mean± semivariance e cient frontier. The use of semivariance r 2 or standard semideviation r in the mean-risk analysis may be considered to be equivalent, with the former easier to implement. In fact, both de ne exactly the same e cient set, since standard deviation is nonneative and the square function is strictly increasin for nonneative aruments. However, our result that the l=r model with trade-o s bounded by one is consistent with the SSD rules cannot be directly applied to the mean±semivariance model. Note that X 2 Q that is maximal by l kr 2 may be not maximal by l kr X r, in eneral. Consider two random variables X and Y with l X ˆ 0, r X ˆ 1 and l Y ˆ 1, r Y ˆ 2, respectively. For k ˆ 0:4, l X 0:4r 2 X ˆ 0:4 > 0:6 ˆ l Y 0:4r 2 Y but l X 0:4r X ˆ 0:4 < 0:2 ˆ l Y 0:4r Y. While comparin two random variables X and Y by the mean±semivariance trade-o analysis the followin relationship is valid: l X kr 2 X P l Y kr 2 Y () l X k r X r Y Šr X P l Y k r X r Y Šr Y : Thus for problems where the l=r 2 e cient set is bounded (like typical portfolio selection problems), there exists an upper bound on the trade-o coe cients which uarantees that for smaller trade-o s the correspondin mean-risk e cient solutions are also e cient under the SSD rules. It explains the hih number of SSD e cient solutions included in the l=r 2 e cient set observed in experiments with real-life portfolio selection problems (Porter, 1974). The upper bound, thouh, may be very tiht. 6. Standard deviation as risk measure After the work of Markowitz (1952) the variance (or the standard deviation) is the most frequently used risk measure in mean-risk models for portfolio selection. The O±R diaram (Fi. 3) shows the variance as a natural area measure of dispersion. Comparison of random variables with equal means leads to uaranteed results (Proposition 6). However, for eneral random variables X and Y with unequal means, no relation involvin their standard deviations is known to be necessary

14 46 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Fi. 9. SSD and the variances: X SSD Y ) 1 2 r2 X r2 Y 1 2 l X l Y M X l X M Y l Y Š. for the second deree stochastic dominance of X over Y. Inp the case of symmetric distributions one has r ˆ 2 r, and r in Proposition 9 can be replaced p with standard deviation r multiplied by factor 2 =2. It turns out, however, that for symmetric distributions the relation between d and r can be described in more detail. Proposition 10. For symmetric random variables X and Y, X SSD Y ) l X P l Y and l X r X P l Y r Y : 23 Proof. If X SSD Y then, by Proposition 5, l X P l Y. Moreover, inequality (21) is valid. Lyapunov inequality (16) for symmetric variables yields r X P 2 d X and r Y P 2 d Y. Usin these inequalities in (21) we et r 2 X r2 Y 6 l X l Y r X r Y : Hence, r X r Y 6 l X l Y, and nally l X r X P l Y r Y which completes the proof. For problems of choice amon risky alternatives in a iven feasible set, Propositions 1 and 6 imply the followin result. Corollary 10. Within the class of symmetric random variables, every random variable X 2 Q that is maximal by l X kr X with 0 < k < 1, is e cient under the SSD rules. The bound on the trade-o coe cient in Corollary 10 is the best in the sense that there exist symmetric random variables X SSD Y such that l X r X ˆ l Y r Y. As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 0:5, PfX ˆ 4 ˆ 0:5; and Y de ned as PfY ˆ 0 ˆ 0:5, PfY ˆ 2 ˆ 0:5. Therefore, the upper bound on the trade-o coe cient k in Corollary 10 cannot be increased. In the eneral case of nonsymmetric random variables, standard deviation is not a symmetric measure and there is no direct analoue of Proposition 9 for the standard deviation. Some similar, but much weaker, necessary conditions for the SSD dominance can be derived for distributions bounded from above (random variables with an upper bounded support). Note that, if X is upper bounded by a real number M X (i.e. PfX > M X ˆ 0), then M X P l X and for P M X, F X ˆ 1. Thus for P M X the function X coincides with its riht asymptote ( X ˆ l X ). Hence

15 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 47 Z lx Z M X r 2 X ˆ 2 X f df 2 X f f l X Š df: l X Consider two random variables X and Y such that PfX > M X ˆ 0 and PfY > M Y ˆ 0 in the common O±R diaram (Fi. 9). If X SSD Y, then, by the de nition of SSD, X is bounded from above by Y and, by Proposition 5, one has l X P l Y. Due to the convexity of X, the area between this function and its asymptotes cannot be reater than the area between Y and its asymptotes plus the area of the trapezoid de ned by the vertices: l Y ; 0, l X ; 0, M X ; M X l X and M Y ; M Y l Y. This is valid for M X > M Y (like in Fi. 9), as well as for M X 6 M Y. Formally, 1 2 r2 X r2 Y 1 2 l X l Y M X l X M Y l Y Š: 24 This inequality is similar but stroner than the inequality derived by Levy (1992), Theorem 9, p. 570, which reads: r 2 X r2 Y 6 l X l Y 2 maxfm X ; M Y l X l Y : Inequality (24) allows for the formulation of the followin necessary conditions for the bicriteria mean-risk model with the standard deviation used as the risk measure. Proposition 11. Suppose that a common upper bound h > 0 is known for X l X =r X and Y l Y =r Y. Then X SSD Y ) l X P l Y and l X 1 h r X P l Y 1 h r Y : Proof. If X SSD Y, then due to Proposition 5, l X P l Y. Further, note that inequality (24) can be rewritten as r 2 X r2 Y 6 h r X r Y l X l Y : This immediately yields the required result. Corollary 11. Within the class of random variables such that PfX > l X hr X ˆ 0, every random variable that is maximal by l X kr X with 0 < k < 1=h, is e cient under the SSD rules. Note that Proposition 11 is applicable to any pair of nite random variables. Similarly, Corollary 11 shows that in the case of a portfolio selection problem with nite random variables (for example de ned by historical data), there exists a positive bound on the trade-o coe cient for the standard deviation which uarantees that for smaller tradeo s the correspondin mean-risk e cient solutions are also e cient under the SSD rules. Analoously to the case of the semivariance discussed in the previous section, an upper bound on the variance trade-o coe cients exists which uarantees that l=r 2 e cient solutions are also e cient under the SSD rules. However, the upper bound may be very tiht. Corollary 11 provides a theoretical explanation for the results of numerous experimental comparisons of mean±variance and SSD e cient sets on real-life portfolio selection problems (Porter, 1974, and references therein). Most of them, like that performed by Porter and Gaumnitz (1972) on over 900 portfolios of securities randomly selected from the Chicao Price Relative File, provided some support for the idea that mean±variance and SSD choices are empirically similar. The main di erence was the tendency of the mean±variance e cient set to include some low mean, low variance portfolios that were eliminated by the SSD rules. Althouh e cient in the mean±variance analysis, they obviously correspond to lare trade-o coe cients for the variance. 7. Concludin remarks The second deree stochastic dominance relation is based on an axiomatic model of risk-averse preferences, but does not provide us with a simple computational recipe. The mean-risk approach quanti es the problem in only two criteria: the mean, representin the expected outcome, and the risk: a scalar measure of the variability of outcomes (usually, a central moment or the correspondin deviation). This is appealin to decision makers and allows a simple trade-o analysis, analytical or eometrical. In the paper we have analyzed the consistency of these two approaches. We have shown that standard semideviation (the square root of the

16 48 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 semivariance) as the risk measure makes the meanrisk model consistent with the second deree stochastic dominance, provided that the trade-o coe cient is bounded by one. Similar results have been obtained for the absolute semideviation as the risk measure. These results are valid for all (possibly nonsymmetric) random variables for which these moments are well-de ned. In the case of symmetric random variables the same results are valid for the standard and absolute deviations, respectively. In many applications, especially in portfolio selection problems, the mean-risk model is analyzed by the critical line alorithm. This is a technique for identifyin the mean-risk e cient frontier via parametric optimization with a varyin trade-o coe cient. Our results uarantee that when risk is measured by the standard or the absolute semideviation (the standard or the absolute deviation in the case of symmetric distributions), the part of the e cient frontier (in the mean-risk imae space) correspondin to trade-o coe cients smaller than one is also e cient under the SSD rules. In some way our analysis justi es the critical line methodoloy for typical risk measures, provided that it is not extended too far in terms of the trade-o coe cient. It also explains some results of experimental comparisons of the SSD and mean-risk e cient sets for portfolio selection problems. In the analysis we have used a new raphical tool, the O±R diaram, which appears to be particularly convenient for comparin uncertain outcomes and examinin second deree stochastic dominance. Typical dispersion statistics, commonly used as risk measures (absolute deviation and semideviation, variance and semivariance) are well depicted in the O±R diaram, and it may be useful for various types of comparisons of uncertain outcomes, especially in computerized decision support systems. Appendix A A.1. Proof of Proposition 2 P1. Simple consequence from de nition of X. P2. Simple consequence from de nition of X. P3. Chanin the order of interation by Fubini's theorem we obtain X ˆ P lim ˆ ˆ ˆ F X f df Z f n P X dn df df P X dn n P X dn ˆ PfX 6 Ef X jx 6 : X! 6 lim! ˆ lim! jnj P X dn ˆ 0; n P X dn because n 6 jnj for n 6 < 0, and EfjX j < 1. P5. X l X ˆ ˆ ˆ n P X dn l X P X dn n P X dn n P X dn ˆ PfX P EfX jx P : P6. Simple consequence from P5. P7.

17 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33± lim X l X Š!1 ˆ lim!1 Z1 6 lim!1 Z1 n P X dn n P X dn ˆ 0; similarly to P4. P8. If < 0, then X 0 X ˆ 0 F X n dn 6 0 supff X n j n < : Interchanin and 0 we obtain the second inequality. A.2. Proof of Proposition 3 By Fubini's theorem, X f df ˆ 2 Z f 4 f n P X dn 5 df f 6 n 6 f 3 Z Z ˆ f n P X dn df ˆ ˆ n 3 7 f n df5 P X dn n 2 P X dn ˆ 1 2 PfX 6 Ef X 2 jx 6 : Analoously, notin that f l X ˆ we obtain f n P X dn ; X f f l X Š df ˆ n f P X dn 5 df f 6 f f 6 n 3 Z Z ˆ n f P X dn df ˆ ˆ Z n 3 7 n f df5 P X dn n 2 P X dn ˆ 1 2 PfX P Ef X 2 jx P : References Bawa, V.S., Stochastic dominance: A research biblioraphy. Manaement Science 28, 698±712. Fishburn, P.C., Decision and Value Theory, Wiley, New York. Fishburn, P.C., Mean-risk analysis with risk associated with below taret returns. The American Economic Review 67, 116±126. Fishburn, P.C., Stochastic dominance and moments of distributions. Mathematics of Operations Research 5, 94± 100. Hanoch, G., Levy, H., The e ciency analysis of choices involvin risk. Review of Economic Studies 36, 335±346. Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities, Cambride University Press, Cambride, MA. Kendall, M.G., Stuart, A., The Advanced Theory of Statistics, vol. 1: Distribution Theory, Gri n, London. Konno, H., Yamazaki, H., Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market. Manaement Science 37, 519±531. Levy, H., Stochastic dominance and expected utility: Survey and analysis. Manaement Science 38, 555±593. Markowitz, H.M., Portfolio selection. Journal of Finance 7, 77±91. Markowitz, H.M., Portfolio Selection. Wiley, New York. Markowitz, H.M., Mean±Variance Analysis in Portfolio Choice and Capital Markets. Blackwell, Oxford. Markowitz, H.M., Todd, P., Xu, G., Yamane, Y., Computation of mean±semivariance e cient sets by the critical line alorithm. Annals of Operations Research 45, 307±317.

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