Risk Probability Functionals and Probability Metrics Applied to Portfolio Theory

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1 Risk Probability Functionals and Probability Metrics Alied to Portfolio Theory Sergio Ortobelli Researcher, Deartment MSIA, Uniersity of Bergamo, Italy. Setlozar T. Rache* Chair Professor, Chair of Econometrics, Statistics and Mathematical Finance, School of Economics and Business Engineering, Uniersity of Karlsruhe (Germany), Alied Probability Uniersity of California at Santa Barbara Haim Shalit Professor of Economics, Deartment of Economics, Ben-Gurion Uniersity Israel Frank J. Fabozzi Adjunct Professor of Finance and Becton Fellow, School of Management, ale Uniersity Abstract: In this aer, we inestigate the imact of seeral ortfolio selection models based on different tracking error measures, erformance measures, and risk measures. In articular, mimicking the theory of ideal robability metrics, we examine ideal financial risk measures in order to sole ortfolio choice roblems. Thus we discuss the roerties of seeral tracking error measures and risk measures and their consistency with the choices of risk-aerse inestors. Furthermore, we roose seeral linearizable allocation roblems consistent with a gien ordering and we show that most of Gini s measures, at less of linear transformations, are linearizable and coherent risk measures. Finally, assuming ellitical distributed returns, we describe the mean-risk efficient frontier using different arametric risk measures reiously introduced. Keywords: Probability metrics, tracking error measures, stochastic orderings, coherent measures, linearizable otimization roblems. ====================================================================== Acknowledgment footnote: Sergio Ortobelli s research has been artially suorted under Murst 60% 2005, Setlozar Rache's research has been suorted by grants from Diision of Mathematical, Life and Physical Sciences, College of Letters and Science, Uniersity of California, Santa Barbara and the Deutsche Forschungsgemeinschaft. *Contact author: Prof. Setlozar T.Rache, Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering, Uniersity of Karlsruhe, Kollegium am Schloss, Bau II, 20.2, R20, Postfach 6980, D-7628, Karlsruhe, Germany; Tel , (s). zari.rache@statistik.uni-karlsruhe.de

2 Risk Probability Functionals and Probability Metrics Alied to Portfolio Theory. Introduction The urose of this aer is to resent a unifying framework for understanding the connection between ortfolio theory, ordering theory, and the theory of robability metrics. To do so, we discuss the use of different tracking error measures, erformance measures, and risk measures consistent with inestors references and we examine the comutational comlexity in deriing otimization roblems. More secifically, we begin by analyzing inestors otimal choices coherently with their references. Then we discuss the arallelism between robability metric theory and the benchmark tracking roblem. Finally, we analyze the use of different measures consistent with stochastic orderings when ortfolio returns are ellitically distributed.. It is within his context that we characterize the efficient choices using different arametric risk measures. In recent years, seeral aers hae roffered alternatie definitions of the best risk measure to emloy in selecting otimal ortfolios. Many studies hae obsered the nonmonotonicity feature of the traditional mean-ariance (MV) criterion and roosed alternatie formulations to correct the limitations of MV analysis. Maccheroni et al (2005), for examle, hae demonstrated recently that the non-monotonicity of MV references can be alleiated using the relatie Gini concentration index. The most notable work on this toic is robably due to Artzner et al. (999) who suggest a minimal set of roerties that a risk measure has to satisfy in order to be a coherent risk measure. These are the axioms of coherency meaning that a coherent measure is a ositiely homogenous, translation inariant, subadditie, and monotone risk measure. For ealuating exosure to market risks, Artzner et al. (999), Martin et al. (2003), and Acerbi (2002) roose, among seeral coherent risk measures, the conditional Value-at-Risk (CVaR). This measure, also called exected shortfall or exected tail loss, is a linear transformation of the Lorenz cure from which it deries its roerties. The Lorenz (905) cure, first used to rank income inequality, is now used in its absolute form to cature the essential descritie features of risk and stochastic dominance (see Shorrocks (983) and, Shalit and itzhaki (994)). In this aer we demonstrate the links between stochastic dominance (SD) orders and dual SD rules based on Lorenz orders (see Aaberge (2005) and Ogryczak and Ruszczynski (2002a. 2

3 2000b)). In this context, we introduce different linearizable ortfolio selection models that are consistent with stochastic orders. In articular, we aly a method referred to as FORS robability functionals introduced by Ortobelli et al (2006) to ortfolio selection roblems. FORS robability functionals and orderings are strictly linked to the theory of robability metrics. In articular, as shown by Rache et al (2005) and Stoyano et al (2006b), a tracking error measure can be associated with each robability metric. Thus we examine and deelo ideal financial risk measures that mimick the theory of ideal robability metrics, a theory that started with the fundamental work of Kolmogoro and Kantoroich (see Kalashniko and Rache (988), Kakosyan et al (987), Rache and Ruschendorf (998, 999), and Rache (99)).. Many recent results on risk measures and ortfolio literature can be seen as articular cases of the results resented in this aer. For examle, a large subclass of sectral measures roosed by Acerbi (2002)) is simly deried from the Lorenz cure. Thus, these measures are coherent and linearizable and can be easily reresented with resect to their consistency with dual orders. Among seeral examles roosed in the theory of robability metrics, we consider Gini-tye measures that are deried from the fundamental studies of Gini and his students (see among others, Gini (92, 94, 965). Salemini (943, 957), and Dall Aglio (956)). Tyically, the extended Gini mean difference (see Shalit and itzhaki (984) and itzhaki (983)) minus the mean is a coherent risk measure because it is a simle deriation from the absolute Lorenz cure. Accordingly, we roose linear ortfolio selection models based on these measures. Moreoer, in order to account for return distributional tails, that reresent the future admissible losses, we generalize the tail extension of Gini mean difference roosed by Ogryczak and Ruszczynski (2002a, 2000b) with the extended Gini mean difference minus the mean being a linearizable coherent risk measure. We also demonstrate that the Gini index of dissimilarity and indexes based on the Lorenz cure define a class of FORS tracking error measures. We also show how to utilize the FORS tye risk measures when the returns are ellitical distributed. In articular, we resent a characterization of the efficient frontier as function of arametric FORS tye risk measures when unlimited short sales are allowed and returns are ellitically distributed. Finally, we discuss the use of different risk measures in terms of rewardrisk erformance ratios (see also Rache et al (2005) and Stoyano et al (2006a)). We hae organized the aer as follows. In Section 2, we summarize continua and inerse stochastic dominance rules and derie ortfolio selection roblems based on risk measures consistent with these orders. In Section 3, we describe tracking error measures based on robability metrics and roose tracking error ortfolio selection roblems based on concentration cures. Portfolio selection roblems based on Gini-tye risk measures are analyzed in Section 4. In 3

4 Section 5, we study ellitical models with FORS risk measures. In Section 6, we discuss rewardrisk models based on different risk/reward measures. In the last section of the aer we briefly summarize our results. 2 Continua and Inerse Stochastic Dominance Rules in Portfolio Theory In this section, we look at the fundamental concets of continua and inerse stochastic dominance rules (see, among others, Ortobelli et al. (2006) and Ley (992)) and study the linearizable otimization roblems that are consistent with these stochastic orders. In financial economics, the main stochastic orders used are () the first-degree stochastic dominance (FSD) for non-satiable agents, (2) the so-called Rothschild-Stiglitz stochastic dominance concae order (RSD) for risk-aerse inestors (Rothschild and Stiglitz (970)), and (3) the second-degree stochastic dominance (SSD) for non-satiable risk-aerse inestors. Gien two risky assets, strictly dominates with resect to FSD if and only if for all and eery increasing utility function u, E(u()) E(u()) and a strict inequality holds for some u. Stated in terms of distribution functions, FSD if and only if F ( t) = Pr( t) F ( t) = Pr( t) and a strict inequality holds for at least a real t. For non-satiable risk-aerters, strictly dominates with resect SSD, if and only if for all and eery increasing, concae utility functions u, E(u()) E(u()) and a strict inequality holds for some u. Exressed in terms of distributions, SSD if and only if t (2) (2) t F () t = F () u du F () t = F () u du for all t and a strict inequality holds for at least a real t. For the Rothschild-Stiglitz dominance, RSD if and only if for all and eery concae utility functions u, E(u()) E(u()) and the inequality is strict for some u, or equialently if and only if E()=E() and SSD. Fishburn (976, 980) ointed out that stochastic dominance rules can be exressed in continuous terms by using the definition of fractional integrals. Thus, we say that dominates with resect to stochastic dominance order (with ) if and only if E u Eu ( ) for all utility functions such that + + { () (), ; σ } () + x u U = u x = c y x d y c x R where is ositie finite measure y d y <, 4

5 if and only if for eery real t t ( ) F () t = ( t y) df( y) F () t Γ( ). In articular, the deriaties of u satisfy the inequalities U k+ () k () u 0where k=,, n- for the integer n that satisfies n < n. Portfolio theory is linked to stochastic dominance theory. Indeed, to select the set of admissible choices that are coherent to a gien category of inestors, we can consider the direct risk measures ρ ( ) associated to random wealth that are consistent with the order relation (i.e., ρ( ) ρ( ) if dominates ). Similarly, we can consider reward measures isotonic with an order relation (i.e., if dominates ). In articular, the risk measure consistent with FSD is generally called a safety-risk measure. Hence a non-satiable or non-satiable riskaerse inestor chooses a ortfolio that minimizes the risk measure that is consistent with the FSD or SSD order. Considering that for eery ( + ) ( ) >, ( ) Γ F t = E t, then we can easily define ortfolio selection models that are consistent with stochastic dominance order. Consider the ortfolio roblem of otimal allocation x = [ x, x2,..., x n ]', between n assets with returns r = [ r,..., r n ]'. No short selling is allowed, (i.e. x i 0 ). To find the otimal ortfolios for inestors with a utility function that belongs to U and >, one soles the following otimization roblem: min x T subject to T k k= Exr ( ' ) m; x = ; x 0; j=,..., n j= 0; t x r, k =,..., T k k i i, k i= n n j j () for a mean greater than m and a gien risk-aersion arameter t max min xr i i, k. n xi 0 k T i= In order to get choices consistent with -bounded RSD, one soles a similar otimization roblem as () adding the further constraints x r, t, k =,..., T. In articular, in order to k n i i k i= get otimal choices for non-satiable risk-aerse inestors, we hae to sole the reious linear rogramming (LP) roblem corresonding to = 2. Furthermore, for > 2 the roblem () is a conex otimization roblem and thus it is linearizable. 5

6 As an alternatie to classic stochastic orders, we can use the dual (also called inerse) reresentations of stochastic dominance rules that we now resent (Shorrocks (983), Dybig (988), Rache (99), and Ogryczak and Ruszczynski (2002a, 2002b)): ) FSD F F 0 2) SSD L L 0 where F (0) = lim F and (0,] 0, F inf { x:pr( x) F ( x) } inerse of F. Furthermore, L u ( ) = () su 0 F t dt = { u F () t dt } = =, is the left is the absolute Lorenz cure (or absolute concentration cure) of asset with resect to distribution function F. Muliere and Scarsini (989), aari (987), and Ortobelli et al (2006) show how inerse stochastic dominance rules can be extended to continuous terms. Thus, dominates with resect to inerse stochastic dominance order u (with ) if and only if for eery [0,], ( ) F = ( u) df ( u) F Γ( ) if and only if φ( x) df 0 0 x φ x df 0 x functions φ V where { + x ( ) τ all utility V = φ() x = ( s x) dτ() s k( x) k 0; τ is a σ finite ositie measure s.. t and (0,): F < the function s x is d ( s) df ( x) integrable in [0,] [0,] }. In the finance literature, the negatie -quantile F is also called Value-at-Risk (VaR) of or ( VaR ( ) = F ). It exresses the maximum loss among the best ercentage cases that could occur for a gien horizon. In contrast, the absolute concentration cure L alued at shows the mean return accumulated u to the lowest ercentage of the distribution. Both measures F and L hae imortant financial and economic interretations and are widely used in the recent risk literature. In articular, the negatie absolute Lorenz cure diided by robability is the conditional Value-at-Risk (CVaR) or exected shortfall, exressed as CVaR L ( ) =. This risk measure is coherent in the sense of Artzner et al (999) because it has the following roerties: ) Subadditiity ( CVaR ( + ) CVaR ( ) + CVaR ); 6

7 2) Positie homogeneity ( 0 CVaR ( ) = CVaR ( )); 3) Monotonicity ( CVaR ( ) CVaR ( ) ) 4) Translation inariance ( t R CVaR ( + t) = CVaR ( ) t ). CVaR is consistent with FSD and SSD stochastic orders. Furthermore, if one uses su { u } L = u F t dt then, u L CVaR ( ) = = inf u + E (( u) ) u +, where the otimal alue u is VaR ( ) = F ( Pflug (2000)). For a gien robability loss, the set of otimal ortfolios for non-satiable and risk-aerse inestors is found by soling the following LP roblem: T min b+ t T xb, t= subject to Exr ( ' ) m; x = ; x 0; j=,..., n j= 0; x r b, t =,..., T t t i i, t i= for some gien m (see, among others, Pflug (2000)). Coherent risk measures using secific functions for the Lorenz cure are easily obtained. In articular, we obsere that some classic Gini tye measures are coherent measures. n n j j (2) Remark The following holds: ) For eery and for eery β (0,) the measure ( ) Γ ( + ) F () ( u) L () udu β β (( + )) 2 β = β β 0 is a coherent risk measure consistent with order that is linearizable ( + ) 2) For eery and for eery β (0,) the measure consistent with RSD order. The roof is gien in the aendix. Γ ( + ) ( ( + )) Γ, β () = E F ( β ) is β A tyical alication of Remark is roided by Acerbi s sectral measures (see Acerbi (2002)). According to his definition, any sectral measure 7

8 φ 0 M = ( u) F ( u) du φ is a coherent risk measure identified by its risk sectrum φ that is an a. e. non-negatie decreasing and integrable function such that 0 φ ( udu ) =. From this definition, it follows that any sectral measure is consistent with FSD. In articular, when F (0) = 0, M φ ( ) ( x) df ( x) where = ( x) dτ ( s) is a secific function that belongs to the set x = 0 τ is a V reiously defined and (s) robability measure on [0,] that is absolutely continuous with resect to the Lebesgue measure whose density is gien by the decreasing risk sectrum function φ. In addition, Acerbi (2002) shows that for any a. e. non-negatie, decreasing function φ (.) and for any N i.i.d. realizations,..., N of the integrable random ariable, the sectral measures Mφ ( ) associated to the standardized risk sectrum can be estimated by the consistent statistic: M N φ N i/ N N i= k= φ ( k/ N) VaR i / N = φ where VaR / ( ) denotes the oosite of the i/n ercentile of (i.e., the oosite of the i-th i N obseration of the ordered,..., N ). Furthermore, when the risk sectrum φ (.) is itself absolutely continuous with resect to the Lebesgue measure on [0,], we can define the linearizable sectral risk measure as: 0 0 GMφ ( ) = φ( u) F ( u) du = φ'( u) L ( u) du φ() E( ). Hence, when all otimal choices are uniquely determined by the mean and the risk measure GMφ ( ) any non-satiable risk-aerse inestor chooses a ortfolio solution by soling the following LP roblem: min i i ' a + φ() E ( x ' r ) subject to T T φ i k, i x,,..., at i T = T T k= n x r a ; 0; t =,..., T; i =,..., T ti, j jt, i ti, j= n j i= Exr ( ' ) m; x = ; x 0; j=,..., n. j for some gien mean m. Ortobelli et al (2006) show that all stochastic dominance and inerse stochastic dominance orders are secific FORS orderings. The sectral measures are FORS measures induced by FSD. Let us recall the basic concet of FORS measures and orderings. We call FORS measure induced by 8

9 order f any robability functional µ : Λ Λ R (where Λ is a sace of real-alued random ariables defined on the robability sace ( ΩI,,P) ) that is consistent with resect to a gien order of references. For examle, that dominates with resect to a gien order of references f on Λ imlies that µ (, Z) µ (, Z) for a fixed and arbitrary benchmark Z. We say that a robability functional µ is a FORS uncertainty measure if it is consistent with RSD orders. Hence a robability functional µ is a FORS risk measure if it is consistent with risk tye orders (for examle, >, > ). Examle of a FORS risk measure is ( F ), for a fixed benchmark (0,) that is induced by > order. Examle of FORS uncertainty measure t, E( t ρ = ) % for a fixed benchmark t R, is induced by -RSD order. Moreoer, suose ρ :[ ab, ] R is a bounded ariation function, for eery random ariable belonging to a gien class Λ and assume that the functional λ, ρ ρ is simle (i.e., for eery Λ,, ρ = ρ F = F). If, for any fixed [ ab, ] λ is a FORS risk measure induced by a risk ordering f, then, we call FORS risk orderings induced by f the following new class of orderings defined for eery >, b, Λ ( ) = Λ t dρ ( t) < a where ρ ( u) ( ), We call FORS f, u ( u t) dρ ( t) if > Γ a =. ρ ( u) if = iff ρ, ( u) ρ, ( u) u [,] ab ρ the FORS risk measure associated with the FORS ordering of random ariables belonging to class Λ. In addition, we say that dominates in the sense of FORS uncertainty ordering induced by f (written as FORS ) if and only if f,unc x ( x s) dρ± () s ( ) + a x x s dρ± () s + x [,] ab a (i.e. when FORS f, and FORS ). f, In the following definition we distinguish classes of FORS measures consistent with a gien ordering of reference that satisfy only some of the coherency axioms. Definition When the simle robability FORS risk measure ρ λ associated with a FORS ordering is ositiely homogeneous and translation inariant for any gien λ [ ab, ], ρ is called a characteristic FORS functional of the associated ordering. If in addition, ρ λ is subadditie 9

10 λ [ ab, ] then ρ ( λ ) is called a coherent FORS functional associated with the underlining ordering. In articular, we obsere that a sectral measure could itself generate arametrically a coherent FORS functional. As a further examle, consider the characteristic functional ρ ρ ( λ ) =VaR λ () that is not coherent (being not subadditie). Instead, the characteristic functionals ( λ ) =CVaR λ () and > and β (0,). g Γ ( + ) ( ( + )) ( β ) = F ( β ) are coherent FORS functional for eery β Moreoer, some characteristic FORS functionals identify the underlining ortfolio distributions only if all risk-returns belong to a articular class of distribution functions. For examle, measures q q λ ρ λ = λ E ( E ) E and λ ρ ( λ ) = E E E are simle robability functionals consistent with SSD, assuming that all the admissible choices deend on the mean and a disersion measure. 3. Probability Distances and Tracking Error Measures Any robability functional µ is called a robability distance with arameter K if it is ositie and it satisfies the following additional roerties: ) Identity f = f µ (, ) = 0; 2) Symmetry µ (, ) = µ (, ) 3) Triangular inequality µ (, Z) K( µ (, ) µ (, Z) ) + for all admissible random ariables,, and Z where f() identifies some characteristics of the random ariable. If the arameter K equals, we hae a robability metric. We can always define the alternatie finite distance µ (, ) = H µ (, ), where H :[0, + ) [0, + ) is a non-decreasing ositie continuous H function such that H(0)=0 and K H H(2 t) = su < + (see Rache (99) for further generalization). t> 0 Ht () Thus, for any robability metric µ, µ H is a robability distance with arameter K H. In this case we distinguish between rimary, simle, and comound robability distances/metrics that deend on certain modifications of the identity roerty (see Rache (99)). Comound robability functionals identify the random ariable almost surely i.e.: µ (, ) = 0 P ( = ) =. Simle 0

11 robability functionals identify the distribution (i.e., µ (, ) = 0 F = F). Primary robability functionals determine only some random ariable characteristics. In essence, robability metrics can be used as tracking error measures. In soling the ortfolio roblem with a robability distance we intend to aroach the benchmark and change the ersectie for different tyes of robability distances. Hence, if the goal is only to control the uncertainty of our ortfolio or to limit its ossible losses, mimicking the uncertainty or the losses of the benchmark can be done using a rimary robability distance. When the objectie for our ortfolio is to mimic entirely the benchmark, a simle or comound robability distance should be used. On the other hand, using a comound distance can be twofold because in addition to their role as tracking error, we can use them as measures of uncertainty. As a matter of fact, if we aly any comound distance µ (, ) to and = that are i.i.d., then we get: µ (, ) = 0 P( = ) = is a constant almost surely. Thus, we call µ (, ) = µ an index concentration measure deried by the comound I distance µ. Similarly, if we aly any comound distance µ (, ) to and = E (either = M, i.e. the median or a ercentile of, if the first moment is not finite), we get: µ (, E ) = 0 P ( = E ) = is a constant almost surely. Thus, we call µ (, E ) = µ a disersion measure deried by the comound distance µ. E Let s consider the following examles of comound distances and the associated concentration/disersion measures. Examles of Probability Comound Metrics: As obsered earlier, for each robability comound metric we can always generate a robability comound distance µ (, ) H( µ (, )) H = with arameter K H. L -metrics: For eery 0 we recall the { } L -metrics: min(,/ ) µ (, ) = E ; µ (, ) = esssu = inf ε > 0: P > ε = 0 µ ( 0, ) = E I( ) = P ; the associated concentration measures are I, ( ) ess su min(,/ ) µ = ; µ = ( ) I P,0 µ I, = E, where is an i.i.d. coy of ; and the associated disersion µ = ; measures are the central moments E, ( ) ess su E( ) min(,/ ) µ, ( E ) = E E ( ). µ E,0 = P E,

12 = > > < and (, ) k2 = E, the + Ky Fan metrics: k (, ) inf{ ε 0/ P ( ε ) ε } resectie concentration measures are k, I inf{ ε 0/ P( ε) ε} = > > <, k2, I E, = + while the associated disersion measures are k, E inf{ ε 0/ P( E ε) ε} E k2, E( ) = E. + E Birnbaum-Orlicz metrics: For eery (, ) P ( < t ) + P ( < t ) 0 = > > <, ( ) min(,/ ) + Θ (, ) = ( < ) + ( < ) P t P t dt ; Θ = I t dt Θ (, ) = sup ( t < ) + P ( t < ) ; the + associated concentration measures are Θ = ( ) [ ] I,0 F () t 0 F () t t R min(,/ ), ( ) 2 F ( t)( F ( t)) dt, I + Θ = I( t) dt Θ, ( ) = su2 F ()( t F ()) t ; while the associated disersion I t R E min(,/ ) E( ) + measures are: ( ) ( ), Θ = F t dt + F t dt + E( ), Θ E, = max su ( F( t)); su F( t) t E t< E, + Θ = I() t,0 [ () 0; ] + I( t) E F t t E [ F (); t t E ] dt. < Generally, any comound robability metric or distance µ (, ) is a articular tracking error measure between and the benchmark. Een if we consider the comound metric/distance as disersion/concentration measure µ (, g( )) (where g() is either a functional of or an indeendent coy of ), then we should obtain a tracking error measure between and using µ (, g( )). In articular, some of these tracking error tye measures (i.e., µ (, g( )) ) recently hae been used in the ortfolio literature (see Roll (992), Rache et al (2005), and Barro and Canestrelli (2004)). Moreoer, een simle robability distances can be used as disersion measures and tracking error measures, but, generally, not as concentration measures. As a matter of fact, when we aly any simle distance µ (, ) to and = E (either = M, i.e., median or a ercentile of, if the first moment is not finite), we get: µ (, E ) = 0 F = FE is a constant almost surely. 2

13 Thus, we call µ (, E ) = µ E a disersion measure deried by the simle distance µ. As for comound metrics, we can generate a simle robability distance µ (, ) = H( µ (, )) with arameter K for any simle robability metric µ (, ). Let s consider the following H examles of simle metrics and the associated disersion measures. H Examles of Simle Probability Metrics: Kolmogoro metric: One of the most used metric in the literature (also called uniform metric) is the Kolmogoro metric gien by: KS(, ) = su F() t F() t and KSE ( ) =Θ E, ( ) = max su F ();su( t F()) t t R t< E t E. Prokhoro metric: π(, ) inf{ ε 0/ P ( A) P ( A ε ) ε A R} ε Borel measurable set and A { x R/ y A: x y ε} E = > + Β where A is oen = <. Its associated disersion measure is ε { R} π = inf ε > 0/ P( A) P( E A ) + ε A Β. Gini-Kantoroich metric: For eery 0, we consider + min(/,) (, ) =, GK (, + ) ( I dt ) 0 [ F ( t ) F ( t )] GK F t F t dt GK (, ) = ess su F F and the associated disersion measures are = and E( ) ( ) min(/ q,) q + q, E( ) GK ( ) =Θ ( ) = F ( t ) dt + F ( t ) dt E( ), q E( ), q +,0 =Θ,0 = + E E F ( t ) 0; t E ( ) F ( t ) ; t E ( ) [ < ] [ ], GK I t I t dt GKE ( ) = KS E ( ) =Θ E, ( ) = max su F( t); su ( F( t) ). A generalization of Gini- t< E t E Kantoroich metric is the following Generalized Zolotare metric (see Rache (99)). Generalized Zolotare metric: For eery q 0, we consider the Generalized Zolotare s metric: b min(/ q,) ( ) ( ) q a, GZM (, b,0, ) ( I dt ) [ a F ( t ) F ( t )] GZM(,, q, ) = F ( t) F ( t) dt GZM (,,, ) ess su F F ( ) ( ) = and the associated disersion measures are =, 3

14 q E( ) q b ( t E( ) ) GZM (, E( ), q, ) = F ( t) dt + F a E( ) t dt Γ( ) min(/ q,), E ( ) b, GZM (, E( ),0, ) = I dt + I dt ( ) ( ) a [ F ( t) 0] E ( ) ( ) t E ( ) [ F ( t) ] Γ ( t E ) ( ) GZM(, E,, ) = max su F ( t); su F ( t). Note that any time t< E t E Γ( ) b ( ) ( ) q ( ) ( ) then ( ) F () t F () t sgn F () t F () t dt 0. Therefore the intuition suggests to extend a the generalized Zolotare s metrics introducing an analogous metric that we call the FORS tracking error metric. FORS tracking error metrics: Let us consider the functional ρ, associated with an FORS order. For eery q 0 and > 0 b b min(/ q,) q ρ ρ a FORSq, (, ) =, ( t), ( t) dt, b FORS (, ) = I dt, and 0, a [ ρ, ( t) ρ, ( t)] FORS ess, (, ) = su ρ, ρ,. Similarly, we describe the associated disersion measures whose definition deend on the definition of the functional ρ, b q ρ, () t, () t sgn, () t, () t dt 0 a ρ ρ ρ.. Clearly, any time FORS, then f, In articular, to the ortfolio roblem we get the following definition. Definition 2 Consider a frictionless economy where a benchmark asset with return r and n 2 risky assets with returns r = [ r,..., r n ]' are traded. Let ρ :[ ab, ] R be a FORS measure associated with a FORS risk ordering defined oer any admissible ortfolio of returns =x r and oer the return = r. Then we define for any the tracking error measures: min(/, ) b ρxr ', ( ) = ρ a xr ' ( λ) ρr ( λ) dλ, ( max ( ',0 )) min(/, ) dsr b ρxr ', ( ) = ρ a xr( λ) ρr ( λ) dλ, that we call ortfolio FORS tracking error measures. 4

15 In order to limit the comutational comlexity of the ortfolio roblem, one uses mostly rimary metrics. These are considered choices consistent with metrics limiting disersion or losses while maximizing exected wealth. Tyically, we can think of some metrics alued exclusiely on the distributional tail. In articular, mimicking the reious consideration we roose the following definition. Definition 3 Consider a frictionless economy where a benchmark asset with return r and n 2 risky assets with returns r = [ r,..., r n ]' are traded. Let ρ :[ ab, ] R be a rimary FORS risk measure associated to a gien FORS risk ordering that identify uniocally the distributional tail i.e. ρ () z = ρ () z z [,] a b F () x = F () x x t for a gien t F ( u) = F ( u) u F ( t) = and it is defined oer any admissible ortfolio of returns =x r and oer the return = r. Then, we call ρ λ -tail FORS risk measure of the ortfolio. In addition, we define min( /, ) ( max ( ',0 )) min( /, ) b ρxr ', ( ) ρxr ' ( λ a ) ρr ( λ) dλ dsr b =, ρxr ', ( ) = ρ a xr( λ) ρr ( λ) dλ, that we call ortfolio tail FORS tracking error measures. Tyical examles of -tail FORS risk measures will be considered in the following sections. Next we roose some ossible ortfolio roblems based on linearizable tracking error measures. 3. Traking Error Measures Based on Concentration Cures Two alternatie ways of using the absolute Lorenz cure consist either in measuring the distance between two random ortfolios or alternatiely in minimizing the downside risk relatie to a gien benchmark. Consider the following two FORS tracking error measures for a gien weight q: q min( / q,) min( / q,) q q L, ( q) = L 0 L d = CVaR 0 ( ) CVaR ( ) d, L and and min ( / q, ) min ( / q, ) dsr q q q L, ( q) = ( max 0 ( L L,0)) d = 0 ( max ( CVaR ( ) CVaR,0)) d. The absolute Lorenz cure can be determined by soling a LP roblem. If one assumes equirobable scenarios for random ortfolios and, minimizing the two measures, i.e., 5

16 dsr L (), for q= leads to a LP roblem To minimize L, T z q t t T = subject to, q, we minimize the function t i T zt ± ( ai bi) + ( k, i + uk, i) 0 i= T T for t =,..., T k= 0, u 0,, x r, a, u, y + b, t, i =,..., T; ti, ti, n ti j jt i j= ti t i where r t = [ r, t,..., rn, t ]' is the ector of returns at time t, y t is the obseration at time t of the random ariable, and the otimal alues - a i and b i are the i -th ercentiles of the ortfolio x r T and, resectiely. dsr Similarly, minimizing the function L, q leads to minimizing the function T z q T t t = t i T i= T T zt 0, for t,..., T k= subject to zt ( bi ai) ( uk, i k, i) n x r + a, u, y b, t, i =,..., T. ti, j jt, i j= ti t i 0, u 0, = ti, ti, By choosing a secific benchmark, these measures could satisfy different roerties that could hae a different imact on the ortfolio selection roblem (see, among others, Szego (2004) and Bigloa et al (2004)). Other examles of FORS tracking error measures and risk measures consistent with FORS orderings are those based on Gini tye measures which are discussed in the next section. 4. Gini-Tye Measures and the Portfolio Selection Problem In this section we roose some new risk measures related to the fundamental work of Gini (92, 94). For this reason we will call all these measures Gini-tye risk measures. 4. Gini Mean Difference and Extensions Gini's mean difference (GMD) is twice the area between the absolute Lorenz cure and the line 6

17 joining the origin with the mean located on the right boundary ertical. Many reresentations for GMD exist. We reort here the most used ones :, E 0 0 Γ (2) = 2 L () = E 2 L ( u) du = E 2 ( u) F ( u) du, (3) Γ (2) = E 2 E ( F ) = 2co, F, (4) Γ (2) = E 2 = E E(min(, 2)), (5) where and 2 are two indeendent coies of. GMD deends on the sread of the obserations among themseles and not on the deiations from some central alue. Consequently, this measure links location with ariability, two roerties that Gini (92) himself argue are distinct and do not deend on each other. While the Gini index, i.e. the ratio GMD/E(), 2 has been used for the ast 80 years as a measure of income inequality, the interest in GMD as a measure of risk in ortfolio selection is relatiely recent (itzhaki (982) and Shalit and itzhaki (984)). On the other hand, the estimator of GMD that resents the best characteristics from a comutational oint of iew, is obtained from formula (5) where ˆ Γ (2) =, T T t k TT ( ) k= t> k t is the t-th obseration of the random wealth. (See Rao Jammalamadaka and Janson (986) and Rache (993) for the asymtotic roerties of all the aboe U-statistic deriing by concentration measures). Therefore, when the returns are uniquely determined by the mean and GMD= Γ (2), all risk aerters will choose a solution for some real m of the following linear otimization roblem: min x k = t > k st.. T n T y tk, y + x ( r r ) 0 for t > k =,..., T tk, i= i it, ik, n y x ( r r ) 0 for t > k =,..., T tk, i it, ik, i= n E( xr ' ) m; x = ; x 0; j=,..., n j= j j (6) See itzhaki (999) for a comlete list. 2 In the income inequality literature, the Gini index is the area between the relatie Lorenz cure and the 45 line exressing comlete equality 7

18 where r, is the k-th obseration of j-th asset. Introduced by Donalson and Weymark (980, j k 983), the extended GMD takes into account degree of risk aersion as reflected by the arameter. itzhaki (983) showed that this index can be exressed as a function of the Lorenz cure. We resent here the most used of the many reresentations for the extended GMD: Γ ( ) = E ( u) F ( u) du = E ( ) ( u) L ( u) du, (7) + Γ () = E ( F ()) x xdf () x = ( ) ( ) = E E ( F ) = co, F From this definition, it follows that the measures, (8) ( ( )) Γ () E = Γ ( + ) F + () which characterize the reious Gini FORS orderings are articular sectral measures. Interest in the otential alications to ortfolio theory of GMD and its extension has been fostered by itzhaki (983, 998) and Shalit and itzhaki (984, 2005), who hae exlained the financial insights of these measures. Moreoer, as obsered reiously, all risk measures Γ ( ) E ( ) are coherent for eery >. In addition, when assuming equally robable scenarios, one can easily linearize the associated ortfolio roblems. Therefore, if all otimal choices are uniquely determined by the mean and the disersion Γ ( ), all risk-aerse inestors will choose a ortfolio with a mean equal or greater than m that soles the LP roblem: T 2 i i T min a i k, i x,,..., at i= T + T T k= n ti, j jt, i ti, j= n j i= subject to x r a ; 0; t =,..., T; i=,..., T Exr ( ' ) m; x = ; x 0; j=,..., n. Thus, when =2 the otimization roblem (9) is an alternatie LP otimization roblem that can determine the otimal choices of all risk-aerse inestors. j (9) 4.2 Gini's Index of Dissimilarity To measure the degree of difference between two random ariables, Gini (94) introduced the index of dissimilarity. The index roerly measures the distance between two ariates and has been intensiely used in mass transortation roblems (see, among others, Rache (99) and Rache 8

19 and Ruschendorf (998, 999)). Many researchers (see, among others, Gini (95, 965), Salemini (943, 957), and Dall Aglio (956)) deoted considerable effort in obtaining the exlicit exressions for this measure of discreancy, its generalizations, and roerties. We resent here some of the many reresentations of Gini's index of dissimilarity: where % = F ( U), G =, () + = 0 F ( x) F ( x) dx F ( u) F ( u) du (0) { } % G, () = inf E % % / F I ( F, F ) = E % % () F F % and U is an uniform (0,), F( xy, ) = min ( F ( x), F( y) ) = F ( U) % is the Hoeffding-Frechet bound of the class of all biariate distribution functions I ( F, F ) with marginals F and F (see Rache (99)). In ortfolio theory, this risk measure changes with resect to the chosen benchmark. For examle, when mean E() is used as benchmark, the index of dissimilarity is the mean absolute deiation of that is a disersion measure consistent with Rothschild-Stiglitz stochastic order. Howeer, when we use the uer stochastic bound of the market as benchmark, Gini s distance is a safety-risk measure consistent with first stochastic dominance order (see Ortobelli and Rache (200)). On the other hand, Gini index of dissimilarity can be used to measure the degree of difference between the ortfolio and a market index. This is the classical tracking error roblem focused on minimizing the ortfolio deiation from a benchmark. The index of dissimilarity can also be extended for a gien weight to roide the extended Gini index of dissimilarity: min( /,) G, ( ) = F ( u) F ( u) du (2) 0 Therefore, when we consider N equally robable scenarios and, an estimator of the extended Gini index of dissimilarity is obtained as: min( /,) ˆ N G, () = VaRk/ N( ) VaRk/ N( ) N. (3) k = If, on the other hand, we are interested in minimizing the downside risk with resect to a gien benchmark, the following tracking error measure can be used: min( /,) dsr G, ( ) = ( max( F ( u) F ( u),0) ) du (9) 0 Unfortunately, ortfolio otimization roblems with these tracking error measures G, (), G dsr, () are not easily linearizable in most cases. Further extensions to the Gini index of 9

20 dissimilarity can be found by minimizing the exected alue of conex ositie functions (or quasi-antitone functions) of % % with resect to all admissible joint biariate distributions (see Cambanis et al (976), Kalashniko and Rache (988), Rache (99), and Rache et al (2005)). Howeer, among many FORS tye tracking error measures, we can use those based on the sectral measures measures: φ 0 M = ( u) F ( u) du. Thus, we identify the class of sectral tracking error φ min( /,) G, ( φ, ) = φ ( u) F ( u) F ( u) du and 0 min( /, ) dsr G, (,) φ = φ ()max( u ( F () u F (),0) u ) du. 0 In articular, by using sectral FORS tye measures obtain the following tracking error measures based on Lorenz cures: that are easily linearizable., 0 0 GMφ ( ) = φ'( u) L ( u) du φ() E( ), we G ( φ,) = φ'( u) L ( u) L ( u) φ() E E du dsr, 0 ( ( )) G (,) φ = max φ'() u L () u L () u φ() E E(),0 du, 4.3 Tail Gini Measures To cature the downside risk of ortfolios, Bigloa et al (2004), among others, roose seeral tail risk measures. Tyically, the Lorenz cure is a tail measure as it is a linear function of CVaR. Alternatiely, we can define tail measures using the Lorenz tye measures for some [0,] : q (, ) =, 0 min(/q, ) L q L u L u du (4) ( ) min(/q, ) dsr q L, ( q, ) = max L 0 ( u) L( u),0 du (5) By minimizing these tail measures, we obtain LP roblems when q= and equally robable scenarios are considered. Using Gini measures, Ogryczak and Ruszczynski (2002a, 2000b) roose the tail Gini measure for a gien β : β β β Γ (2) = ( Eu L ( u) ) du= E ( β u) F ( udu ) = E L ( udu ), β β 0 β 0 β 0 20

21 Ogryczak and Ruszczynski s analysis can also be deeloed to the extended GMD by using the tail measure: β ( ) β 2, β β β β 0 β 0 Γ ( ) = E ( u) F ( u) du = E ( u) L ( u) du (6) Γ ( ) = E ( u) F ( u) du = E ( ) ( u) L ( u) du (7) + Γ () = E ( F ()) x xdf () x = ( ) ( ) = E E ( F ) = co, F Γ ( + ) (( + )) for some β [0,]. As a result of Remark, all measures Γ, β () E = F ( β) are β coherent that can be linearized by considering equally robable scenarios. s Assuming equally robable T scenarios and β = minimizing the risk Γxr ', β () E( x') r T of ortfolio x ' r with a mean equal or greater than m is equialent to soling the LP roblem: (8) 2 s s i i T min i t, i xb,,..., b b + s i= T T T t= β T xer ' m; x = ; x 0; j=,..., n j= 0; x r b ; t =,..., T; i =,..., s ti, ti, j jt, i j= n subject to n j j (9) Moreoer, we can consider the following tail extensions to linearizable FORS tracking error measures for some [0,] : φ G ( φ,,) = φ'( u) ( L ( u) L ( u) ) ( L L ) du, (20), 0 dsr φ G, ( φ,,) = max φ'( u) ( L( u) L( u) ) ( L L ),0 du. (2) 0 All these measures take into account only the tail behaior of ortfolio distributions. 5 Efficient Frontier with Ellitical Distributions and FORS-tye Measures We now study the risk measures for jointly ellitical distributed returns r = [ r,..., r n ]'. The 2

22 ortfolio return x ' r belongs to a family of ellitical distributions with finite mean as determined by the non-negatie integrable function g(y) if asset returns hae the following distribution: λ 2 ( t x' µ ) Fxr ' ( λ) = Pr( x' r λ) = g dt 2 KU xqx ' K x' Qx where µ = Er () is the ector of exected returns; Q is a ositie-definite disersion matrix that differs by a ositie constant factor from the coariance matrix when all assets returns hae finite + 2 ( t x' µ ) ariance; U = g dt; and K is a constant. 2 K x' Qx 2 K x' Qx Chamberlain (983) has shown that the ellitical families with finite ariance are all ossible families of distributions deemed necessary and sufficient for the exected utility of final wealth to be a function of only the mean and ariance. Bawa (975, 978), Owen and Rabinoitch (983), Ingersoll (987), and Ortobelli (200) hae shown that the mean-disersion dominance rule is equialent to the SSD rule if ortfolio returns belong to the same ellitical family of unbounded random ariables. Een if ellitical families are the natural best candidate to study the mean-risk ortfolio roblem, any risk measure roosed for the ortfolio selection roblem has to be roerly used taking into account its secific characteristics (see Ortobelli et al (2005a)). In articular, when we assume that returns are ellitical distributed, the extended Gini mean difference and its tail xrβ extensions Γ xr ' (), Γ ', () for a gien ortfolio x ' r are roortional to the resectie Ginitye measures alied to the standard ellitical distribution Ell(0,), the constant of roortionality is the disersion x ' Qx, i.e. Γ xr ' () = x' QxΓ Ell(0,) (), and Γ xr ', β() = x' QxΓ Ell(0,), β(). Thus, with ellitical distributions, we hae exchangeability among assets (see Shalit and itzhaki (2005)). These distribution families are scalar and translation inariant. Any element of the family Ell ( µ i, σ i) with mean µ i and disersion σ i has the same distribution of σ Ell(0,) + µ. This roerty of the ellitical families imlies that one can characterize the i i ortfolio efficient frontier with resect to the risk measure one uses. We now state the following roosition and corollaries. Proosition Suose there are n 2 risky assets with returns r = [ r,..., r n ]' traded in a frictionless economy with unlimited short selling. If returns belong to an ellitical multiariate family Ell with finite mean µ = E() r and non-singular disersion matrix Q, then, for eery characteristic FORS risk measure ρ associated with a FORS ordering induced either by or by 22

23 order (with ), all ortfolios satisfying the first-order conditions of the constrained roblem: are ortfolios of the mean-disersion frontier min ρ x xe ' = x' r (22) σ ( AC B ) m C + 2mB A = 0 (23) whose ortfolio weights are gien by ( Ell(0,) ) ( ρell(0,) ) sgn ρ ( CQ µ BQ e) Q e x( ρell(0,) ) = C C C AC + B (24) where e ' = [,,...,] ; m= x ' µ ; σ = x ' Qx ; A= µ ' Q µ ; B = e' Q µ ; and C = e' Q e. Corollary Assuming Proosition, we exress the mean m = x ' µ, the disersion σ = x ' Qx, and the risk measure ρ x' r of the otimal ortfolios satisfying the first-order conditions of the constrained roblem (22) as: ( (0,) ) AC B B m = x' µ = sgn ρell C C C AC + B ( ρell(0,) ) 2 (25) σ = xqx ' = ρ Ell(0,) ( ρell(0,) ) 2 2 C AC + B (26) ρ xr ' Ell(0,) 2 2 ρ (0,) ρ (0,) sgn Ell C Ell AC + B B = σρ m = (27) C Consequently, we characterize the otimal ortfolios satisfying the first-order conditions of constrained roblem (22) by arying the arameter λ. More secifically, we can find the otimal ortfolios as function of the arameter λ that minimize articular safety-risk measures such as: ρ ( λ ) =VaR λ () ρ ( λ ) =CVaR λ () or those obtained by ositiely homogeneous and Gaioronsky-Pflug (G-P) translation inariant measures minus the exected mean (such as λ ρ ( λ ) = λ E E E, ρ ( λ) = Γ ( λ) E, ρ ( λ) =Γ, ( λ) E, β 23

24 q q ρ ( λ) = λ E E E and many others). This feature is imortant because often the arameter λ oints out the inestor s aersion to risk. In articular, let s look at the following examles. Assume that returns follow one of the three joint multiariate ellitical distribution: () a Gaussian distribution, (2) a Student s t distribution with u> degrees of freedom, and (3) stable sub-gaussian distribution with index of stability (, 2). Under these distributional assumtions we hae the following ρ (0,) ( λ ) =CVaR λ (Ell(0,)) functions (see Rache and Mittnik (2000) and Stoyano et al. (2006a)): Ell ) 2) 3) where CVaR λ (N(0,))= ex λ 2π ( VaR ( N(0,) )) 2 λ 2 ; u 2 2 u + Γ u 2 ( VaR ( tu (0,))) λ CVaR λ ( tu (0,))= + u u λ( u ) Γ π 2 ( S (,0,0)) ; /( ) ( ) VaRλ π /2 CVaR λ(s (,0,0))= g( θ ) ex ( θ) VaR S (,0,0) d 0 λ θ λ( ) π ( θ 2 ) θ ( θ ) 2 sin ( θ ) sin ( 2) cos g( θ ) = and sin ( θ) θ cos( θ) sin ( θ) cos cos ( θ ) = λ Similarly, the safety-first functions λ ρ ( λ ) = E E E uniocally. determine the ellitical distributions. Thus, for stable sub-gaussian distributed and (, 2] ( = 2 reresents the Gaussian case), we obtain the following set of arametric ositiely homogeneous and translation inariant safety-risk measures: ρ Ell(0,) λ + λ Γ 2 if Ell(0,) = N(0,) 2 π ( λ) = λ, λ λ 2 + Γ Γ if Ell(0,) = S (,0,0) λ 2 Γ π 2 24

25 for λ (0, s] s (0, ) (see Ortobelli et al (2005b)). As a consequence of the reious corollaries, the otimal ortfolios solutions for (22) are characterized by the functions (25), (26), and (27). [Insert Figure about here] In Figure, different efficient frontiers are been used to distinguish between the different risk aersions of inestors. The figure shows the extended Gini mean difference Γ (0,) (), the - mean (formula (25)), and the -ariance (formula (26)) efficient frontiers assuming either Gaussian returns or Student s t returns with 5 degrees of freedom. On the other hand, Figure confirms the intuitie result that otimal ortfolios minimizing Γ ( ) E ( ) with the lowest risk Ell aersion resent the highest mean and ariance. Howeer, inestors who assume Student s t distributed returns are more conseratie because they account for the risk in the heaier tails. As a matter of fact, for the same (for examle, = 2) Student s t inestors choose ortfolios with lower mean and ariance than Gaussian inestors. The mean-tracking error ariance frontier has been analyzed by Roll (992). Consequently we can define the mean- tracking error disersion frontier with resect to different risk measures when returns belong to an ellitical family. Corollary 2 Suose there are n 2 risky assets with returns r = [ r,..., r n ]' traded in a frictionless economy where unlimited short selling is allowed. Let us assume as benchmark return r = x' r a articular ortfolio (i.e. x ' e = ) of these returns and assume that the returns belong to an ellitical multiariate family Ell with finite mean µ = Er () and non-singular disersion matrix Q. Then, for eery ositiely homogeneous measure ρ consistent either with or with order ( ), that is either translation inariant (i.e. ρ t = ρ t t R ) or translation inariant in the sense of G-P (i.e. ρ t = ρ t R ), all the ortfolios satisfying the first-order conditions of the following constrained tracking error roblem: + min ρ( x x) ' r x ( x x) ' e= 0; ( x x) ' µ = g are ortfolios of the mean-tracking error disersion frontier whose ortfolio weights are gien by + (28) g µ e x= x + Q A/ B B/ C B C (29) where e ' = [,,...,] ; A= µ ' Q µ ; B = e' Q µ and C = e' Q e. 25

26 Moreoer, we can also characterize the FORS tracking error measures as we roe in the following roosition. Proosition 2 Suose, in a frictionless economy, are traded n 2 risky assets with returns r r r n = [,..., ]' and a benchmark asset with return r that is not a ortfolio of the other returns. Assume that all admissible ortfolios of returns x ' r belong to an ellitical family Ell with finite mean and non-singular disersion matrix Q and een r belongs to the same family of ellitical distributions. Let ρ (0,) ( λ ) be a continuous and monotone function that defines a (-tail) FORS Ell characteristic risk measure on a comact real interal [a,b], then: b xr ', a xr ' r t b ( σx' r σr )( ρell(0,) ( λ) dλ ρell(0,) ( λ) dλ) ( 2t a b)( mr mx' r) a ρ = ρ λ ρ λ dλ = = + dsr b xr ', () max ( xr ' r,0) d a t ( σxr ' σr ) (0,) ( ρell λ λ a )( r xr ' ) b ( σxr ' σr ) ρell(0,) λ λ ( r xr ' ) t ρ = ρ λ ρ λ λ = d + t a m m if s> 0 = d + b t m m if s< 0 t 0 otherwise (30) (3) where m, σ are resectiely the mean and the disersion of, of Ell(0,) ρ, s ( σx' r σr ) ρell(0,) ( λ) ( mr mx' r) ρ Ell(0,) = + for a gien λ ( at, ), and is the inerse function m m m m ρ λ ρ ( λ) = t = b otherwise r x' r r x' r Ell (0,) if ( a, b): Ell (0,) σ x' r σ r σx' r σr. Obsere that if b is oortunely small (say b=t), s=s(λ,x)>0 for eery λ ( at, ) and for eery dsr xr ortfolio x, then when we minimize ρ ', () subject to x ' e= ; x' µ = m, we get ortfolios of the mean-disersion frontier. Tyical alications of Proosition 2 are the Gini-tye tracking error dsr dsr measures L ( q ), L ( q ), G ( q ), G,,,, q and their tail extensions. Howeer, we can aly Proosition 2 to many other FORS tracking error measures. For examle, if we consider a - tail characteristic risk measure ρ ( λ) = Γ, ( λ) E, then we can aly Proosition 2 to the new FORS tracking error measures (see Ortobelli et al (2006)). In articular, it is not difficult to roe that for any unbounded ellitical family the index of dissimilarity is gien by: 26

27 m m m m G () = 2 L F + 2F m m r x' r r x' r ( σ σ ), x' r r Ell(0,) Ell(0,) Ell(0,) r x' r σxr ' σ r σxr ' σ r if σ x' r σ r and G, () = mr m x' r if σ x' r = σ r. This formula generalizes the analogous one obtained by Salemini (957) with Gaussian distributions. 6. Reward-Risk Analysis with FORS-Tye Risk Measures The imortance of including the inestor s reference toward reward in ortfolio analysis is well founded. To consider both risk and reward, the so-called erformance measures use reward/risk ratios. In articular, when we maximize a gien reward/risk ratio as / ρ, we could determine non-dominated choices that are consistent with exected utility maximization. When the erformance ratio / ρ is maximized and we get choices that are non-dominated with resect to a gien stochastic dominance law, we say that the ratio / ρ is coherent with the inestors choices ordered with resect to the underlined dominance law. The following roosition roes conditions that guarantee the coherency with more than one ordering. Proosition 3 Consider a frictionless economy where a benchmark asset with return r and n 2 risky assets with returns r = [ r,..., r n ]' are traded. Let, ρ be two robability functionals defined on a sace of random ortfolios with weights that belong to { n / ' ; ; n k ;, k } V = x R x e = Lb Ax Ub A R Lb Ub R, (32) where we assume they are strictly ositie. Suose that is a ositiely homogeneous concae FORS reward measure induced by a risk ordering and ρ is ositiely homogeneous conex FORS xr measure that is consistent with another stochastic order. If we maximize ratio ( ' ), or ρ( x ' r) xr ( ' r ) or ρ( x ' r) xr ( ' r ) ρ( x ' r r ) non-dominated ortfolios and ρ ). subject to the ortfolio weights that belong to the sace V, we obtain x ' r (or x ' r r ) with resect to both the reious stochastic orders (of 27