The Proper Use of Risk Measures in Portfolio Theory

Transcription

1 The Proper Use of Rsk Measures n Portfolo Theory Sergo Ortobell a, Svetlozar T. Rachev b, Stoyan Stoyanov c, Frank J. Fabozz d,* and Almra Bglova e a Unversty of Bergamo, Italy b Unversty of Calforna, Santa Barbara and Unversty of Karlsruhe, Germany c FnAnalytca Inc. d Yale Unversty, Connectcut e Unversty of Karlsruhe, Germany * Correspondng author: E-mal: [email protected] The authors thank Andrew Chen for helpful comments of an earler draft of ths paper. Rachev's research has been supported by grants from Dvson of Mathematcal, Lfe and Physcal Scences, College of Letters and Scence, Unversty of Calforna, Santa Barbara and the Deutschen Forschungsgemenschaft. Ortobell's research has been partally supported under Murst 4%, 6% 23, 24, 25.

2 The Proper Use of Rsk Measures n Portfolo Theory Abstract Ths paper dscusses and analyzes rsk measure propertes n order to understand how a rsk measure has to be used to optmze the nvestor s portfolo choces. In partcular, we dstngush between two admssble classes of rsk measures proposed n the portfolo lterature: safety rsk measures and dsperson measures. We study and descrbe how the rsk could depend on other dstrbutonal parameters. Then, we eamne and dscuss the dfferences between statstcal parametrc models and lnear fund separaton ones. Fnally, we propose an emprcal comparson among three dfferent portfolo choce models whch depend on the mean, on a rsk measure, and on a skewness parameter. Thus, we assess and value the mpact on the nvestor s preferences of three dfferent rsk measures even consderng some dervatve assets among the possble choces. Key words: skewness, safety rsk measures, rsk averson, dsperson measures, portfolo selecton, nvestors preference, fund separaton. JEL Classfcaton: G14, G15 2

3 1. INTRODUCTION Many possble defntons of rsk have been proposed n the lterature because dfferent nvestors adopt dfferent nvestment strateges n seekng to realze ther nvestment objectves. In some sense rsk tself s a subjectve concept and ths s probably the man characterstc of rsk. Thus, even f we can dentfy some desrable features of an nvestment rsk measure, probably no unque rsk measure ests that can be used to solve every nvestor s problem. Loosely speakng, one could say that before the publcaton of the paper by Artzner, Delbaen, Eber, and Heath (2) on coherent rsk measures, t was hard to dscrmnate between good and bad rsk measures. However, the analyss proposed by Artzner, et al.(2) was addressed to pont out the value of the rsk of future wealth, whle most of portfolo theory has based the concept of rsk n strong connecton wth the nvestor s preferences and ther utlty functon. From an hstorcal pont of vew, the optmal nvestment decson always corresponds to the soluton of an epected utlty mamzaton problem. Therefore, although rsk s a subjectve and relatve concept (see Balzer (21), Rachev et al (25)) we can always state some common rsk characterstcs n order to dentfy the optmal choces of some classes of nvestors, such as non-satable and/or rsk-averse nvestors. In partcular, the lnk between epected utlty theory and the rsk of some admssble nvestments s generally represented by the consstency of the rsk measure wth a stochastc order. 1 Thus, ths property s fundamental n portfolo theory to classfy the set of admssble optmal choces. On the other hand, there est many other rsk propertes that could be used to characterze nvestor s choces. For ths reason, n ths paper, we classfy several rsk measure propertes for ther fnancal nsght and then dscuss how these propertes characterze the dfferent use of a rsk measure. In partcular, we descrbe three rsk measures (MnMa, mean-abolute devaton, and standard devaton) and we show that these rsk measures (as many others) can be 1 Recall that the wealth X frst order stochastcally domnates the rsky wealth Y (X FSD Y) f and only f for every ncreasng utlty functon u, E(u(X)) E(u(Y)) and the nequalty s strct for some u. Analogously, we say that X second order stochastcally domnates Y (X SSD Y), f and only f for every ncreasng, concave utlty functons u, E(u(X)) E(u(Y)) and the nequalty s strct for some u. We also say that X Rothschld Stgltz stochastcally domnates Y 3

4 consdered equvalent by rsk-averse nvestors, although they are formally dfferent. Then we dscuss the mult-parameter dependence of rsk and show how we could determne the optmal choces of non-satable and/or rsk-averse nvestors. In partcular, we observe that when asset returns present heavy tals and asymmetres, fund separaton does not hold. However, f we consder the presence of the rskless asset, then two fund separaton holds among portfolos wth the same skewness and kurtsoss parameters. Fnally, we propose an emprcal comparson among dfferent portfolo allocaton problems n a three parameter contet n order to understand the mpact that MnMa, mean-absolute devaton, and standard devaton could have for some non-satable and rsk-averse nvestors. In ths framework we also consder the presence of some contngent clams and compare the optmal choces of several nvestors n a mean-rsk-skewness space. 2. RISK MEASURES AND THEIR PROPERTIES Let us consder the problem of optmal allocaton among n assets wth vector of returns r=[r 1,,r n ] where r Pt, + 1 Pt, = whle P t, s the prce of -th asset at tme t. No short P t, sellng s allowed,.e., the wealth y nvested n the -th asset s non negatve for every =1,...,n. Thus consderng an ntal wealth W, magne that the followng optmzaton problem: mn p y ( W + y' r) n y = W y = 1,..., n (1) = 1 EW ( + yr ' ) µ s equvalent to mamzng the epected utlty EUW ( ( y )) of the future wealth y W : = W + y' r nvested n the portfolo of assets. Then, we mplctly assume that the y (X R-S Y) f and only f for every concave utlty functons u, E(u(X)) E(u(Y)) and the nequalty s strct for some u. (See, among others, Levy (1992) and the references theren). 4

5 epected utlty of the future wealth W y has a mean greater than µ y and the epected utlty depends only on the mean and the rsk measure p. In ths case, we say that the rsk measure p s consstent wth the order relaton nduced by the utlty functon U. More generally, a rsk measure s consstent wth an order relaton (Rothschld-Stgltz stochastc order, frst-order stochastc domnance, second-order stochastc domnance) f E(U( W )) E(U( W )) (for all utlty functons U belongng to a gven category of y functons: ncreasng; concave; ncreasng and concave) mples that p( W ) p( W ) for all admssble future wealths W, W y. Consstency s absolutely necessary for a rsk measure to make sense. It ensures us that we can characterze the set of all the optmal choces when ether wealth dstrbutons or epected utlty depend on a fnte number of parameters. 2 Although, when we assume that ether wealth dstrbutons or epected utlty depend on more than two parameters (the mean, the rsk, and other skewness and/or kurtoss parameters see Secton 4), the complety of the optmzaton problem could ncrease dramatcally. As a consequence of consstency, all the best nvestments of a gven category of nvestors (non-satable, rsk-averse, non-satable and rsk-averse) are among the less rsky ones. But the converse s not generally true; that s, we cannot guarantee that all the less rsky choces are the best ones even f the rsk measure s consstent wth some stochastc orders. In fact, any rsk measure assocates only a real number to a random wealth, whle the stochastc orders compare all cumulatve dstrbuton functons. Then, ntutvely, a unque number cannot summarze the nformaton derved from the whole wealth dstrbuton functon. y 2 See Ortobell (21). 5

6 Ths s the man reason why every rsk measure s ncomplete and other parameters have to be consdered. The standard devaton 2 (( ) ) 1/2 STD( W ) = E W E( W ) (2) y y y s the typcal eample of a rsk measure consstent wth Rothschld-Stgltz (R-S) stochastc order (concave utlty functons). It was also the frst measure of uncertanty proposed n portfolo theory for controllng portfolo rsk (see Markowtz ( ) and Tobn (1958)). Another eample of a rsk measure consstent wth Rothschld-Stgltz stochastc order s the mean-absolute devaton (MAD) MAD( W ) = E( W E( W )), (3) y y y where the rsk s based on the absolute devatons from the mean rather than the squared devatons as n the case of the standard devaton. The MAD s more robust wth respect to outlers and proposed as a measure to order the nvestor s choces (see Konno and Jamazak (1991), Speranza (1993), and Ogryczak and Ruszczynsk (1999)). Artzner et al (2) have defned another type of consstency, called monotony, that s p( W ) p( W ) for the rsky wealths y W y and W that satsfy Wy W. Sometmes there s only a partal consstency between a rsk measure and a stochastc order. For eample, we say that a rsk measure s consstent wth frst-order stochastc domnance wth respect to addtve shfts f p( W ) p( W ) when W = W +t, for some constant t. In ths case, the wealth y W s consdered less rsky than W y by any nvestor that prefers more than less. An eample of a monotone rsk measure proposed by Young (1998) for portfolo theory that s consstent wth frst and second order stochastc domnance s the MnMa (MM) rsk measure, { y } MM ( W ) = sup c R P( W c) =. (4) y Consderng and realzng that the utlty mamzaton problem can be dffcult to solve, many researchers have sought and proposed equvalent formulatons wth ncer y 6

7 numercal propertes. Ths leads to the defnton of the followng propertes. The frst property whch should be fulflled by a rsk measure s postvty. Ether there s rsk, ths means p( W )> or there s no rsk (p( W )=). Negatve values (less rsk than no rsk) y y does not make sense. Partcularly, we mpose the condton that p( W ) = holds f and only f nvestment W y s non-stochastc. Ths property s called postvty. Clearly, dfferent rsk measures could have a dfferent mpact on the complety of the problem gven by (1). In partcular, we must take nto account the computatonal complety when solvng large-scale portfolo selecton problems. Under some crcumstances, t mght happen that the resultng mnmzaton problem mght be lnearzable, whch mples easy soluton algorthms; n ths case, we call the rsk measure lnearzable. Hence, the success of some rsk measures s due to the computatonal practcablty of the relatve lnearzable optmzaton problems. Another mportant property whch should be accounted for by the rsk measure s the effect of dversfcaton: f the wealth W bears rsk p( W ) and nvestment y y y W bears rsk p( W ), then the rsk of nvestng half of the money n the frst portfolo and half of the money n the second one should be not be greater than the correspondng weghted sum of the rsks. Formally, we have: p( λw + (1 λ) W ) λpw ( ) + (1 λ) pw ( ) for all y y λ [,1]. A rsk measure p fulfllng ths equaton s called conve. The property of convety can also be deduced f the rsk measure fulflls two other propertes whch are called subaddtvty and postve homogenety: (1) p s subaddtve f p( W + W ) p( W ) + p( W ) and y y (2) t s called postve homogeneous f p( αw) = α p( W) for all random wealth W and real α >. The last property of rsk measures s called translaton nvarance. There are dfferent defntons of translaton nvarance. We obtan the so-called Gavoronsky-Pflug (G-P) translaton nvarance (see Gavoronsky and Plfug (21)) f for all real t: 7

8 p( W +t)=p( W ). Ths property can be nterpreted as follows: the rsk of a portfolo cannot be reduced or ncreased by smply addng a certan amount of rskless money. Ths property s, for eample, fulflled by the standard devaton but not fulflled by the MnMa measure or the Condtonal Value at Rsk (CVaR) measure that has recently been suggested for rsk management. Alternatvely, translaton nvarance holds f p( W +t) = p( W )-t for all real t. Furthermore, we can generalze all the prevous defntons of translaton nvarance consderng the so-called functonal translaton nvarance, f for all real t and any rsky wealth W, the functon f(t)=p( W +t) s a contnuous and non-ncreasng functon. Ths property summarzes not only the dfferent defntons of translaton nvarance, but t consders also the consstency wth frst-order stochastc domnance wth respect to addtve shfts. In order to take nto account the temporal dependence of rsk, the above statc propertes can be generalzed to an ntertemporal framework assumng the same defntons at each moment of tme (see, among others, Artzner et al (23)). Artzner, et al (2) have called a coherent rsk measure any translaton nvarant, monotonous, subaddtve, and postvely homogeneous rsk measure. In partcular the MnMa measure can be seen as an etreme case of condtonal value at rsk (CVaR), that s a coherent rsk measure. Other rsk measure classfcatons have been proposed recently. In partcular, Rockafeller et al (23) (see also Ogryczak and Ruszczynsk (1999)) defne devaton measure as a postve, subaddtve, postvely homogeneous, G-P translaton nvarant rsk measure and epectaton-bounded rsk measure as any translaton nvarant, subaddtve, postvely homogeneous rsk measure p that assocates the value p( W ) > E( W ) wth a non-constant wealth W. Typcal eamples of devaton measures are the standard devaton gven by (2) and the MAD gven by (3), whle the MnMa measure gven by (4) s a coherent epectaton-bounded rsk measure. The most mportant feature of these new classfcatons s that there ests a correspondng one-to-one relatonshp between devaton measures and epectaton-bounded rsk measures. As a matter of fact, gven a devaton measure p, then the measure defned q( W ) = 8

9 p( W )-E( W ) for any rsky wealth W s an epectaton-bounded rsk measure. Conversely, gven an epectaton-bounded rsk measure q, then the measure defned p( W )=q( W-E( W )) s a devaton measure. Thus, the devaton measure assocated wth MnMa s gven by MM ( W E( W )). 3. MEASURES OF UNCERTAINTY AND PROPER RISK MEASURES From the dscusson above, some propertes are substantally n contrast wth others. For eample, t s clear that a G-P translaton nvarant measure cannot be translaton nvarant and/or consstent wth frst-order stochastc domnance (FSD) due to addtve shfts. As a matter of fact, G-P translaton nvarance mples that the addton of certan wealth does not ncrease the uncertanty. Thus, ths concept s lnked to uncertanty. Conversely the translaton nvarance and consstency wth FSD due to addtve shfts mply that the addton of certan wealth decreases the wealth under rsk even f t does not ncrease uncertanty. Artzner et al (2) have dentfed n the coherent property the rght prce of rsk. However, n the prevous analyss, we have dentfed some propertes whch are mportant to measure the uncertanty and other propertes whch are typcal of the proper rsk measures because they are useful to value wealth under rsk. Clearly, coherency s typcal of proper rsk measures. Instead, a postve rsk measure p does not dstngush between two certan wealths W 1 and W 2 because pw ( 1) = pw ( 2) = even f W1 < W2 and the second wealth s preferred to the frst one. That s, f wealth uncertanty, then p( W )>, otherwse no uncertanty s allowed and p( W )=. W presents We meet an analogous dfference between the two categores of rsk measures f we consder the rsk percepton of dfferent nvestors. So, rsk averson characterzes nvestors who want to lmt the uncertanty of ther wealth. Instead, non-satable nvestors want to ncrease wealth, thus they mplctly reduce the wealth under rsk. Therefore, the consstency wth Rothschld-Stglz stochastc order s typcal of uncertanty measures and the consstency wth FSD order or the monotony characterzes the proper rsk measures. 9

10 In contrast, there are some propertes that are useful n order to measure uncertanty and wealth under rsk. For eample, convety s a property that dentfes the mportance of dversfcaton. Undversfed portfolos present a greater grade of uncertanty and a larger wealth under rsk. Smlarly, postve homogenety mples that when wealth under rsk s multpled by a postve factor, then rsk and uncertanty must also grow wth the same proportonalty. In addton, t s possble to show that postvty, functonal translaton nvarance, and postve homogenety are suffcent to characterze the uncertanty of any reasonable famly of portfolo dstrbutons. 3 Thus, we wll generally requre that at least these propertes are satsfed by any uncertanty measure. Moreover, consderng that consstency s the most mportant property n portfolo theory, we requre that any measure of wealth under rsk s at least consstent wth FSD. Table 1 summarzes the propertes of uncertanty measures and proper rsk measures of wealth under rsk. However, ths classfcaton s substantally known n the lterature despte the fact that researchers have labeled the two categores of rsk measures dfferently and have not dentfed all ther propertes and characterstcs. As a matter of fact, accordng to the portfolo theory lterature, we can defne these two dsjont categores of rsk measures as dsperson measures and safety-rsk measures. Typcally, a dsperson measure values the grade of uncertanty, and a safety-frst measure values wealth under rsk. In very general terms, we say that a dsperson measure s a strctly ncreasng functon of a functonal translaton nvarant, postve and postvely homogeneous rsk measure, whle a safety-rsk measure s consstent wth FSD. The two categores are dsjonted snce a dsperson measure s never consstent wth FSD. More precsely, gven a postve rsky wealth W and a postve α < 1, then t s not dffcult to verfy that W FSD α W. Thus, any safety-frst rsk measure q presents less rsk for the domnant random varable, that s qw ( ) q( αw ). In contrast, a dsperson measure p s a strctly ncreasng functon of a postve and postvely homogeneous rsk measure p 1, that s p = f(p 1 ) wth f strctly ncreasng functon. Therefore, p satsfes the relaton p( W ) = f( p ( W )) > f( α p ( W )) = p( αw ) See Ortobell (21). 1

11 In partcular, Tables 2 and 3 recall the defntons and the propertes of some of the dsperson measures and safety-frst rsk measures proposed n the portfolo lterature (for a revew, see also Gacomett and Ortobell (24)). Observe that X domnates Y for a gven stochastc order (Rothschld-Stgltz stochastc order, frst-order stochastc domnance, and second-order stochastc domnance), f and only f α X + b domnates α Y + b for the same stochastc order, for any postve α and real b. Ths s the man reason why we can nterchange wealth and return n problems of type (1) wth consstent rsk measures. Let us refer to A as the class of optmal choces that we obtan solvng the optmzaton problem (1) and varyng µ y for a gven consstent rsk measure p. Then, the class A s practcally the same (up to an affne transformaton) to the one that we obtan by solvng the same problem but consderng ether W / W or W / W 1 nstead of the fnal wealth W : = W + y' r. In y ths case, the varables are the portfolo weghts y y = (=1,...,n) that represent the W percentage of wealth nvested n the -th asset. Besdes, the future wealth of one unt nvested today s gven by 1+ ' r. Thus, the optmzaton problem (1) can be rewrtten as: mn p(1 + r ' ) st.. = 1, (1 ) E(1 + ' r) m y for an opportune level m. For ths reason, n the followng we deal and study smplfed selecton problems wth the gross returns 1+ ' r, nstead of the fnal wealth W : = W + y' r. y 4. LIMITS AND ADVANTAGES OF RISK MEASURES IN PORTFOLIO OPTIMIZATION 4.1 How to Use Uncertanty Measures 11

12 In the prevous analyss, we eplaned that the most wdely used rsk measure, the varance, s n realty a measure of uncertanty. Thus, the queston s: When and how can we use an uncertanty measure to mnmze rsk? When we mnmze the rsk measure at a fed mean level, we are not tryng to ncrease our future wealth (because the mean s fed), but we are only lmtng the uncertanty of future wealth. Thus we can obtan a portfolo that could be optmal for a rsk-averse nvestor, but not necessarly for a non-satable one. However, we do not have to mnmze uncertanty n order to mnmze rsk. For eample, suppose that future wealth s unquely determned by the mean and a dsperson measure p. Assumng that no short sales are allowed, every non-satable nvestor wll choose a portfolo among the solutons of the followng problem. 4 ma p(1 + r ' ) st.. = 1, 1 + E ( r) = h p(1 + r ' ) (5) where the rato between the mean and the uncertanty measure must be greater than an opportune level h. That s, we mamze the uncertanty for an opportune level of wealth under rsk. The level of wealth under rsk s measured assumng that the epected future wealth s proportonal to ts uncertanty,.e. 1 + Er ( ) = hp(1 + r ' ). Therefore, even f returns are unquely determned from the mean and the varance, there are some optmal portfolos from the Markowtz pont of vew whch cannot be consdered optmal for a non-satable nvestors. In fact, Markowtz analyss s theoretcally justfed only f dstrbutons are unbounded ellptcal (normal, for eample) or nvestors have quadratc utlty functons. Fgure 1 shows the optmal choces n a mean-dsperson plane. All the admssble choces have mean and dsperson n the closed area. In Fgure 1, we mplctly assume 4 See Ortobell (21). 12

13 that future wealth s postve because wealth s not unbounded from below (n the worst case t s equal to zero when we lose everythng). Thus, t s generally unrealstc to assume return dstrbutons that are unbounded from below such as the normal one. Portfolos on the arc EA (n a neghborhood of the global mnmum dsperson portfolo) are not optmal because there are other ones wth greater uncertanty that are preferred by every non-satable nvestor. Observe that the quadratc utlty s not always ncreasng and t dsplays the undesrable sataton property. Thus, an ncrease n wealth beyond the sataton pont decreases utlty. Then, there could est some quadratc utlty functons whose mamum epected utlty s attaned at portfolos n the arc EA, but for any ncreasng utlty functon, the epected utlty of portfolos on the arc EA s lower than the epected utlty of some portfolos on the arc AB. From ths eample we see that although dsperson measures are uncertanty measures, we can opportunely use them n order to fnd optmal choces for a gven class of nvestors. Moreover, mnmum dsperson portfolos are not always optmal for non-satable nvestors. 4.2 Two Fund Separaton and Equvalence Between Rsk Measures Generally, we say that two rsk measures are consdered equvalent by a gven category of nvestors f the correspondng mean-rsk optmzaton problems generate one and the same soluton. From the analyss of rsk measure propertes, we cannot deduce f there ests the best rsk measure. In fact, under some dstrbutonal assumptons, t has been proven that all dsperson measures are equvalent. In partcular, when we assume that choces depend on the mean and a G-P translaton nvarant, postve and postvely homogeneous rsk measure, then any other G-P translaton nvarant postve and postvely homogeneous rsk measure dffers from the frst one by a multplcatve postve factor. 5 Ths result mples that t n theory one s ndfferent when decdng to employ one or any other estng G-P translaton nvarant postve and postvely homogeneous rsk measure (n a mean-rsk framework). Furthermore, consderng the equvalence between epectaton-bounded rsk measures and devaton measures, we have to epect the same results mnmzng ether a G-P translaton nvarant dsperson 5 See Ortobell (21). 13

14 measure, or an epectaton-bounded safety rsk measure for any fed mean level. 6 Thus epectaton-bounded safety rsk measures are equvalent to the G-P translaton nvarant dsperson measures from the perspectve of rsk-averse nvestors. However, a comparson among several allocaton problems, whch assume varous equvalent-rsk measures, has shown that there est sgnfcant dfferences n the portfolo choces. 7 There are two logcal consequences of these results. Frst, practcally, the portfolo dstrbutons depend on more than two parameters and optmal choces cannot be determned only by the mean and a sngle rsk measure. Ths s also confrmed by emprcal evdence. Return seres often show dstrbutonal anomales such as heavy tals and asymmetres. Then, t could be that dfferent rsk measures penalze/favor the same anomales n a dfferent way. For ths reason, t makes sense to dentfy those rsk measures that mprove the performance of nvestors strateges. Second, most of the mean-varance theory can be etended to other mean G-P translaton nvarant dsperson models and/or mean-epectaton-bounded rsk models. On the other hand, assume that the portfolo returns are unquely determned by the mean and a G-P translaton nvarant postve and postvely homogeneous rsk measure σ ' r. Thus, we obtan an analogous captal asset prcng model (CAPM) for any opportune mean-rsk parameterzaton of the portfolo famly. In partcular, we can use the etended Sharpe measure Er ( ' r ) σ r ' to dentfy superor, ordnary, and nferor performance of portfolo ecess return ' r r where r s the rskless return. If ' r s the rsky portfolo whch mamzes the etended Sharpe measure, then, for any λ (,1), an optmal portfolo wth the same mean and lower rsk than z = λr + (1 λ) ' r cannot est because Ez ( r) Er ( ' r) =. Therefore, the portfolos r and σ σ z ' r ' r span the effcent fronter and two fund separaton holds. However, as t follows from the net 6 See Rockafeller et al. (23), Ogryczak and Ruszczynsk (1999), and Tokat et al (23). 7 See Gacomett and Ortobell (24). 14

15 dscusson, we cannot generally guarantee that k fund separaton holds when the portfolo of returns depend on k statstcal parameters. 4.3 Mult-parameter Effcent Fronters and Non-lnearty To take nto account the dstrbutonal anomales of asset returns, we need to measure the skewness and kurtoss of portfolo returns. In order to do ths, statstcans typcally use the so called Pearson-Fsher skewness and kurtoss ndees whch provde a measure of the departure of the emprcal seres from the normal dstrbuton. A postve (negatve) nde of asymmetry denotes that the rght (left) tal of the dstrbuton s more elongated than that mpled by the normal dstrbuton. The Pearson-Fsher coeffcent of skewness s gven by γ ( r ' ) = 3 E( ( ' r E( ' r) ) ) 2 (( ' ( ' )) ) ( E r E r ) 1 3/2 The Pearson-Fsher kurtoss coeffcent for a Gaussan dstrbuton s equal to 3. Dstrbutons whose kurtoss s greater (smaller) than 3 are defned as leptokurtc (platkurtc) and are characterzed by fat tals (thn tals). The Pearson-Fsher kurtoss coeffcent s gven by γ ( r ' ) = 4 (( ' ( ' )) ) 2 (( ' ( ' )) ) E r E r ( E r E r ) 2 2. Accordng to the analyss proposed by Ortobell (21), t s possble to determne the optmal choce for an nvestor under very weak dstrbutonal assumptons. For eample, when all admssble portfolos of gross returns are unquely determned by the frst k moments, under nsttutonal restrctons of the market (such as no short sales and lmted lablty), all rsk-averse nvestors optmze ther portfolo choosng among the solutons of the followng constraned optmzaton problem: 15

16 mn p(1 + ' r) subject to ' Er ( ) = m; = 1; = 1,..., n (6) E(( ' r E( ' r)) ) = q ; 3,..., 2 = k ( Q ' ) for some mean m and q =3,, k, where p(1 + r ' ) s a gven dsperson measure of the future portfolo wealth 1 + ' r and Q s the varance covarance matr of the return vector r = [ r1,..., r n ]'. Moreover, all non-satable nvestors wll choose portfolo weghts, solutons of the followng optmzaton problem ma p(1 + ' r) subject to 1 + Er ' ( ) h; = 1; = 1,..., n; (7) p(1 + ' r) E(( ' r E( ' r)) ) = q 3,..., 2 = k ( Q ' ) for some q =3,, k, and an opportune h. Smlarly, all non-satable rsk-averse nvestors wll choose portfolo weghts that are solutons to the followng optmzaton problem ma Er ( ' ) subject to 1 + Er ' ( ) h; = 1; = 1,..., n; (8) p(1 + ' r) E(( ' r E( ' r)) ) = q 3,..., 2 = k ( Q ' ) for some q =3,, k, and an opportune h. Moreover, n solvng the above constraned problems, we can dentfy the optmal choces respect to other nvestor s atttude. As a matter of fact, t has been argued n the lterature that decson makers have ambguous skewness atttudes, whle others say that nvestors are skewness-prone or prudent. 8 8 See, among others, Horvarth and Scott (198), Gamba and Ross (1998), and Pressacco and Stucch (2). 16

17 For eample, accordng to the defnton gven n Kmball (199), we can recognze the nonsatable, rsk-averse nvestors who dsplay prudence,.e. the agents that dsplay a skewness preference for fed mean and dsperson. In any case, f we assume the standard devaton as a rsk measure, we fnd that the Markowtz mean-standard devaton fronter s contaned n the set of the solutons to problem (6) obtaned by varyng the parameters m and q. In spte of ths, the mean-varance optmal portfolos are generally chosen by rsk-averse nvestors who do not dsplay prudence. Gamba and Ross (1998), n fact, have shown that n a three fund separaton contet, prudent nvestors choose optmal portfolos wth the same mean and greater skewness and varance of a mnmum varance portfolo. Thus, the present analyss s substantally a generalzaton of the Markowtz one that permts one to determne the non-lnearty aspect of rsk. For ths reason, we contnue to refer to the effcent fronter (for a gven category of nvestors) as the whole set of optmal choces (of that category of nvestors). Moreover, as recently demonstrated by Athayde and Flôres (24, 25), when unlmted short sales are allowed and the rsk measure s the varance, we can gve an mplct analytcal soluton to the above problems usng the tensoral notaton for the hgher moments. From these mplct solutons, we observe that the non-lnearty of the above problems represents the bggest dfference wth the mult-parameter lnear models proposed n the portfolo choce lterature (see Ross (1976, 1978)). As a matter of fact, factor prcng models are generally well justfed for large stock market aggregates. In ths case, some general economc state (centered) varables Y 1,..., Yk 1 nfluence the prcng (see Chen, Roll, Ross (1986)). Recall that, most of the portfolo selecton models dependng on the frst moments proposed n lterature are k-fund separaton models (see, among others, the three-moments based models proposed by Kraus and Ltzenberg (1976), Ingersoll (1987), Smaan (1993), and Gamba and Ross (1998)). Thus, they assume that each return follows the lnear equaton: r = µ + b Y b Y + ε =1,,n, (9),1 1, k 1 k 1 where generally the zero mean vector ε = ( ε1,..., ε n )' s ndependent of Y1,..., Yk 1 and the famly of all conve combnatons ' ε s a translaton and scale nvarant famly 17

18 dependng on a G-P translaton nvarant dsperson measure p( ' ε ). Then, when we requre the rank condton 9 (see Ross (1978) and Ingersoll (1987)), k+1 fund separaton holds. Hence, f the rskless r s allowed, every rsk-averse nvestor chooses a portfolo among the soluton of the followng constraned problem: mn p ( ' ε ) subject to Er ' ( ) + (1 ) r = m; = 1;., j j = 1 ; = 1,..., n; ' b = c ; j = 1,..., k 1 n (1) for some c j j=1,, k-1, and an opportune mean m. Furthermore, f unlmted short sellng s allowed and the vector ε = ( ε1,..., ε n )' s ellptcal dstrbuted wth defnte postve dsperson matr V (see Owen and Rabnowtch (1983)), then all the solutons of (7) are gven by: ( 1) ( ) 1 1 k rv ' Er ( ) r k 1 rv ' b., j λj r λ1 λ 1 j 1 j= 1 1' V E( r) r 1 j= 1 1' V b., j 1 + +, where 1 = [1,1,...,1]' s a vector composed of ones and λ =1,,k represent the weghts n the k funds that together wth the rskless asset span the effcent fronter (see Ortobell (21), Ortobell et al. (24)). Therefore, coherently wth the classc arbtrage prcng theory the mean returns can be appromated by the lnear prcng relaton Er ( ) = µ = r + b% δ b% δ,1 1, k k where δ j for j=1,...,k, are the rsk premums relatve to the dfferent factors. In partcular, when we consder a three-fund separaton model whch depend on the frst three moments, we obtan the so called Securty Market Plane (SMP) (see, among others, Ingersoll (1987), Pressacco and Stucch (2), and Adcock et al (25)). However, the approaches (6), (7), and (8) generalze the prevous fund separaton approach. As a matter of fact, f (9) s satsfed and all the portfolos are unquely determned from the frst k 9 Ths further condton s requred n order to avod that the above model degenerates nto a s-fund separaton model wth s<k+1. 18

19 moments, then the prevous optmal solutons also can be parameterzed wth the frst k moments. However, the converse s not necessarly true. Let s assume that the portfolo returns ' r are unquely determned by the mean and a G-P translaton nvarant postve and postvely homogeneous rsk measure p(1 + r ' ) and the skewness parameter γ 1 ( ' r). Also suppose ' r = ' r( q3 ) s the rsky portfolo that mamzes the etended Sharpe measure for a fed γ 1( ' r) = q3. Then, for any λ (,1), an optmal portfolo wth the same mean, skewness, and lower rsk than Ez ( r) Er ( ' r) z = λr + (1 λ) ' r cannot est because = and γ 1 ( z) = q 3. Thus, p(1 + z) p(1 + ' r) when unlmted short sales are allowed, 1 all the optmal choces are a conve combnaton of the rskless return and the solutons of the constraned problem Er ' ( ) r ma subject to p(1 + ' r) j j E(( ' r E( ' r)) ) = 1; = q 3 2 ( Q ' ) 3 3 (11) varyng the parameter q 3. However, we cannot guarantee that fund separaton holds because the solutons of (11) are not generally spanned by two or more optmal portfolos. As typcal eample, we refer to the analyss by Athayde and Flôres (24) and (25) that assumes the varance as the rsk measure. As for the three-moments framework, we can easly etend the prevous analyss to a contet where all admssble portfolos are unquely determned by a fnte number of moments (parameters). Therefore, when returns present heavy tals and strong asymmetres, we cannot accept the k fund separaton assumpton. However, f we consder the presence of the rskless asset, then two-fund separaton holds among portfolos wth the same asymmetry parameters. On the other hand, the mplementaton of nonlnear portfolo selecton models should be evaluated on the bass of the trade-off 1 When no short sales are allowed, we have to add the condton =1,,n at problem (11). 19

20 between costs and benefts. As a matter of fact, even the above moment analyss presents some non-trval problems whch are: 1) Estmates of hgher moments tend to be qute unstable, thus rather large samples are needed n order to estmate hgher moments wth reasonable accuracy. In order to avod ths problem, Ortobell et al (23, 24) proposed the use of other parameters to value skewness, kurtoss, and the asymptotc behavor of data. 2) We do not know how many parameters are necessary to dentfy the mult-parameter effcent fronter. However, ths s a common problem on every mult-parameter analyss proposed n lterature. 3) Even f the above optmzaton problems determne the whole class of the nvestor s optmal choces, those problems are computatonally too comple to be solved for large portfolos, n partcular when no short sales are allowed. Thus, we need to smplfy the portfolo problems by reducng the number of parameters. When we smplfy the optmzaton problem, for every rsk measure we fnd only some among all optmal portfolos. Hence, we need to determne the rsk measure that better characterzes and captures the nvestor s atttude. 5 AN EMPIRICAL COMPARISON AMONG THREE-PARAMETER EFFICIENT FRONTIERS Let us assume, for eample, that the nvestors choces depend on the mean, on the Pearson-Fsher skewness coeffcent, and on a rsk measure equvalent to a dsperson measure. Then, all rsk-averse nvestors optmze ther choces selectng the portfolos among the solutons of the followng optmzaton problem: mn p(1 + ' r) subject to ' Er ( ) = m; = 1; =,1,..., n (12) E(( ' r E( ' r)) ) = q; 3 2 ( Q ' ) 3 2

21 for some mean m and skewness q. In ths portfolo selecton problem, we also consder the rskless asset that has weght. The questons we wll try to answer are the followng: Is the rsk measure used to determne the optmal choces stll mportant? If t s, whch rsk measure ehbts the best performance? What s the mpact of skewness n the choces when we consder very asymmetrc returns? In order to answer to these questons, we consder the three rsk measures dscussed earler: the MnMa, the MAD, and standard devaton. These three measures are equvalent when portfolo dstrbutons depend only on two parameters. In addton, when three parameters are suffcent to appromate nvestors optmal choces, the optmal portfolo solutons of problem (12) wth the three rsk measures lead to the same effcent fronter (see Ortobell (21)). 5.1 Portfolo selecton wth and wthout the rskless return In the emprcal comparson, we consder 84 observatons of daly returns from 1/3/1995 to 1/3/1998 on 23 rsky nternatonal ndees converted nto U.S. dollars (USD) wth the respectve echange rates. 11 In addton, we consder a fed rskless asset of 6% annual rate. Solvng the optmzaton problem (12) for dfferent rsk measures, we obtan Fgure 2 on the mean-rsk-skewness space. Here, we dstngush the effcent fronters wthout the rskless asset (on the left) and wth the rskless (on the rght). Thus we can geometrcally observe the lnear effect obtaned by addng a rskless asset to the admssble choces. As a matter of fact, when the rskless asset s allowed, all the optmal choces are appromately represented by a curved plane, even f no short sales are allowed. These effcent fronters are composed of 5, optmal portfolos found by varyng n problem (12) the mean m and the skewness q between the mnmum (mean; skewness) and the mamum (mean; skewness). 11 We consder daly returns on DAX 3, DAX 1 Performance, CAC 4, FTSE all share, FTSE 1, FTSE actuares 35, Reuters Commodtes, Nkke 225 Smple average, Nkke 3 weghted stock average, Nkke 3 smple stock average, Nkke 5, Nkke 225 stock average, Nkke 3, Brent Crude Physcal, Brent current month, Corn No2 Yellow cents, Coffee Brazlan, Dow Jones Futures1, Dow Jones Commodtes, Dow Jones Industrals, Fuel Ol No2, and Goldman Sachs Commodty, S&P 5. 21

22 Generally, we cannot compare the three effcent fronters because they are developed on dfferent three-dmensonal spaces. Thus, Fgure 2 serves only to show that we could obtan dfferent representatons of the effcent fronters when usng dfferent rsk measures. Moreover, from Fgure 2 we can also dstngush the optmal portfolos of rsk averse, nonsatable, prudent nvestors,.e. the portfolos wth the smallest rsk and the hghest mean and skewness. If three parameters are suffcent to descrbe the nvestor s optmal choces, then the optmal portfolo compostons obtaned as soluton of (12), correspondng to the three rsk measures and fed mean m and skewness q, must be equal. In ths case, all three-parameter effcent fronters represented on the same space must be equal. However, we have found that for any fed mean m and skewness q the soluton to the optmzaton problem (12) does not correspond to the same portfolo composton when we use dfferent rsk measures. From ths dfference, we deduce that three parameters are stll nsuffcent to descrbe all the effcent portfolo choces. Now, we ntroduce a comparson among mean-rsk-skewness models from the perspectve of some non-satable rsk-averse nvestors. We assume that several nvestors want to mamze ther epected (ncreasng and concave) utlty functon. For every mean-rsk-skewness effcent fronter, each nvestor wll choose one of the 5, effcent portfolos. Thus, we obtan three optmal portfolos that mamze the epected utlty on the three effcent fronters. Comparng the three epected utlty values, we can determne whch effcent fronter better appromates the nvestor s optmal choce wth that utlty functon. In partcular, we assume that each nvestor has one among the followng utlty functons: 1) U( ' r) = log(1 + ' r) ; ( 1 + ' r) α 2) U( ' r) = wth α = 5, 1, 15, 5 ; 3) U( ' r) = ep( k(1 + ' r)) α wth k = 8, 1, 11, 12, 13, 5. In order to emphasze the dfferences n the optmal portfolo composton we denote by: 22

23 =,,...,, the optmal portfolo that realzes the mamum a) best best, best,1 best,23 epected utlty among the three dfferent approaches; = the optmal portfolo that realzes the lowest b) worst worst,, worst,1,..., worst,23 epected utlty among the three approaches. Then we consder the absolute dfference between the two vectors of portfolo composton,.e. 23 = best, worst,. Ths measure ndcates n absolute terms how much change the portfolo consderng dfferent approaches. From a quck comparson of the estmated epected utlty, major dfferences are not observed. However, the portfolo composton changes when we adopt dstnct rsk measures n the portfolo selecton problems. That s, the portfolo composton s hghly senstve to small changes n the epected utlty. For eample, even f the dfference between the hghest and lowest optmal value of the eponental epected utlty U( ' r) = ep( 5(1 + ' r)) s of order 1-22, the correspondng optmal portfolo composton obtaned n mean-standard devaton-skewness space s sgnfcantly dfferent (about 37%) from that obtaned n a mean-mnma-skewness space. Table 4 summarzes the comparson among the three mean-rsk-skewness approaches. In partcular, we denote by "B" cases where the epected utlty s the hghest among the three models, "M" where the epected utlty s the "medum value" among the three models, and "W" when the model presents the lowest epected utlty. Table 4 shows that the optmal solutons are ether on the mean-standard devaton-skewness fronter or on the mean-mnma-skewness fronter. Hence, nvestors wth greater rsk averson obtan the best performance on the mean-standard devaton-skewness fronter, whle less rsk-averse nvestors mamze ther epected utlty on the mean-mnma-skewness effcent fronter. Although we consder nternatonal ndees whch lack substantal asymmetres, we observe some sgnfcant dfferences n the optmal portfolo compostons of nvestors wth greater rsk averson. Instead, we do not observe very bg dfferences n the optmal choces of less rsk-averse nvestors. As a matter of fact, portfolo compostons of less 23

24 rsk-averse nvestors present dfferences of order 1-6 (that we appromate at %). On the other hand, even f the varance cannot be consdered the unque ndsputable rsk measure that t has been characterzed by n portfolo theory, ths former emprcal analyss confrms the good appromaton of epected utlty obtaned n a mean, varanc,e and skewness contet (see Levy and Markowtz (1979) and Markowtz and van Djk (25)). Thus, we net nvestgate the effects of very asymmetrc returns n portfolo choce. 5.2 An emprcal comparson among portfolo selecton models wth dervatve assets As observed by Bookstaber and Clarke (1985), Mulvey and Zemba (1999), and Iaqunta et al. (23), the dstrbuton of contngent clam returns present heavy tals and asymmetres. For ths reason, t has more sense to propose a three-parameter portfolo selecton comparson consderng some contngent clam returns. Generally, we cannot easly obtan the hstorcal observatons of the same contngent clam. Thus, n order to capture the jont dstrbutonal behavor of asset dervatves, we need to appromate the hstorcal observatons of dervatve returns. In partcular, mmckng the RskMetrcs' appromaton of dervatve's returns even for hstorcal data (see, Longestaey and Zangar (1996)), we can descrbe the returns of a European opton wth value Vt = V( Pt, K, τ, r, σ ) where P t s the spot prce of the underlyng asset at tme t, K, the opton's eercse prce, τ, the tme to maturty of the opton, r, the rskless rate, and σ, the standard devaton of the log return. Now, the value of the contngent clam can be wrtten n terms of the Taylor appromaton 1 2 Vt+ 1 Vt = Γ ( Pt+ 1 Pt ) + ( Pt+ 1 Pt ) +Θ, 2 where we have used the Greeks V Γ=, 2 t 2 Pt V t = and P t V Θ= t. Hence, the opton t return R t Vt+ 1 Vt = over the perod [t,t+1] s appromated by the quadratc relaton: V t 2 t t t R = Ar + Br + C, (13) 24

25 where t Pt+ 1 Pt rt = s the return of the underlned asset, whle P t 2 t P Γ A =, 2V t Pt B = and V Θ C =. The man advantage of ths appromaton conssts that we can analyze, descrbe V and evaluate the dependence structure of contngent clam portfolos. Moreover, as shown n Longerstaey and Zangar (1996), the relatve errors of these appromatons are reasonably low when optons are not too close to the epraton date. In ths emprcal analyss, we consder a subset of 1 of the rsky nternatonal ndees used n the prevous emprcal analyss 12 and a fed rskless asset of 6% annual rate. We appromated hstorcal returns on s European calls and s European puts on the correspondng ndees. We assume that the optons were purchased on 1/3/98 wth a three months epraton. Thus, f we assume that non-lnear appromaton (13) holds wth A, B and C fed, then we can derve mplct appromatons of a contngent clam return seres consderng..d. observatons of asset return r t. Generally speakng, n order to obtan a better appromaton of contngent clam returns, we follow the advse of RskMetrcs emprcal analyss. Consderng ths portfolo composton, t s dffcult to beleve that three-fund separaton holds and that the nvestors wll all hold combnaton of no more than two mutual funds and the rskless asset. Then, we perform an analyss smlar to the prevous one based on the optmzaton problem (12), n order to value the mpact and the dfferences of strongly asymmetrc returns n the optmal nvestors choces. Fgure 3 shows the effcent fronters we obtan by solvng the optmzaton problem (12) for dfferent rsk measures. In ths case, dfferences from the fgures obtaned prevously are evdent. In partcular, the mean m and the skewness q of problem (12) vary n a larger nterval and consequently we used 1, portfolos to appromate the effcent fronters. Even n ths case, we nclude a comparson among mean-rsk-skewness models from the perspectve of some non-satable rsk-averse nvestors. In addton, we want to t 12 We consder daly returns from 1/3/1995 to 1/3/1998 on DAX 3, DAX 1 Performance, CAC 4, FTSE all share, FTSE 1, Nkke 225 Smple average,, Nkke 225 stock average, Dow Jones Industrals, Fuel Ol No2, S&P 5, and we consder puts and calls on DAX 3, CAC 4, FTSE 1, Nkke 225, Dow Jones Industrals, and S&P 5. We convert all the returns nto U.S. dollars wth the respectve echange rates. 25

26 value the dfference between the optmal choces obtaned wth the best of the three parameter models and the mean-varance optmal choces. Thus, we assume that each nvestor has one among the followng utlty functons: 1) U( ' r) = log(1 + ' r) ; ( ) α 1 + ' r 2) U( ' r) = wth α = 15, 25, 45, 55 ; α 3) U( ' r) = ep( k(1 + ' r)) wth k = 1, 2, 3, 55, 65, 75. Then, as n the prevous analyss, we compute the absolute dfference between the two optmal portfolo that realzes the best and the worst performance among the three dfferent approaches,.e. best,, worst. In addton, we calculate the absolute dfference between the portfolo that realzes the best performance best and the optmal portfolo that mamzes the epected utlty on the mean varance effcent fronter that we pont out wth MV = MV,, MV,1,..., MV,22. Thus, the measure best,, ndcates n absolute terms how much the portfolo composton changes consderng ether a three parametrc approach or the two parametrc one. Table 5 summarzes ths emprcal comparson. An analyss of the results substantally confrms the prevous fndngs. In fact, the optmal solutons are ether on the mean-standard devaton-skewness fronter or on the mean-mnma-skewness fronter. However, as we could epect, we observe much greater dfferences n the portfolo composton. Moreover, there est sgnfcant dfferences between the meanvarance model and the three parametrc ones. In partcular, our emprcal analyss suggests that: 1) The skewness parameter has an mportant mpact n the portfolo choces when contngent clams are ncluded n the optmzaton problem. MV 26

27 2) In the presence of returns wth heavy tals and asymmetres, three parameters are stll nsuffcent to evaluate the complety of the portfolo choce problem,. 3) More rsk-averse nvestors appromate ther optmal choces on the meanvarance-skewness effcent fronter, whle less rsk-averse agents choose nvestments on the mean-mnma-skewness effcent. 6. CONCLUDING REMARKS In ths paper we demonstrate that rsk measures propertes characterze the use of a rsk measure. In partcular, dsperson measures must be mamzed at a fed level of wealth under rsk n order to obtan optmal portfolos for non-satable nvestors. Thus, standard devaton, as wth every dsperson measure, s not a proper rsk measure. We observe that most of the rsk measures proposed n the lterature can be consdered equvalent when the returns depend only on the mean and the rsk. In ths case, two-fund separaton holds. However, when the return dstrbutons present heavy tals and skewness, the returns cannot be generally characterzed by lnear models. In ths case, we can only say that two-fund separaton holds among portfolos wth the same asymmetry parameters when the rskless asset s present. Fnally, a prelmnary emprcal analyss shows that there are stll motvatons to analyze the mpact of dfferent rsk measures and of skewness n portfolo theory and that three parameters are stll nsuffcent to evaluate the complety of a portfolo choce problem, n partcular when we consder contngent clam returns. Further analyss, comparson, and dscusson are stll necessary to decde whch rsk measure gves the best performance. Probably, for ths purpose t s better to compare only mean-rsk models because the mpact that a rsk measure has n portfolo choce s much more evdent. On the other hand, many other aspects of dstrbutonal behavor of asset returns should be consdered. As a matter of fact, several studes on the emprcal behavor of returns have reported evdence that condtonal frst and second moments of stock returns are tme varyng and potentally persstent, especally when returns are measured over long horzons. Therefore, t s not the uncondtonal return dstrbuton whch s of nterest but the condtonal dstrbuton whch s condtoned on nformaton 27

28 contaned n past return data, or a more general nformaton set. In addton, the assumpton of condtonal homoskedastcty s often volated n fnancal data where we often observe volatlty clusterng and the class of auto-regressve (movng average) wth auto-regressve condtonal heteroskedastc AR(MA)-GARCH models s a natural canddate for condtonng on the past of return seres. In ths contet the complety of portfolo selecton problems could grow enormously (see, among others, Tokat et al (23), Bertocch et al (25)). However, n some cases, t can be reduced by ether consderng the asymptotc behavor of asset returns (see for eample Rachev and Mttnk (2) and Ortobell et al. (23, 24) and the reference theren) or consderng alternatve equvalent optmzaton problems that reduce the computatonal complety. (see Rachev et al (24, 25), Bglova et al. (24)). 28

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32 35. Rachev S. T., Ortobell S., Stoyanov S., Fabozz F. and A., Bglova 25 Desrable propertes of an deal rsk measure n portfolo theory, to appear as Techncal Report, Unversty of Karlsruhe (Internatonal Summer School n Rsk Measurement and Management Rome, June 25). 36. Rockafeller R.T., Uryasev S., and M., Zabarankn 23. Portfolo analyss wth general devaton measures, Research Report, 23-8, Unversty of Florda. 37. Ross, S The arbtrage theory of captal asset prcng, Journal of Economc Theory 13, Ross, S Mutual fund separaton n fnancal theory: The separatng dstrbutons, Journal of Economc Theory 17, Rothschld, M. and J., Stgltz 197. Increasng rsk: I. defnton, Journal of Economc Theory 2, Smaan, Y Portfolo selecton and asset prcng: Three parameter framework, Management Scence 5, Speranza, M. G Lnear programmng models for portfolo optmzaton, Fnance 14, Tobn, J Lqudty preference as behavor toward rsk, Revew of Economc Studes 25, Tokat, Y., S. T. Rachev, and E., Schwartz 23. The stable non-gaussan asset allocaton: A comparson wth the classcal Gaussan approach, Journal of Economc Dynamcs & Control 27, Young, M.R A mnma portfolo selecton rule wth lnear programmng soluton, Management Scence 44,

33 Propertes Proper rsk Measures Uncertanty Measures Consstency wth respect to frst order stochastc yes no domnance due to addtve shfts Consstency wth respect to frst order stochastc yes no domnance Monotony yes no Consstency wth respect to second order stochastc yes no domnance Consstency wth respect to Rothschld-Stgltz no yes stochastc order Postvely homogeneous yes yes Convety yes yes Subaddtve yes yes Postve no yes Gavoronsky-Pflug translaton nvarant no yes Translaton nvarant yes no Functonal translaton nvarant yes yes Coherent rsk measure yes no Devaton measure no yes Epectaton-bounded rsk measure yes no Table 1. Propertes of uncertanty measures and proper rsk measures. 33

34 RISK MEASURES VaRα ( W ) Value at Rsk nf z Pr( W z) > α { } CVaRα ( W ) Condtonal Value at Rsk E( W W VaR ( W )) α MM( W ) MnMa sup c R Pr( W c) = Safety Frst Pr ( W λ ) Lower Partal Moment { } (( ) q ) q E W Y, where q 1s the power nde, Y s the target wealth. CVaRα, q( W) Power CVaR q ( / α ( ) ) E W W VaR W where q 1 s the power nde. PROPERTIES Safety rsk measure that s monotone; consstent wth FSD stochastc order; postvely homogeneous; and translaton nvarant. Safety rsk measure that s monotone; consstent wth FSD, SSD, R-S stochastc orders; postvely homogeneous; conve; sub-addtve; lnearzable; coherent; translaton nvarant and epectaton-bounded. Safety rsk measure that s monotone; consstent wth FSD, SSD, R-S stochastc orders; postvely homogeneous; conve; sub-addtve; lnearzable; coherent; translaton nvarant and epectaton-bounded. Safety rsk measure that s consstent wth FSD stochastc order and monotone. Safety rsk measure that s monotone; consstent wth FSD, SSD, R-S stochastc orders; conve; and sub-addtve. Safety rsk measure that s monotone; consstent wth FSD, SSD, R-S stochastc orders; conve; and sub-addtve. Table 2 Propertes of safety rsk measures 34

35 RISK MEASURES Standard Devaton ( ( )) 2 E W E W ( ) MAD EW ( EW ( )). Mean-absolute moment q ( EW ( ( ) )) 1/ EW q PROPERTIES Devaton measure that s postve; consstent w.r.t. R-S stochastc order; postvely homogeneous; conve; subaddtve; and G-P translaton nvarant. Devaton measure that s postve; consstent w.r.t. R-S stochastc order; postvely homogeneous; conve; subaddtve; lnearzable; and G-P translaton nvarant. Devaton measure that s postve; consstent w.r.t. R-S stochastc order; conve; postvely homogeneous; G-P translaton nvarant and sub-addtve., where q 1. Gn's mean dfference EW ( Y), Devaton measure that s postve; consstent w.r.t. R-S stochastc order; postvely homogeneous; conve; subaddtve; lnearzable; and G-P translaton nvarant. where Y ponts out an..d. copy of wealth W. Eponental entropy (only for wealth that admt a densty dstrbuton) ( log ( )) W e E f t where fw ( t ) s the densty of wealth W. Colog of W EW ( log( W)) EW ( ) E(log( W )). Devaton measure that s postve; consstent w.r.t. R-S stochastc order; postvely homogeneous; conve; subaddtve; and G-P translaton nvarant. Rsk measure that s postve; consstent w.r.t. FSD due to addtve shfts and R-S stochastc order; postvely homogeneous; conve and sub-addtve. Table 3 Propertes of Dsperson Rsk Measures 35

36 Fgure 1. Effcent portfolos for non-satable nvestors;- - - Non-optmal portfolos. 36

37 .1 1 MEAN.2 SKEWNESS MEAN SKEWNESS STANDARD.4.6 DEVIATION.8 STANDARD.1 DEVIATION Mean-Standard Devaton-Skewness optmal choces wthout and wth the rskless. MEAN Skewness Mean SKEWNESS MAD MAD Mean-MAD-Skewness optmal choces wthout and wth the rskless. MEAN SKEWNESS MEAN SKEWNESS MINIMAX MINIMAX Mean-Mnma-Skewness optmal choces wthout and wth the rskless.. Fgure 2. Three-parameter effcent fronters for rsk-averse nvestors (wthout and wth the rskless asset). 37

38 .1.1 MEAN.75 MEAN SKEWNESS SKEWNESS STANDARD DEVIATION.1.15 MAD.5.1 Mean-Standard devaton-skewness optmal choces. Mean-MAD-Skewness optmal choces MEAN SKEWNESS MINIMAX Mean-Mnma-Skewness optmal choces. Fgure 3. Three-parameter effcent fronters for rsk-averse nvestors consderng portfolos of dervatves. 38

39 Epected Utlty Mean Standard Devaton- Skewness Mean-MAD- Skewness Mean- Mnma Skewness Dfference between portfolo composton E(log(1+ r)) M W B % ( 1+ ) 5 1 E r p 5 1 E ( 1 r ) 1 + p 1 1 E ( 1 r ) 15 + p 15 1 E ( 1 r ) 5 + p 5 -E(ep(-8(1+ r p ))) -E(ep(-1(1+ r p ))) 23 = M W B % best, worst, M W B 8.5% B W M 18.1% B M W 38.1% M W B % W M B 4.3% -E(ep(-11(1+ r p ))) W M B 6.9% -E(ep(-12(1+ r p ))) B W M 6.4% -E(ep(-13(1+ r p ))) B W M 9.1% -E(ep(-5(1+ r p ))) B M W 37.3% Table 4 Atttude to rsk of some nvestors on three parametrc effcent fronters and analyss of the models performance. We mamze the epected utlty on the effcent fronters consderng daly returns from 1/3/1995 to 1/3/1998 on 23 rsky nternatonal ndees and a fed rskless return. We wrte "B" when the epected utlty s the hghest among the three models, we wrte "M" when the epected utlty s the "medum value" among the three models and we wrte "W" when the model presents the lowest epected utlty. 39

40 Epected Utlty Mean- Standard Devaton- Skewness Mean- MAD- Skewness Mean- MnMa- Skewness Dfference between portfolo composton 22 = best, worst, Dfference between portfolo composton 22 = best, MV, E(log(1+ r)) W M B 6.21% 81.33% ( 1+ ) 15 1 E r p 15 1 E ( 1 r ) 25 + p 25 1 E ( 1 r ) 45 + p 45 1 E ( 1 r ) 55 + p 55 -E(ep(-1(1+ r p ))) -E(ep(-2 (1+ r p ))) W M B 8.2% 45.16% M W B 14.52% 25.1% B W M 27.73% 38.37% B M W 48.17% 52.9% W M B 6.59% 86.33% W M B 1.52% 5.9% -E(ep(-3(1+ r p ))) M W B 17.49% 26.72% -E(ep(-55(1+ r p ))) B W M 26.44% 28.6% -E(ep(-65(1+ r p ))) B W M 35.94% 43.22% -E(ep(-75(1+ r p ))) B M W 42.31% 46.33% Table 5 Atttude to rsk of some nvestors on three parametrc effcent fronters and analyss of the models performance when we consder portfolos of asset dervatves. We mamze the epected utlty on the effcent fronters consderng a fed rskless return, the appromated hstorcal daly returns of 12 asset dervatves and daly returns on 1 rsky nternatonal ndees. We wrte "B" when the epected utlty s the hghest among the three models, we wrte "M" when the epected utlty s the "medum value" among the three models and we wrte "W" when the model presents the lowest epected utlty. 4