Kimball's prudence and two-fund separation as determinants of mutual fund performance evaluation Breuer, Wolfgang; Gürtler, Marc

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1 Kimball's prudence and two-und separation as determinants o mutual und perormance evaluation Breuer, Wolgang; Gürtler, Marc Arbeitspapier / working paper Zur Verügung gestellt in Kooperation mit / provided in cooperation with: SSG Sozialwissenschaten, USB Köln Empohlene Zitierung / Suggested Citation: Breuer, Wolgang ; Gürtler, Marc ; Technische Universität Braunschweig, Department Wirtschatswissenschaten, Institut ür Finanzwirtschat (Ed.): Kimball's prudence and two-und separation as determinants o mutual und perormance evaluation. Braunschweig, 005 (IF Working aper Series FW17V4). URN: hdl.handle.net/10419/5555 Nutzungsbedingungen: Dieser Text wird unter einer Deposit-Lizenz (Keine Weiterverbreitung - keine Bearbeitung) zur Verügung gestellt. Gewährt wird ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht au Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich ür den persönlichen, nicht-kommerziellen Gebrauch bestimmt. Au sämtlichen Kopien dieses Dokuments müssen alle Urheberrechtshinweise und sonstigen Hinweise au gesetzlichen Schutz beibehalten werden. Sie düren dieses Dokument nicht in irgendeiner Weise abändern, noch düren Sie dieses Dokument ür öentliche oder kommerzielle Zwecke vervielältigen, öentlich ausstellen, auühren, vertreiben oder anderweitig nutzen. Mit der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an. Terms o use: This document is made available under Deposit Licence (No Redistribution - no modiications). We grant a non-exclusive, nontranserable, individual and limited right to using this document. This document is solely intended or your personal, noncommercial use. All o the copies o this documents must retain all copyright inormation and other inormation regarding legal protection. You are not allowed to alter this document in any way, to copy it or public or commercial purposes, to exhibit the document in public, to perorm, distribute or otherwise use the document in public. By using this particular document, you accept the above-stated conditions o use.

2 Working aper Series Kimball s rudence and Two-Fund Separation as Determinants o Mutual Fund erormance Evaluation by Wolgang Breuer and Marc Gürtler No.: FW17V4/05 First Drat: This Version: Technical University at Braunschweig Institute or Economics and Business Administration Department o Finance Abt-Jerusalem-Str. 7 D Braunschweig

3 Kimball s rudence and Two-Fund Separation as Determinants o Mutual Fund erormance Evaluation by Wolgang Breuer * and Marc Gürtler ** Abstract. We consider investors with mean-variance-skewness preerences who aim at selecting one out o F dierent unds and combining it optimally with the riskless asset and direct stock holdings. Direct stock holdings are either exogenously or endogenously determined. In our theoretical section, we derive and discuss several perormance measures or the investor s decision problems with a central role o Kimball s (1990) prudence and o several variants o Sharpe and Treynor measures. In our empirical section, we show that the distinction between exogenous and endogenous stock holding is less important than the issue o skewness preerences. The latter are most relevant or und rankings, when an investor s skewness preerences are not derived rom cubic HARA utility so that the two-und separation theorem is not valid. Keywords: investor speciic perormance measure, perormance evaluation, prudence, skewness preerences JEL classiication: G11 * roessor Dr. Wolgang Breuer ** roessor Dr. Marc Gürtler RWTH Aachen University Technical University at Braunschweig Department o Finance Department o Finance Templergraben 64, 5056 Aachen, Germany Abt-Jerusalem-Str. 7, Braunschweig, Germany hone: Fax: hone: Fax: wolgang.breuer@rwth-aachen.de marc.guertler@tu-bs.de

4 I. Introduction The main task o perormance measures or investment unds is to help investors in identiying the most suitable und or given preerence structures. In general, there are two possible ways to tackle this problem. On the one hand, one can choose a partial-analytical ramework, thereby ocussing on the decision problem o a given investor or given expectations and neglecting any kind o general capital market considerations. On the other hand, one can analyze capital market price ormation processes in order to derive conclusions with respect to the attractiveness o certain unds. For example, the well-known capital asset pricing model (CAM) as introduced by Sharpe (1964) may deine such a setting. One may conclude rom this equilibrium description that the same perormance measure o zero should be assigned to all investment unds, just expressing that the holding o shares o any und is irrelevant or any capital market participant. Another prominent example o a market-based approach has been developed by Leland (1999) on the basis o power utility unctions and lognormal return distributions. Although such analyses on capital market levels certainly are apt to create interesting general insights, or practical application we preer the partial-analytical ramework ocussing on the view o a single investor with given preerence structures and expectations who typically acts as a price-taker. I or such an investor the CAM in its original version or in the modiied setting applied by Leland (1999) in act held, we would learn this rom his or her speciic expectations. But i this is not true, the CAM (as any other capital market model) is not o immediate relevance or the investor under consideration. In what ollows, we thus examine an investor with a one-period horizon who aces at time t = 0 the problem o selecting just one out o F dierent unds in order to combine this investment with the direct holding o a given (reerence) portolio o equity shares and riskless lending or borrowing until time t = 1. As a consequence, or any und under consideration we are searching or optimal ractions x 0, x, and x o the investor s initial wealth optimally invested in the riskless asset, the und and the equity portolio. Ater this, resulting preerence values or any und are used to generate a und ranking which can be utilized as a recommendation or und selection. Certainly, the examination o a situation where only one out o F dierent unds can be chosen is somewhat restrictive. Nevertheless such a scenario can be interpreted as a classical asset allocation problem with three classes o assets (a und, direct stock holding and riskless 1

5 lending or borrowing). As an illustration, this decision problem corresponds to the important case o institutional investors relying only on a single und manager, a not uncommon practice in many countries. In addition, it is necessary to deine dierent unds as alternative investments i perormance measures or single unds shall be derived. Moreover, the analysis o situations with the selection o only one und at a time may be used as a starting point or the examination o more complex portolio selection problems in uture work. In act, our derivations remain valid i we reinterpret = 1,, F not as single unds but as F dierent given portolios o unds. Only the analysis o the determination o the optimal combination o a certain set o unds must then be the object o urther research. One recent numerical approach that is devoted to this latter task has been introduced by Davies/Kat/Lu (006). However, because o the complexity o their decision-problem they not even attempt to derive general results that could be interpreted as perormance measures. Moreover, owed to computational problems they have to rely on non-standard ways o describing investors preerences. As a consequence, or their approach it does not seem to be possible to derive any connection to expected utility maximizing behavior, as is done or our approach in one o the ollowing sections. Recently, or simple µ-σ-preerences the decision-problem sketched above has been analyzed or two dierent settings (see also Figure 1). In the irst one which may be called the endogenous case, all three ractions x 0, x, and x are indeed variable. In Breuer/Gürtler (1999, 000) it has been demonstrated that in such a situation unds can be ranked according to an optimized Sharpe measure which coincides with the conventional Sharpe ratio 1 o the optimal risky portolio o und and equity portolio. Additionally, based on previous indings by Jobson/Korkie (1984) it could be shown that or inner solutions the optimized Sharpe measure is identical to the Treynor/Black appraisal ratio while in the case o short sales restrictions the optimized Sharpe measure may lead to border solutions that coincide with the original Sharpe ratio, the Treynor ratio 3, or Jensen s alpha 4. As mean-variance preerences are in particular the result o quadratic utility, in what ollows we simply speak o the quadratic Sharpe measure, Treynor measure, Jensen measure, and Treynor/Black measure. The optimized Sharpe measure in situations with short sales restrictions will be named the optimized restricted Sharpe measure. 1 See Sharpe (1966). See or the Treynor/Black appraisal ratio in particular Treynor/Black (1973). 3 See Treynor (1965). 4 See Jensen (1968).

6 >>> Insert Figure 1 about here <<< In the second setting, one may reasonably argue that the raction x o an investor s direct stock holding is exogenously ixed and only x 0 and x can be optimized any more, or example, because o ormer transactions and corresponding transaction costs considerations. This exogenous case has intensively been examined by Scholz/Wilkens (003), and in Breuer/Gürtler (005) both approaches have been analyzed with respect to their theoretical and empirical relationships. Quite remarkably, it could be proven that, in general, a und g is unambiguously preerred to a und h in the endogenous case as well as in the exogenous case i it exhibits both a higher quadratic Sharpe measure and a higher quadratic Treynor measure. Nevertheless, besides this inding, theoretical relationships between und rankings in the endogenous case and the exogenous one seem to be quite loose, while empirical evidence suggests that at least or simple µ-σ-preerences the distinction between both scenarios is negligible. >>> Insert Table 1 about here <<< In this paper, we want to extend the analysis by the explicit consideration o skewness preerences o investors, i.e. preerences regarding the third central moment o uncertain wealth or return, as recent approaches like the ones by Harvey/Siddique (000), Dittmar (00) or Fletcher/Kihanda (005) are in particular stressing the relevance o preerences or higherorder return moments in asset pricing models. As sketched in Table 1, this extension can be done or the endogenous case (cell (3) in Table 1) as well as or the exogenous one (cell (4) in Table 1). Moreover, while Breuer/Gürtler (005) ocus on the relationship between cases (1) and () o Table 1, we will examine in more depth the relationship between und rankings or the cases () and (4), thus contrasting und rankings in the exogenous case or mean-variance preerences and mean-variance-skewness preerences. For the endogenous case and with a restriction to the case o cubic utility unctions with hyperbolic absolute risk aversion (HARA), such a comparison is presented by Breuer/Gürtler (006). It is shown that unds can be unambiguously ranked according to an optimized cubic perormance measure which only depends on two arguments: the optimized quadratic Sharpe measure o the und under consideration and a newly introduced perormance measure which may be called an optimized cubic Sharpe measure. The latter is deined as the quotient o the (third root o the) skewness o the return o the optimal ( preerence-independent, i.e. being valid or the whole class o cubic HARA utility unctions) combination o a und with the reerence portolio 3

7 and the corresponding variance o this portolio return. Thereby, the possibility o preerenceindependent und ranking is a consequence o the two-und separation theorem introduced by Tobin (1958) and later on extended by Hakansson (1969) and Cass/Sitglitz (1970). However, the two-und separation theorem holds only in the endogenous case with HARA utility with the latter certainly being a relevant restriction o skewness preerences in itsel. Against this background, we start our theoretical exposition in the ollowing Section II with a general discussion o mean-variance-skewness preerences. Certainly, preerence parameters are least restricted when we only exclude ineicient solutions rom the analysis, i.e. solutions with mean-variance-skewness characteristics that are dominated by other admissible portolios. However, as is known rom simple mean-variance analysis, not every eicient solution may be the outcome o expected utility maximizing behavior. This additional requirement narrows the set o admissible mean-variance-skewness preerences. Moreover, or expected utility maximizing behavior, we are able to show that an investor s optimal portolio selection is mainly determined by Kimball s prudence, i.e. the negative relation between the third and the second derivative o his or her utility unction, as this value governs the relationship between the subjective evaluation o portolio return skewness (being mainly determined by the third derivative o an investor s utility unction) and o portolio return variance (being mainly determined by the second derivative o an investor s utility unction). The range o admissible mean-variance-skewness-preerences under consideration becomes even smaller, when only cubic utility unctions o the HARA type are examined. Based on such a general discussion o mean-variance-skewness preerences, the main theoretical contribution o our paper in Section III aims at the derivation o perormance measures or the exogenous case with skewness preerences. In this context, we rerain rom restricting ourselves to the analysis o only cubic HARA utility, as the advantage o the HARA property (validity o the two-und separation theorem) does not hold or the exogenous case. Moreover, in Section IV, we will be able to extend the analysis o the endogenous case to non- HARA skewness preerences as well. We do not know o any other approach attempting to derive general perormance measures or arbitrary skewness preerences in the endogenous case or in the exogenous one deined above. We are able to identiy several simple submeasures o perormance which serve as arguments or our general perormance measures and can be interpreted as variants o a und s (cubic and quadratic) Sharpe or Treynor measures and thus as a straightorward extension o Scholz/Wilkens (003) and Breuer/Gürtler (005) or the simple mean-variance case. Thereby, or our analysis including skewness preerences, a 4

8 preerence parameter becomes relevant that is directly related to Kimball s prudence in order to deine the relative importance o portolio return skewness and portolio return variance in perormance measurement. Based on our theoretical derivations, our empirical analysis in Section V addresses the ollowing two issues: 1) Does the empirical inding by Breuer/Gürtler (005) o the irrelevancy o the distinction between the endogenous case and the exogenous one in a mean-variance context carry over to a situation with mean-variance-skewness considerations? ) Which role does a possible restriction to only cubic HARA utility play or the relevance o perormance measures recognizing skewness preerences? We ind that the distinction between the exogenous case and the endogenous one indeed remains to be o only minor importance even i we allow or mean-variance-skewness preerences, while the empirical relevance o skewness preerences seems to be depending on the validity o the two-und separation. As is well-known, a simple mean-variance approach ceases to be o good approximative quality even in cases with non-quadratic utility when an investor s risk aversion is suiciently high. Nevertheless, this circumstance can only lead to variations in und perormance in cases without two-und separation, because otherwise optimal und rankings are not inluenced by variations o an investor s risk aversion. Thereore, it does not seem to be too surprising that skewness preerences aect und rankings in our empirical example most when mean-variance preerences are not restricted to such parameter constellations that are in line with cubic HARA utility, as derived in Section II o our paper. In any case, this inding sheds additional light on the empirical relevance o preerences or higher-order return moments in perormance evaluation or mutual unds. Section VI tackles the problem o possible ways to practical application o the perormance measures developed in this paper. In particular, empirical indings regarding typical values o Kimball s prudence may be an adequate starting point to speciy the preerence parameter in our perormance measures. Section VII concludes. Because o space constraints, all mathematical derivations have been deerred to separate appendices. For the same reasons, several tables (numbered rom Ad 1 to Ad 7 ) have been omitted that are not absolutely necessary 5

9 or the understanding o our exposition. Moreover, Table oers a synopsis o the most relevant symbols utilized in this paper. >>> Insert Table about here <<< II. Decision-theoretical background 1. Mean-variance-skewness preerences The skewness o a wealth distribution can be characterized as its third central moment 3 3 (1) γ W : = E[(W E(W)) ], with W as the investor s uncertain terminal wealth. As a generalization o the basic meanvariance case, we consider investors who are aiming at the maximization o a µ-σ-γpreerence unction Φ W with 3 3 () ΦW W σw γ w = W κw σ w λw γ w (µ,, ) µ. κ W and λ W are positive preerence-depending parameters, as risk-averse investors are characterized by negative variance preerences and (typically 5 ) by positive skewness preerences. For initial wealth W 0 we can deine r: = (W/W) 0 1 as the investor s uncertain portolio return and introduce µ, σ, and γ 3 as the relevant moments o the investor s return distribution. Then with given initial wealth W 0, the maximization o the preerence unction Φ W is equivalent to the maximization o Φ := Φ W /W 0 1: 3 3 (3) Φ(µ, σ, γ ) = µ κ σ λ γ, with κ= : W0 κ W and λ= : W 0 λ W. For W 0 = 1, κ and κ W as well as λ and λ W are identical. Without loss o generality we thereore will rom now on assume W 0 = 1. Moreover, (3) can be expressed equivalently as a unction o the relevant moments o the investor s excess return u: = r r0 or given riskless interest rate r 0 and with expectation value u, as we have µ = u r 0, while the second and the third central moment or r and u are identical. Analogously to µ-σ-dominance and µ-σ-eiciency it is possible to introduce the concept o µ- σ-γ-dominance and µ-σ-γ-eiciency: An alternative 1 is (strictly) dominated by an alternative, i we have µ 1 µ, σ 1 σ as well as γ 1 γ with at least one inequality being strict. An alternative is µ-σ-γ-eicient, unless it is (strictly) µ-σ-γ-dominated by at least one alternative. 5 Among other things, it is well-known that or an expected utility maximizing individual positive skewness preerences are a necessary condition or decreasing absolute risk aversion which in turn seems to be typical or individuals attitudes towards risk. See, or example, Arrow (1971). 6

10 Certainly, or preerences according to (3) only µ-σ-γ-eicient alternatives have to be regarded as potential optimal solutions o an investor s decision problem. In order to solve a portolio selection problem or given preerence unction () or (3) one has to ix parameters κ and λ. Unortunately, this straightorward approach does not lead to meaningul general results. We thereore ollow another way o derivation, whereby we assume the investor to deine a desired (positive) expected overall excess return u o his or her portolio which he or she wants to achieve. Since all portolios under consideration are just characterized by the same desired overall expected rate o return, preerence unction (3) reduces to 3 3 (4) Φ σ γ = ω σ γ with ω = κ/λ >0. u (, ):, As a consequence o this modiied approach the determination o preerence parameters κ and λ is thus replaced by the speciication o u and ω. Instead o some absolute preerence levels regarding σ and γ 3, only the relative relevance o variance aversion in comparison to skewness loving (as expressed by ω) remains relevant. Such an approach seems to be irst suggested by Breuer/Gürtler (1998). Later on Berényi (00) coined the term variance equivalent risk measure or the unctional orm Φ (, )/ /. u σ γ ω=σ γ ω However, neither Breuer/Gürtler (1998) nor Berényi (00) have examined the exogenous case or the endogenous case as deined in this study. Moreover, Breuer/Gürtler (1998) present no utiliytheoretical analysis, while Berényi (00) ails to explicitly consider any portolio selection problem at all and thus is not able to derive perormance measures endogenously. Apparently, one might wonder about the relationship between optimizing (4) or given portolio excess return u and preerence parameter ω and the optimization o (3) (or ()) or given values o κ and λ. This is not a trivial issue. In particular, it should be emphasized that u is endogenously determined by the investor in question and as such the trade-o between expected (excess) returns and risk properties o return distributions is not neglected at all when applying (4) or means o portolio optimization. Nevertheless, it remains to be analyzed whether any possible pair ( u, ω) is admissible in that sense that there is another ( reasonable ) pair o preerence parameters (κ, λ) that leads to the same optimal portolio selection. 6 See also Berényi (003) and Onorato (004). 7

11 In order to answer this question we irst have to clariy the utility-theoretical background o preerence unction () (or (3)) to some larger extent.. Relationships between u and preerence parameters In the same way, as mean-variance preerences can be derived rom the assumption o quadratic utility, it is possible to justiy the preerence unction described by () via a cubic von Neumann-Morgenstern utility unction U(W) or uncertain terminal wealth W with 3 (5) U(W) = aˆ W aˆ W aˆ W a ˆ Using a Taylor expansion around µ W, expected utility in the case o (5) can be computed as (6) 1 1 E[U(W)] = U( µ ) U '( µ ) ( µ µ ) U ''( µ ) σ U '''(µ ) γ = U( µ W) U ''( µ W) σ W U '''(µ W) γw. 6 3 W W W W W W W W As in the case o preerence unction () we restrict ourselves to situations with positive skewness preerences which obviously requires U'''(µ W) = 6 aˆ3 > 0 aˆ3 > 0. Consequently, the raction : = U '''(µ W) / U ''(µ W) becomes positive, too. Actually, Kimball (1990) introduced the term absolute prudence or this raction. A positive prudence implies that an investor will increase ceteris paribus his or her riskless lending, when uncertain returns become riskier: The greater the prudence, the more sensitive an investor s reaction by increasing his or her precautionary saving. ositive skewness preerences thus coincide with a positive prudence and mere mean-variance preerences imply a prudence o zero. Moreover, and rather interestingly, or given value u o u (and thus given µ W ) and with given value W 0 = 1, (6) yields ω=κw / λ W = 3 U ''(µ W) / U '''(µ W) = 3/ and hence the preerence parameter ω o section II.1 can be interpreted as (three times) the reciprocal value o an investor s pru- dence or an excess return realization u with u = u. Moreover, we have κ/λ = 3/(W 0 ) which also simpliies to 3/ because o our assumption W 0 = 1. According to Kimball (1990), the product µ W is called the relative prudence or an expected excess return realization u = u. In what ollows we simply speak o prudence when we mean the absolute one, but will return to the concept o relative prudence in Section VI below. Because o the cardinality o von Neumann-Morgenstern utility unctions we can reduce (5) by â 0 and then divide it by â. 3 Deining a : = a ˆ/aˆ3 and a 1 : = a ˆ 1 /a ˆ 3, (5) can thus be rewrit- 3 ten as U(W) = W a W a W so that (6) becomes 1 8

12 E[U(W)] a a (3 a ). 3 3 (7) = µ W µ W 1 µ W µ W σ W γ W As long as we restrict ourselves to situations with positive, but diminishing marginal utility (and positive prudence), it is easy to show that the maximization o (7) results in the selection o a µ-σ-γ-eicient alternative. 7 However, not every µ-σ-γ-eicient alternative can be the outcome o the maximization o (7) i we hold on to the requirement o positive, but decreasing marginal utility. 8 In act, this result is already well-known or simple mean-variance preerences, i.e. the case λ = 0. 9 It thus seems reasonable to explicitly allow or the requirement o a positive irst and a negative second derivative o the utility unction. As a necessary condition or the ulilment o these properties which is independent 10 o the speciic return distribution these signs o the derivatives must be given at least or expected return µ W, i.e. a 1 U'(µ W) > 0 3 µ W a µ W a1 > 0 a > 1.5 µ W, (8) µ W > 0 µ U ''(µ ) < 0 a < 3 µ. W W Apparently, (or µ W > 0) both conditions o (8) can only be simultaneously valid or a > 3 µ. 1 W W 3 With respect to U(W) = W a W a1 W a, the special case o a 1 = deserves particular 3 attention, as this leads to a cubic utility unction that can be written as (9) U(W) = (W a) a = W 3 a W 3 a W, with a > 0 and a a 1 (10) a = 3 a a =,a1 = 3 a a =. 3 3 Such a cubic utility unction exhibits the property o hyperbolic absolute risk aversion mentioned previously, i.e. we have U ''(W) 1 (11) =, U'(W) aˆ b W with risk aversion parameters â = 0.5 a and b = See Appendix 1. 8 See Appendix. 9 See, or example, Breuer/Gürtler/Schuhmacher (004), p It is not diicult to derive stricter restrictions or given domains o uncertain excess returns. However, (8) must be valid in any case and even i we only know the relevant moments o excess returns and not their domains. 9

13 As already stated, in order to apply preerence unction (4), an investor has to determine a pair (u, ω ) o desired expected overall excess return u and preerence parameter ω = κ/λ. We are now able to return to the issue o which pairs (u, ω ) are actually consistent with preerence or utility unctions (), (5), and (9). To be more speciic, a consistent speciication o (u, ω ) by an investor requires or the case o preerence unction () that there exists at least one corresponding pair o preerence parameters κ and λ so that the resulting optimal overall portolio o the best und, reerence portolio, and the riskless asset leads to an overall ex- pected excess return o u. I such a pair (κ, λ) does not exist, then the resulting ranking or (u, ω ) lacks any relevance and the pair (u, ω ) can be called not admissible. Certainly, () imposes the ewest restrictions on admissible pairs (u, ω ), but even or (), not all, but only suiciently great values o expected excess returns u can be the result o portolio optimization. Things get even worse, i we require a cubic von Neumann-Morgenstern utility unction according to (5), as this implies additional lower or upper bounds or admissible values o ω or given expected excess return. As a consequence o the urther restriction o HARA util- ity, there will be at most two admissible values or ω or any given expected excess return u. These indings are made more precise in Result 1: 1) In the case o general mean-variance-skewness preerences according to () or (3), or any given exogenous value o x and given preerence parameter ω, it will be possible to justiy any desired overall expected excess return u as preerence maximizing when choosing the best und, as long as u is not smaller than the expected excess return o the portolio that maximizes γ 3 ω σ. ) Deine σ and 3 γ as the variance and the skewness o the return o the investor s overall portolio or x ˆ : = (u x u )/u, i.e. the necessary share o und as part o the investor s overall portolio in order to attain an overall expected excess return u. Then, in the case o expected utility maximizing behavior with a general cubic utility unction according to (5) only preerence parameters ω satisying (1) 1.5 [ σ u (x γ x xˆ γ x ˆ γ )] (a) ω>, i x σ xˆ σ > 0, 3 x ˆ σ x σ 3 x ˆ σ x σ 1.5 σ [ u (x γ x xˆ γ x ˆ γ )] (b) ω<, i x σ xˆ σ < 0, 10

14 are in line with decreasing positive marginal utility at least with respect to an investor s expected terminal wealth and are consistent with an expected utility maximizing choice o u regarding the best und under consideration. Additionally, we need ω > 0 because o our requirement o positive skewness preerences. 3) For cubic HARA utility as described by (9) conditions (1a) and (1b) simpliy to (13) 3 σ σ γ ω= 1.5 ± σ 3 u u u with the additional requirement o ω being positive. roo: See Appendix 3. The considerations o this subsection highlight the relationships between the dierent approaches to justiy skewness preerences. We avor the application o the preerence unction () (or (3)), or this objective unction encompasses the maximization o expected cubic (HARA) utility as a special case. As in a situation with mean-variance preerences, a utilitytheoretic oundation o mean-variance-skewness preerences does not seem to be a sine qua non or the application o (). 11 As a last point it should be noted that there is just one drawback o the approach applied in this paper. In what ollows we will utilize the preerence unction (4) with given parameter combination (u, ω ) to solve F dierent portolio selection problems or the exogenous case and or the endogenous case. In each o them, one und is optimally combined with the (possibly exogenously given holding o) reerence portolio and riskless lending or borrowing. Subsequently, unds are ranked according to the corresponding maximum preerence values they oer and these preerence values ater some algebraic manipulations are interpreted as perormance values. In contrast to the analysis sketched above, we are thus examining not just one, but F dierent portolio selection problems with ixed values or u and ω. Unortunately, or a given preerence unction () or (3) it is not suicient to be the best und based on (4) and a given expected return u or being the best und at all, because another und may be better than that und or another value o u and it might be that dierent values or u describe optimal portolio selection behavior or dierent unds. This is a problem typically not discussed in the literature, although there are other approaches that rely on similar stan- 11 For such an argument in the case o pure mean-variance preerences see Löler (1996). 11

15 dardization techniques like the ones by Graham/Harvey (1997) and Modigliani/Modigliani (1997). In act, only in the endogenous case with mean-variance preerences it is apparent that there is not any problem, because the und ranking here is identical or any given desired ex- pected excess return u, problems. a eature which is not generally shared by more complex decision However, being the best und or at least one value u is a necessary condition or being the best overall und or given preerences. Thereore, the investor only has to choose among those unds which are best or at least one achievable expected excess return u. Typically, we will expect only a ew unds to emerge as candidates and among them an investor should be able to choose without urther ormal assistance. We will return to this issue in Section VI. III. The exogenous case 1. Some basic variables At irst glance, the replacement o equation () by ormula (4) does not seem to be too great an alleviation o the original decision problem. Nevertheless, this approach enables us to derive a measure o perormance evaluation that consists o several easily understandable basic elements. In order to do so, we additionally have to introduce the notion o subportolio Q() which consists o the riskless asset as well as o the investor s holding o a und and thus describes the variable part o his or her overall portolio in the exogenous case (see also Figure 1). Correspondingly, R() stands or an investor s risky subportolio consisting o a relative investment y := x /(x x ) in a und and o y := x /(x x ) or direct stock holdings. Furthermore, we deine u Q( ) : = u x u as the contribution o portolio Q() to overall expected excess return u. It should be noted that in the exogenous case we have u = u (x ) = : u = const. or all unds = 1,..., F, since u, Q( ) Q Q exogenously given or any und. x = xˆ as well as u are From now on, we assume all expected excess returns to be nonnegative, as investments with negative expected excess returns are generally not preerable. Moreover, we introduce γ Q()Q() and γ Q() as symbols or the two co-skewnesses E[(uQ() u Q() ) (u u ) ] E[(u u ) (u u )] and Q() Q(). In addition, we need symbols b Q()Q() and b Q() or ractions γ Q( )Q( ) / γ and 3 p 3 γq( ) / γ p, respectively. Finally, we deine Q( ) σ as the covariance between excess returns u Q( ) and u and β : =σ / σ as the corresponding regression Q( ) Q( ) 1

16 coeicient. All relevant (co-) moments regarding excess returns u and u are named analogously and indexed by an or a.. The investor-speciic cubic perormance measure As already mentioned in Section I, Scholz/Wilkens (003) analyzed the exogenous case or mean-variance preerences. This means that their approach can be interpreted as i examining F portolio selection problems according to the setting o Figure 1 based on (4) with preerence parameter λ = 0 (i.e. ω ). From the resulting optimal preerence values or each und under consideration, they derived a so-called (quadratic) investor-speciic perormance measure ( qim, henceorth), because und rankings turn out to be depending on investors speciic preerences, since the two-und separation theorem does not apply. From the analysis in Scholz/Wilkens (003) and Breuer/Gürtler (005) we know that qim only depends on the quadratic Sharpe measure qsm and the quadratic Treynor measure qtm as deined in Table. This inding is intuitive appealing, as the irst measure applies or the special case (exg) y = 0, i.e no direct stock holdings at all, and the latter or the special situation (exg) y 1, i.e. only marginal und investments. Up to now, or perormance evaluation with meanvariance-skewness preerences, we can only reer to Breuer/Gürtler (006). As has already been mentioned in Section I, they showed that or the endogenous case with cubic HARA utility unctions each und is evaluated on the basis o two basic perormance measures. The irst one is the optimized quadratic Sharpe measure, that is, the value o qsm or the best combination o a und and the reerence portolio. The second one can be interpreted as an optimized cubic Sharpe measure. We may deine a cubic Sharpe measure by replacing the original numerator or denominator o the quadratic Sharpe measure u/σ by γ, i.e. the third root o the respective return skewness. Since this leads to two dierent versions o a cubic Sharpe measure (see also Table ), we call the one with γ in the numerator the cubic Sharpe measure o type 1, and the one with γ in the denominator the cubic Sharpe measure o type. For the endogenous case with cubic HARA utility Breuer/Gürtler (006) showed the relevance o the cubic Sharpe measure o type 1. However, we will shortly see that in the exogenous case the cubic Sharpe measure o type becomes relevant. In a similar way, one may distinguish more than just one cubic Treynor measure. In the quadratic case there is just one covariance σ between u and u. Nevertheless, there are at least two co-skewnesses γ and γ as deined in Table and consequently there are two Treynor measures: type 1, deined as u / b and type, deined as u/b. 13

17 In act, as is revealed by ormula (T1) and Result T1 i) o Table 3, a repetition o the analysis o Scholz/Wilkens (003) or λ > 0, i.e. or an investor with positive skewness preerences, leads to a cubic investor-speciic perormance measure cim (exg) (x ˆ ) or the exogenous case that is a unction not only o the quadratic Sharpe and Treynor measure, but also o the cubic Sharpe measure o type and o both cubic Treynor measures. 1 Other und speciic parameters are not relevant or perormance evaluation in the exogenous case. The perormance measure (T1), although lengthy, can thus be traced back to only a ew und-dependent deter- minants. To be more precise, or the typical case o a positive value o u Q( ) (x ˆ ), the perormance o a und is the better, the greater its quadratic Sharpe measure qsm. Moreover, in our empirical example o Section V all unds under consideration as well as the reerence portolio exhibit negative return skewnesses and co-skewnesses and positive return covariances. For such a situation und perormance is ceteris paribus improving with a higher quadratic Treynor measure as well as a lower cubic Sharpe perormance measure () csm o type and becoming better with greater cubic Treynor measures. The negative impact o the cubic Sharpe measure may appear somewhat surprising, but it is simply caused by the act that higher values o γ, i.e. o the denominator o the cubic Sharpe measure, lead to higher preerence values, while this coincides with a lower cubic Sharpe measure. In any case, it should be clear that or certain relationships between their respective Sharpe measures on the one side and their Treynor measures on the other, two unds can be unambiguously ranked regardless o which pair (u, ω ) is in eect. Result T1 ii) o Table 3 thus tells us under which conditions an investor does not need to bother much about the precise speciication o his or her preerence parameters. In addition, according to Result T1 ii) it then even plays no role at all, i the exogenous or the endogenous case is considered. Furthermore, the relevance o the und-dependent submeasures in cim (exg) (x ˆ ) may become clearer, i we examine some special cases, as is done in the ollowing subsection. >>> Insert Table 3 about here <<< 1 See Appendix 4 or a proo o Result T1. 14

18 3. Some special cases Special cases arise or extreme values o ω and ˆx. Some o them are described in Table Case a) (ω ) describes a situation with mere mean-variance preerences, while ω = 0 implies a situation with mere mean-skewness preerences. In this context, it should be noted that ω = 0 does not necessarily imply that the investor is not variance averse at all. It simply means that the relevance o his or her skewness loving exceeds the relevance o variance considerations by an ininite amount. Rather interestingly, taking together cases a) and b) gives Result T o Table 3 which leads to a second possibility to assess potential und rankings without the precise speciication o ω: For given desired expected excess return u a und g is better than a und h or any preerence parameter ω, i its perormance measure cim (exg) g (x ˆ ) is greater than that o und h, cim (exg) h (x ˆ ), or both extreme scenarios ω and or ω = 0. To put it another way: For given overall expected excess return, only unds with greater perormance measures cim (exg) g (x ˆ ) or one o these extreme scenarios can be better than a certain und h even or any other preerence parameter ω with 0 < ω <. The reason or these indings is that the resulting perormance measure or values o ω with 0 < ω < is a linear combination o the perormance measures or the two extreme cases ω and ω = 0. We will use Result T o Table 3 in our empirical analysis presented later on, but now turn to special cases described by extreme values o ˆx. In act, we are more interested in ractions y and y o und and reerence portolio as parts o the risky subportolio R() than in the raction ˆx in itsel. Allowing or short sales restrictions we just have to consider situations with (exg) y = 1 and (exg) y =ε with ε > 0, but small. The irst case coincides with y = 0 and thus requires ˆx = 0. Case c) in Table 3 reers to this situation. According to the last sentence o Result T3, it is even possible to conclude that (or all return skewnesses being o the same sign) a und g is better than a und h in the endogenous case in situations with border * * solutions y = y = 1, i und s g cubic Sharpe measure is smaller and its quadratic g h Sharpe measure is greater than the corresponding measure o und h. This is quite remarkable, as according to Breuer/Gürtler (006), in the endogenous case with cubic HARA utility, border solutions with no investment in the reerence portolio o direct stock holdings at all 13 The cases a) and b) immediately ollow rom (T1). See Appendix 5 or the derivation o the special perormance measures o situations c) and d). 15

19 imply that und rankings are only (positively) depending on the quadratic Sharpe measure qsm and the cubic Sharpe measure (1) csm o type 1. Only in situations with γ g > 0 and γ h > 0 it is possible to always derive a greater cubic Sharpe measure 1 or a und g in comparison with a und h exhibiting both a greater quadratic Sharpe measure as well as a smaller cubic Sharpe measure ). 14 Obviously, the perormance submeasures according to (T6) thus oer new opportunities or straightorward perormance assessments not at hand beore. Now consider the second limiting case described by (exg)* y =ε with ε > 0, but small. For such a situation, portolio Q() just converges to the sole holding o the riskless asset and we thus arrive at a situation with u Q (x ˆ ) 0 (i.e. ˆx u /u ). For this, we get the special per- ormance measure according to case d) o Table 3. In act, the limiting case ˆx u /u has also been analyzed in Breuer/Gürtler (006) as a possible border solution or the endogenous case with HARA utility and has also led to the derivation o some kind o cubic Treynor measure, because or the special case o mean-variance preerences this cubic measure collapses to the (negative inverse o the) quadratic Treynor measure. Actually, this cubic Treynor measure o Breuer/Gürtler (006) is a special case o the perormance measure (T8) o this paper. 15 We thus once again have been able to generalize our indings. IV. The endogenous case In the endogenous case, or any given und the investor optimizes all three relative portions x 0, x, and x, simultaneously. Let thereore ( )* x stand or the optimal investment in reerence portolio when combining this portolio with the riskless asset and und, and deine optimal ractions ( )* x 0 and ( )* x, analogously. Then, in the endogenous case, each und will be evaluated according to the perormance measure (end) cim o Table 3. This ollows immediately rom (T1) o Table 3 i one replaces ˆx with ( )* x. Moreover, in the case o pure mean-variance preerences (ω ), the best und according to the optimized quadratic Sharpe measure as discussed, or example, in Breuer/Gürtler (1999) is always also the best one as well according to (T10) o Table 3 or arbitrary desired overall expected excess return u. 16 It should be noted that such a relationship between (T10) and the 14 See Appendix 6 or a proo o this statement. 15 See Appendix 7 or a proo o this statement. 16 See Appendix 8 or a proo o this statement. 16

20 optimized cubic perormance measure o Breuer/Gürtler (006) does not generally exist, as the latter perormance measure is only based on cubic HARA utility. With the indings o Sections III and IV, we now turn to the empirical investigation o the relevance o skewness preerences or und rankings and the importance o the distinction between the endogenous case and the exogenous one. V. Empirical example In order to ensure comparability with the results o Breuer/Gürtler (005) we ollow their steps o analysis by considering (post tax) return data or 45 mutual unds investing in German equity shares 17 over a period rom July 1996 to August 1999 which are calculated on the basis o the development o the respective monthly repurchase prices per share. We assume that all earnings paid out to the investors by a und are reinvested in this und. The riskless interest rate r 0 can be approximated by the expected return o German time deposit running or one month and covering the respective period o time to be observed. We use the DAX 100 as a broadly diversiied reerence portolio. Based on this historical return data, or all 45 unds and the DAX 100 unbiased estimators or the relevant moments o one-monthly returns are calculated and listed in Table >>> Insert Table 4 about here <<< 1. Dierences in rankings in the exogenous case and the endogenous one From the analysis in Section III we know that, in the case o short sales restrictions with all skewnesses and co-skewnesses o und returns being negative, a und g with a higher quadratic Sharpe measure and a higher quadratic Treynor measure as well as a lower cubic Sharpe measure o type and higher cubic Treynor measures than a und h simultaneously exhibits a greater restricted optimized cubic perormance measure and a greater cubic investor speciic perormance measure (exg) IM (Result T1 o Table 3). While or the case o simple meanvariance preerences, i.e. with the neglection o all cubic submeasures, in Breuer/Gürtler (005) it has been possible to identiy 8 o our 45 unds or which the ranking according to their quadratic Sharpe measure and their quadratic Treynor measure, respectively, was identical, a similarly strong result or mean-variance-skewness preerences cannot be obtained. 17 In what ollows we briely speak o German unds, though we do not mean their country o origin, but the geographical ocus o their investments. 18 See Rohatgi (1976) or the unbiased estimators o the expectation value and the second central moment. Unbiased estimators o other moments can be worked out in just the same manner. 17

21 These 8 unds are listed irst in Table 4, but only unds # 1 to # 17 can be unambiguously ranked in a situation with skewness preerences as well. The number o each o these irst 17 unds in the irst column coincides with their ranking position when compared to each other, while the last column o Table 4 is relevant or the irst 8 unds and gives their unambiguous corresponding ranking place or mean-variance preerences. When taking into account simultaneously all 45 unds, there is no unambiguous und ranking, i.e. a comparison o the last 17 unds (# 9 to # 45) with unds # 1 to # 8 depends on the speciic parameter constellation (ω, u, ˆx ) under consideration. According to this result, the recognition o skewness preerences may lead to a greater variety in und ranking depending on the given raction ˆx o the reerence portolio and the desired overall expected excess return u. In order to better assess resulting dierences in rankings we ollow Breuer/Gürtler (005) by calculating Spearman ranking correlation coeicients ρ S between rankings according to the exogenous cubic investor-speciic perormance measure (in what ollows: exogenous cubic IM-rankings ) or given identical desired overall expected excess returns (1) () u = u = u with u U := { %, 1.9 %,.0 %,,.7 %, 10 % 19 } and dierent values (1) x and () x with (1) x, () x X = {0, 5 %,, 95 %, %}. To assure comparability o our results with those o Breuer/Gürtler (005), we thereby restrict ourselves in the same way as Breuer/Gürtler (005) to the analysis o the unds # 9 to # 45 o Table 4. Moreover, we must allow or dierent intensities o skewness preerences. Thereby, because o space constraints we only consider the two extreme cases ω = 0 (mere mean-skewness preerences, i.e. an ininite prudence) and ω = 100,000 (mere mean-variance preerences, i.e. a zero prudence). For any given expected excess return u U we compute 1 dierent und rankings, as this is the number o exogenous values (1) x and () x taken into account. This leads to an overall sum o 10 = 40 dierent und rankings or all ten desired expected overall excess returns u under consideration with 10 o them (or ω = 100,000; i.e., a situation with mere meanvariance preerences) already calculated by Breuer/Gürtler (005). 19 We add u = 10 % as an extreme value in order to better assess the stability o our results. Since we rerain rom considering situations with short sales o stocks or unds, the minimum accessible value or u amounts to % because u (x ) = u x u > 0 (and thus x > 0) is only ulilled or all 0 x 1 i u > u %. Q 18

22 As has already been pointed out by Breuer/Gürtler (005) or the case o mean-variance preerences and given desired expected overall excess return u, resulting correlation coeicients between two und rankings do not change much, i diering pairs (x, x ) o exogenous (1) () direct stock holdings are considered, as long as we have a constant value or x : = x x. In act, this inding carries over to situations with (even extreme) skewness (1) () preerences. For example, or the special case o a desired expected excess return u =.3 % with ω = 0, varying values x (1) and x () with x x = 10 % imply ranking correlation (1) () coeicients rom % (or the respective two cubic IM-rankings in the case o 50 % and () x = 40 % as well as in the case o or the respective two cubic IM-rankings in the case o in the case o (1) x 100 % and (1) x = 55 % and (1) x = 0 % and (1) x = () x = 45 %) to 100 % (e.g., () x = 10 % as well as () x = 90 %) thus leading to a variation o ρ S o only percentage points. 0 Variations o ρ S or other expected excess returns u and given dierences x are o similar scale. Hence, as in Breuer/Gürtler (005), it suices to take a closer look at average correlation coeicients between exogenous quadratic or cubic IM-rankings, respectively, or dierent identical values o desired expected returns u U and varying dierences x X between exogenously given holdings o the reerence portolio. Once again, the results o Breuer/Gürtler (005) or the case o mean-variance preerences also hold true or situations with skewness preerences: We ind out that average correlations between two und rankings with given dierence return u x and given identical desired expected are rather high, even i we restrict ourselves to unds which cannot be unambiguously ranked according to the quadratic and cubic submeasures regardless o the intensity o skewness preerences. In act, or u U and x X, smallest average values o ρ S amount to % or ω = 100,000 ( u = % and x 100 %) and % or ω = 0 ( u = % and x = 95 % or x 100 %). 1 Moreover, average ranking correlation coeicients are slightly decreasing with alling value or u. Finally, average ranking correlation coeicients are smallest or high dierences x which is intuitively appealing. Since x = 1 ( ε ) implies either (x = 1 ε, x = 0) or (1) () (x = 0, x = 1 ε ) as (1) () 0 See also Table Ad 1. 1 See also Tables Ad a and Ad b. 19