Single period Markowitz portfolio selection, performance gauging and duality : a variation on Luenberger's shortage function

Documents de treball Single Period Markowitz Portfolio Selection, Performance Gauging and Duality: A variation on Luenberger’s Shortage Function Walter Briec Kristiaan Kerstens Jean Baptiste Lesourd Document de treball núm. 2002/3 Departament d'economia de l'empresa  Walter Briec, Kristiaan Kerstens, Jean Baptiste Lesourd Coordinador documents de treball: Pere Ortín Ángel http://selene.uab.es/dep-economia-empresa/document.htm e-mail: [email protected] Telèfon: 93 581451 Edita: Departament d'economia de l'empresa http://selene.uab.es/dep-economia-empresa/ Universitat Autònoma de Barcelona Facultat de Ciències Econòmiques i Empresarials Edifici B 08193-Bellaterra (Barcelona), Spain Tel. 93 5811209 Fax 93 5812555 Abril, 2002 Single Period Markowitz Portfolio Selection, Performance Gauging and Duality: A variation on Luenberger’s Shortage Function Walter Briec Kristiaan Kerstens Jean Baptiste Lesourd Document de treball núm. 2002/3 La serie documents de treball d'economia de l'empresa presenta los avances y resultados de investigaciones en curso que han sido presentadas y discutidas en este departamento, no obstante las opiniones son responsabilidad de los autores. El documento no debe ser reproducido total ni parcialmente sin el consentimiento del autor o autores. Dirigir los comentarios y sugerencias directamente al autor, cuya dirección aparece en la página siguiente. A Working Paper in the documents de treball d'economia de l'empresa series is intended as a mean whereby a faculty researcher's thoughts and findings may be communicated to interested readers for their comments. Nevertheless, the ideas put forwards are responsibility of the author. Accordingly a Working Paper should not be quoted nor the data referred to without the written consent of the author. Please, direct your comments and suggestions to the author, which address shows up in the next page. Single Period Markowitz Portfolio Selection, Performance Gauging and Duality: A Variation on Luenberger’s Shortage Function Walter Briec Kristiaan Kerstens Jean Baptiste Lesourd* Abstract: Markowitz portfolio theory (1952) has induced research into the efficiency of portfolio management. This paper studies existing nonparametric efficiency measurement approaches for single period portfolio selection from a theoretical perspective and generalises currently used efficiency measures into the full mean-variance space. Therefore, we introduce the efficiency improvement possibility function (a variation on the shortage function), study its axiomatic properties in the context of Markowitz efficient frontier, and establish a link to the indirect mean-variance utility function. This framework allows distinguishing between portfolio efficiency and allocative efficiency. Furthermore, it permits retrieving information about the revealed risk aversion of investors. The efficiency improvement possibility function thus provides a more general framework for gauging the efficiency of portfolio management using nonparametric frontier envelopment methods based on quadratic optimisation. Keywords: shortage function, efficient frontier, risk aversion, mean-variance portfolios. December 2001 We are grateful to comments made by E. Clark and A. Coen. The usual disclaimer applies. * JEREM, Université de Perpignan, 52 Avenue Villeneuve, F-66000 Perpignan, France. Tel: ++/33/(0)468662264; Fax: ++/33/(0)468662201; [email protected]. CNRS-LABORES (URA 362), Université Catholique de Lille, B.P. 109, 60 Boulevard Vauban, F-59016 Lille Cédex, France. Tel: ++/33/(0)320134080; Fax: ++/33/(0)320134070; kris[email protected].asso.fr. Corresponding author. GREQAM, Centre de la Vieille Charité, 2 Rue de la Charité, F-13002 Marseille, France. Tel: ++/33/(0)491140748; Fax: ++/33/(0)491900227; le[email protected]. 1. Introduction Markowitz portfolio theory (1952), based on the idea of a trade-off between portfolio risk (as measured by its variance) and portfolio expected return, is generally considered as a cornerstone of modern portfolio theory. This approach is essentially based on the efficient frontier concept, defined as the Pareto-optimal subset of portfolios, i.e., a set of portfolios such that their expected returns may not increase unless their variances increase. In addition to its strong maintained assumptions on probability distributions and on Von Neumann-Morgenstern utility functions, the main problem with Markowitz model at the time was its computational cost.1 Though Farrar (1962) was seemingly the first to empirically test the full-covariance Markowitz model, computing costs motivated Sharpe (1963) to formulate a simplification known as the “diagonal model”. Later, Sharpe (1964) and Lintner (1965) introduced a capital asset pricing model (CAPM), an equilibrium model assuming that all agents have similar expectations about the market. Under these circumstances, it is not necessary to compute the efficient frontier.2 Tools for gauging the efficiency of portfolios, such as the Sharpe (1966) and Treynor (1965) ratios and the Jensen (1968) alpha, have mainly been developed with reference to the above developments (in particular CAPM).3 Despite these developments, the static Markowitz model remains the more general framework. Our contribution focuses on integrating an efficiency measure in this single period Markowitz model and to develop a dual framework for assessing the degree of satisfaction of investors’ preferences, starting from –seemingly forgottenideas in Farrar (1962). This leads to decomposing portfolio performance into allocative and portfolio efficiency components. In addition, duality allows revealing information about investors’ risk aversion. This is an issue of great practical significance that, to the best of our knowledge, is novel. There are both theoretical and practical motivations guiding these developments. Theoretically, this contribution brings portfolio theory in line with developments in mainly 1 A problem largely alleviated by today’s computing power. 2 Surveys on the history of these developments are, e.g., Constantinides and Malliaris (1995) and Philippatos (1979). 3 General surveys on tools for measuring the performance of managed portfolios are Grinblatt and Titman (1995) or Shukla and Trzcinka (1992). 1 production theory, where distance functions have proven useful tools to derive efficiency measures and to develop dual relations with economic (e.g., profit) support functions (Chambers, Chung and Färe (1998)). From a practical viewpoint, one can list the following advantages. First, the integration of efficiency measures into portfolio theory responds to the needs for rating tools. Second, instead of tracing the whole efficient portfolio frontier using a critical line search method, each asset or fund is projected onto the relevant part of the frontier according to a meaningful efficiency measure. This may lead to computational gains, depending on the number of assets or funds to evaluate and the aimed fineness of the portfolio frontier representation. Third, the possibility of measuring portfolio performance using a dual approach not only allows to gauge assets or funds using given information about risk aversion, but it also allows to reveal the (shadow) risk aversion minimising portfolio inefficiency. Therefore, our contribution enriches the empirical toolbox of practitioners. In particular, we introduce a variation of the shortage function, a distance function introduced in production theory by Luenberger (1995) that is dual to the profit function. This function accomplishes four goals: (i) it gauges the performance of portfolios by measuring a distance between a portfolio and an optimal portfolio projection on the Markowitz efficient frontier; (ii) it leads to a nonparametric estimation of this efficient frontier; (iii) it judges simultaneously mean return expansions and risk contractions –in fact, performance can be gauged in any directionand thereby generalises existing approaches; and (iv) it provides a new, dual interpretation of our portfolio efficiency distance. Given the investment context, our efficiency measure is called the Efficiency Improvement Possibility (EIP) function. To develop the fourth point somewhat, the paper establishes a link between the EIP function and mean-variance utility functions, thereby offering an integrated framework for assessing portfolio efficiency from the dual standpoint. To each efficient portfolio corresponds a particular utility function, whose optimal value is the indirect utility function. This approach provides a dual interpretation of the EIP function through the structure of risk preferences. Technically, this result is easily derived from Luenberger (1992, 1996). Along this line, we also establish a link to some kind of “Slutsky matrix”, defined as a matrix of 2 derivatives with respect to risk aversion (based on the structure of the mean-variance utility function). To situate our contribution more precisely, it is possible to distinguish between several approaches to test for portfolio efficiency. It is common to develop statistical tests based on certain parametric distributional assumptions (e.g., Jobson and Korkie (1989), Gouriéroux and Jouneau (1999), Philippatos (1979a)). However, right from the start (Markowitz (1952)) there has also been attention to simple nonparametric approaches to test for portfolio efficiency. Our contribution is situated within the latter tradition. Our work can best be contrasted with Varian (1983) and some later developments in the nonparametric test tradition using economic restrictions (Matzkin (1994)). Varian (1983) developed nonparametric nonstatistical tests checking whether the observed investment behaviour is consistent with the expected utility and the mean variance models. However, his formulation only allows to infer whether certain data are either consistent or not with the tested hypothesis. This regularity test lacks an indication about the degree of goodness of fit between data and models. In this respect, it is similar to the early nonparametric test literature in production (Diewert and Parkan (1983)) and consumption (Varian (1982)). Sengupta (1989) is probably the first to link the Varian (1983) portfolio test approach to the nonparametric efficiency literature by explicitly introducing an efficiency measure. Färe and Grosskopf (1995) establish a link between the above literature on regularity tests in general and the growing number of efficiency contributions employing distance functions (or their inverses, efficiency measures) as an explicit (nonstatistical) indicator of goodness of fit. Recently, Morey and Morey (1999) presented a nonparametric, quadratic programming approach to measure investment fund performance focusing on radial potentials for either risk contraction or mean return expansion. By contrast, our approach gauges portfolio performance simultaneously in terms of risk contraction and mean return augmentation. Among the obvious advantages of a nonparametric approach to production, consumption and investment one can mention: (i) it avoids the necessity to postulate specific functional forms, (ii) it is related to “revealed preference” conditions of some sort that are finite in nature and that are directly tested on any finite amount of observations, (iii) it leads 3 to the determination of inner and outer approximations of choice sets that contain the true but unknown frontier of the set, (iv) these approximations are based on (most often piecewise linear) functions that are directly spanned by the observations in the sample at hand, (v) the computational cost is rather low (frequently limited to solving mathematical programming problems), etc. (see, e.g., Matzkin (1994), Morey and Morey (1999), Varian (1983)). Aside from the investment setting, the problem of estimating monotone concave boundaries has recently been extensively studied and widely applied in production. Following the seminal article of Farell (1957), nonparametric efficiency methods estimate an inner bound approximation of the true, unknown production frontier using piecewise linear envelopments of the data, instead of traditional parametric, econometric estimation methods that suffer from the risk of specification error.4 Our contribution can then be interpreted as an envelopment method for estimating an inner bound of the true but unknown Markowitz efficient frontier. While most nonparametric efficiency methods in production rely on linear programming, the portfolio context requires quadratic optimisation.5 The paper is organised as follows. Section 2 lays down the groundwork for our analysis. Section 3 introduces the EIP function and studies its axiomatic properties. Section 4 studies the link between the EIP function and the direct and indirect mean-variance utility functions. Section 5 presents mathematical programs to compute the EIP. A simple empirical illustration using a small sample of 26 investment funds is provided in Section 6. Conclusions and possible extensions are formulated in a final section. 2. Efficient Frontier and Portfolio Management In developing our basic definitions, we consider n financial assets. Assets are characterised by an expected return ( ) i RE for i = 1...n, and, since returns of assets are correlated, by a covariance matrix ( ) ji RRCov , ji,=Ω for { } nji ,...,1, ∈ . A portfolio x is composed by a 4 Briec and Lesourd (2000) study mutual fund performance employing stochastic parametric frontiers. This study extends their work in an effort to avoid specification errors. 5 Nonparametric efficiency methods in production are known in operations research as Data Envelopment Analysis models. By analogy, our method could therefore be termed “Portfolio Envelopment Analysis” (PEA). 4 efficient frontier maximising the decision maker’s direct mean-variance utility function. Furthermore, Farrar (1962) suggested to trace the set of efficient portfolios by solving this dual problem for different sets of parameters (ρ,µ). Note that more elaborate dual frameworks exist in the literature. For instance, Varian (1983) describes nonparametric test procedures verifying whether a suitable mean-variance utility function rationalises observed portfolio choices and asset prices. Our contribution adheres to the before mentioned tradition and does not depend on asset price information. To apprehend duality in our framework, it is useful to distinguish between overall, allocative and portfolio efficiency when evaluating the scope for improvements in portfolio management. The following definition clearly distinguishes between these concepts. Definition 4.2. Let be the EIP function defined on g S ℑ . We call: 1) Overall Efficiency (OE) index, the quantity: () ( )()() ( ) ( ) ( )({}) µ ρ δ ρ δ µ δ µ ρ ,*UgxRVgxRE;sup,OE VE ≤ − − + = 2) Allocative Efficiency(AE) index, the quantity: () ( ) ( ) xS,OE,,xAE g − = µ ρ µ ρ 3) Portfolio Efficiency (PE) index, the quantity: ( ) ( ) xSxPE g = This definition immediately implies: () ( ) () ( ) EV , gg xU,*U ,OE µρ µ ρ µρ µρ + − = (11) Thus, OE is simply the ratio between (i) the difference between (maximum) indirect mean-variance utility (Definition 4.1) and the value of the direct mean-variance utility function for the observation evaluated and (ii) the normalised value of the direction vector for given parameters (ρ,µ). ( EV ggg ,−= ) Expanding on the decomposition introduced in Definition 4.2, Portfolio Efficiency only guarantees reaching a point on the portfolio frontier, not necessarily a point on the frontier maximising the investor’s indirect mean-variance utility function. In this sense, it is similar to the notion of technical efficiency in production theory. Allocative Efficiency, by 11 contrast, measures the needed portfolio reallocation, along the portfolio frontier, to achieve the maximum of the indirect mean-variance utility function. This requires adjusting an eventual Portfolio Efficient portfolio in function of relative prices, i.e., the parameters of the mean-variance utility function. Overall Efficiency ensures that both these ideals are achieved simultaneously. Obviously, the following additive decomposition identity holds: () ( ) ( ) xPExAEOE + = µ ρ µ ρ ,,, (12) Notice that changes in the risk-aversion parameters (ρ,µ) alter the slope of the indirect utility function. While the amount of PE is invariant to these changes, the relative importance of AE and OE normally changes. In Figure 2, this decomp d for a portfolio denoted by point A. For simplicity, assume that osition is illustrate () 1=−= EV g,gg , where . , is the usual Euclidean metric. In terms of this figure, it is easy to see that AC −=OE , while AB −=PE and BC −=AE . The indirect mean-variance utility function turns out to be a useful tool to characterise the representation set ℜ . In particular, by using duality one can state the following property. Proposition 4.1. The representation set ℜ admits the following dual characterisation: () ( ) { } 22 =+ ∩≤−∈ℜ R,*UVE;RE,V µρρµ Proof. By definition, ( ) ( ){ 2 +++ ∩−×+ℵ=ℜ RRR ()()() } . However, if () , then 2 ,+ ∉R µρ () () ()() ( ) ( ) { } +∞=−×+ℵ∈ ++ RRxRE,xRV;xUsup , µρ () . Since for any variance-mean vector we have ( )() ++ − ×+ℵ RR∈E,V , we deduce that ( ) ,* VEU ρ µ µ ρ −≥ . Now, assume that () ℜ ∉E,V . From Proposition 3.1, ℜ is convex. From the separation theorem, there exists () such that 2 + ∈R, µρ ( ) µ ρ ρ µ ,*UVE > − . Consequently, U () E,* V ρ µ µ ρ − ≥ implies () and Proposition 4.1 follows. Q.E.D. ℜ∈E,V This proposition allows establishing a link between the shortage function and the indirect utility function in the next proposition. In particular, the duality result in Proposition 4.2 shows that the EIP function can be derived from the indirect mean-variance utility function, 12 and conversely. It is inspired by Luenberger (1995), who established duality between the expenditure function and the shortage function. Proposition 4.2. Let be the EIP function defined on g S ℑ . has the following properties: g S 1) () ( ) () () }0,0,1;,*{inf ,≥≥ = + −= ρ µ ρ µ µ ρ µρ VEg ggxUUxS U2) () () () };)({sup,* , ℑ ∈ −= xxSxU g µρ µ ρ Proof. The proof is a straightforward consequence of Luenberger (1995). Q.E.D. This result proves that the EIP function can be computed over the dual of the mean-variance space. The support function of the representation set is the indirect utility function U*. We are now interested in studying the properties of the EIP function that presume differentiability at the point where the function is evaluated. Therefore, we introduce the adjusted risk aversion function: ()() () () ( ) { } 0,0,1;,*minarg, ,≥≥ = + − = ρ µ ρ µ µ ρ µ ρ µρ VE ggxUUx (13) that implicitly characterises the agent’s risk aversion. It could also be labelled a shadow indirect mean-variance utility function, since it adopts a reverse approach by searching for the parameters (ρ,µ) defining a shadow risk aversion that renders the current portfolio optimal for the investor. This function is similar to the adjusted price function defined by Luenberger (1995) in consumer theory, whence our naming of the adjusted risk aversion function. Proposition 4.3. Let be the EIP function defined on g S ℑ . At the point where is differentiable, it has the following properties: g S 1) () ()() ( )() ()() RxIx x xU x xS xg Ω−= ∂ ∂ = ∂ ∂ ρµ µρ 2 , 2) () ()() ()() () x xRV xS CtexRE g ρ = ∂ ∂ = and ( ) ()() ()() () x xRE xS CtexRV g µ −= ∂ ∂ = where R denotes the vector of expected asset returns and I is a unit vector of appropriate dimensions. 13 Proof. 1) The proof is obtained by the standard envelope theorem. We have obviously the relationship () ()() () x xU x xS xg ∂ ∂ = ∂ ∂ µρ ,. Since ()() ( )() () RxRx x xU xΩ−= ∂ ∂ ρµ µρ 2 ,, we deduce the result. The proof for 2) is obtained in a similar way. Q.E.D. Result 1) shows that the variations of the shortage function with respect to x are identical to the variation of the indirect utility function, but calculated with respect to the adjusted risk aversion function. Moreover, it can be directly linked to the return of each asset and the covariance matrix. Furthermore, result 2) shows that the shortage function decreases when the expected return increases. As shown below, one can link the adjusted risk aversion function and some kind of Marshalian demand for each asset. First, let us introduce the matrix of derivatives: [] ji ji x x B , ,          ∂ ∂ ∂ ∂ = µ ρ (14) Moreover, given a risk aversion vector ( ) µ ρ ,, we define “Marshalian” demand for assets by: () ()() ( ) { } ℑ ∈ =xxUm x;maxarg, , µρ µ ρ (15) One can then define some kind of Slutsky matrix: [] () ( ) j,i j,i ,m , ,m S      ∂ ∂ ∂ ∂ = µ µρ ρ µρ (16) As shown in the next proposition, this Slutsky matrix can be linked to the matrix B. Proposition 4.4. Let be the EIP function defined on g S ℑ . At the point where is differentiable, it has the following properties: g S 1) () ()        ×         + − + =EV EV EV g,g gg I gg BS µ ρ µρ µρ 2 11 2) ()()        ×         + − + = µρ µρ µρ , g g gg I gg BS E V EV EV TT 2 11 3) () () EV E V Ev g,g g g gg IBB ×         + −= + 2 1 µρ 14 Proof. 1) Let us consider () ( ) EV gg , , µρ µ ρ µρ + =. We have the equalities: () 2 1E Vk n...k kg m x µ 1 V EV g g gg ρ ρ µρ + − + ρ ρ ρ ρ = ∂ ∂ ∂ ∂ = ∂ ∂∑ = ; () 2 1 1 EV E EV k n...k kgg g gg m x µρ µ µρµ µ µ µ + − + = ∂ ∂ ∂ ∂ = ∂ ∂∑ = ; () 2 1EV E k n...k kgg g m x µρ ρ ρ ρ µ ρ + −= ∂ ∂ ∂ ∂ = ∂ ∂∑ = ; and () 2 1EV Vk n...k kgg gm x µρ µ ρ ρ ρ µ + −= ∂ ∂ ∂ ∂ = ∂ ∂∑ = . Now, since             ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =∑∑ ∑∑ == == µ µ ρ µ ρ ρ ρ ρ k nk k k nk k k nk k k nk k m x m x m x m x BS ...1...1 ...1...1 we deduce the result. 2) is obtained by taking the transpose of 1). 3) follows by combining 1) and 2). Q.E.D. This proof can also be derived from Luenberger (1996). This result states that the Slutstky matrix, characterizing the “Marshalian” demand for each asset, is a type of skewed pseudo-inverse of the matrix B. 5. Computational Aspects of the EIP Function The representation set ℜ , defined by expression (6), can be used directly to compute the EIP function by recourse to standard quadratic optimisations methods. Assume a sample of m portfolios (or investment funds) . Now, consider a specific portfolio y m yyy ,...,, 21 k for whose performance needs to be gauged. The shortage function for this portfolio y { m,...,k 1∈ } k under evaluation is computed by solving the following quadratic program: ()() ()() ()() ()() nixx bAx xRVgyRV xREgyREts i ni i V k E k ...1,0,1 .. max ...1 =≥= ≤ ≥− ≤+ ∑ = δ δ δ (P1) From equations (2) and (3), program (P1) can be rewritten as follows: 15 ()() () ()() nixx bAx xxgyRV RExgyREts i ni i ji jijiV k ni iiE k ...1,0,1 .. max ...1 , , ...1 =≥= ≤ Ω≥− ≤+ ∑ ∑ ∑ = = δ δ δ (P2) Thus, one quadratic program is solved for each portfolio to assess its performance. To obtain the entire decomposition from Definition 4.2, one only needs to compute the additional quadratic program from Definition 4.1. Then, applying expression (11) and Definition 4.2 itself, the components OE and AE follow suit. All of the above programs can be seen as special cases of the following standard form: () () p kk jj T Rz rkzQ qjzLts zc ∈ =≤ == ...1, ...1,.. min β α (P3) where is a linear map for j = 1...q and Q is a positive semi-definite quadratic form for k = 1...r. In the case of program (P2), q=1, and r=n+3, the latter because there are n non-negativity constraints. Program (P3) is a standard quadratic optimisation problem (see Fiacco and McGormick (1968), Luenberger (1984)). j Lk A novel result of some practical significance is that the adjusted risk aversion function (13) can be derived from the Kuhn-Tucker multipliers in program (P2). This is shown in the next proposition. Proposition 5.1. Let such that program (P2) has a regular optimal solution. Let and be respectively the Kuhn-Tucker multipliers of the first two constraints in program (P2). If the EIP function is differentiable at point , then: { m,...,k 1∈ } 0≥ E λ 0≥ V λ ℑ∈ k y 1) We have: () ()() ()() ()() and yRV yS V yREyRE yy g k k λ = ∂ ∂ = = ( ) ()() ()() ()() E yRVyRV yy g k k yRE yS λ −= ∂ ∂ = = 2) The adjusted price function is identical to the Kuhn-Tucker multipliers: 16 ( ) ( ) ( ) EV k,y, λλµρ = Proof. 1) The proof is based on the sensitivity theorem (e.g., Luenberger (1984)). A solution of Program (P2) is immediately obtained solving the program: () ()() ()() nixx bAx yRVgxx yREgRExts i ni i k V ji jiji k E ni ii ...1,0,1 .. min ...1 , , ...1 =≤−= ≤ ≤+Ω −≤+− − ∑ ∑ ∑ = = δ δ δ (P4) Remark: all constraint functions on the left hand side in the two first inequalities are convex. Therefore, (P4) has the standard form described in Luenberger (1984). Now, let us consider the parametric program: () nixx bAx cgxx cgRExts i ni i VV ji jiji EE ni ii ...1,0,1 .. min ...1 , , ...1 =≥= ≤ ≤+Ω ≤+− − ∑ ∑ ∑ = = δ δ δ (P5) Since (P2) has a regular optimal solution, the bordered Hessian of (P4) at the optimum is non-singular. Consequently, the sensitivity theorem applies. Let ( ) EV ccx ,* be the optimal solution of the parametric program (P5). Let us denote ( ) ( ) EV c,c*x* δ − the corresponding optimal value function. By definition, the Kuhn-Tucker multipliers of programs (P2) and (P4) are identical. From the sensitivity theorem, we have: ()()() ()() and V yRVc V EV k V c c,c*x* λ δ −= ∂ −∂ = ( ) ( ) ( ) ()() E yREc E EV k E c c,c*x* λ δ −= ∂ −∂ −= We immediately deduce that () ()() ()() ()() k k yREyRE yy g yRV yS = = ∂ ∂ ( ) ( ) ( ) ()() V yRVc V EV k V c c,c*x* λ δ = ∂ −∂ −= = Moreover: 17 () ()() ()() ()() k k yRVyRV yy g yRE yS = = ∂ ∂ ()()() () ()() ( ) ( ) ( ) ()() E yREc E EV yREc E EV k E k E c c,c*x* c c,c*x* λ δδ −= ∂ −∂ = −∂ −−∂ −= −==− . This ends the proof. 2) This result is immediate from Proposition 4.3, 2). Q.E.D. It may seem that the interest of our approach based on quadratic programming concerns essentially the original Markowitz model with short sales excluded. Our models would then simply provide a novel approach to the efficiency of portfolio management in the case of regulated investment funds, such as mutual funds in the United States, unit trusts in the UK, or EU-regulated UCITS7, which under current regulation cannot invest in uncovered derivative instruments. Of course, if the possibility of short sales is not excluded or if there exists a riskless asset with zero variance and non-zero positive return, then the efficient frontier is straightforwardly determined by simpler, analytical solutions without recourse to quadratic optimisation (e.g., Elton, Gruber and Padberg (1979)). However, the quadratic programming approach remains valid in general. In particular, since quadratic program P3 can be derived from P4, it does not require a positive definite covariance matrix. Therefore, our models remain equally valid under these cases, with practical applications to measuring asset management efficiency for, e.g., regulated funds of futures and unregulated hedge funds. Computation of these quadratic programs provides an inner bound approximation of the true, unknown portfolio frontier. This envelopment frontier is akin to the production frontiers alluded to in the introduction. This estimator is a nonparametric method, inasmuch as no functional form is specified for the Pareto frontier. Figure 3 illustrates this logic behind the performance gauging of portfolios using program (P2). We evaluate the technically inefficient observations (V0,E0) to (V3,E3) and project them onto the portfolio frontier using the same direction vector g. By adding fictitious points or by implementing a critical line 7 UCITS is the acronym for “Undertakings for Collective Investment in Transferable Securities”, as regulated under a European Community directive of 1985 which, just as the American law of 1940 regulating mutual funds, precludes any form of leveraging, whether from borrowed funds or the use of uncovered derivatives. 18 search following Markowitz (1959), it is possible to refine the approximation of the efficient set of portfolios until it coincides with the Markowitz frontier. <FIGURE 3 ABOUT HERE> We end with two general remarks. One is concerned with the possibility of weakly efficient portfolios. The other observation focuses on the selection of a direction vector in all these mathematical programs. ( EV ggg ,−= ) First, the projection of (V0,E0) onto a vertical segment of the set of weakly efficient portfolios illustrates the scope for further removing inefficiencies until one reaches the global minimum variance portfolio. A pragmatic solution is to substitute the global minimum variance portfolio, that provides a better expected return for the same risk, for projection points representing weakly efficient portfolios (identifiable by positive slack variables). Theoretical solutions that could be developed require sharpening the definition of the efficient frontier, or formulating doubts about the choice of direction ( ) EV ggg , − = for weakly efficient portfolios (e.g., selecting a direction that guarantees at least a projection onto the global minimum variance portfolio). We deem such developments beyond the scope of this contribution. Furthermore, assuming one is interested in estimating the OE decomposition (OE implying strongly efficient portfolios), the problem of weakly efficient portfolios is limited to the PE component and only leads to a slight change in the relative importance of both components (AE versus PE). Second, some remarks on the choice of the direction vector may prove useful. In principle, various alternative directions are possible (e.g., Chambers, Chung and Färe (1998)). For instance, it is possible to choose a common direction for all portfolios, as illustrated in Figure 3 above. This has a clear economic meaning in consumer theory where, for instance, utility may be measured using a type of distance function with respect to a common basket of goods (see the benefit function in Luenberger (1992)). But the economic interpretation of a common direction g in production and investment theory is not evident to us. A far more straightforward choice for investment theory is to use the observation under evaluation itself (i.e., ( ) ( ) ( ) ( )( xRE,xRVg −= ) ). Then, the shortage function measures 19 the maximum percentage of risk reduction and expected return improvement. The dual formulation of the shortage function leads to a simpler interpretation: () ( ) () ( ) { } () () () ()() ()() {} () () () () () {} 001 001 001 ≥≥=−= ≥≥=+−−= ≥≥ = + −= ρµµρ ρµρµµρ ρ µ ρ µ µ ρ µρµρ µρ µρ ,,xU;xU,*Uinf ,,xRVxRE;xU,*Uinf ,,gg;xU,*UinfxS ,, , VE,g (17) Now, by a simple normalization scheme (see Chambers, Chung and Färe (1998)), we can equivalently write: () () () ( ) () ()          ≥≥ − =00 ','; xU xU','*U infxS ',' ',' g ρµ µρ µρ µρ (18) Thus, the shortage function is now interpreted as the minimum percentage improvement in the direction to reach the maximum of the utility function (i.e., the indirect utility function). Since we work in mean-variance space, the shadow risk-aversion minimising this percentage provides a general efficiency index. 6. Empirical Illustration: Investment Funds To show the ease of implementing the basic framework developed in this contribution, we compute the decomposition of overall efficiency for a small sample of 26 investment funds earlier analysed in Morey and Morey (1999). Return and risk are computed over a 3-year time horizon between July 1992 and June 1995 (see their Tables 1 and 8). Computing program (P2), the quadratic program in Definition 4.1 for parameters µ = 1 and ρ = 2, and applying the decomposition in Definition 4.2, we obtain the results summarised in Table 1. To save space, we do not report portfolio weights and slack variables. Risk aversion is based on conventional values for ρ that often range between 0.5 and 10 (e.g., Uysal, Trainer and Reis (2001)). To underline the ease of interpretation of our performance measure, we briefly comment on the decomposition results of a single fund: “44 Wall Street Equity”. It could improve its overall efficiency by 40%, both in terms of improving its return and reducing its risk. In terms of the decomposition, 22.5% of this rather poor performance is due to portfolio inefficiency, i.e., operating below the portfolio frontier, while 17% is due to allocative inefficiency, i.e., choosing a wrong mix of return and risk given the postulated risk attitudes. 20 Treynor, J.L. (1965) How to Rate Management of Investment Funds, Harvard Business Review, 43, 63-75. Uysal, E., F.H. Trainer, J. Reis (2001) Revisiting Mean-Variance Optimization: From a Scenario Analysis Perspective, Journal of Portfolio Management, 27, 71-81. Varian, H. (1982) The Nonparametric Approach to Demand Analysis, Econometrica, 50, 945-973. Varian, H. (1983) Nonparametric Tests of Models of Investment Behavior, Journal of Financial and Quantitative Analysis, 18, 269-278. 27 Figure 1: Finding the Optimal Portfolio ∂ M ℑ ( ) ℵ ℜ (V ~,R ~ ) µ E(R(x)) - ρ V(R(x)) V(R) E (R) (V(R(x)),E(R(x))) 0 Figure 2: Efficiency Improvement Possibility Function & Decomposition 0V(R) E (R) A = (V(R(x)),E(R(x))) ∂ M ℑ ( ) ℜ µ E(R(x)) - ρ V(R(x)) C B L egend: OE = ||C– A|| P E = ||B –A|| A E = ||C– B|| Assume: || g ||=1 (-gV,gE) Figure 3: Portfolio Efficiency Analysis: Projections onto the Nonparametric Frontier 0 (V1,E1) • • • • M 1 M 2 M3 M 0 ∂ M ℑ ( ) • • • • • ( V0,E0) (V2,E2) ( V3,E3) (V ~,R ~ ) L egend: True frontier N on-Parametric frontier ℜ V(R) E (R) (-gV,gE) 28 Figure 4: Portfolio Frontier: Observed Portfolios and Decomposition Results 0.000 0.500 1.000 1.500 2.000 2.500 3.000 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Variance Return Evaluated mutual funds Frontier projections (PE) Optimum (OE) Figure 5: Sensitivity of Portfolio Efficiency Decomposition Results for ρ 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 0.0112345678910 Rho (ρ) Efficiency OEF AEF PEF 0.0000 0.2000 0.4000 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 29 Table 1: Decomposition Results for Morey and Morey (1999) Sample Observations OE PE AE φ* 20th Century Ultra Investors 0.718 0.433 0.285 0.095 44 Wall Street Equity 0.398 0.225 0.172 0.166 AIM Aggressive Growth 0.606 0.000 0.606 0.072 AIM Constellation 0.627 0.274 0.353 0.097 Alliance Quasar A 0.616 0.550 0.066 0.205 Delaware Trend A 0.610 0.351 0.259 0.116 Evergreen Aggressive Grth A 0.742 0.538 0.204 0.108 Founders Special 0.589 0.439 0.150 0.152 Fund Manager Aggressive Grth 0.366 0.357 0.009 0.330 IDS Strategy Aggressive B 0.593 0.583 0.011 0.314 Invesco Dynamics 0.543 0.274 0.269 0.122 Keystone Amer Omega A 0.521 0.448 0.073 0.213 Keystone Small Co Grth (S-4) 0.722 0.331 0.391 0.079 Oppenheimer Target A 0.402 0.320 0.082 0.219 Pacific Horizon Aggr Growth 0.700 0.619 0.081 0.175 PIMCo Adv Opportunity C 0.742 0.304 0.438 0.000 Putnam Voyager A 0.541 0.323 0.218 0.135 Security Ultra A 0.559 0.503 0.057 0.225 Seligman Capital A 0.573 0.564 0.009 0.319 Smith Barney Aggr Growth A 0.726 0.485 0.241 0.102 State St. Research Capital C 0.643 0.245 0.399 0.089 SteinRoe Capital Opport 0.588 0.317 0.272 0.116 USAA Aggressive Growth 0.708 0.545 0.162 0.128 Value Line Leveraged Gr Inv 0.481 0.319 0.163 0.161 Value Line Spec Situations 0.687 0.517 0.170 0.129 Winthrop Focus Aggr Growth 0.026 0.014 0.011 0.332 Mean 0.578 0.380 0.198 0.162 Standard deviation 0.155 0.159 0.152 0.087 Maximum 0.742 0.619 0.606 0.332 * Absolute risk aversion derived from the adjusted risk aversion function. 30 Issues: 95/1 Productividad del trabajo, eficiencia e hipótesis de convergencia en la industria textilconfección europea Jordi López Sintas 95/2 El tamaño de la empresa y la remuneración de los máximos directivos Pedro Ortín Ángel 95/3 Multiple-Sourcing and Specific Investments Miguel A. García-Cestona 96/1 La estructura interna de puestos y salarios en la jerarquía empresarial Pedro Ortín Ángel 96/2 Efficient Privatization Under Incomplete Contracts Miguel A. García-Cestona Vicente Salas-Fumás 96/3 Institutional Imprinting, Global Cultural Models, and Patterns of Organizational Learning: Evidence from Firms in the Middle-Range Countries Mauro F. Guillén (The Wharton School, University of Pennsylvania) 96/4 The relationship between firm size and innovation activity: a double decision approach Ester Martínez-Ros (Universitat Autònoma de Barcelona) José M. Labeaga (UNED & Universitat Pompeu Fabra) 96/5 An Approach to Asset-Liability Risk Control Through Asset-Liability Securities Joan Montllor i Serrats María-Antonia Tarrazón Rodón 97/1 Protección de los administradores ante el mercado de capitales: evidencia empírica en España Rafael Crespí i Cladera 97/2 Determinants of Ownership Structure: A Panel Data Approach to the Spanish Case Rafael Crespí i Cladera 97/3 The Spanish Law of Suspension of Payments: An Economic Analysis From Empirical Evidence Esteban van Hemmen Almazor 98/1 Board Turnover and Firm Performance in Spanish Companies Carles Gispert i Pellicer 98/2 Libre competencia frente a regulación en la distribución de medicamentos: teoría y evidencia empírica para el caso español Eva Jansson 98/3 Firm’s Current Performance and Innovative Behavior Are the Main Determinants of Salaries in Small-Medium Enterprises Jordi López Sintas y Ester Martínez Ros 98/4 On The Determinants of Export Internalization: An Empirical Comparison Between Catalan and Spanish (Non-Catalan) Exporting Firms Alex Rialp i Criado 98/5 Modelo de previsión y análisis del equilibrio financiero en la empresa Antonio Amorós Mestres 99/1 Avaluació dinàmica de la productivitat dels hospitals i la seva descomposició en canvi tecnològic i canvi en eficiència tècnica Magda Solà 99/2 Block Transfers: Implications for the Governance of Spanish Corporations Rafael Crespí, and Carles Gispert 99/3 The Asymmetry of IBEX-35 Returns With TAR Models M.ª Dolores Márquez, César Villazón 99/4 Sources and Implications of Asymmetric Competition: An Empirical Study Pilar López Belbeze 99/5 El aprendizaje en los acuerdos de colaboración interempresarial Josep Rialp i Criado 00/1 The Cost of Ownership in the Governance of Interfirm Collaborations Josep Rialp i Criado, i Vicente Salas Fumás 00/2 Reasignación de recursos y resolución de contratos en el sistema concursal español Stefan van Hemmen Alamazor 00/3 A Dynamic Analysis of Intrafirm Diffusion: The ATMs Lucio Fuentelsaz, Jaime Gómez, Yolanda Polo 00/4 La Elección de los Socios: Razones para Cooperar con Centros de Investigación y con Proveedores y Clientes Cristina Bayona, Teresa García, Emilio Huerta 00/5 Inefficient Banks or Inefficient Assets? Emili Tortosa-Ausina 01/1 Collaboration Strategies and Technological Innovation: A Contractual Perspective of the Relationship Between Firms and Technological Centers Alex Rial, Josep Rialp, Lluís Santamaria 01/2 Modelo para la Identificación de Grupos Estratégicos Basado en el Análisis Envolvente de Datos: Aplicación al Sector Bancario Español Diego Prior, Jordi Surroca. 01/3 Seniority-Based Pay: Is It Used As a Motivation Device? Alberto Bayo-Moriones. 01/4 Calidad de Servicio en la Enseñanza Universitaria: Desarrollo y Validación de una Escala de Medida. Joan-Lluís Capelleras, José Mª Veciana. 01/5 Enfoque Estructural vs. Recursos y Capacidades: Un Estudio Empírico de los Factores Clave de Éxito de las Agencias de Viajes en España. Fabiola López-Marín, José M.ª Veciana. 01/6 Opción de Responsabilidad Limitada y Opción de Abandonar: Una Integración para el Análisis del Coste de Capital. Neus Orgaz. 01/7 Un Modelo de Predicción de la Insolvencia Empresarial Aplicado al Sector Textil y Confección de Barcelona (1994-1997). Antonio Somoza López 01/8 La Gestión del Conocimiento en Pequeñas Empresas de Tecnología de la Información: Una Investigación Exploratoria. Laura E. Zapata Cantú. 01/9 Marco Institucional Formal de Creación de Empresas en Catalunya: Oferta y Demanda de Servicios de Apoyo David Urbano y José María Veciana. 02/1 Access as a Motivational Device: Implications for Human Resource Management. Pablo Arocena, Mikel Villanueva. 02/2 Efficiency and Quality in Local Government. The Case of Spanish Local Authorities M.T. Balaguer, D. Prior, J.M. Vela 02/3 Single Period Markowitz Portfolio Selection, Performance Gauging and Duality: A variation on Luenberger’s Shortage Function Walter Briec, Kristiaan Kerstens, Jean Baptiste Lesourd.