Description
Data envelopment analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers[clarification needed]. It is used to empirically measure productive efficiency of decision making units (or DMUs). Non-parametric approaches have the benefit of not assuming a particular functional form/shape for the frontier, however they do not provide a general relationship (equation) relating output and input. There are also parametric approaches which are used for the estimation of production frontiers (see Lovell & Schmidt 1988 for an early survey).
On the Use of Data Envelopment Analysis in Hedge Fund Performance Appraisal
Huyen Nguyen-Thi-Thanh?† This Draft: December 2006
Abstract This paper aims to show that Data Envelopment Analysis (DEA) is an ef?cient tool to assist investors in multiple criteria decision-making tasks like assessing hedge fund performance. DEA has the merit of offering investors the possibility to consider simultaneously multiple evaluation criteria with direct control over the priority level paid to each criterion. By addressing main methodological issues regarding the use of DEA in evaluating hedge fund performance, this paper attempts to provide investors suf?cient guidelines for tailoring their own performance measure which re?ect successfully their own preferences. Although these guidelines are formulated in the hedge fund context, they can also be applied to other kinds of investment funds. JEL CLASSIFICATION: G2, G11, G15 KEYWORDS: hedge fund, mutual fund, data envelopment analysis, performance measures, Sharpe ratio.
d’Economie d’Orl´ eans (LEO), Universit´ e d’Orl´ eans, Rue de Blois, B.P. 6739, 45 067 Orl´ eans Cedex 2. E-mail: [email protected]. † I am grateful to Georges Gallais-Hamonno and Michel Picot for their valuable guidance and support. I thank all participants in the 2006 MMF Conference (York, the United-Kingdom), the 2006 International Conference of the French Finance Association (Poitiers, France) and LEO members for their helpful comments. I also thank the company Standard & Poor’s, especially Marina Ivanoff and Zahed Omar, for providing hedge fund data used in this study. Naturally, all errors remain mine alone.
? Laboratoire
1
Introduction
The highly successful performance of the so-called hedge funds over the past two decades, notably during the long bull equity market of the 1990s, has made them quickly wellknown to ?nancial communities as well as to the public. While hedge funds still manage only $1 trillion at the end of 2004, a fraction of the $8 trillion invested by mutual funds, their assets have ballooned from only about $150 billion a decade ago. With over 8,000 hedge funds now available, fund selecting is quite challenging for investors. Hence, before any due-diligence process, investors ?rst need an ef?cient tool to assist them in screening task in which the most important evaluation is undoubtedly fund (historical) performance. In general, the historical performance of funds is de?ned as their return adjusted for risk. According to traditional ?nancial theories, the risk is measured either by the standard deviation of returns or by the correlation of fund returns with market factors via different betas1 . Most of these measures, even though validated in ”buy-and-hold” portfolios of mutual funds and pension funds, are irrelevant within the context of hedge funds. On the one hand, hedge fund returns are documented as usually asymmetric and kurtotic, a feature largely imputed to the intensive use of short sales, leverage, derivative instruments and to the free call-option like incentive structure, all speci?c to only the hedge fund industry. On the other hand, their short-term movements across diverse asset categories and the market neutral absolute investment objective of hedge fund managers make it really delicate to identify market factors necessary to the use of multi-factorial models2 . Recent techniques enlarge the evaluation dimension to the skewness (Stutzer 2000), the skewness and the kurtosis (Gregoriou & Gueyie 2003) or to the whole distribution of returns (Keating & Shadwick 2002) in order to take into account the non normality of return distributions. Despite this signi?cant progress, these measures do not allow considering after-net-returns fees paid by investors if only. Besides, most of them are restrictive in the sense that they often assume very simplistic decision-making rules which are common to all investors. Yet, it is well documented that actual evaluation criteria, in fact, may be more complicated and differ signi?cantly from theoretical formulations. Not only are there many
fund’s risk is measured by betas, fund performance is simply the alpha. other kinds of investment funds, hedge funds are loosely regulated, and in many cases, are largely exempted from legal obligations as the case of offshore hedge funds. Hedge fund managers thus have a broad ?exibility in determining the proportion of securities they hold, the type of positions (long or short) they take and the leverage level they make. As a consequence, they are free to make very short-term movements across diverse asset categories involving frequent use of short sales, leverage and derivatives to attempt to time the market.
2 Unlike 1 When
2
attributes to consider, each one being associated with a priority level, but also these attributes and their importance level are usually quite speci?c to each investor. The need to consider simultaneously multiple criteria while incorporating investors’ own preferences is natural since they are do not always share the same ?nancial objective, risk aversion, investment horizon, etc. From such perspectives, the Data Envelopment Analysis approach (hereafter, DEA) seems particularly appealing as it provides the possibility of incorporating many criteria at the same time, together with a direct control over the importance level paid to each criterion by means of a tailor-made optimizing system. DEA can be roughly de?ned as a mathematical optimizing technique ?rst introduced by Charnes, Cooper & Rhodes (1978), based on Farrell (1957)’s ef?ciency concept, to measure the ef?ciency (technical, allocative, economic, etc.) of decision-making units (hereafter, DMU) whose objective consists in transforming multiple inputs into multiple outputs. The merits of the DEA method lies in providing an unique aggregate measure for each DMU from a system of multiple inputs and multiple outputs and in putting emphasis on the ”best observed practices” in a comparative perspective. In addition, DEA allows considering inputs and outputs whose measure units are different, a property known as ”units invariance”3 . Furthermore, it makes no assumption on the form of the relation between inputs and outputs. Because of its many advantages, DEA has been applied in various ?elds including public administration (to evaluate hospitals, administrative of?ces, educational establishments or to resolve siting problems), engineering (to evaluate airplanes and engines), commerce (to evaluate supermarkets), ?nance (to evaluate bank branches, micro-?nance institutions, assurance companies, to identify dominant ?nancial assets and recently to assess investment funds’ performance). The application of DEA is generally proceeded in two main perspectives: (1) to evaluate the ef?ciency of DMUs whose activities are to employ inputs to produce outputs; and (2) to solve decision-making problems with multiple criteria. It is in the second perspective that DEA can be applied to assess hedge fund performance. Initiated by Murthi et al. (1997) to evaluate empirically the performance of mutual funds, this idea has been applied and revisited by several studies, including those on hedge fund performance. However, this literature is composed essentially of empirical applications, methodological issues remain either ignored or discussed in a simplistic and super?cial manner with little directive value. To the best of my knowledge, none of methodological studies investigates the use of DEA in the hedge fund context.
3 This is true provided that unit measures are the same for all DMUs in the sample. For example, one person can measure outputs in mile and inputs in gallons of gasoline and quarts of oil while another measures these same outputs and inputs in kilometers and liters with the same collection of automobiles .
3
Following this literature, this paper is devoted to methodological issues in applying DEA to hedge fund performance appraisal. Speci?cally, I focus on the choice of evaluation criteria (DEA’s inputs and outputs), the choice of DEA models with and without negative data on returns and performance, and on ”transcribing” speci?c evaluating preferences of investors into mathematical constraints. By doing so, this study attempts to offer investors suf?cient guidelines in order to apply successfully the DEA method to assessing hedge fund performance. Although it only addresses the hedge fund context, the whole framework is completely applicable to mutual funds, pension funds, ethical funds, etc. The remainder of the paper is organized as follows. Section 1 reviews brie?y the literature related to this study. Section 2 introduces basic concepts of the DEA method. Section 3 addresses methodological issues of applying DEA to screening hedge funds via their performance. Section 4 provides several numerical illustrations on a sample including 38 hedge funds. The last section summaries and concludes the paper.
1 Related literature
This study emanates from two main streams of literature. The ?rst one concerns DEA’s use in making a selection when decision-makers have multiple criteria. The second evolves evaluating the performance of investment funds by means of the DEA method. With respect to the ?rst literature, three studies can be enumerated: Thompson et al. (1986), Tone (1999) and Powers & McMullen (2000). Thompson et al. (1986) dealt with identifying feasible sites among six candidate sites for location of a very high-energy physics lab in Texas. A comparative analysis between six sites was conducted by applying the basic DEA model, incorporating project cost, user time delay, and environmental impact data as selection criteria. These criteria are those evaluators want to minimize, they thus form exclusively the DEA’s inputs. Being absent, the output is assumed to be unique and equal to unity so that DEA can be applied. This setting is naturally plausible as it is equivalent to considering inputs per one unity of output4 . In the same spirit, Tone (1999) described a japanese governmental project applying DEA to select a city to take over some political functions of Tokyo as a new capital. In this study, the selection criteria are composed of distance from Tokyo, safety indexes (regarding earthquakes and volcanoes), access to an international airport, ease of land acquisition, landscape, water supply, matters with historical associations; they form exclusively DEA’s outputs. The
4 Inputs
(outputs) include all criteria that evaluators want to minimize (maximize).
4
input is thus set to be equal to unity5 . It is important to note that in these studies, only inputs (outputs) are available and thus output (input) is assumed to be unique and equal to 1. Another common interesting point is that the evaluators, with prior expert knowledge about the relative importance of chosen criteria, ?xe lower and upper bounds to the weights associated with each criterion in the mathematical optimization. In ?nance, Powers & McMullen (2000) suggested using DEA to select dominant stocks among the 185 american largest capitalization stocks because this technique makes it possible to incorporate multiple selection attributes such as the Price-Earnings Ratio, the systematic risk and the total risk (DEA’s inputs), the Earnings Per Share ratio and the mean return over 1 year, 3 years, 5 years and 10 years (DEA’s outputs). The second literature relates to studies using DEA to evaluate the performance of mutual funds, ethical funds and more recently hedge funds. Studies on mutual funds include Murthi et al. (1997), McMullen & Strong (1998), Choi & Murthi (2001), Basso & Funari (2001), Tarim & Karan (2001) and Sengupta (2003). All these studies assume that fund performance is a combination of multiple attributes such as mean returns (DEA’s outputs), total or systematic risk, expenses6 , and sometimes even fund size, turnover speed and minimum initial investment (DEA’s inputs). In the same vein, Basso & Funari (2003) suggested putting in the DEA’s outputs, together with the mean return, an indicator measuring funds’ ethical level ful?llments since according to them, ”the solidarity and social responsibility features that characterize the ethical funds satisfy the ful?llment of humanitarian aims, but may lower the investment pro?tability”. The application of DEA in evaluating hedge funds emerged from the work of Gregoriou (2003). It was then supported by Gregoriou et al. (2005)7 and discussed in Kooli et al. (2005). A common feature of these studies is that they only consider risk–return performance without referring to fees. Besides, risks and returns are approximated respectively by lower variations (what investors seek to minimize) and upper variations (what investors seek to maximize) compared to a threshold de?ned by mean return. Speci?cally, the inputs are composed of lower mean monthly semi-skewness, lower mean monthly semi-variance and mean monthly lower return; the outputs include upper mean monthly semi-skewness, upper mean monthly semi-variance and mean monthly upper return. Another common feature is that they put emphasis on fund’s absolute rankings by
did not have access to documents related to this project. All the information mentioned here is extracted from Cooper et al. (2000, p.169). 6 The concept of expenses differs from study to study. It might include transaction costs and administration fees (totaled in expense ratio) and loads (subscription or/and redemption costs). 7 Gregoriou et al. (2005) is an extended version of Gregoriou (2003) and more complete while employing the same DEA methodology with Gregoriou (2003). Therefore, I refer only to Gregoriou et al. (2005).
5I
5
employing modi?ed DEA techniques: super–ef?ciency (Andersen & Petersen 1993) and cross–ef?ciency (Sexton et al. 1986). By comparing DEA results with rankings provided by Sharpe and modi?ed Sharpe ratios via rank correlation coef?cients, they observed a weak consistency between DEA and these measures. In particular, Kooli et al. (2005) found quite low correlation between DEA rankings and rankings given by the stochastic dominance technique and concluded to a weak relevancy of DEA to fund performance evaluation context. With regard to super-ef?ciency and cross-ef?ciency models, despite their appealing property, i.e. providing fund absolute rankings, their technical caveats cast doubts about their ef?cacy8 . Hence, in what follows, I will only introduce the basic DEA model and its dichotomic classi?cation into assessing hedge fund performance.
2 DEA’s approach
2.1 DEA as a measure of technical ef?ciency
Before introducing the general approach of DEA and the basic DEA model, it is important to distinguish the ”technical ef?ciency”, on which is based this study, from the ”economic ef?ciency” usually applied in production context. According to Fried, Lovell & Schmidt (1993, p.9-10), ”productive ef?ciency has two components. The purely technical, or physical, component refers to the ability to avoid waste by producing as much output as input usage allows, or by using as little input as output production allows. . . . The allocative, or price, or economic, component refers to the ability to combine inputs and outputs in optimal proportions in light of prevailing prices.”. Consequently, technical ef?ciency measurement is based solely on quantity information on the inputs and the outputs whereas the economic ef?ciency necessitates the recourse to information on prices as well as on economic behavioral objectives of producers (cost minimization, pro?t maximization or revenue maximization). Conceptually, the ef?ciency of each DMU under evaluation is determined by the distance from the point representing this DMU to the ef?cient frontier (production frontier in the case of technical ef?ciency; cost, revenue or pro?t frontier in the case of cost, revenue or pro?t ef?ciency respectively). In ?gure 1, the isoquant L
represents various combinations of two inputs that a perfectly ef?cient
8 The super-ef?ciency model has two main caveats. First, it allocates so excessively high ef?ciency score to ef?cient DMUs having extreme values of inputs and outputs that optimal values can sometimes ”explode”. Second, it is infeasible in some circumstances (Zhu 1996, Seiford & Zhu 1999). The pitfall of the crossef?ciency model is that it penalizes DMUs whose the combination of inputs and outputs is different from the others while it highly praises average DMUs. Besides, the use of the mean, the variance, the mode or the median, etc. of scores to completely rank DMUs is too ambiguous, especially when different indicators provide different rankings.
6
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¤
¨¦ ¡¢
Figure 1: Technical ef?ciency versus economic ef?ciency with two inputs (Farrell, 1957)
?rm like Q or Q? might use to produce an unit of output. The line CC ? whose slope is equal to the ratio of the prices of the two inputs represents the price constraint that all the ?rms must face. Farrell (1957) de?ned OQ/OP as the technical ef?ciency level, OR/OQ as the price (cost) ef?ciency and OR/OP as the overall ef?ciency of the ?rm P. In DEA, the production frontier against which the technical ef?ciency of each DMU is derived is empirically constructed from observed DMUs, and thus without any assumption on the functional relation between inputs and outputs9 . In other words, it is formed by a set of best practices (the most ef?cient DMUs) and the other DMUs are enveloped by this frontier, which explains the origin of the name ”Data Envelopment Analysis” of this method. For the shake of brevity, hereafter I will use the term ”ef?ciency” to refer to the technical ef?ciency and the term ”ef?ciency frontier” to denote the production frontier.
2.2
DEA’s basic model — CCR (1978)
2.2.1 The general formulation Consider n DMUs under evaluation that use m inputs (X ) to produce s outputs (Y ) with X and Y are semipositive10 . The ef?ciency score hk assigned to the DMU k is the solution
9 In econometric methods, the ef?cient frontier is estimated by supposing a particular form of the production function (e.g., Cobb-Douglas, translog, etc.). 10 The semipositivity signi?es that all data are nonnegative but at least one component of every input and output vector is positive.
7
of the following optimizing system: ? ur yrk (1) ? vi xik
s
max
u,v
hk =
s
r =1 m
i =1
subject to:
r =1 m
? ur yrj ? vi xij
? 1, j = 1, . . . , n
(2) (3)
i =1
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m
where k is the DMU under evaluation, yrj is the amount of the output r of the DMU j , xij is the amount of the input i of the DMU j , ur and vi (also called ”absolute weights”) are the weights assigned respectively to the output r and the input i, ? is an in?nitesimal positive number imposed to assure that no input or output is ignored in the optimization, vi xij and ur yrj are called ”virtual weights” of respectively the input i and the output r of the DMU j . Mathematically, the model’s objective is to seek for the most favorable (positive) weight system associated with each input and each output which maximizes the weighted sum of the outputs over the weighted sum of the inputs of the DMUk (hk ), provided that this ratio does not exceed 1 for any DMU in the sample (re?ected by constraint (2)). Given that the ef?ciency frontier contains ef?cient DMUs and envelopes inef?cient ones, and that the ef?ciency level of each DMU is, by de?nition, the distance from its position to the ef?ciency frontier, it is natural to ?xe the maximal value of the objective function to unity11 . Thus ef?cient DMUs will obtain a score of 1 and inef?cient DMUs a score smaller than 1. Conceptually, each DMU is free to choose its own combination of inputs and outputs so that it is as desirable as possible compared to other DMUs in the same category. Obviously, this combination must also be technically ”feasible” for others, that is the ef?ciency level of any other DMU using this combination should not exceed the maximum attainable bounded by the ef?ciency curve (the constraint (2) is thus also applied to j = 1, . . . , n with j = k). The idea is that if one DMU can not attain an ef?ciency rating of 100% under this set of weights, then it can never be attained from any other set. It should be noted that in practice, more constraints on weight systems can be imposed to take into account
the maximal value of the objective function can be given any other number without changing the relative ef?ciency of the DMUs. The choice of unity is to assure the coherence between mathematical calculations and ef?ciency de?nitions.
11 Mathematically,
8
speci?c preferences of decision-makers. This point will be illustrated further. Alternatively, the DEA original problem can be formulated as the following system: ? vi xik (4)
m
min
u,v
hk =
m
i =1 s r =1
? ur yrk
subject to
i =1 s r =1
? vi xij
? ur yrj
? 1, j = 1, . . . , n
(5) (6)
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m
where the objective is to seek for optimal weights so as to minimize the ratio of the weighted sum of inputs to the weighted sum of outputs. The smaller this ratio, the better. In this case, ef?cient DMUs have a score of 1 and inef?cient ones have a score greater than 1. Note however that the system (4-6) is less familiar within DEA’s applications in ?nance than the system (1-3). It is important to keep in mind that basic DEA models do provide a dichotomic classi?cation, not a complete ranking of DMUs as all ef?cient DMUs have the same score equal to 1. Besides, ef?ciency or inef?ciency of DMUs is solely relative to the sample under consideration. Hence, once the sample is modi?ed, results may be very different.
2.2.2 The primal program The optimizing systems (1-3) and (4-6) are fractional problems, non convex with fractional constraints, which are quite dif?cult to solve. According to Charnes & Cooper (1962, 1973) and Charnes et al. (1978), the fractional problem (1-3) (or 4-6) can be conveniently converted into an equivalent linear programming problem by normalizing the denominator to 1 and maximizing (minimizing) the nominator. By doing so, we obtain the input-oriented version (system (7-10)) and the output-oriented version (system (11-
9
14)) of the so-called CCR model — the seminal model of the DEA method: Input-oriented: max
u,v
hk =
m
? ur yrk
r =1
s
(7) (8) (9) (10)
subject to:
? vi xik = 1
i =1 s r =1
? ur yrj ? ? vi xij , j = 1, . . . , n
i =1
m
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m or Output-oriented: min
u,v
hk =
s
? vi xik
i =1
m
(11) (12) (13) (14)
subject to
? ur yrk = 1
r =1 s r =1
? ur yrj ?
? vi xij , j = 1, . . . , n
i =1
m
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m
The input-oriented (output-oriented) version assumes that only inputs (outputs) can be adjusted, outputs (inputs) being ?xed.
2.2.3 The dual program According to linear programming theories, each primal program is associated with a dual program which provides the same optimal value of the objective function as the primal. The system (7-10) thus has a dual below: Input-oriented: min
? ,?
? ? xik ? yrk ?
(15)
subject to:
? ? j xij , i = 1, . . . , m
j =1 n
n
(16) (17) (18)
? ? j yrj , r = 1, . . . , s
j =1
? j ? 0, ? unconstrained in sign with ? and ? are dual variables. Note that ? can not, by construction, exceed unity12 .
12 We can easily see that ? = 1, ? = 1, ? = 0 ( j = k) is a feasible solution to (15-18). Hence, the optimal j k value of ? can not be greater than 1. Besides, the constraint (16) implies that ? must be positive as X is
10
In a similar fashion, the dual of the system (11-14) is de?ned by: Output-oriented: max
? ,?
? xik ?
(19)
subject to:
? ?j xij , i = 1, . . . , m
j =1
n
(20) (21) (22)
? yrk ?
? ?j yrj , r = 1, . . . , s
j =1
n
? j ? 0, ? unconstrained in sign where ? and ? are dual variables and ? can not be, by construction, lower than 1.
In fact, the primal program can be solved directly to obtain the optimal ef?ciency score. However, the dual program is usually preferred for the following reasons. On the one hand, it is mathematically easier to ?nd the optimal solution via the dual because of a considerable reduction of constraints: from n + s + m + 1 constraints in the primal to only s + m constraints in the dual. This calculating parsimony is of particularly appealing when dealing with large samples. On the other hand, the dual formulation has an interesting economic interpretation. In economic terms, under the input-oriented form (output-oriented form), the dual looks for a feasible activity — a virtual DMU which is a linear combination of the best practices — that guarantees (uses) the output level yk (the input level xk ) of the DMUk in all components while using only a proportion of the Hence, ? (or ? ) is de?ned as a measure of the ef?ciency level of the DMUk . Graphically, in the input-output plan depicted in ?gure 2, under the input-oriented or input contraction setting, ? of the DMU A is the ratio DC / DA, with C being the virtual DMU which serves as benchmark to measure the ef?ciency of A; under the output-oriented or output expansion setting, ? of the DMU A is measured by FH / AH with F being the reference DMU for A now. In order to obtain the ef?ciency scores of n DMUs, the optimizing system (primal or dual) must be run n times with each time the DMU under evaluation changes. Theorem 1 (Connexion between the CCR input-oriented version and the CCR output-oriented version) Let (? ? , ?? ) be an optimal solution for the CCR input-oriented version. Then (1/? ? , ?? /? ? ) = DMUk ’s inputs ? xik (producing higher outputs than DMUk ’s outputs, ? yrk with ? ? 1 ).
(? ? , ?? ) is optimal for the corresponding CCR output-oriented version. Similarly, if (? ? , ?? ) is
optimal for the CCR output-oriented version, then (1/? ? , ?? /? ? ) = (? ? , ?? ) is optimal for the corresponding CCR output-oriented version (Seiford et al. 2004, p.17).
assumed to be semipositive.
11
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.
Q P R I U %
T
&'()0123(4 5(663738329 6@2
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Figure 2: CCR ef?ciency frontier
3 A DEA framework for hedge fund performance appraisal
In ?nancial literature, funds’ historical performance is often measured by the ratio of return to risk. Traditionally founded on the ”mean-variance” basis, the evaluation dimension has been recently extended to the skewness (Stutzer index — Stutzer (2000)), to the skewness and the kurtosis (the modi?ed Sharpe ratio — Gregoriou & Gueyie (2003)), even to the whole distribution of returns (Omega index — Keating & Shadwick (2002)) in an attempt to take into account the non normality features of returns. Despite this improvement, most of these measures are highly restrictive in the sense that they usually assume simplistic decision-making rules common to all investors. Yet, there are suggestions that actual individual decisions differ signi?cantly from theoretical formulations since they are much more complicated and quite speci?c to investors. Often there are more attributes to consider and for each investor, each attribute does not necessarily have the same priority level. While some investors are more concerned with central tendencies (mean, variance), others may care more about extreme values (skewness, kurtosis). One kind of such preferences is summarized by the positive preference for skewness ?rst invoked by Arditti (1967) and then supported by Jean (1971), Kraus & Litzenberger (1976), Francis & Archer (1979), Scott & Horvath (1980), Kane (1982), Broihanne et al. (2004). It implies that individuals prefer portfolio A to portfolio B with higher mean return if both portfolios have the same variance, and if portfolio A has greater positive skewness, all higher moments being the same. In other words, individuals may attach more importance to the skewness than to the mean of returns. Despite the diversity of preferences for moments of returns, most measures assume the
12
same preference structure for all investors. Consider for example the modi?ed Sharpe ratio (hereafter, M-Sharpe) (Gregoriou & Gueyie 2003) computed by the following equation: M ? Sharpe = r ? rf = 2 MVAR W µ ? zc + 1 6 ( z c ? 1) S + r ? rf
1 24
( z3 c ? 3z c ) K ?
1 36
2 ? (2z3 c ? 5z c ) S (23)
where r is the mean return, r f is the average risk-free rate, W is the amount of portfolio at risk, µ is the mean return and naturally equal to r, ? is the standard deviation of returns, S is the skewness, K is the kurtosis excess, zc is the critical value for probability by Favre & Galeano (2002). According to Favre & Galeano (2002) and Gregoriou & Gueyie (2003), all investors are certainly concerned about the skewness and the kurtosis of returns but they share the same preference structure which is necessarily in the form of MVAR. This rigidity is not only restrictive but might bias signi?cantly investors’ choice of funds as their true evaluation criteria are not considered at all or considered but in a biased manner. In addition to that, investors may need to take account of sales loads charged by the fund on their entrance into (front-end sales load) or/and on their exit of the fund (backend or deferred sales load). Unlike management fees which are directly deducted from the fund’s value, sales loads are charged on the net returns paid to investors13 . As a result, a fund with good performance and a high percentage of loads is not necessarily more attractive than another fund which has lower performance but charges lower loads. Moreover, as argued by McMullen & Strong (1998), Morey & Morey (1999) and Powers & McMullen (2000), investors may also be concerned about fund’s performances over various time horizons (over the last year, the last 3 years, the last 5 years and sometimes the last 10 years). Such information is undoubtedly valuable as it provides much more informative insight into fund’s perspective than the performance over only one horizon. Furthermore, even when investors care about the return and the risk, or the performance of funds over only one horizon, it is often quite dif?cult to choose an absolutely
et al. (1997), McMullen & Strong (1998), Tarim & Karan (2001), Choi & Murthi (2001) and Sengupta (2003) advocated incorporating also expense ratio (in percentage of fund assets, covering various operating expenses incurred by the fund management such as management fees, administrative fees, advisory fees) in evaluating fund performance. This element which is obviously necessary to appraise the performance of funds in a productivity perspective, i.e. their capacity to exploit ef?ciently input resources (fund expenses are considered here as a production factor), is irrelevant in this context where inputs and outputs are selection criteria chosen by investors. In this regard, investors are not likely concerned by these expenses as they are directly deducted before calculating funds’ net asset value — the real value of investors’ investments. Hence, such expenses are generally invisible to investors.
13 Murthi
(1 ? ?) (zc = ?1.96 for a 95% probability), MVAR (modi?ed value-at-risk) is introduced
13
suitable measure among a wide range of existing measures in the literature. This dif?culty is particularly true for the choice of risk and performance indicators because of inexistence or de?ciency of mechanisms to validate empirically them. Consequently, investors are sometimes in need of considering simultaneously several measures. Here again, they do not necessarily share the same preferences for such and such measures. Given these speci?cities in performance evaluating practices, the DEA’s approach seems very appealing. In fact, the application of DEA into hedge fund performance appraisal can be made in two perspectives. The ?rst one consists in evaluating the productive performance of funds where the latter are considered as a particular type of production units which employs multiple resources (risks, various operating expenses, turnover speed, etc.) to realize pro?ts (returns). The second, which is in the spirit of Thompson et al. (1986), Tone (1999) and Powers & McMullen (2000), aims to assess funds as decision–making units whose inputs and outputs are evaluation criteria chosen by decision-makers. It is the second perspective that interests investors as DEA, with its broad ?exibility, allows investors to tailor their own evaluation tools corresponding the most to their own preferences. Since each investor naturally has different risk aversion levels, performance objectives and other distinct constraints, the tailor-made possibility is essential to correctly screen fund. In this context, the DEA method can be applied to evaluate either the ”local” performance or the ”global” performance of hedge funds. By the ”local performance”, I imply the performance measured by the weighted sum of several criteria of gain (or return) on the weighted sum of several criteria of risk and possibly certain types of expenses. In contrast, the term ”global performance” denotes the performance synthesized from either several measures of ”local” performance, or elementary performances over several temporal horizons. Within this framework, the application of DEA (in its basic form) raises four main questions: (1) how to choose inputs and outputs, (2) what version to choose (input-oriented or output-oriented), (3) how to deal with negative values in the inputs or/and the outputs if they exist14 , and (4) how to incorporate more speci?c preferences of investors into the mathematical formulation. If the ?rst, the second and the fourth questions are relevant to any application ?elds, the third one is quite speci?c to data of returns and performances. These issues will be addressed successively in what follows.
14 Inputs
and outputs of DEA are originally assumed to be semipositive.
14
3.1
Evaluation criteria and the choice of inputs and outputs
Unlike applications of DEA in production ?elds where inputs and outputs are tangible elements, the choice of inputs and outputs is not straightforward when dealing with fund performance. Nevertheless, in a multiple criteria decision-making framework, it is logical to consider inputs as criteria that investors want to minimize and outputs as those they want to maximize. Hence, if investors seek to evaluate the funds’ ”local” performance, i.e. returns15 to risks, the inputs can be (1) several measures of risk (standard deviation, kurtosis, beta, various measures of value-at-risk) over one (or several) horizon(s), (2) possibly the sales loads; the outputs can be composed of (1) several measures of returns (mean, skewness) — over one (or several) horizon(s). The difference between the con?guration suggested here (to evaluate fund ”local” performance) and that assumed by standard performance indicators is that according to the former, each investor knows perfectly his relevant evaluation criteria but does not know the functional relation between these criteria as well as the exact trade-off between them, which is not the case of the modi?ed Sharpe ratio as previously described. The case where investors know the relative trade-off between these criteria will be discussed further. Otherwise, if investors want to evaluate funds by considering several elementary performances simultaneously, they can calculate the global performance by (1) including in the outputs either the performances measured by the same technique on several periods, or the performances on the same period but measured by several indicators, (2) setting the input equal to one. It is important to notice that in this setting, all selection criteria are those investors want to maximize, they thus form exclusively DEA’s outputs; meanwhile, there is no input. Assuming the presence of one input equal to 1 makes it possible to apply DEA without any modi?cation of results. As explained earlier, we are in a basic con?guration in which there is one input and several outputs and the quantity of each output is often ”standardized” by the quantity of the input to obtain the unit outputs (per one unit of the input) in order to facilitate calculations. This setting is employed by Thompson et al. (1986) and Tone (1999). Following the principles evoked above, each investor will determine, according to his own preferences, the inputs and outputs for DEA while complying with general rules as: – Inputs and outputs must be criteria indispensable to the appraisal of fund perforterm ”return” should be understood here in broad sense. In traditional language of portfolio theories, the concept of return is always associated to the arithmetic mean of elementary returns over a given period. By ”return”, I imply in what follows any measure, in addition to the mean return, that is indicative of fund’s expected returns such as the skewness.
15 The
15
mance. – The number of inputs and outputs should be lower than the number of funds. In general, the number of funds should be at least three times larger than the number of inputs and outputs. Any violation of these rules will lead to a de?ciency of the discriminatory power of DEA. As a result, we risk obtaining an excessive number of dominant (ef?cient) funds whereas some of them are not rightly so16 .
3.2
Input-oriented or output-oriented versions ?
In general, when inputs and outputs are semipositive, the choice between the CCR inputoriented version and the CCR output-oriented version can be simply made at users’ discretion following their preferences. Note that the input-oriented version (output-oriented version) assumes that outputs (inputs) are ?xed, only inputs (outputs) can be adjusted. This assumption conditions the reference fund on the ef?cient frontier to which is compared the target fund and thus determines the distance between the former and the latter, this distance measuring the ef?ciency level of the latter. The theorem 1 describes the correspondence between the optimal solutions of the two versions. We can easily see that the two versions of the CCR model provides the same classi?cation of inef?cient DMUs17 (ef?cient DMUs always obtain a full score of 1 under any version). Nevertheless, it is interesting to notice that all studies which apply DEA to evaluating fund performance adopted the input-oriented version whatever the DEA model is used. This popularity is undoubtedly due to the fact that this mathematical form shares the same logic as Markowitz’s ef?cient frontier construction, that is to minimize the risks (inputs) for a de?ned level of returns (outputs). However, when there are only outputs (inputs), the input-oriented (output-oriented) version is required as in this case, we assume the existence of one input (output) whose quantity is ?xed equal to 1.
16 The terms ”dominant funds” and ”ef?cient funds” will be used interchangeably hereafter to indicate funds having a full ef?ciency score of 1. 17 It is important to specify that this equivalence between the input-oriented version and the outputoriented version is only valid under the constant returns-to-scale technology assumed by the CCR model.
16
3.3
Dealing with negative inputs and outputs
DEA models as originally designed require that inputs and outputs are semipositive, i.e. all inputs and all outputs are non negative and at least one input and one output are positive. In many application ?elds like production economics, negative inputs and outputs naturally make no sense. However, in fund performance appraisal context, it is likely that we sometimes have negative values like mean, skewness of returns, or some performance indicators, etc. Although in the CCR model, or more generally in basic DEA models, inputs and outputs are systematically required to be semipositive, we can easily see that negative values in inputs and outputs are tolerated in following ways without any incidence on the solubility of DEA optimizing systems (Cooper et al. 2000, p.304-305): – If there are at least one input and one output positif, either the input-oriented version or the output-oriented version can be used; – If all outputs (inputs) are negative and at least one input (output) is positive, the input-oriented (output-oriented) version is required; – If there is no (effective) input (output) and all outputs (inputs) are negative, the input-oriented (output-oriented) version is required; – The case where all inputs and all outputs are negative at the same time, which is extremely rare in fund performance appraisal context, can not be dealt with within the DEA framework. Note that in the second and the third cases, the optimal value of the objective function will be negative.
3.4
Taking account of investors’ more speci?c preferences
The CCR model as presented earlier allows a quasi-absolute freedom in the determination input and output level. Speci?cally, {u, v} are only required to be equal to or greater than of the weights {u, v} so that each funds obtains a maximum score of ef?ciency, given its
an in?nitesimal positive number ?. This constraint is essential to assure that all selected evaluation criteria are considered in the evaluating process. Nevertheless, such ?exibility level also implies that important, even excessive, weights can be assigned to the input(s) or/and the output(s) which make the funds as ef?cient as possible compared to others. 17
As a result, this setting is only plausible when investors have no idea about the tradeoff between the selected criteria. When such information is available, it can be easily virtual weights {uy, vx } associated with each input and each output. incorporated in DEA optimizing systems by restricting the absolute weights {u, v} or the An investor in full knowledge of the ”price” range for each evaluation criterion — e.g. the coef?cient of aversion to the mean, the variance, the skewness or the kurtosis of returns — can have recourse to constraints like: ur ? ? r vi ? ?i ?r ? ur ? ? r ?i ? vi ? ?i (24) (25) (26) (27)
An investor who knows more or less his personal or conventional trade-off or substitution rate between evaluation criteria can add following constraints into original DEA program: ur ?? vi vi ?? v i +1 ? i v i + ? i +1 v i +1 ? v i +2 (28) (29) (30)
An investor who wants to control the relative importance of each criterion in the performance appraisal process will formulate additional constraints on the virtual weights as follows: ar ? ci ? ur yrj s ?r=1 ur yrj vi xij m ?i=1 vi xij
? br ? di
(31) (32) (33) (34)
ur yrj ? ur+1 yr+1, j vi xij ? vi+1 xi+1, j
where ? , ?, ?, ?, ? , ? , a, b, c, d are values pre-de?ned by investors to bound absolute and virtual weights. There are certainly many other forms of additional constraints because of a broad
18
variety of investors’ preferences. The constraints mentioned above are to give examples of ”transcribing” more speci?c preferences into mathematical formulations. A numerical illustration will be provided further. This possibility of exerting a direct control on the relative importance of each evaluation criteria in assessing fund performance, along with the choice of evaluation criteria (inputs and outputs), makes it possible for each investor to conceive a customized measure corresponding to his preferences. With such quality, the DEA approach is an ef?cient and complementary tool to other existing measures.
4 Illustrative applications
4.1 Data
To illustrate the use of DEA in assessing the performance of hedge funds, I used a sample of 38 hedge funds belonging to the category Equity Hedge18 . Data includes 60 monthly returns covering the period of January 2000 to December 2004. Table 1 reports some descriptive statistics of these funds. As we can see, return distributions of many funds show highly positive (negative) skewness signifying higher probability of extreme positive (negative) values compared to that implied by the normal distribution. Besides, many of them possess high kurtosis excess, which indicates more returns close to the central value but also more regular large positive or negative returns than in a normal distribution. The normality assumption of return distributions is tested by means of three tests: Shapiro-Wilk, KolmogorovSmirnov, and Jarque-Bera. Results provided by the Shapiro-Wilk and Jarque-Bera tests are quite similar although they are rather different from those provided by the KolmogorovSmirnov test. This divergence is likely due to the sample’s limit size as the KolmogorovSmirnov test is more appropriate to large samples. According to the Shapiro-Wilk test, documented as the most reliable for small samples, the normality assumption is rejected in 14 out of 38 cases at the con?dence level of 95%. These ?ndings imply much higher return or risk of these funds than those approximated under normality assumption. They thus highlight the importance of incorporating moments of order higher than the mean and the variance when appraising funds’ return and risk pro?les.
38 funds are extracted from a database provided by the company Standard & Poor’s. Equity Hedge covers several different strategies whose investments are focused on the equity markets. Its two large categories are Global Macro and Relative Value.
18 These
19
Table 1: Descriptive statistics
Funds Min Max Me SD SK KU S-W K-S J-B (%) (%) (%) (%) 1 -6.16 10.20 0.68 3.77 0.34 -0.35 0.98 0.11 1.43 2 -7.89 7.69 0.16 3.33 -0.07 -0.25 0.99 0.06 0.20 3 -12.51 19.66 -0.30 5.59 0.32 1.86 0.95** 0.12 9.72*** 4 -7.25 5.63 0.25 3.04 -0.27 -0.49 0.98 0.07 1.31 5 -11.37 11.95 0.10 4.65 -0.10 0.09 0.99 0.06 0.12 6 -6.50 5.92 0.06 2.34 -0.36 0.29 0.98 0.08 1.50 7 -14.67 24.36 -0.11 6.23 0.91 3.37 0.95*** 0.11 36.60*** 8 -22.96 33.89 -0.18 8.86 0.71 2.69 0.96** 0.06 23.09*** 9 -7.87 8.59 -0.01 3.92 -0.01 -0.43 0.98 0.07 0.46 10 -11.84 13.05 0.03 5.40 0.16 -0.12 0.99 0.08 0.29 11 -8.19 17.11 1.08* 4.95 1.18 2.13 0.92*** 0.16* 25.40*** 12 -13.49 9.11 0.01 4.35 -0.43 0.66 0.98 0.08 2.98 13 -6.77 7.23 -0.57 3.36 0.12 -0.75 0.98 0.07 1.53 14 -40.85 19.45 -0.73 9.02 -1.42 5.88 0.90*** 0.15 106.6*** 15 -12.04 14.17 0.23 5.53 -0.05 0.06 0.98 0.07 0.03 16 -5.76 6.58 0.24 2.83 0.13 -0.22 0.99 0.06 0.29 17 -7.10 6.27 0.33 3.25 -0.15 -0.49 0.98 0.06 0.83 18 -6.33 5.94 0.15 2.65 0.00 -0.33 0.99 0.08 0.28 19 -7.21 8.18 0.25 3.43 0.17 -0.46 0.99 0.06 0.81 20 -9.90 14.45 0.12 4.54 0.25 0.71 0.99 0.05 1.91 21 -6.81 9.77 0.64* 2.75 0.48 2.08 0.96* 0.10 13.19*** 22 -9.20 7.57 -0.23 3.95 -0.09 -0.38 0.99 0.05 0.44 23 -5.31 6.82 0.33 2.48 0.62 0.59 0.96** 0.15 4.66*** 24 -13.75 15.03 -0.81 5.72 -0.10 0.30 0.98 0.09 0.33 25 -9.78 19.59 0.09 5.46 1.35 3.85 0.90*** 0.12 55.13*** 26 -16.34 25.90 1.34 * 5.31 1.35 10.02 0.72*** 0.24*** 269.3*** 27 -1.24 5.45 0.52*** 1.10 1.64 5.81 0.88 *** 0.17* 111.2*** 28 -2.37 15.86 0.74 ** 2.39 4.36 27.27 0.63 *** 0.19** 2050*** 29 -15.48 22.37 0.64 5.07 1.23 7.06 0.85 *** 0.16* 139.7*** 30 -13.76 17.90 -0.35 5.29 0.50 1.86 0.97 0.09 11.22*** 31 -14.30 17.93 0.28 5.00 0.21 2.69 0.95** 0.10 18.58*** 32 -6.93 11.54 0.00 3.15 0.73 2.47 0.96** 0.08 20.64*** 33 -7.88 11.53 0.08 4.07 0.58 0.71 0.97 0.08 4.57 34 -7.12 8.67 -0.07 4.12 0.20 -0.64 0.97 0.08 1.42 35 -5.68 10.93 1.10*** 2.97 0.38 1.30 0.96 * 0.12 5.66* 36 -10.13 7.95 0.58 3.75 -0.29 -0.05 0.98 0.08 0.82 37 -6.48 11.01 0.38 2.60 0.71 4.25 0.93 *** 0.10 50.14*** 38 -9.93 12.48 -0.13 4.10 0.04 0.71 0.98 0.07 1.26 Me = Mean, SD = Standard deviation, SK = Skewness, KU = Kurtosis excess relatively to the normal distribution. S-W = Shapiro-Wilk, K-S = Kolmogorov-Smirnov, J-B = Jarque-Bera are normality tests on return distributions. ***, **, ** denote the rejection of the normality assumption at respectively the 99%, 95% and 90% con?dence levels.
20
4.2
Assessing hedge fund local performance
4.2.1 Settings Due to unavailable data on sales loads charged by the funds in the sample, illustrations are limited to considering their historical return and risk pro?les. Since the distribution of hedge fund returns is documented as usually non gaussian, it is important to incorporate these features into the selection of evaluation criteria (DEA’s inputs and outputs). Several settings are likely. The ?rst setting assumes the case where investors have a positive preference for odd moments and a negative preference for even moments. Given these preference, it is logical to include in the inputs the standard deviation and the kurtosis of returns, and in the outputs the mean and the skewness. In this con?guration, the problem of negative outputs raises. More speci?cally, 11 out of 38 funds under consideration have a negative mean, 12 funds have a negative skewness, 4 funds among them have simultaneously negative mean and negative skewness. The second setting is designed in the spirit of Gregoriou et al. (2005) and Kooli et al. (2005), following which it is more clever to reason in terms of partial variations. As documented in the literature, investors are likely to be averse only to volatility under the Minimum Accepted Return (MAR)19 , which are called lower variations. In contrast, they appreciate volatility above this value, which are called upper variations. Thus, the composition of inputs and outputs can be determined in the following manner. The inputs include lower mean, lower semi-standard deviation, lower semi-skewness and lower semi-kurtosis which are obtained from returns lower than the MAR represented by the average rate over the period january 2000 to december 2004 of the US 3-month T-bill. The outputs contain upper mean, upper semi-standard deviation, upper semi-skewness and upper semi-kurtosis obtained from returns greater than the MAR. In addition to his ?nancial ?nesse, this con?guration has the clear-cut advantage to avoid the problem of negative inputs and outputs. Now assume furthermore that investors are more concerned for extreme values than central ones. Hence, they naturally pay more attention to the skewness and the kurtosis than to the mean and the standard deviation. Mathematically, they will require that the contribution of the upper (lower) skewness and kurtosis to the ef?ciency score of the fund must be greater than or equal to the contribution of the upper (lower) mean and
19 The determination of the Minimum Accepted Return is purely subjective and speci?c to each investor. It can be a risk-free rate or any rate required by investors.
21
standard deviation. This preference can be taken into consideration by adding four more constraints on virtual weights into the optimization system: y3 j u3 y3 j u3 y4 j u4 y4 j u4 y1 j u1 ; x3 j v3 y2 j u2 ; x3 j v3 y1 j u1 ; x4 j v4 y2 j u2 ; x4 j v4 x1 j v1 x2 j v2 x1 j v1 x2 j v2
where y1 j , y2 j , y3 j , y4 j are the amount of upper mean, upper standard deviation, upper skewness and upper kurtosis of the fund j under consideration; x1 j , x2 j , x3 j , x4 j are the amount of its lower mean, lower standard deviation, lower skewness and lower kurtosis; u1 , u2 , u3 , u4 , v1 , v2 , v3 , v4 are the weights associated respectively with these outputs and inputs. Otherwise, if investors are more or less markowitzian, i.e. they rely essentially on the mean and standard deviation to assess fund performance, the following constraints are necessary so that this preference is incorporated: y1 j u1 y1 j u1 y2 j u2 y2 j u2 y3 j u3 ; x1 j v1 y4 j u4 ; x1 j v1 y3 j u3 ; x2 j v2 y4 j u4 ; x2 j v2 x3 j v3 x4 j v4 x3 j v3 x4 j v4
The third setting illustrates another case where investors need to reconcile funds’ local performance over several horizons, from a long period to a more recent period in the past. To this end, DEA inputs are modeled by the MVAR (described by the denominator of equation 23) representing the loss limits over three horizons: 1 year, 3 years and 5 years; outputs are the mean returns over these three horizons. Again, many cases of negative outputs are found: 12 cases over the one-year horizon, 22 cases over the three-year horizon and 11 cases over the ?ve-year horizon, among them 5 funds have all negative outputs. It is important to keep in mind that the above settings are only some standard con?gurations used by investors. Given the diversity of investors’ preferences, many other con?gurations are also expected.
22
4.2.2 Choice of CCR version After inputs and outputs corresponding to investors’ preferences are speci?ed, the next step consists in running the foregoing inputs and outputs under the CCR model. Then what version to choose, input-oriented or output-oriented? Following principles highlighted in the section 3.3, we are constraint to adopt the input-oriented version for the ?rst and the third settings where outputs are sometimes all negative. Regarding the second setting, either version is possible. However, in this study, the input-oriented version is chosen for all settings. Its primal and dual programs are described respectively by the systems (7-10) and (15-18). Note that the weights assigned to each output and input are constrained to be equal to or greater than 0.001 (? = 0.001)20 to assure that all criteria are considered in the optimization program.
4.2.3 Results Table 2 displays detailed results on DEA score, absolute weights (u, v) and virtual weights (uy, vx) obtained under a CCR input-oriented setting with mean and skewness as outputs, standard deviation and kurtosis as inputs. Funds with negative scores are those having simultaneously negative mean and negative skewness. Given the difference of measure scale between mean, standard deviation on the one hand and skewness, kurtosis on the other hand, virtual weights rather than absolute weights are more informative about key factors (inputs and outputs) that make some funds dominant compared to others in the sample. Each of the ?ve funds quali?ed as dominant (1, 11, 27, 28, 35) has its own combination of evaluation criteria to attain the full ef?ciency. For fund 27 and fund 35, the virtual weights associated with the mean and the standard deviation are much higher than those associated with the skewness and the kurtosis. By referring to the statistics of returns given in table 1, we ?nd that they effectively have fairly high mean and small standard deviation in comparison with the others. Their pro?les are thus well adapted to markowitzian investors. By contrast, the dominance of fund 28 is primarily due to its positive skewness. In fact, this fund has the highest positive skewness in the sample. With fund 1, the dominance is mainly based on the mean and the kurtosis while with fund 11, dominant factors are the mean, the skewness and the kurtosis. These ?ndings imply that not all dominant funds are necessarily adapted to an investor having a precise
fact, all calculations were already tested with several values of ?: 0, 0.0001, 0.001 and 0.01. However, performance scores changed very slightly while the relative rank between funds remains unchanged. Thus, ? was ?xed to be equal to 0.001 to facilitate result presentation.
20 In
23
preference. Consequently, when no additional constraint is formulated like in this setting, an investor who is not completely indifferent among evaluation criteria should identify the factors determining the ef?ciency of dominant funds and select only those whose pro?les correspond the most to his preferences. Results on DEA scores across various settings are summarized in table 3. Note that in the ?rst and the third settings (respectively in columns 2 and 6), funds with negative scores are those whose all outputs are negative. Several points are noteworthy. In general, results are rather sensitive to the speci?cation of evaluation criteria and supplementary constraints. Not only the number of dominant funds varies (from 1 to 5) but also dominant members differ across settings. Look at for example fund 26 which is quali?ed as dominant only when its returns and risks over three horizons are considered simultaneously. Related to the second setting, as would be expected, the introduction of additional constraints on virtual weights naturally deteriorates ef?ciency scores and the short list of dominant funds becomes more selective. When preferences for extreme values (represented by the skewness and the kurtosis) are explicitly formulated, only fund 28 (among ?ve funds 1, 8, 11, 25, 28 quali?ed as dominant without any additional constraints) satis?es this requirement. Similarly, when more importance is explicitly attached to central values (represented by the mean and the standard deviation), there are only three funds 8, 11, 28 in the dominant list. These results highlight the importance of correct speci?cation of relevant DEA inputs and outputs as well as additional constraints which re?ect best investors’ evaluation preferences. At empirical level, one may notice persistent dominance of several funds across settings like the case of fund 28, which stays dominant whatever preferences are considered. This feature can be regarded as a sign of the robustness of fund 28’s performance relatively to other funds in the sample. Since we are examining funds’ local performance without sales loads, it could be interesting at this point to contrast DEA results in the ?rst and the second settings with fund rankings provided by the traditional Sharpe ratio (Sharpe 1966) and the M-Sharpe ratio. The latter is computed following equation 23 while the former is calculated by the formula below: Sharpe = r ? rf ? (35)
where r is the average return of the fund, r f is the average risk-free rate approximated here by the US 3-month T-bill rate, ? is the standard deviation of fund returns. Note that
24
Table 2: Performance with standard deviation-kurtosis as inputs, mean-skewness as outputs
Absolute weights (u, v)b Virtual weights (uy, vx) Me SK SD KU Me SK SD KU (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 1 147.427 0.001 0.167 0.375 0.9997 0.0003 0.0063 0.9937 2 0.228 142.003 0.001 0.161 0.361 0.2284 -0.0001 0.0054 0.9946 3 0.286 0.001 0.891 0.001 0.206 0.0000 0.2864 0.0001 0.9999 4 0.389 155.891 0.001 0.176 0.396 0.3888 -0.0003 0.0054 0.9946 5 0.127 126.330 0.001 0.001 0.323 0.1270 -0.0001 0.0000 1.0000 6 0.077 119.256 0.001 0.136 0.303 0.0774 -0.0004 0.0032 0.9968 7 0.618 0.001 0.681 0.001 0.157 0.0000 0.6181 0.0001 0.9999 8 0.537 0.001 0.762 0.001 0.176 0.0000 0.5374 0.0001 0.9999 9 -0.000 0.001 0.001 25.456 0.001 -0.0000 -0.0000 0.9974 0.0026 10 0.240 0.001 1.507 0.001 0.348 0.0000 0.2403 0.0001 0.9999 11 1 66.707 0.237 1.281 0.182 0.7193 0.2807 0.0635 0.9365 12 0.006 106.752 0.001 0.122 0.272 0.0062 -0.0004 0.0053 0.9947 13 0.222 0.001 1.925 0.001 0.444 0.0000 0.2219 0.0000 1.0000 14 -0.001 0.001 0.001 10.984 0.001 0.0000 -0.0014 0.9911 0.0089 15 0.295 127.735 0.001 0.001 0.327 0.2947 -0.0001 0.0001 0.9999 16 0.347 122.726 0.436 2.357 0.336 0.2899 0.0566 0.0668 0.9332 17 0.514 155.511 0.001 0.176 0.396 0.5140 -0.0002 0.0057 0.9943 18 0.215 146.828 0.001 0.166 0.373 0.2148 0.0000 0.0044 0.9956 19 0.412 126.397 0.556 0.001 0.394 0.3183 0.0937 0.0000 1.0000 20 0.297 0.001 1.168 0.001 0.269 0.0000 0.2966 0.0000 1.0000 21 0.684 68.216 0.508 15.009 0.116 0.4380 0.2463 0.4122 0.5878 22 -0.000 0.001 0.001 25.250 0.001 -0.0000 -0.0001 0.9974 0.0026 23 0.793 0.001 1.289 6.883 0.231 0.0000 0.7929 0.1706 0.8294 24 -0.000 0.001 0.001 17.422 0.001 -0.0000 -0.0001 0.9967 0.0033 25 0.888 0.001 0.659 3.518 0.118 0.0000 0.8877 0.1922 0.8078 26 0.695 29.615 0.221 6.516 0.050 0.3973 0.2977 0.3461 0.6539 27 1 193.390 0.001 90.268 0.001 0.9984 0.0016 0.9912 0.0088 28 1 33.048 0.173 40.581 0.001 0.2437 0.7563 0.9697 0.0303 29 0.592 0.001 0.482 2.576 0.086 0.0000 0.5918 0.1306 0.8694 30 0.448 0.001 0.891 0.001 0.206 0.0000 0.4482 0.0001 0.9999 31 0.212 60.499 0.215 1.162 0.165 0.1664 0.0460 0.0581 0.9419 32 0.639 0.001 0.871 4.652 0.156 0.0000 0.6393 0.1463 0.8537 33 0.675 0.001 1.170 0.001 0.270 0.0000 0.6747 0.0000 1.0000 34 0.361 0.001 1.840 0.001 0.424 0.0000 0.3607 0.0000 1.0000 35 1 91.251 0.001 31.662 0.014 0.9996 0.0004 0.9390 0.0610 36 0.773 132.376 0.001 0.150 0.337 0.7735 -0.0003 0.0056 0.9944 37 0.502 55.541 0.414 12.220 0.094 0.2083 0.2940 0.3175 0.6825 38 0.044 0.001 1.170 0.001 0.270 0.0000 0.0439 0.0000 1.0000 Note: Me = Mean, SK = Skewness, SD = Standard deviation, KU = Kurtosis. Values in italics are approximative. Funds with negative scores are those whose mean and skewness are simultaneously negative. b u and v are required to be equal to or greater than 0.001 (? = 0.001).
a Funds
Scorea
25
Table 3: Local performance
DEA scores Partial momentsb Horizonsc Standard Preference Preference SK & KU Me & SD (1) (2) (3) (4) (5) (6) 1 1 1 0.52 0.93 0.46 2 0.23 0.71 0.38 0.69 0.46 3 0.29 0.83 0.69 0.71 -0.00 4 0.39 0.66 0.37 0.64 0.19 5 0.13 0.78 0.36 0.77 0.08 6 0.08 0.64 0.35 0.58 0.93 7 0.62 0.86 0.67 0.81 0.21 8 0.54 1 0.62 1 0.11 9 0.00 0.72 0.44 0.67 0.22 10 0.24 0.85 0.36 0.84 0.14 11 1 1 0.60 1 0.83 12 0.01 0.68 0.32 0.66 0.24 13 0.22 0.79 0.38 0.74 -0.00 14 0.00 0.62 0.21 0.61 1 15 0.29 0.89 0.45 0.75 0.10 16 0.35 0.72 0.41 0.70 0.24 17 0.51 0.80 0.32 0.78 0.63 18 0.21 0.71 0.39 0.68 0.14 19 0.41 0.78 0.44 0.76 0.19 20 0.30 0.85 0.45 0.84 1 21 0.68 0.75 0.44 0.72 0.70 22 0.00 0.85 0.31 0.82 -0.00 23 0.79 0.88 0.37 0.83 0.26 24 0.00 0.72 0.46 0.70 0.48 25 0.89 1 0.72 0.97 0.04 26 0.70 0.72 0.65 0.59 1 27 1 0.92 0.92 0.81 0.96 28 1 1 1 1 1 29 0.59 0.75 0.59 0.66 0.41 30 0.45 0.89 0.45 0.85 0.35 31 0.21 0.83 0.43 0.82 0.71 32 0.64 0.70 0.58 0.66 0.17 33 0.67 0.94 0.45 0.91 0.26 34 0.36 0.91 0.44 0.88 -0.00 35 1 0.76 0.52 0.74 1 36 0.77 0.84 0.30 0.82 0.37 37 0.50 0.63 0.58 0.59 0.44 38 0.04 0.80 0.43 0.76 -0.00 Rank correlation (Sharpe & M-Sharpe) Note: Results are obtained from the CCR input-oriented version with ? = negative scores are those whose all outputs are simultaneously negative. Skewness, SD = Standard deviation, KU = Kurtosis. Funds Standarda moments
a In
Rank Sharpe
Rank M-Sharpe
(8) 7 18 34 15 20 28 29 26 27 23 4 25 38 31 17 16 12 21 14 19 6 35 11 37 22 5 2 1 8 36 13 32 24 30 3 9 10 33 0.995 0.001. Funds with Me = Mean, SK =
(7) 7 18 33 14 21 29 27 25 28 23 5 26 38 34 17 16 12 22 15 19 6 36 11 37 20 4 2 3 9 35 13 30 24 31 1 8 10 32
the ?rst setting, inputs are standard deviation and kurtosis, outputs are mean and skewness. b In the second setting, inputs are composed of lower mean, lower semi standard deviation, lower skewness and lower kurtosis; outputs contain upper mean, upper semi standard deviation, upper skewness and upper kurtosis. c In the third setting, inputs include the MVAR over the previous year, the 3 previous years and the 5 previous years; outputs include mean returns over these three periods.
26
the Sharpe ratio is based on the mean-variance paradigm while the modi?ed Sharpe ratio takes account of the skewness and the kurtosis. Fund rankings according to these two ratios are reported in the columns 7 and 8 of table 3. Several main observations can be drawn from these results. We can see easily that despite differences in the approach taken by the two measures, fund rankings are surprisingly quite similar, both in terms of correlation coef?cient (0.995) and in terms of direct contrasting from fund to fund. Does this strong similarity imply that the return distribution of all funds is quite close to the normal one? The answer according to the Shapiro-Wilk normality test is rather negative because the normality assumption is rejected in 14 among 38 cases at the con?dence level of 95% (see table 1). However, ?nding explanations to such problem is beyond the scope of this paper. Regarding the connection of DEA classi?cations (except for the third setting) with Sharpe and M-Sharpe rankings, the results show that most dominant funds are generally among the seven funds the most highly ranked by Sharpe and M-Sharpe ratios. Nevertheless, funds 8 (dominant once) and 25 (dominant twice) in the second setting are only placed respectively at the 25th and 20th rank by Sharpe, 26th and 22th by M-Sharpe. This disfavor is certainly related to the slightly negative mean of fund 28 (-0.18%) and to the quite low positive mean of fund 25 (0.09%). A closer examination of their return distributions reveals much wider dispersal of returns and higher frequency of extreme positive values in these two distributions than in those of other funds ranked before them by Sharpe and M-Sharpe ratios. It is undoubtedly the reason why these funds are highly praised by the second setting of DEA. An investor who likes good surprises would ?nd his interests in these pro?les. Yet, if he used only Sharpe and M-Sharpe ratios, he would have missed his chance, at least in the case of this sample. Such result provides evidence that DEA can be an ef?cient supplementary tool to assist investors in selecting correctly funds satisfying their preferences.
4.3
Assessing hedge fund global performance
4.3.1 Settings As argued previously, investors may sometimes want to evaluate local performance of funds (1) on several horizons simultaneously or (2) by using several measures at the same time. In these cases, how will they reconcile between elementary performances ? On which basis they can assign a ?nal note to each fund so as to rank them ? This choice is
27
particularly dif?cult when elementary performances provide divergent rankings of funds. Meanwhile, by means of optimizing the weighted sum of elementary performances, DEA offers an aggregate measure allowing investors to identify funds having the best combination of these performances. In other words, by combining multiple performance criteria simultaneously, DEA provides an exhaustive image of funds. In order to illustrate the ?rst setting, the M-Sharpe performance ratio over three horizons — the previous year, the three previous years, the ?ve previous years — are used as elementary performances. In the second setting, three performance measures which consider the non normal features of returns are selected, namely modi?ed Stutzer index (Stutzer 2000, Kaplan & Knowles 2001), M-Sharpe ratio and Omega index (Keating & Shadwick 2002). The formulas to compute the modi?ed Stutzer and the Omega indices are given below:
? M-Stutzer = sign(r ) 2Stutzer with
Stutzer = max ? ln
?
(36) (37) (38)
1 T
? e ? (r ?r
t
T
f t)
t =1
Omega =
? [1 ? F (r )]dr ? ? ?? F (r ) dr
where rt is fund return on month t, sign(r ) is the sign of the mean return, ? is a negative number, T is the number of monthly returns, r f t is risk-free rate (approximated by the US 3-month T-bill rate) on month t, ? is the MAR pre-determined by investors (approximated by the US 3-month T-bill rate’s average over the study period). It is necessary to specify that the modi?ed Stutzer index (hereafter, M-Stutzer) considers up to the skewness of returns, the M-Sharpe ratio takes into account both the skewness and the kurtosis while the Omega index regards the whole (empirical) distribution of returns. These two settings are characterized as having only outputs. Since there are no inputs, it is possible to assume existence of one input equal to one so that DEA can be applied. Given this feature, the input-oriented version is required. Besides, investors are also assumed to be indifferent among horizons and performance indicators so that no additional constraints are needed.
4.3.2 Results Table 4 reports detailed results of the two settings under consideration. In panel A presenting fund classi?cation over three horizons, empirical results con?rm that fund performance varies strongly from one horizon to another. Indeed, coef?cients of rank corre-
28
Table 4: Global performance
Panel A: Perf. over 3 horizons Panel B: Perf. over 5 years with 3 measures M-Sh over Global Perf. M-St ? M-Sh Global Perf. Funds 5 years 3 years 1 year Rank Score Rank Score (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 1 7 21 27 16 0.2028 7 7 7 7 0.657 2 18 17 11 13 0.2854 22 18 18 18 0.430 3 34 26 35 33 -0.0003 33 34 34 34 0.334 4 15 28 28 29 0.0153 24 13 15 13 0.466 5 20 25 36 31 -0.0002 19 21 20 21 0.416 6 28 8 3 6 0.6827 11 29 28 29 0.374 7 29 23 18 22 0.0806 29 30 29 30 0.365 8 26 14 23 27 0.0419 28 28 26 28 0.380 9 27 11 25 24 0.0734 30 26 27 26 0.389 10 23 16 20 25 0.0568 17 23 23 23 0.409 11 4 22 17 12 0.3600 5 5 4 5 0.745 12 25 10 26 21 0.0951 13 25 25 25 0.391 13 38 37 37 37 -0.0005 38 38 38 38 0.260 14 31 6 1 1 1 35 36 31 36 0.310 15 17 24 31 30 0.0034 27 16 17 16 0.450 16 16 29 19 23 0.0749 26 15 16 15 0.459 17 12 27 6 9 0.5109 16 12 12 12 0.494 18 21 35 34 36 -0.0003 18 19 21 19 0.419 19 14 32 30 28 0.0162 25 14 14 14 0.461 20 19 13 2 4 0.9856 21 20 19 20 0.416 21 6 7 8 7 0.5718 6 6 6 6 0.661 22 35 33 32 35 -0.0003 36 33 35 33 0.341 23 11 12 22 20 0.1046 14 11 11 11 0.497 24 37 18 7 11 0.4276 37 37 37 37 0.278 25 22 31 29 32 -0.0002 20 24 22 24 0.408 26 5 1 4 1 1 4 4 5 4 0.877 27 2 5 13 8 0.5565 1 2 2 1 1 28 1 3 9 1 1 3 3 1 1 1 29 8 15 16 15 0.2217 9 9 8 9 0.530 30 36 20 14 17 0.1691 34 35 36 35 0.318 31 13 4 10 10 0.4409 23 17 13 17 0.438 32 32 36 21 26 0.0504 10 31 32 31 0.358 33 24 34 15 18 0.1415 15 22 24 22 0.409 34 30 38 38 38 -0.0006 31 27 30 27 0.382 35 3 2 5 5 0.8112 2 1 3 1 1 36 9 19 24 19 0.1411 8 8 9 8 0.568 37 10 9 12 14 0.2675 12 10 10 10 0.500 38 33 30 33 34 -0.0003 32 32 33 32 0.355 Correlation M-Sh 3 years 0.71 M-Sh 5 years 0.49 0.34 0.82 0.99 ? 0.82 Note: M-Sh = modi?ed Sharpe ratio, M-St = modi?ed Stutzer index, ? = Omega index. Funds with negative scores are those whose elementary performances are all negative.
29
lation between classi?cations are rather weak: 0.49 (5 years versus 3 years), 0.34 (5 years versus 1 year) and 0.71 (1 year versus 3 years). The most striking example is fund 14 classi?ed at the 31th position over ?ve-year horizon but at 6th and 1th ranks over respectively three-year and one-year horizons. According to DEA, this fund is classi?ed as dominant with an aggregate score of 1. In detail, it is found to have the performance over one-year horizon not only classi?ed at the 1st rank but also suf?ciently high to compensate for the slightly negative performance over ?ve-year horizon so as to arrive ?nally at the head of the sample. Unlike fund 14, two other dominant funds according to the DEA global performance score — funds 26 and 28 — have very stable performance pro?les over time. With regard to fund rankings according to the three selected performance indicators (Panel B), contrary to the preceding case, classi?cations are overall very coherent between them. This coherence is con?rmed by high coef?cients of rank correlation: 0.99 (M-Sharpe with Omega), 0.82 (M-Sharpe with M-Stutzer) and 0.82 (Omega with M-Stutzer). Such ?nding certainly does not provide an ideal illustration of the problem which this study aims to illustrate here. Nevertheless, in detail, rankings given by the M-Stutzer index and those provided by the M-Sharpe ratio and the Omega index are quite divergent on several occasions. It is particularly true for fund 6 which is ranked at the 11th place by the M-Stutzer index but ranked only at the 29th and the 28th places in the classi?cation of respectively the Omega index and the M-Sharpe ratio. The other examples are funds 4, 10, 12, 15, 16, 19 and 32. In such cases, applying DEA to determine de?nitive ranks of these funds presents an undeniable interest. Finally, three funds globally quali?ed as dominant are funds 27, 28 and 35. All of them are the most highly ranked of the sample, no matter what measure is used. In these cases, the dominance of these funds is obvious, DEA can only con?rm it. Only when performance measures disagree on fund rankings that DEA proves its perspicacity by providing for each fund an aggregate indicator of performance allowing a global and de?nitive classi?cation. Such is the case of funds 14, 18, 22, 25, 31, 33 and 34. Consider for example fund 25. It is ranked 20th by the M-Stutzer index, 24th by the Omega index and 22th by the M-Sharpe ratio. However, according to DEA, it is globally placed only 24th when all three performance indicators are considered. Similarly, fund 34 is at the 31th position of the list according to M-Stutzer, 27th according to Omega, 30th according to M-Sharpe but globally ranked 27th by DEA.
30
Conclusion
Previous studies documented that DEA could be a good tool to solve decision-making problems with multiple criteria, including investment fund performance evaluating task. This paper shows that DEA is particularly adapted to assess hedge fund performance for the following reasons. First, it can incorporate multiple risk-return attributes of non normal returns in an unique aggregate score so as to rank funds. Hence, DEA can be used to evaluate local and global performances of hedge funds. The local performance is obtained when evaluation criteria include risks, eventually sales loads (DEA’s inputs) and returns (DEA’s outputs). The global performance is de?ned as the aggregate score of several elementary performances which could be performances over several temporal horizons, or performances over one temporal horizon but measured by different indicators. Second, unlike other performance measures, DEA offers investors the possibility to exert direct control on the importance level paid to each evaluation criteria. Thus, each investor can tailor his own performance measure to select funds corresponding the most to his own preferences. This ?exibility is very important as in reality, each investor usually has his own preferences and constraints. Third, by putting emphasis on the best observed funds, DEA makes no assumption on the functional relation between evaluation criteria. To this end, this paper focuses on the most important methodological issues concerning the application of the basic CCR model to hedge fund performance appraisal, namely (1) the choice of evaluation criteria as DEA’s inputs and outputs, (2) the choice between input-oriented or output-oriented version of the CCR model, (3) dealing with negative inputs and outputs, and (4) transcribing investors’ speci?c preferences into mathematical constraints. These elements are presented in such a way to provide investors with a general framework to apply DEA in assessing fund performance. In order to make these guidelines more intuitive, several numerical illustrations with thorough discussion of results are provided on a sample of 38 hedge funds. The illustrations also highlight the importance of correct speci?cation of evaluation criteria and preference structure for ef?cient application of DEA. A comparison between DEA classi?cation and rankings provided by traditional Sharpe and modi?ed Sharpe ratios indicates that they are sometimes radically inconsistent. Further examination of funds’ return distributions suggests that these latter two measures might not price properly good surprises (extremely high positive returns). In such case, DEA proves to be a good supplement to improve the precision of selection tasks. Although this paper only addresses the application of DEA in the hedge fund context, its guidelines are also applicable to other types of investment funds like mutual funds, pension funds or ethical funds, etc.
31
Like any other tools, DEA also has its caveats. One of the main weaknesses arises from the fact that DEA basic models do not provide complete rankings of dominant funds. Nevertheless, this weakness can be mitigated by either adding more restrictive preferences (additional mathematical constraints) so that the short list becomes more and more selective, or applying other qualitative and quantitative criteria on dominant funds so as to rank them. Besides, the dominance or ef?ciency of funds is only relative to the other funds in the sample and thus can be changed once the sample is modi?ed. However, relative evaluation is a well-established concept in economic literature (Holmstrom 1982). In addition, the relative property of fund evaluation is still quite valuable because in the investment industry, funds are often rated relatively to others in the same category. A broad literature documented that the investment fund market is a tournament and the managers compete against each other in the same category to attract investors (Brown et al. 1996, Agarwal et al. 2003, Kristiansen 2005).
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doc_854825880.pdf
Data envelopment analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers[clarification needed]. It is used to empirically measure productive efficiency of decision making units (or DMUs). Non-parametric approaches have the benefit of not assuming a particular functional form/shape for the frontier, however they do not provide a general relationship (equation) relating output and input. There are also parametric approaches which are used for the estimation of production frontiers (see Lovell & Schmidt 1988 for an early survey).
On the Use of Data Envelopment Analysis in Hedge Fund Performance Appraisal
Huyen Nguyen-Thi-Thanh?† This Draft: December 2006
Abstract This paper aims to show that Data Envelopment Analysis (DEA) is an ef?cient tool to assist investors in multiple criteria decision-making tasks like assessing hedge fund performance. DEA has the merit of offering investors the possibility to consider simultaneously multiple evaluation criteria with direct control over the priority level paid to each criterion. By addressing main methodological issues regarding the use of DEA in evaluating hedge fund performance, this paper attempts to provide investors suf?cient guidelines for tailoring their own performance measure which re?ect successfully their own preferences. Although these guidelines are formulated in the hedge fund context, they can also be applied to other kinds of investment funds. JEL CLASSIFICATION: G2, G11, G15 KEYWORDS: hedge fund, mutual fund, data envelopment analysis, performance measures, Sharpe ratio.
d’Economie d’Orl´ eans (LEO), Universit´ e d’Orl´ eans, Rue de Blois, B.P. 6739, 45 067 Orl´ eans Cedex 2. E-mail: [email protected]. † I am grateful to Georges Gallais-Hamonno and Michel Picot for their valuable guidance and support. I thank all participants in the 2006 MMF Conference (York, the United-Kingdom), the 2006 International Conference of the French Finance Association (Poitiers, France) and LEO members for their helpful comments. I also thank the company Standard & Poor’s, especially Marina Ivanoff and Zahed Omar, for providing hedge fund data used in this study. Naturally, all errors remain mine alone.
? Laboratoire
1
Introduction
The highly successful performance of the so-called hedge funds over the past two decades, notably during the long bull equity market of the 1990s, has made them quickly wellknown to ?nancial communities as well as to the public. While hedge funds still manage only $1 trillion at the end of 2004, a fraction of the $8 trillion invested by mutual funds, their assets have ballooned from only about $150 billion a decade ago. With over 8,000 hedge funds now available, fund selecting is quite challenging for investors. Hence, before any due-diligence process, investors ?rst need an ef?cient tool to assist them in screening task in which the most important evaluation is undoubtedly fund (historical) performance. In general, the historical performance of funds is de?ned as their return adjusted for risk. According to traditional ?nancial theories, the risk is measured either by the standard deviation of returns or by the correlation of fund returns with market factors via different betas1 . Most of these measures, even though validated in ”buy-and-hold” portfolios of mutual funds and pension funds, are irrelevant within the context of hedge funds. On the one hand, hedge fund returns are documented as usually asymmetric and kurtotic, a feature largely imputed to the intensive use of short sales, leverage, derivative instruments and to the free call-option like incentive structure, all speci?c to only the hedge fund industry. On the other hand, their short-term movements across diverse asset categories and the market neutral absolute investment objective of hedge fund managers make it really delicate to identify market factors necessary to the use of multi-factorial models2 . Recent techniques enlarge the evaluation dimension to the skewness (Stutzer 2000), the skewness and the kurtosis (Gregoriou & Gueyie 2003) or to the whole distribution of returns (Keating & Shadwick 2002) in order to take into account the non normality of return distributions. Despite this signi?cant progress, these measures do not allow considering after-net-returns fees paid by investors if only. Besides, most of them are restrictive in the sense that they often assume very simplistic decision-making rules which are common to all investors. Yet, it is well documented that actual evaluation criteria, in fact, may be more complicated and differ signi?cantly from theoretical formulations. Not only are there many
fund’s risk is measured by betas, fund performance is simply the alpha. other kinds of investment funds, hedge funds are loosely regulated, and in many cases, are largely exempted from legal obligations as the case of offshore hedge funds. Hedge fund managers thus have a broad ?exibility in determining the proportion of securities they hold, the type of positions (long or short) they take and the leverage level they make. As a consequence, they are free to make very short-term movements across diverse asset categories involving frequent use of short sales, leverage and derivatives to attempt to time the market.
2 Unlike 1 When
2
attributes to consider, each one being associated with a priority level, but also these attributes and their importance level are usually quite speci?c to each investor. The need to consider simultaneously multiple criteria while incorporating investors’ own preferences is natural since they are do not always share the same ?nancial objective, risk aversion, investment horizon, etc. From such perspectives, the Data Envelopment Analysis approach (hereafter, DEA) seems particularly appealing as it provides the possibility of incorporating many criteria at the same time, together with a direct control over the importance level paid to each criterion by means of a tailor-made optimizing system. DEA can be roughly de?ned as a mathematical optimizing technique ?rst introduced by Charnes, Cooper & Rhodes (1978), based on Farrell (1957)’s ef?ciency concept, to measure the ef?ciency (technical, allocative, economic, etc.) of decision-making units (hereafter, DMU) whose objective consists in transforming multiple inputs into multiple outputs. The merits of the DEA method lies in providing an unique aggregate measure for each DMU from a system of multiple inputs and multiple outputs and in putting emphasis on the ”best observed practices” in a comparative perspective. In addition, DEA allows considering inputs and outputs whose measure units are different, a property known as ”units invariance”3 . Furthermore, it makes no assumption on the form of the relation between inputs and outputs. Because of its many advantages, DEA has been applied in various ?elds including public administration (to evaluate hospitals, administrative of?ces, educational establishments or to resolve siting problems), engineering (to evaluate airplanes and engines), commerce (to evaluate supermarkets), ?nance (to evaluate bank branches, micro-?nance institutions, assurance companies, to identify dominant ?nancial assets and recently to assess investment funds’ performance). The application of DEA is generally proceeded in two main perspectives: (1) to evaluate the ef?ciency of DMUs whose activities are to employ inputs to produce outputs; and (2) to solve decision-making problems with multiple criteria. It is in the second perspective that DEA can be applied to assess hedge fund performance. Initiated by Murthi et al. (1997) to evaluate empirically the performance of mutual funds, this idea has been applied and revisited by several studies, including those on hedge fund performance. However, this literature is composed essentially of empirical applications, methodological issues remain either ignored or discussed in a simplistic and super?cial manner with little directive value. To the best of my knowledge, none of methodological studies investigates the use of DEA in the hedge fund context.
3 This is true provided that unit measures are the same for all DMUs in the sample. For example, one person can measure outputs in mile and inputs in gallons of gasoline and quarts of oil while another measures these same outputs and inputs in kilometers and liters with the same collection of automobiles .
3
Following this literature, this paper is devoted to methodological issues in applying DEA to hedge fund performance appraisal. Speci?cally, I focus on the choice of evaluation criteria (DEA’s inputs and outputs), the choice of DEA models with and without negative data on returns and performance, and on ”transcribing” speci?c evaluating preferences of investors into mathematical constraints. By doing so, this study attempts to offer investors suf?cient guidelines in order to apply successfully the DEA method to assessing hedge fund performance. Although it only addresses the hedge fund context, the whole framework is completely applicable to mutual funds, pension funds, ethical funds, etc. The remainder of the paper is organized as follows. Section 1 reviews brie?y the literature related to this study. Section 2 introduces basic concepts of the DEA method. Section 3 addresses methodological issues of applying DEA to screening hedge funds via their performance. Section 4 provides several numerical illustrations on a sample including 38 hedge funds. The last section summaries and concludes the paper.
1 Related literature
This study emanates from two main streams of literature. The ?rst one concerns DEA’s use in making a selection when decision-makers have multiple criteria. The second evolves evaluating the performance of investment funds by means of the DEA method. With respect to the ?rst literature, three studies can be enumerated: Thompson et al. (1986), Tone (1999) and Powers & McMullen (2000). Thompson et al. (1986) dealt with identifying feasible sites among six candidate sites for location of a very high-energy physics lab in Texas. A comparative analysis between six sites was conducted by applying the basic DEA model, incorporating project cost, user time delay, and environmental impact data as selection criteria. These criteria are those evaluators want to minimize, they thus form exclusively the DEA’s inputs. Being absent, the output is assumed to be unique and equal to unity so that DEA can be applied. This setting is naturally plausible as it is equivalent to considering inputs per one unity of output4 . In the same spirit, Tone (1999) described a japanese governmental project applying DEA to select a city to take over some political functions of Tokyo as a new capital. In this study, the selection criteria are composed of distance from Tokyo, safety indexes (regarding earthquakes and volcanoes), access to an international airport, ease of land acquisition, landscape, water supply, matters with historical associations; they form exclusively DEA’s outputs. The
4 Inputs
(outputs) include all criteria that evaluators want to minimize (maximize).
4
input is thus set to be equal to unity5 . It is important to note that in these studies, only inputs (outputs) are available and thus output (input) is assumed to be unique and equal to 1. Another common interesting point is that the evaluators, with prior expert knowledge about the relative importance of chosen criteria, ?xe lower and upper bounds to the weights associated with each criterion in the mathematical optimization. In ?nance, Powers & McMullen (2000) suggested using DEA to select dominant stocks among the 185 american largest capitalization stocks because this technique makes it possible to incorporate multiple selection attributes such as the Price-Earnings Ratio, the systematic risk and the total risk (DEA’s inputs), the Earnings Per Share ratio and the mean return over 1 year, 3 years, 5 years and 10 years (DEA’s outputs). The second literature relates to studies using DEA to evaluate the performance of mutual funds, ethical funds and more recently hedge funds. Studies on mutual funds include Murthi et al. (1997), McMullen & Strong (1998), Choi & Murthi (2001), Basso & Funari (2001), Tarim & Karan (2001) and Sengupta (2003). All these studies assume that fund performance is a combination of multiple attributes such as mean returns (DEA’s outputs), total or systematic risk, expenses6 , and sometimes even fund size, turnover speed and minimum initial investment (DEA’s inputs). In the same vein, Basso & Funari (2003) suggested putting in the DEA’s outputs, together with the mean return, an indicator measuring funds’ ethical level ful?llments since according to them, ”the solidarity and social responsibility features that characterize the ethical funds satisfy the ful?llment of humanitarian aims, but may lower the investment pro?tability”. The application of DEA in evaluating hedge funds emerged from the work of Gregoriou (2003). It was then supported by Gregoriou et al. (2005)7 and discussed in Kooli et al. (2005). A common feature of these studies is that they only consider risk–return performance without referring to fees. Besides, risks and returns are approximated respectively by lower variations (what investors seek to minimize) and upper variations (what investors seek to maximize) compared to a threshold de?ned by mean return. Speci?cally, the inputs are composed of lower mean monthly semi-skewness, lower mean monthly semi-variance and mean monthly lower return; the outputs include upper mean monthly semi-skewness, upper mean monthly semi-variance and mean monthly upper return. Another common feature is that they put emphasis on fund’s absolute rankings by
did not have access to documents related to this project. All the information mentioned here is extracted from Cooper et al. (2000, p.169). 6 The concept of expenses differs from study to study. It might include transaction costs and administration fees (totaled in expense ratio) and loads (subscription or/and redemption costs). 7 Gregoriou et al. (2005) is an extended version of Gregoriou (2003) and more complete while employing the same DEA methodology with Gregoriou (2003). Therefore, I refer only to Gregoriou et al. (2005).
5I
5
employing modi?ed DEA techniques: super–ef?ciency (Andersen & Petersen 1993) and cross–ef?ciency (Sexton et al. 1986). By comparing DEA results with rankings provided by Sharpe and modi?ed Sharpe ratios via rank correlation coef?cients, they observed a weak consistency between DEA and these measures. In particular, Kooli et al. (2005) found quite low correlation between DEA rankings and rankings given by the stochastic dominance technique and concluded to a weak relevancy of DEA to fund performance evaluation context. With regard to super-ef?ciency and cross-ef?ciency models, despite their appealing property, i.e. providing fund absolute rankings, their technical caveats cast doubts about their ef?cacy8 . Hence, in what follows, I will only introduce the basic DEA model and its dichotomic classi?cation into assessing hedge fund performance.
2 DEA’s approach
2.1 DEA as a measure of technical ef?ciency
Before introducing the general approach of DEA and the basic DEA model, it is important to distinguish the ”technical ef?ciency”, on which is based this study, from the ”economic ef?ciency” usually applied in production context. According to Fried, Lovell & Schmidt (1993, p.9-10), ”productive ef?ciency has two components. The purely technical, or physical, component refers to the ability to avoid waste by producing as much output as input usage allows, or by using as little input as output production allows. . . . The allocative, or price, or economic, component refers to the ability to combine inputs and outputs in optimal proportions in light of prevailing prices.”. Consequently, technical ef?ciency measurement is based solely on quantity information on the inputs and the outputs whereas the economic ef?ciency necessitates the recourse to information on prices as well as on economic behavioral objectives of producers (cost minimization, pro?t maximization or revenue maximization). Conceptually, the ef?ciency of each DMU under evaluation is determined by the distance from the point representing this DMU to the ef?cient frontier (production frontier in the case of technical ef?ciency; cost, revenue or pro?t frontier in the case of cost, revenue or pro?t ef?ciency respectively). In ?gure 1, the isoquant L
represents various combinations of two inputs that a perfectly ef?cient8 The super-ef?ciency model has two main caveats. First, it allocates so excessively high ef?ciency score to ef?cient DMUs having extreme values of inputs and outputs that optimal values can sometimes ”explode”. Second, it is infeasible in some circumstances (Zhu 1996, Seiford & Zhu 1999). The pitfall of the crossef?ciency model is that it penalizes DMUs whose the combination of inputs and outputs is different from the others while it highly praises average DMUs. Besides, the use of the mean, the variance, the mode or the median, etc. of scores to completely rank DMUs is too ambiguous, especially when different indicators provide different rankings.
6
¡£ © ¨ ¥ § ¥¦ !"#$ ¨!" "$
¤
¨¦ ¡¢
Figure 1: Technical ef?ciency versus economic ef?ciency with two inputs (Farrell, 1957)
?rm like Q or Q? might use to produce an unit of output. The line CC ? whose slope is equal to the ratio of the prices of the two inputs represents the price constraint that all the ?rms must face. Farrell (1957) de?ned OQ/OP as the technical ef?ciency level, OR/OQ as the price (cost) ef?ciency and OR/OP as the overall ef?ciency of the ?rm P. In DEA, the production frontier against which the technical ef?ciency of each DMU is derived is empirically constructed from observed DMUs, and thus without any assumption on the functional relation between inputs and outputs9 . In other words, it is formed by a set of best practices (the most ef?cient DMUs) and the other DMUs are enveloped by this frontier, which explains the origin of the name ”Data Envelopment Analysis” of this method. For the shake of brevity, hereafter I will use the term ”ef?ciency” to refer to the technical ef?ciency and the term ”ef?ciency frontier” to denote the production frontier.
2.2
DEA’s basic model — CCR (1978)
2.2.1 The general formulation Consider n DMUs under evaluation that use m inputs (X ) to produce s outputs (Y ) with X and Y are semipositive10 . The ef?ciency score hk assigned to the DMU k is the solution
9 In econometric methods, the ef?cient frontier is estimated by supposing a particular form of the production function (e.g., Cobb-Douglas, translog, etc.). 10 The semipositivity signi?es that all data are nonnegative but at least one component of every input and output vector is positive.
7
of the following optimizing system: ? ur yrk (1) ? vi xik
s
max
u,v
hk =
s
r =1 m
i =1
subject to:
r =1 m
? ur yrj ? vi xij
? 1, j = 1, . . . , n
(2) (3)
i =1
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m
where k is the DMU under evaluation, yrj is the amount of the output r of the DMU j , xij is the amount of the input i of the DMU j , ur and vi (also called ”absolute weights”) are the weights assigned respectively to the output r and the input i, ? is an in?nitesimal positive number imposed to assure that no input or output is ignored in the optimization, vi xij and ur yrj are called ”virtual weights” of respectively the input i and the output r of the DMU j . Mathematically, the model’s objective is to seek for the most favorable (positive) weight system associated with each input and each output which maximizes the weighted sum of the outputs over the weighted sum of the inputs of the DMUk (hk ), provided that this ratio does not exceed 1 for any DMU in the sample (re?ected by constraint (2)). Given that the ef?ciency frontier contains ef?cient DMUs and envelopes inef?cient ones, and that the ef?ciency level of each DMU is, by de?nition, the distance from its position to the ef?ciency frontier, it is natural to ?xe the maximal value of the objective function to unity11 . Thus ef?cient DMUs will obtain a score of 1 and inef?cient DMUs a score smaller than 1. Conceptually, each DMU is free to choose its own combination of inputs and outputs so that it is as desirable as possible compared to other DMUs in the same category. Obviously, this combination must also be technically ”feasible” for others, that is the ef?ciency level of any other DMU using this combination should not exceed the maximum attainable bounded by the ef?ciency curve (the constraint (2) is thus also applied to j = 1, . . . , n with j = k). The idea is that if one DMU can not attain an ef?ciency rating of 100% under this set of weights, then it can never be attained from any other set. It should be noted that in practice, more constraints on weight systems can be imposed to take into account
the maximal value of the objective function can be given any other number without changing the relative ef?ciency of the DMUs. The choice of unity is to assure the coherence between mathematical calculations and ef?ciency de?nitions.
11 Mathematically,
8
speci?c preferences of decision-makers. This point will be illustrated further. Alternatively, the DEA original problem can be formulated as the following system: ? vi xik (4)
m
min
u,v
hk =
m
i =1 s r =1
? ur yrk
subject to
i =1 s r =1
? vi xij
? ur yrj
? 1, j = 1, . . . , n
(5) (6)
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m
where the objective is to seek for optimal weights so as to minimize the ratio of the weighted sum of inputs to the weighted sum of outputs. The smaller this ratio, the better. In this case, ef?cient DMUs have a score of 1 and inef?cient ones have a score greater than 1. Note however that the system (4-6) is less familiar within DEA’s applications in ?nance than the system (1-3). It is important to keep in mind that basic DEA models do provide a dichotomic classi?cation, not a complete ranking of DMUs as all ef?cient DMUs have the same score equal to 1. Besides, ef?ciency or inef?ciency of DMUs is solely relative to the sample under consideration. Hence, once the sample is modi?ed, results may be very different.
2.2.2 The primal program The optimizing systems (1-3) and (4-6) are fractional problems, non convex with fractional constraints, which are quite dif?cult to solve. According to Charnes & Cooper (1962, 1973) and Charnes et al. (1978), the fractional problem (1-3) (or 4-6) can be conveniently converted into an equivalent linear programming problem by normalizing the denominator to 1 and maximizing (minimizing) the nominator. By doing so, we obtain the input-oriented version (system (7-10)) and the output-oriented version (system (11-
9
14)) of the so-called CCR model — the seminal model of the DEA method: Input-oriented: max
u,v
hk =
m
? ur yrk
r =1
s
(7) (8) (9) (10)
subject to:
? vi xik = 1
i =1 s r =1
? ur yrj ? ? vi xij , j = 1, . . . , n
i =1
m
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m or Output-oriented: min
u,v
hk =
s
? vi xik
i =1
m
(11) (12) (13) (14)
subject to
? ur yrk = 1
r =1 s r =1
? ur yrj ?
? vi xij , j = 1, . . . , n
i =1
m
ur , vi ? ?, r = 1, . . . , s; i = 1, . . . , m
The input-oriented (output-oriented) version assumes that only inputs (outputs) can be adjusted, outputs (inputs) being ?xed.
2.2.3 The dual program According to linear programming theories, each primal program is associated with a dual program which provides the same optimal value of the objective function as the primal. The system (7-10) thus has a dual below: Input-oriented: min
? ,?
? ? xik ? yrk ?
(15)
subject to:
? ? j xij , i = 1, . . . , m
j =1 n
n
(16) (17) (18)
? ? j yrj , r = 1, . . . , s
j =1
? j ? 0, ? unconstrained in sign with ? and ? are dual variables. Note that ? can not, by construction, exceed unity12 .
12 We can easily see that ? = 1, ? = 1, ? = 0 ( j = k) is a feasible solution to (15-18). Hence, the optimal j k value of ? can not be greater than 1. Besides, the constraint (16) implies that ? must be positive as X is
10
In a similar fashion, the dual of the system (11-14) is de?ned by: Output-oriented: max
? ,?
? xik ?
(19)
subject to:
? ?j xij , i = 1, . . . , m
j =1
n
(20) (21) (22)
? yrk ?
? ?j yrj , r = 1, . . . , s
j =1
n
? j ? 0, ? unconstrained in sign where ? and ? are dual variables and ? can not be, by construction, lower than 1.
In fact, the primal program can be solved directly to obtain the optimal ef?ciency score. However, the dual program is usually preferred for the following reasons. On the one hand, it is mathematically easier to ?nd the optimal solution via the dual because of a considerable reduction of constraints: from n + s + m + 1 constraints in the primal to only s + m constraints in the dual. This calculating parsimony is of particularly appealing when dealing with large samples. On the other hand, the dual formulation has an interesting economic interpretation. In economic terms, under the input-oriented form (output-oriented form), the dual looks for a feasible activity — a virtual DMU which is a linear combination of the best practices — that guarantees (uses) the output level yk (the input level xk ) of the DMUk in all components while using only a proportion of the Hence, ? (or ? ) is de?ned as a measure of the ef?ciency level of the DMUk . Graphically, in the input-output plan depicted in ?gure 2, under the input-oriented or input contraction setting, ? of the DMU A is the ratio DC / DA, with C being the virtual DMU which serves as benchmark to measure the ef?ciency of A; under the output-oriented or output expansion setting, ? of the DMU A is measured by FH / AH with F being the reference DMU for A now. In order to obtain the ef?ciency scores of n DMUs, the optimizing system (primal or dual) must be run n times with each time the DMU under evaluation changes. Theorem 1 (Connexion between the CCR input-oriented version and the CCR output-oriented version) Let (? ? , ?? ) be an optimal solution for the CCR input-oriented version. Then (1/? ? , ?? /? ? ) = DMUk ’s inputs ? xik (producing higher outputs than DMUk ’s outputs, ? yrk with ? ? 1 ).
(? ? , ?? ) is optimal for the corresponding CCR output-oriented version. Similarly, if (? ? , ?? ) is
optimal for the CCR output-oriented version, then (1/? ? , ?? /? ? ) = (? ? , ?? ) is optimal for the corresponding CCR output-oriented version (Seiford et al. 2004, p.17).
assumed to be semipositive.
11
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.
Q P R I U %
T
&'()0123(4 5(663738329 6@2
A4502
Figure 2: CCR ef?ciency frontier
3 A DEA framework for hedge fund performance appraisal
In ?nancial literature, funds’ historical performance is often measured by the ratio of return to risk. Traditionally founded on the ”mean-variance” basis, the evaluation dimension has been recently extended to the skewness (Stutzer index — Stutzer (2000)), to the skewness and the kurtosis (the modi?ed Sharpe ratio — Gregoriou & Gueyie (2003)), even to the whole distribution of returns (Omega index — Keating & Shadwick (2002)) in an attempt to take into account the non normality features of returns. Despite this improvement, most of these measures are highly restrictive in the sense that they usually assume simplistic decision-making rules common to all investors. Yet, there are suggestions that actual individual decisions differ signi?cantly from theoretical formulations since they are much more complicated and quite speci?c to investors. Often there are more attributes to consider and for each investor, each attribute does not necessarily have the same priority level. While some investors are more concerned with central tendencies (mean, variance), others may care more about extreme values (skewness, kurtosis). One kind of such preferences is summarized by the positive preference for skewness ?rst invoked by Arditti (1967) and then supported by Jean (1971), Kraus & Litzenberger (1976), Francis & Archer (1979), Scott & Horvath (1980), Kane (1982), Broihanne et al. (2004). It implies that individuals prefer portfolio A to portfolio B with higher mean return if both portfolios have the same variance, and if portfolio A has greater positive skewness, all higher moments being the same. In other words, individuals may attach more importance to the skewness than to the mean of returns. Despite the diversity of preferences for moments of returns, most measures assume the
12
same preference structure for all investors. Consider for example the modi?ed Sharpe ratio (hereafter, M-Sharpe) (Gregoriou & Gueyie 2003) computed by the following equation: M ? Sharpe = r ? rf = 2 MVAR W µ ? zc + 1 6 ( z c ? 1) S + r ? rf
1 24
( z3 c ? 3z c ) K ?
1 36
2 ? (2z3 c ? 5z c ) S (23)
where r is the mean return, r f is the average risk-free rate, W is the amount of portfolio at risk, µ is the mean return and naturally equal to r, ? is the standard deviation of returns, S is the skewness, K is the kurtosis excess, zc is the critical value for probability by Favre & Galeano (2002). According to Favre & Galeano (2002) and Gregoriou & Gueyie (2003), all investors are certainly concerned about the skewness and the kurtosis of returns but they share the same preference structure which is necessarily in the form of MVAR. This rigidity is not only restrictive but might bias signi?cantly investors’ choice of funds as their true evaluation criteria are not considered at all or considered but in a biased manner. In addition to that, investors may need to take account of sales loads charged by the fund on their entrance into (front-end sales load) or/and on their exit of the fund (backend or deferred sales load). Unlike management fees which are directly deducted from the fund’s value, sales loads are charged on the net returns paid to investors13 . As a result, a fund with good performance and a high percentage of loads is not necessarily more attractive than another fund which has lower performance but charges lower loads. Moreover, as argued by McMullen & Strong (1998), Morey & Morey (1999) and Powers & McMullen (2000), investors may also be concerned about fund’s performances over various time horizons (over the last year, the last 3 years, the last 5 years and sometimes the last 10 years). Such information is undoubtedly valuable as it provides much more informative insight into fund’s perspective than the performance over only one horizon. Furthermore, even when investors care about the return and the risk, or the performance of funds over only one horizon, it is often quite dif?cult to choose an absolutely
et al. (1997), McMullen & Strong (1998), Tarim & Karan (2001), Choi & Murthi (2001) and Sengupta (2003) advocated incorporating also expense ratio (in percentage of fund assets, covering various operating expenses incurred by the fund management such as management fees, administrative fees, advisory fees) in evaluating fund performance. This element which is obviously necessary to appraise the performance of funds in a productivity perspective, i.e. their capacity to exploit ef?ciently input resources (fund expenses are considered here as a production factor), is irrelevant in this context where inputs and outputs are selection criteria chosen by investors. In this regard, investors are not likely concerned by these expenses as they are directly deducted before calculating funds’ net asset value — the real value of investors’ investments. Hence, such expenses are generally invisible to investors.
13 Murthi
(1 ? ?) (zc = ?1.96 for a 95% probability), MVAR (modi?ed value-at-risk) is introduced
13
suitable measure among a wide range of existing measures in the literature. This dif?culty is particularly true for the choice of risk and performance indicators because of inexistence or de?ciency of mechanisms to validate empirically them. Consequently, investors are sometimes in need of considering simultaneously several measures. Here again, they do not necessarily share the same preferences for such and such measures. Given these speci?cities in performance evaluating practices, the DEA’s approach seems very appealing. In fact, the application of DEA into hedge fund performance appraisal can be made in two perspectives. The ?rst one consists in evaluating the productive performance of funds where the latter are considered as a particular type of production units which employs multiple resources (risks, various operating expenses, turnover speed, etc.) to realize pro?ts (returns). The second, which is in the spirit of Thompson et al. (1986), Tone (1999) and Powers & McMullen (2000), aims to assess funds as decision–making units whose inputs and outputs are evaluation criteria chosen by decision-makers. It is the second perspective that interests investors as DEA, with its broad ?exibility, allows investors to tailor their own evaluation tools corresponding the most to their own preferences. Since each investor naturally has different risk aversion levels, performance objectives and other distinct constraints, the tailor-made possibility is essential to correctly screen fund. In this context, the DEA method can be applied to evaluate either the ”local” performance or the ”global” performance of hedge funds. By the ”local performance”, I imply the performance measured by the weighted sum of several criteria of gain (or return) on the weighted sum of several criteria of risk and possibly certain types of expenses. In contrast, the term ”global performance” denotes the performance synthesized from either several measures of ”local” performance, or elementary performances over several temporal horizons. Within this framework, the application of DEA (in its basic form) raises four main questions: (1) how to choose inputs and outputs, (2) what version to choose (input-oriented or output-oriented), (3) how to deal with negative values in the inputs or/and the outputs if they exist14 , and (4) how to incorporate more speci?c preferences of investors into the mathematical formulation. If the ?rst, the second and the fourth questions are relevant to any application ?elds, the third one is quite speci?c to data of returns and performances. These issues will be addressed successively in what follows.
14 Inputs
and outputs of DEA are originally assumed to be semipositive.
14
3.1
Evaluation criteria and the choice of inputs and outputs
Unlike applications of DEA in production ?elds where inputs and outputs are tangible elements, the choice of inputs and outputs is not straightforward when dealing with fund performance. Nevertheless, in a multiple criteria decision-making framework, it is logical to consider inputs as criteria that investors want to minimize and outputs as those they want to maximize. Hence, if investors seek to evaluate the funds’ ”local” performance, i.e. returns15 to risks, the inputs can be (1) several measures of risk (standard deviation, kurtosis, beta, various measures of value-at-risk) over one (or several) horizon(s), (2) possibly the sales loads; the outputs can be composed of (1) several measures of returns (mean, skewness) — over one (or several) horizon(s). The difference between the con?guration suggested here (to evaluate fund ”local” performance) and that assumed by standard performance indicators is that according to the former, each investor knows perfectly his relevant evaluation criteria but does not know the functional relation between these criteria as well as the exact trade-off between them, which is not the case of the modi?ed Sharpe ratio as previously described. The case where investors know the relative trade-off between these criteria will be discussed further. Otherwise, if investors want to evaluate funds by considering several elementary performances simultaneously, they can calculate the global performance by (1) including in the outputs either the performances measured by the same technique on several periods, or the performances on the same period but measured by several indicators, (2) setting the input equal to one. It is important to notice that in this setting, all selection criteria are those investors want to maximize, they thus form exclusively DEA’s outputs; meanwhile, there is no input. Assuming the presence of one input equal to 1 makes it possible to apply DEA without any modi?cation of results. As explained earlier, we are in a basic con?guration in which there is one input and several outputs and the quantity of each output is often ”standardized” by the quantity of the input to obtain the unit outputs (per one unit of the input) in order to facilitate calculations. This setting is employed by Thompson et al. (1986) and Tone (1999). Following the principles evoked above, each investor will determine, according to his own preferences, the inputs and outputs for DEA while complying with general rules as: – Inputs and outputs must be criteria indispensable to the appraisal of fund perforterm ”return” should be understood here in broad sense. In traditional language of portfolio theories, the concept of return is always associated to the arithmetic mean of elementary returns over a given period. By ”return”, I imply in what follows any measure, in addition to the mean return, that is indicative of fund’s expected returns such as the skewness.
15 The
15
mance. – The number of inputs and outputs should be lower than the number of funds. In general, the number of funds should be at least three times larger than the number of inputs and outputs. Any violation of these rules will lead to a de?ciency of the discriminatory power of DEA. As a result, we risk obtaining an excessive number of dominant (ef?cient) funds whereas some of them are not rightly so16 .
3.2
Input-oriented or output-oriented versions ?
In general, when inputs and outputs are semipositive, the choice between the CCR inputoriented version and the CCR output-oriented version can be simply made at users’ discretion following their preferences. Note that the input-oriented version (output-oriented version) assumes that outputs (inputs) are ?xed, only inputs (outputs) can be adjusted. This assumption conditions the reference fund on the ef?cient frontier to which is compared the target fund and thus determines the distance between the former and the latter, this distance measuring the ef?ciency level of the latter. The theorem 1 describes the correspondence between the optimal solutions of the two versions. We can easily see that the two versions of the CCR model provides the same classi?cation of inef?cient DMUs17 (ef?cient DMUs always obtain a full score of 1 under any version). Nevertheless, it is interesting to notice that all studies which apply DEA to evaluating fund performance adopted the input-oriented version whatever the DEA model is used. This popularity is undoubtedly due to the fact that this mathematical form shares the same logic as Markowitz’s ef?cient frontier construction, that is to minimize the risks (inputs) for a de?ned level of returns (outputs). However, when there are only outputs (inputs), the input-oriented (output-oriented) version is required as in this case, we assume the existence of one input (output) whose quantity is ?xed equal to 1.
16 The terms ”dominant funds” and ”ef?cient funds” will be used interchangeably hereafter to indicate funds having a full ef?ciency score of 1. 17 It is important to specify that this equivalence between the input-oriented version and the outputoriented version is only valid under the constant returns-to-scale technology assumed by the CCR model.
16
3.3
Dealing with negative inputs and outputs
DEA models as originally designed require that inputs and outputs are semipositive, i.e. all inputs and all outputs are non negative and at least one input and one output are positive. In many application ?elds like production economics, negative inputs and outputs naturally make no sense. However, in fund performance appraisal context, it is likely that we sometimes have negative values like mean, skewness of returns, or some performance indicators, etc. Although in the CCR model, or more generally in basic DEA models, inputs and outputs are systematically required to be semipositive, we can easily see that negative values in inputs and outputs are tolerated in following ways without any incidence on the solubility of DEA optimizing systems (Cooper et al. 2000, p.304-305): – If there are at least one input and one output positif, either the input-oriented version or the output-oriented version can be used; – If all outputs (inputs) are negative and at least one input (output) is positive, the input-oriented (output-oriented) version is required; – If there is no (effective) input (output) and all outputs (inputs) are negative, the input-oriented (output-oriented) version is required; – The case where all inputs and all outputs are negative at the same time, which is extremely rare in fund performance appraisal context, can not be dealt with within the DEA framework. Note that in the second and the third cases, the optimal value of the objective function will be negative.
3.4
Taking account of investors’ more speci?c preferences
The CCR model as presented earlier allows a quasi-absolute freedom in the determination input and output level. Speci?cally, {u, v} are only required to be equal to or greater than of the weights {u, v} so that each funds obtains a maximum score of ef?ciency, given its
an in?nitesimal positive number ?. This constraint is essential to assure that all selected evaluation criteria are considered in the evaluating process. Nevertheless, such ?exibility level also implies that important, even excessive, weights can be assigned to the input(s) or/and the output(s) which make the funds as ef?cient as possible compared to others. 17
As a result, this setting is only plausible when investors have no idea about the tradeoff between the selected criteria. When such information is available, it can be easily virtual weights {uy, vx } associated with each input and each output. incorporated in DEA optimizing systems by restricting the absolute weights {u, v} or the An investor in full knowledge of the ”price” range for each evaluation criterion — e.g. the coef?cient of aversion to the mean, the variance, the skewness or the kurtosis of returns — can have recourse to constraints like: ur ? ? r vi ? ?i ?r ? ur ? ? r ?i ? vi ? ?i (24) (25) (26) (27)
An investor who knows more or less his personal or conventional trade-off or substitution rate between evaluation criteria can add following constraints into original DEA program: ur ?? vi vi ?? v i +1 ? i v i + ? i +1 v i +1 ? v i +2 (28) (29) (30)
An investor who wants to control the relative importance of each criterion in the performance appraisal process will formulate additional constraints on the virtual weights as follows: ar ? ci ? ur yrj s ?r=1 ur yrj vi xij m ?i=1 vi xij
? br ? di
(31) (32) (33) (34)
ur yrj ? ur+1 yr+1, j vi xij ? vi+1 xi+1, j
where ? , ?, ?, ?, ? , ? , a, b, c, d are values pre-de?ned by investors to bound absolute and virtual weights. There are certainly many other forms of additional constraints because of a broad
18
variety of investors’ preferences. The constraints mentioned above are to give examples of ”transcribing” more speci?c preferences into mathematical formulations. A numerical illustration will be provided further. This possibility of exerting a direct control on the relative importance of each evaluation criteria in assessing fund performance, along with the choice of evaluation criteria (inputs and outputs), makes it possible for each investor to conceive a customized measure corresponding to his preferences. With such quality, the DEA approach is an ef?cient and complementary tool to other existing measures.
4 Illustrative applications
4.1 Data
To illustrate the use of DEA in assessing the performance of hedge funds, I used a sample of 38 hedge funds belonging to the category Equity Hedge18 . Data includes 60 monthly returns covering the period of January 2000 to December 2004. Table 1 reports some descriptive statistics of these funds. As we can see, return distributions of many funds show highly positive (negative) skewness signifying higher probability of extreme positive (negative) values compared to that implied by the normal distribution. Besides, many of them possess high kurtosis excess, which indicates more returns close to the central value but also more regular large positive or negative returns than in a normal distribution. The normality assumption of return distributions is tested by means of three tests: Shapiro-Wilk, KolmogorovSmirnov, and Jarque-Bera. Results provided by the Shapiro-Wilk and Jarque-Bera tests are quite similar although they are rather different from those provided by the KolmogorovSmirnov test. This divergence is likely due to the sample’s limit size as the KolmogorovSmirnov test is more appropriate to large samples. According to the Shapiro-Wilk test, documented as the most reliable for small samples, the normality assumption is rejected in 14 out of 38 cases at the con?dence level of 95%. These ?ndings imply much higher return or risk of these funds than those approximated under normality assumption. They thus highlight the importance of incorporating moments of order higher than the mean and the variance when appraising funds’ return and risk pro?les.
38 funds are extracted from a database provided by the company Standard & Poor’s. Equity Hedge covers several different strategies whose investments are focused on the equity markets. Its two large categories are Global Macro and Relative Value.
18 These
19
Table 1: Descriptive statistics
Funds Min Max Me SD SK KU S-W K-S J-B (%) (%) (%) (%) 1 -6.16 10.20 0.68 3.77 0.34 -0.35 0.98 0.11 1.43 2 -7.89 7.69 0.16 3.33 -0.07 -0.25 0.99 0.06 0.20 3 -12.51 19.66 -0.30 5.59 0.32 1.86 0.95** 0.12 9.72*** 4 -7.25 5.63 0.25 3.04 -0.27 -0.49 0.98 0.07 1.31 5 -11.37 11.95 0.10 4.65 -0.10 0.09 0.99 0.06 0.12 6 -6.50 5.92 0.06 2.34 -0.36 0.29 0.98 0.08 1.50 7 -14.67 24.36 -0.11 6.23 0.91 3.37 0.95*** 0.11 36.60*** 8 -22.96 33.89 -0.18 8.86 0.71 2.69 0.96** 0.06 23.09*** 9 -7.87 8.59 -0.01 3.92 -0.01 -0.43 0.98 0.07 0.46 10 -11.84 13.05 0.03 5.40 0.16 -0.12 0.99 0.08 0.29 11 -8.19 17.11 1.08* 4.95 1.18 2.13 0.92*** 0.16* 25.40*** 12 -13.49 9.11 0.01 4.35 -0.43 0.66 0.98 0.08 2.98 13 -6.77 7.23 -0.57 3.36 0.12 -0.75 0.98 0.07 1.53 14 -40.85 19.45 -0.73 9.02 -1.42 5.88 0.90*** 0.15 106.6*** 15 -12.04 14.17 0.23 5.53 -0.05 0.06 0.98 0.07 0.03 16 -5.76 6.58 0.24 2.83 0.13 -0.22 0.99 0.06 0.29 17 -7.10 6.27 0.33 3.25 -0.15 -0.49 0.98 0.06 0.83 18 -6.33 5.94 0.15 2.65 0.00 -0.33 0.99 0.08 0.28 19 -7.21 8.18 0.25 3.43 0.17 -0.46 0.99 0.06 0.81 20 -9.90 14.45 0.12 4.54 0.25 0.71 0.99 0.05 1.91 21 -6.81 9.77 0.64* 2.75 0.48 2.08 0.96* 0.10 13.19*** 22 -9.20 7.57 -0.23 3.95 -0.09 -0.38 0.99 0.05 0.44 23 -5.31 6.82 0.33 2.48 0.62 0.59 0.96** 0.15 4.66*** 24 -13.75 15.03 -0.81 5.72 -0.10 0.30 0.98 0.09 0.33 25 -9.78 19.59 0.09 5.46 1.35 3.85 0.90*** 0.12 55.13*** 26 -16.34 25.90 1.34 * 5.31 1.35 10.02 0.72*** 0.24*** 269.3*** 27 -1.24 5.45 0.52*** 1.10 1.64 5.81 0.88 *** 0.17* 111.2*** 28 -2.37 15.86 0.74 ** 2.39 4.36 27.27 0.63 *** 0.19** 2050*** 29 -15.48 22.37 0.64 5.07 1.23 7.06 0.85 *** 0.16* 139.7*** 30 -13.76 17.90 -0.35 5.29 0.50 1.86 0.97 0.09 11.22*** 31 -14.30 17.93 0.28 5.00 0.21 2.69 0.95** 0.10 18.58*** 32 -6.93 11.54 0.00 3.15 0.73 2.47 0.96** 0.08 20.64*** 33 -7.88 11.53 0.08 4.07 0.58 0.71 0.97 0.08 4.57 34 -7.12 8.67 -0.07 4.12 0.20 -0.64 0.97 0.08 1.42 35 -5.68 10.93 1.10*** 2.97 0.38 1.30 0.96 * 0.12 5.66* 36 -10.13 7.95 0.58 3.75 -0.29 -0.05 0.98 0.08 0.82 37 -6.48 11.01 0.38 2.60 0.71 4.25 0.93 *** 0.10 50.14*** 38 -9.93 12.48 -0.13 4.10 0.04 0.71 0.98 0.07 1.26 Me = Mean, SD = Standard deviation, SK = Skewness, KU = Kurtosis excess relatively to the normal distribution. S-W = Shapiro-Wilk, K-S = Kolmogorov-Smirnov, J-B = Jarque-Bera are normality tests on return distributions. ***, **, ** denote the rejection of the normality assumption at respectively the 99%, 95% and 90% con?dence levels.
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4.2
Assessing hedge fund local performance
4.2.1 Settings Due to unavailable data on sales loads charged by the funds in the sample, illustrations are limited to considering their historical return and risk pro?les. Since the distribution of hedge fund returns is documented as usually non gaussian, it is important to incorporate these features into the selection of evaluation criteria (DEA’s inputs and outputs). Several settings are likely. The ?rst setting assumes the case where investors have a positive preference for odd moments and a negative preference for even moments. Given these preference, it is logical to include in the inputs the standard deviation and the kurtosis of returns, and in the outputs the mean and the skewness. In this con?guration, the problem of negative outputs raises. More speci?cally, 11 out of 38 funds under consideration have a negative mean, 12 funds have a negative skewness, 4 funds among them have simultaneously negative mean and negative skewness. The second setting is designed in the spirit of Gregoriou et al. (2005) and Kooli et al. (2005), following which it is more clever to reason in terms of partial variations. As documented in the literature, investors are likely to be averse only to volatility under the Minimum Accepted Return (MAR)19 , which are called lower variations. In contrast, they appreciate volatility above this value, which are called upper variations. Thus, the composition of inputs and outputs can be determined in the following manner. The inputs include lower mean, lower semi-standard deviation, lower semi-skewness and lower semi-kurtosis which are obtained from returns lower than the MAR represented by the average rate over the period january 2000 to december 2004 of the US 3-month T-bill. The outputs contain upper mean, upper semi-standard deviation, upper semi-skewness and upper semi-kurtosis obtained from returns greater than the MAR. In addition to his ?nancial ?nesse, this con?guration has the clear-cut advantage to avoid the problem of negative inputs and outputs. Now assume furthermore that investors are more concerned for extreme values than central ones. Hence, they naturally pay more attention to the skewness and the kurtosis than to the mean and the standard deviation. Mathematically, they will require that the contribution of the upper (lower) skewness and kurtosis to the ef?ciency score of the fund must be greater than or equal to the contribution of the upper (lower) mean and
19 The determination of the Minimum Accepted Return is purely subjective and speci?c to each investor. It can be a risk-free rate or any rate required by investors.
21
standard deviation. This preference can be taken into consideration by adding four more constraints on virtual weights into the optimization system: y3 j u3 y3 j u3 y4 j u4 y4 j u4 y1 j u1 ; x3 j v3 y2 j u2 ; x3 j v3 y1 j u1 ; x4 j v4 y2 j u2 ; x4 j v4 x1 j v1 x2 j v2 x1 j v1 x2 j v2
where y1 j , y2 j , y3 j , y4 j are the amount of upper mean, upper standard deviation, upper skewness and upper kurtosis of the fund j under consideration; x1 j , x2 j , x3 j , x4 j are the amount of its lower mean, lower standard deviation, lower skewness and lower kurtosis; u1 , u2 , u3 , u4 , v1 , v2 , v3 , v4 are the weights associated respectively with these outputs and inputs. Otherwise, if investors are more or less markowitzian, i.e. they rely essentially on the mean and standard deviation to assess fund performance, the following constraints are necessary so that this preference is incorporated: y1 j u1 y1 j u1 y2 j u2 y2 j u2 y3 j u3 ; x1 j v1 y4 j u4 ; x1 j v1 y3 j u3 ; x2 j v2 y4 j u4 ; x2 j v2 x3 j v3 x4 j v4 x3 j v3 x4 j v4
The third setting illustrates another case where investors need to reconcile funds’ local performance over several horizons, from a long period to a more recent period in the past. To this end, DEA inputs are modeled by the MVAR (described by the denominator of equation 23) representing the loss limits over three horizons: 1 year, 3 years and 5 years; outputs are the mean returns over these three horizons. Again, many cases of negative outputs are found: 12 cases over the one-year horizon, 22 cases over the three-year horizon and 11 cases over the ?ve-year horizon, among them 5 funds have all negative outputs. It is important to keep in mind that the above settings are only some standard con?gurations used by investors. Given the diversity of investors’ preferences, many other con?gurations are also expected.
22
4.2.2 Choice of CCR version After inputs and outputs corresponding to investors’ preferences are speci?ed, the next step consists in running the foregoing inputs and outputs under the CCR model. Then what version to choose, input-oriented or output-oriented? Following principles highlighted in the section 3.3, we are constraint to adopt the input-oriented version for the ?rst and the third settings where outputs are sometimes all negative. Regarding the second setting, either version is possible. However, in this study, the input-oriented version is chosen for all settings. Its primal and dual programs are described respectively by the systems (7-10) and (15-18). Note that the weights assigned to each output and input are constrained to be equal to or greater than 0.001 (? = 0.001)20 to assure that all criteria are considered in the optimization program.
4.2.3 Results Table 2 displays detailed results on DEA score, absolute weights (u, v) and virtual weights (uy, vx) obtained under a CCR input-oriented setting with mean and skewness as outputs, standard deviation and kurtosis as inputs. Funds with negative scores are those having simultaneously negative mean and negative skewness. Given the difference of measure scale between mean, standard deviation on the one hand and skewness, kurtosis on the other hand, virtual weights rather than absolute weights are more informative about key factors (inputs and outputs) that make some funds dominant compared to others in the sample. Each of the ?ve funds quali?ed as dominant (1, 11, 27, 28, 35) has its own combination of evaluation criteria to attain the full ef?ciency. For fund 27 and fund 35, the virtual weights associated with the mean and the standard deviation are much higher than those associated with the skewness and the kurtosis. By referring to the statistics of returns given in table 1, we ?nd that they effectively have fairly high mean and small standard deviation in comparison with the others. Their pro?les are thus well adapted to markowitzian investors. By contrast, the dominance of fund 28 is primarily due to its positive skewness. In fact, this fund has the highest positive skewness in the sample. With fund 1, the dominance is mainly based on the mean and the kurtosis while with fund 11, dominant factors are the mean, the skewness and the kurtosis. These ?ndings imply that not all dominant funds are necessarily adapted to an investor having a precise
fact, all calculations were already tested with several values of ?: 0, 0.0001, 0.001 and 0.01. However, performance scores changed very slightly while the relative rank between funds remains unchanged. Thus, ? was ?xed to be equal to 0.001 to facilitate result presentation.
20 In
23
preference. Consequently, when no additional constraint is formulated like in this setting, an investor who is not completely indifferent among evaluation criteria should identify the factors determining the ef?ciency of dominant funds and select only those whose pro?les correspond the most to his preferences. Results on DEA scores across various settings are summarized in table 3. Note that in the ?rst and the third settings (respectively in columns 2 and 6), funds with negative scores are those whose all outputs are negative. Several points are noteworthy. In general, results are rather sensitive to the speci?cation of evaluation criteria and supplementary constraints. Not only the number of dominant funds varies (from 1 to 5) but also dominant members differ across settings. Look at for example fund 26 which is quali?ed as dominant only when its returns and risks over three horizons are considered simultaneously. Related to the second setting, as would be expected, the introduction of additional constraints on virtual weights naturally deteriorates ef?ciency scores and the short list of dominant funds becomes more selective. When preferences for extreme values (represented by the skewness and the kurtosis) are explicitly formulated, only fund 28 (among ?ve funds 1, 8, 11, 25, 28 quali?ed as dominant without any additional constraints) satis?es this requirement. Similarly, when more importance is explicitly attached to central values (represented by the mean and the standard deviation), there are only three funds 8, 11, 28 in the dominant list. These results highlight the importance of correct speci?cation of relevant DEA inputs and outputs as well as additional constraints which re?ect best investors’ evaluation preferences. At empirical level, one may notice persistent dominance of several funds across settings like the case of fund 28, which stays dominant whatever preferences are considered. This feature can be regarded as a sign of the robustness of fund 28’s performance relatively to other funds in the sample. Since we are examining funds’ local performance without sales loads, it could be interesting at this point to contrast DEA results in the ?rst and the second settings with fund rankings provided by the traditional Sharpe ratio (Sharpe 1966) and the M-Sharpe ratio. The latter is computed following equation 23 while the former is calculated by the formula below: Sharpe = r ? rf ? (35)
where r is the average return of the fund, r f is the average risk-free rate approximated here by the US 3-month T-bill rate, ? is the standard deviation of fund returns. Note that
24
Table 2: Performance with standard deviation-kurtosis as inputs, mean-skewness as outputs
Absolute weights (u, v)b Virtual weights (uy, vx) Me SK SD KU Me SK SD KU (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 1 147.427 0.001 0.167 0.375 0.9997 0.0003 0.0063 0.9937 2 0.228 142.003 0.001 0.161 0.361 0.2284 -0.0001 0.0054 0.9946 3 0.286 0.001 0.891 0.001 0.206 0.0000 0.2864 0.0001 0.9999 4 0.389 155.891 0.001 0.176 0.396 0.3888 -0.0003 0.0054 0.9946 5 0.127 126.330 0.001 0.001 0.323 0.1270 -0.0001 0.0000 1.0000 6 0.077 119.256 0.001 0.136 0.303 0.0774 -0.0004 0.0032 0.9968 7 0.618 0.001 0.681 0.001 0.157 0.0000 0.6181 0.0001 0.9999 8 0.537 0.001 0.762 0.001 0.176 0.0000 0.5374 0.0001 0.9999 9 -0.000 0.001 0.001 25.456 0.001 -0.0000 -0.0000 0.9974 0.0026 10 0.240 0.001 1.507 0.001 0.348 0.0000 0.2403 0.0001 0.9999 11 1 66.707 0.237 1.281 0.182 0.7193 0.2807 0.0635 0.9365 12 0.006 106.752 0.001 0.122 0.272 0.0062 -0.0004 0.0053 0.9947 13 0.222 0.001 1.925 0.001 0.444 0.0000 0.2219 0.0000 1.0000 14 -0.001 0.001 0.001 10.984 0.001 0.0000 -0.0014 0.9911 0.0089 15 0.295 127.735 0.001 0.001 0.327 0.2947 -0.0001 0.0001 0.9999 16 0.347 122.726 0.436 2.357 0.336 0.2899 0.0566 0.0668 0.9332 17 0.514 155.511 0.001 0.176 0.396 0.5140 -0.0002 0.0057 0.9943 18 0.215 146.828 0.001 0.166 0.373 0.2148 0.0000 0.0044 0.9956 19 0.412 126.397 0.556 0.001 0.394 0.3183 0.0937 0.0000 1.0000 20 0.297 0.001 1.168 0.001 0.269 0.0000 0.2966 0.0000 1.0000 21 0.684 68.216 0.508 15.009 0.116 0.4380 0.2463 0.4122 0.5878 22 -0.000 0.001 0.001 25.250 0.001 -0.0000 -0.0001 0.9974 0.0026 23 0.793 0.001 1.289 6.883 0.231 0.0000 0.7929 0.1706 0.8294 24 -0.000 0.001 0.001 17.422 0.001 -0.0000 -0.0001 0.9967 0.0033 25 0.888 0.001 0.659 3.518 0.118 0.0000 0.8877 0.1922 0.8078 26 0.695 29.615 0.221 6.516 0.050 0.3973 0.2977 0.3461 0.6539 27 1 193.390 0.001 90.268 0.001 0.9984 0.0016 0.9912 0.0088 28 1 33.048 0.173 40.581 0.001 0.2437 0.7563 0.9697 0.0303 29 0.592 0.001 0.482 2.576 0.086 0.0000 0.5918 0.1306 0.8694 30 0.448 0.001 0.891 0.001 0.206 0.0000 0.4482 0.0001 0.9999 31 0.212 60.499 0.215 1.162 0.165 0.1664 0.0460 0.0581 0.9419 32 0.639 0.001 0.871 4.652 0.156 0.0000 0.6393 0.1463 0.8537 33 0.675 0.001 1.170 0.001 0.270 0.0000 0.6747 0.0000 1.0000 34 0.361 0.001 1.840 0.001 0.424 0.0000 0.3607 0.0000 1.0000 35 1 91.251 0.001 31.662 0.014 0.9996 0.0004 0.9390 0.0610 36 0.773 132.376 0.001 0.150 0.337 0.7735 -0.0003 0.0056 0.9944 37 0.502 55.541 0.414 12.220 0.094 0.2083 0.2940 0.3175 0.6825 38 0.044 0.001 1.170 0.001 0.270 0.0000 0.0439 0.0000 1.0000 Note: Me = Mean, SK = Skewness, SD = Standard deviation, KU = Kurtosis. Values in italics are approximative. Funds with negative scores are those whose mean and skewness are simultaneously negative. b u and v are required to be equal to or greater than 0.001 (? = 0.001).
a Funds
Scorea
25
Table 3: Local performance
DEA scores Partial momentsb Horizonsc Standard Preference Preference SK & KU Me & SD (1) (2) (3) (4) (5) (6) 1 1 1 0.52 0.93 0.46 2 0.23 0.71 0.38 0.69 0.46 3 0.29 0.83 0.69 0.71 -0.00 4 0.39 0.66 0.37 0.64 0.19 5 0.13 0.78 0.36 0.77 0.08 6 0.08 0.64 0.35 0.58 0.93 7 0.62 0.86 0.67 0.81 0.21 8 0.54 1 0.62 1 0.11 9 0.00 0.72 0.44 0.67 0.22 10 0.24 0.85 0.36 0.84 0.14 11 1 1 0.60 1 0.83 12 0.01 0.68 0.32 0.66 0.24 13 0.22 0.79 0.38 0.74 -0.00 14 0.00 0.62 0.21 0.61 1 15 0.29 0.89 0.45 0.75 0.10 16 0.35 0.72 0.41 0.70 0.24 17 0.51 0.80 0.32 0.78 0.63 18 0.21 0.71 0.39 0.68 0.14 19 0.41 0.78 0.44 0.76 0.19 20 0.30 0.85 0.45 0.84 1 21 0.68 0.75 0.44 0.72 0.70 22 0.00 0.85 0.31 0.82 -0.00 23 0.79 0.88 0.37 0.83 0.26 24 0.00 0.72 0.46 0.70 0.48 25 0.89 1 0.72 0.97 0.04 26 0.70 0.72 0.65 0.59 1 27 1 0.92 0.92 0.81 0.96 28 1 1 1 1 1 29 0.59 0.75 0.59 0.66 0.41 30 0.45 0.89 0.45 0.85 0.35 31 0.21 0.83 0.43 0.82 0.71 32 0.64 0.70 0.58 0.66 0.17 33 0.67 0.94 0.45 0.91 0.26 34 0.36 0.91 0.44 0.88 -0.00 35 1 0.76 0.52 0.74 1 36 0.77 0.84 0.30 0.82 0.37 37 0.50 0.63 0.58 0.59 0.44 38 0.04 0.80 0.43 0.76 -0.00 Rank correlation (Sharpe & M-Sharpe) Note: Results are obtained from the CCR input-oriented version with ? = negative scores are those whose all outputs are simultaneously negative. Skewness, SD = Standard deviation, KU = Kurtosis. Funds Standarda moments
a In
Rank Sharpe
Rank M-Sharpe
(8) 7 18 34 15 20 28 29 26 27 23 4 25 38 31 17 16 12 21 14 19 6 35 11 37 22 5 2 1 8 36 13 32 24 30 3 9 10 33 0.995 0.001. Funds with Me = Mean, SK =
(7) 7 18 33 14 21 29 27 25 28 23 5 26 38 34 17 16 12 22 15 19 6 36 11 37 20 4 2 3 9 35 13 30 24 31 1 8 10 32
the ?rst setting, inputs are standard deviation and kurtosis, outputs are mean and skewness. b In the second setting, inputs are composed of lower mean, lower semi standard deviation, lower skewness and lower kurtosis; outputs contain upper mean, upper semi standard deviation, upper skewness and upper kurtosis. c In the third setting, inputs include the MVAR over the previous year, the 3 previous years and the 5 previous years; outputs include mean returns over these three periods.
26
the Sharpe ratio is based on the mean-variance paradigm while the modi?ed Sharpe ratio takes account of the skewness and the kurtosis. Fund rankings according to these two ratios are reported in the columns 7 and 8 of table 3. Several main observations can be drawn from these results. We can see easily that despite differences in the approach taken by the two measures, fund rankings are surprisingly quite similar, both in terms of correlation coef?cient (0.995) and in terms of direct contrasting from fund to fund. Does this strong similarity imply that the return distribution of all funds is quite close to the normal one? The answer according to the Shapiro-Wilk normality test is rather negative because the normality assumption is rejected in 14 among 38 cases at the con?dence level of 95% (see table 1). However, ?nding explanations to such problem is beyond the scope of this paper. Regarding the connection of DEA classi?cations (except for the third setting) with Sharpe and M-Sharpe rankings, the results show that most dominant funds are generally among the seven funds the most highly ranked by Sharpe and M-Sharpe ratios. Nevertheless, funds 8 (dominant once) and 25 (dominant twice) in the second setting are only placed respectively at the 25th and 20th rank by Sharpe, 26th and 22th by M-Sharpe. This disfavor is certainly related to the slightly negative mean of fund 28 (-0.18%) and to the quite low positive mean of fund 25 (0.09%). A closer examination of their return distributions reveals much wider dispersal of returns and higher frequency of extreme positive values in these two distributions than in those of other funds ranked before them by Sharpe and M-Sharpe ratios. It is undoubtedly the reason why these funds are highly praised by the second setting of DEA. An investor who likes good surprises would ?nd his interests in these pro?les. Yet, if he used only Sharpe and M-Sharpe ratios, he would have missed his chance, at least in the case of this sample. Such result provides evidence that DEA can be an ef?cient supplementary tool to assist investors in selecting correctly funds satisfying their preferences.
4.3
Assessing hedge fund global performance
4.3.1 Settings As argued previously, investors may sometimes want to evaluate local performance of funds (1) on several horizons simultaneously or (2) by using several measures at the same time. In these cases, how will they reconcile between elementary performances ? On which basis they can assign a ?nal note to each fund so as to rank them ? This choice is
27
particularly dif?cult when elementary performances provide divergent rankings of funds. Meanwhile, by means of optimizing the weighted sum of elementary performances, DEA offers an aggregate measure allowing investors to identify funds having the best combination of these performances. In other words, by combining multiple performance criteria simultaneously, DEA provides an exhaustive image of funds. In order to illustrate the ?rst setting, the M-Sharpe performance ratio over three horizons — the previous year, the three previous years, the ?ve previous years — are used as elementary performances. In the second setting, three performance measures which consider the non normal features of returns are selected, namely modi?ed Stutzer index (Stutzer 2000, Kaplan & Knowles 2001), M-Sharpe ratio and Omega index (Keating & Shadwick 2002). The formulas to compute the modi?ed Stutzer and the Omega indices are given below:
? M-Stutzer = sign(r ) 2Stutzer with
Stutzer = max ? ln
?
(36) (37) (38)
1 T
? e ? (r ?r
t
T
f t)
t =1
Omega =
? [1 ? F (r )]dr ? ? ?? F (r ) dr
where rt is fund return on month t, sign(r ) is the sign of the mean return, ? is a negative number, T is the number of monthly returns, r f t is risk-free rate (approximated by the US 3-month T-bill rate) on month t, ? is the MAR pre-determined by investors (approximated by the US 3-month T-bill rate’s average over the study period). It is necessary to specify that the modi?ed Stutzer index (hereafter, M-Stutzer) considers up to the skewness of returns, the M-Sharpe ratio takes into account both the skewness and the kurtosis while the Omega index regards the whole (empirical) distribution of returns. These two settings are characterized as having only outputs. Since there are no inputs, it is possible to assume existence of one input equal to one so that DEA can be applied. Given this feature, the input-oriented version is required. Besides, investors are also assumed to be indifferent among horizons and performance indicators so that no additional constraints are needed.
4.3.2 Results Table 4 reports detailed results of the two settings under consideration. In panel A presenting fund classi?cation over three horizons, empirical results con?rm that fund performance varies strongly from one horizon to another. Indeed, coef?cients of rank corre-
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Table 4: Global performance
Panel A: Perf. over 3 horizons Panel B: Perf. over 5 years with 3 measures M-Sh over Global Perf. M-St ? M-Sh Global Perf. Funds 5 years 3 years 1 year Rank Score Rank Score (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 1 7 21 27 16 0.2028 7 7 7 7 0.657 2 18 17 11 13 0.2854 22 18 18 18 0.430 3 34 26 35 33 -0.0003 33 34 34 34 0.334 4 15 28 28 29 0.0153 24 13 15 13 0.466 5 20 25 36 31 -0.0002 19 21 20 21 0.416 6 28 8 3 6 0.6827 11 29 28 29 0.374 7 29 23 18 22 0.0806 29 30 29 30 0.365 8 26 14 23 27 0.0419 28 28 26 28 0.380 9 27 11 25 24 0.0734 30 26 27 26 0.389 10 23 16 20 25 0.0568 17 23 23 23 0.409 11 4 22 17 12 0.3600 5 5 4 5 0.745 12 25 10 26 21 0.0951 13 25 25 25 0.391 13 38 37 37 37 -0.0005 38 38 38 38 0.260 14 31 6 1 1 1 35 36 31 36 0.310 15 17 24 31 30 0.0034 27 16 17 16 0.450 16 16 29 19 23 0.0749 26 15 16 15 0.459 17 12 27 6 9 0.5109 16 12 12 12 0.494 18 21 35 34 36 -0.0003 18 19 21 19 0.419 19 14 32 30 28 0.0162 25 14 14 14 0.461 20 19 13 2 4 0.9856 21 20 19 20 0.416 21 6 7 8 7 0.5718 6 6 6 6 0.661 22 35 33 32 35 -0.0003 36 33 35 33 0.341 23 11 12 22 20 0.1046 14 11 11 11 0.497 24 37 18 7 11 0.4276 37 37 37 37 0.278 25 22 31 29 32 -0.0002 20 24 22 24 0.408 26 5 1 4 1 1 4 4 5 4 0.877 27 2 5 13 8 0.5565 1 2 2 1 1 28 1 3 9 1 1 3 3 1 1 1 29 8 15 16 15 0.2217 9 9 8 9 0.530 30 36 20 14 17 0.1691 34 35 36 35 0.318 31 13 4 10 10 0.4409 23 17 13 17 0.438 32 32 36 21 26 0.0504 10 31 32 31 0.358 33 24 34 15 18 0.1415 15 22 24 22 0.409 34 30 38 38 38 -0.0006 31 27 30 27 0.382 35 3 2 5 5 0.8112 2 1 3 1 1 36 9 19 24 19 0.1411 8 8 9 8 0.568 37 10 9 12 14 0.2675 12 10 10 10 0.500 38 33 30 33 34 -0.0003 32 32 33 32 0.355 Correlation M-Sh 3 years 0.71 M-Sh 5 years 0.49 0.34 0.82 0.99 ? 0.82 Note: M-Sh = modi?ed Sharpe ratio, M-St = modi?ed Stutzer index, ? = Omega index. Funds with negative scores are those whose elementary performances are all negative.
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lation between classi?cations are rather weak: 0.49 (5 years versus 3 years), 0.34 (5 years versus 1 year) and 0.71 (1 year versus 3 years). The most striking example is fund 14 classi?ed at the 31th position over ?ve-year horizon but at 6th and 1th ranks over respectively three-year and one-year horizons. According to DEA, this fund is classi?ed as dominant with an aggregate score of 1. In detail, it is found to have the performance over one-year horizon not only classi?ed at the 1st rank but also suf?ciently high to compensate for the slightly negative performance over ?ve-year horizon so as to arrive ?nally at the head of the sample. Unlike fund 14, two other dominant funds according to the DEA global performance score — funds 26 and 28 — have very stable performance pro?les over time. With regard to fund rankings according to the three selected performance indicators (Panel B), contrary to the preceding case, classi?cations are overall very coherent between them. This coherence is con?rmed by high coef?cients of rank correlation: 0.99 (M-Sharpe with Omega), 0.82 (M-Sharpe with M-Stutzer) and 0.82 (Omega with M-Stutzer). Such ?nding certainly does not provide an ideal illustration of the problem which this study aims to illustrate here. Nevertheless, in detail, rankings given by the M-Stutzer index and those provided by the M-Sharpe ratio and the Omega index are quite divergent on several occasions. It is particularly true for fund 6 which is ranked at the 11th place by the M-Stutzer index but ranked only at the 29th and the 28th places in the classi?cation of respectively the Omega index and the M-Sharpe ratio. The other examples are funds 4, 10, 12, 15, 16, 19 and 32. In such cases, applying DEA to determine de?nitive ranks of these funds presents an undeniable interest. Finally, three funds globally quali?ed as dominant are funds 27, 28 and 35. All of them are the most highly ranked of the sample, no matter what measure is used. In these cases, the dominance of these funds is obvious, DEA can only con?rm it. Only when performance measures disagree on fund rankings that DEA proves its perspicacity by providing for each fund an aggregate indicator of performance allowing a global and de?nitive classi?cation. Such is the case of funds 14, 18, 22, 25, 31, 33 and 34. Consider for example fund 25. It is ranked 20th by the M-Stutzer index, 24th by the Omega index and 22th by the M-Sharpe ratio. However, according to DEA, it is globally placed only 24th when all three performance indicators are considered. Similarly, fund 34 is at the 31th position of the list according to M-Stutzer, 27th according to Omega, 30th according to M-Sharpe but globally ranked 27th by DEA.
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Conclusion
Previous studies documented that DEA could be a good tool to solve decision-making problems with multiple criteria, including investment fund performance evaluating task. This paper shows that DEA is particularly adapted to assess hedge fund performance for the following reasons. First, it can incorporate multiple risk-return attributes of non normal returns in an unique aggregate score so as to rank funds. Hence, DEA can be used to evaluate local and global performances of hedge funds. The local performance is obtained when evaluation criteria include risks, eventually sales loads (DEA’s inputs) and returns (DEA’s outputs). The global performance is de?ned as the aggregate score of several elementary performances which could be performances over several temporal horizons, or performances over one temporal horizon but measured by different indicators. Second, unlike other performance measures, DEA offers investors the possibility to exert direct control on the importance level paid to each evaluation criteria. Thus, each investor can tailor his own performance measure to select funds corresponding the most to his own preferences. This ?exibility is very important as in reality, each investor usually has his own preferences and constraints. Third, by putting emphasis on the best observed funds, DEA makes no assumption on the functional relation between evaluation criteria. To this end, this paper focuses on the most important methodological issues concerning the application of the basic CCR model to hedge fund performance appraisal, namely (1) the choice of evaluation criteria as DEA’s inputs and outputs, (2) the choice between input-oriented or output-oriented version of the CCR model, (3) dealing with negative inputs and outputs, and (4) transcribing investors’ speci?c preferences into mathematical constraints. These elements are presented in such a way to provide investors with a general framework to apply DEA in assessing fund performance. In order to make these guidelines more intuitive, several numerical illustrations with thorough discussion of results are provided on a sample of 38 hedge funds. The illustrations also highlight the importance of correct speci?cation of evaluation criteria and preference structure for ef?cient application of DEA. A comparison between DEA classi?cation and rankings provided by traditional Sharpe and modi?ed Sharpe ratios indicates that they are sometimes radically inconsistent. Further examination of funds’ return distributions suggests that these latter two measures might not price properly good surprises (extremely high positive returns). In such case, DEA proves to be a good supplement to improve the precision of selection tasks. Although this paper only addresses the application of DEA in the hedge fund context, its guidelines are also applicable to other types of investment funds like mutual funds, pension funds or ethical funds, etc.
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Like any other tools, DEA also has its caveats. One of the main weaknesses arises from the fact that DEA basic models do not provide complete rankings of dominant funds. Nevertheless, this weakness can be mitigated by either adding more restrictive preferences (additional mathematical constraints) so that the short list becomes more and more selective, or applying other qualitative and quantitative criteria on dominant funds so as to rank them. Besides, the dominance or ef?ciency of funds is only relative to the other funds in the sample and thus can be changed once the sample is modi?ed. However, relative evaluation is a well-established concept in economic literature (Holmstrom 1982). In addition, the relative property of fund evaluation is still quite valuable because in the investment industry, funds are often rated relatively to others in the same category. A broad literature documented that the investment fund market is a tournament and the managers compete against each other in the same category to attract investors (Brown et al. 1996, Agarwal et al. 2003, Kristiansen 2005).
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