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British Accounting Review (1995) 27, 127–138

THE EVALUATION OF MANAGED FUNDPERFORMANCE

DR JONATHAN FLETCHERGlasgow Caledonian University

This paper reviews the theoretical conditions under which the Jensen (1968) per-formance measure provides valid inferences about fund performance. The key as-sumptions are the unconditional mean-variance efficiency of the benchmarkportfolio(s), the existence of a riskless asset, no binding constraints on investors andinvestors only possess selectivity information. The paper discusses the empiricalsignificance of these conditions and notes that only some of the conditions will havesignificant empirical consequences if they fail to hold in practice in interpreting theJensen measure.

INTRODUCTION

Since the development of the Capital Asset Pricing Model (CAPM) in themid-1960s, risk-adjusted performance measures of managed fund per-formance have been widely used. Of these the most popular is the Jensen(1968) performance measure which compares the performance of the fundto what we expect from the security market line given the level of systematicrisk for the fund. Ippolito (1993) reviews the widespread use of the Jensenmeasure in US mutual fund performance studies. However, the Jensenmeasure has not been without its critics e.g. Roll (1978), Dybvig & Ross(1985a,b), Green (1986) and Grauer (1991) amongst others. The problemsof interpreting the Jensen measure have been partly countered by Grinblatt& Titman (1989).

The purpose of this paper is to review the existing literature of performancemeasurement to identify the assumptions and conditions under which theJensen measure provides valid inferences about fund performance. Thepaper also discusses the empirical significance when some of the theoreticalconditions fail to hold. There appears to be three key issues in interpretingthe Jensen performance. These relate to the mean-variance efficiency of the

Comments from P. Draper and anonymous referees are gratefully acknowledged.Correspondence should be addressed to Dr J. Fletcher, Department of Finance and Accounting,

Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK.

Received 1 December 1993; revised 1 February 1995; accepted 17 February 1995

0890–8389/95/020127+12 $08.00 1995 Academic Press Limited

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benchmark portfolio, dealing with asymmetric information between investorsand the impact of binding investment constraints. Grauer (1991) raisesadditional issues when we estimate Jensen performance measures over manytime intervals.

The paper is organized as follows. Section I introduces the Jensenperformance measure and the framework of Grinblatt & Titman (1989).This will identify the conditions under which the Jensen measure willcorrectly assign uninformed investors with zero performance. Section IIexamines the conditions under which the Jensen performance measurewill correctly assign informed investors with positive performance. It alsodiscusses the empirical significance of the theoretical timing biases in theJensen measure. Sections III and IV consider the impact of the Jensenmeasure when the benchmark portfolio is mean-variance inefficient andwhen investors face constraints on the investments they can make. The finalsection contains concluding comments.

EVALUATING FUND PERFORMANCE

The Jensen (1968) performance measure can be estimated from the followingexcess returns regression;

rpt=ap+bprmt+ept (1)

where rpt and rmt are the excess returns (total returns minus the risklessreturn) on fund p and the benchmark portfolio m in period t, ept is a randomerror term with E(ept)=0 and E(ept, rmt)=0, bp is the estimated systematicrisk for the fund. The intercept ap is the estimated Jensen performance forfund p1. The null hypothesis of no performance ability is that ap=0. Apositive ap is deemed to reflect superior performance since the fund hasearned a higher return that we expect. A negative ap reflects inferiorperformance because the fund has earned a lower return than expected.The interpretation of positive Jensen performance has proved to be highlycontroversial in the literature. Grinblatt & Titman (1989) point out that asuitable performance measure should assign uninformed investors with zeroperformance and informed investors with positive performance. Grinblatt& Titman (1989) present a general framework within which to evaluateperformance which is helpful in discussing the key issues in correctlyinterpreting performance. Two types of investors are modelled in the analysis,either uninformed or informed.

It is assumed that N risky assets trade in frictionless markets i.e. notransactions costs, taxes, short sales restrictions and that a riskless assetexists. Excess returns as viewed by the uninformed investor are assumed tobe normal, independently and identically distributed (i.i.d.) and computedfrom the perspective of the uninformed investor. The stationarity assumption

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is necessary in order to distinguish between performance and changes inthe parameters of the return generating process. The uninformed investoris assumed to be a mean-variance optimizer. Given that it is possible totrade in N risky assets and a riskless asset exists, the uninformed investorwill choose the portfolio of risky assets m that is ex ante mean-varianceefficient2 with respect to the unconditional (on information) efficient frontier.With the assumption of i.i.d. excess returns the uninformed investor willhave the same expectations through time of the efficient frontier. Thisimplies that the expected return and variance of portfolio m and theinvestment weights in m remain unchanged through time.

The assumption of constant weights in the benchmark portfolio is im-portant. Grauer (1991) shows that when the weights in the benchmarkportfolio change over time (as will happen with most market proxies), thenexpected returns, variances, betas will all be time-varying. This can furthercomplicate the analysis of evaluating fund performance. MacKinlay (1987)argues that given the large number of assets in most benchmark portfolios,this should be a reasonably good approximation to a constant weightsportfolio. However, it still remains an empirical question. For benchmarkportfolios such as the Financial Times 100 Index, this will probably be lessaccurate since the proportions and the composition of the index may varygreatly over time. For large benchmark portfolios e.g. Financial Times AllShare Index or an equally weighted index, the approximation will be moreaccurate.

We can write the excess returns on the individual assets as:

rit=birmt+eit (2)

Where rit is the excess return on asset i in period t, rmt is the period t excessreturn on the unconditional efficient portfolio m, bi=cov(rit,rmt)/r

2(rm) andE(eit)=0 since portfolio m is efficient. Similarly the excess return on aportfolio of the risky assets can be written as;

rpt=bptrmt+ept (3)

with

bpt=;N

j=1xitbi

and

ept=;N

j=1xiteit

where xit are the portfolio weights of asset i in period t. Although uninformed

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investors will have constant expected returns, variances and covariances ofasset returns, informed investors can change their expectations in responseto information. This implies that informed investors will change theirportfolio weights in response to information which results in time-varyingbetas for such portfolios. It also implicity assumes that the forecasting andtrading horizons of informed investors are of the same length as the returnobservation intervals.

Grinblatt & Titman define information as arising from two sources:

(1) Selectivity information – where E(emt) does not equal zero for any asseti in any period t.

(2) Timing information – where E(rmt) does not equal E(rm) for any periodt.

Grinblatt & Titman derive the large sample decomposition of the Jensenperformance measure as:

ap=(biT−bp)r∗m+[1/T;bpt

T

t=1(rmt−r∗m)]+ep (4)

where biT is the average dynamic portfolio beta from (3) and can be viewedas the target beta of the fund, bp is the least squares estimate from equation(1), r∗m is the sample mean excess return of portfolio m, T is the numberof time series observations. Grinblatt & Titman define the three terms asthe bias in beta, timing and selectivity components of performance. Theyshow that when an investor has no timing information, bp is a consistentestimator of biT. Additionally when an investor has no selectivity or timinginformation, the Jensen measure will assign an uninformed investor withzero performance. This is because the second and third terms in equation(4) will equal zero. It is important to note that this result assumes the mean-variance efficiency of portfolio m, a riskless asset exists, no restrictions oninvestments and i.i.d. excess returns. A lot of the controversy surroundingthe validity of the Jensen measure arises when these assumptions fail tohold.

ASYMMETRIC INFORMATION

The previous section discussed the conditions under which the Jensenmeasure will correctly identify uninformed investors with zero performance.The aim of a performance measure is to try and identify superior informationon the part of informed investors. The essential usefulness of the Jensenmeasure is the extent to which it can correctly attribute informed investorswith superior information. There has been considerable debate in theliterature as to whether the Jensen measure can actually do this. The

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controversy largely centres on the type of information that managers receiveand use.

Dybvig & Ross (1985a) and Grinblatt & Titman (1989) have derived theconditions under which the Jensen measure will correctly assign informedinvestors with positive performance. The key assumptions are that a risklessasset exists, rational expectations, informed investors are mean-varianceoptimizers and are unable to forecast either the expected return or varianceon the benchmark portfolio i.e. only possess selectivity information. Dybvig& Ross show that such an investor will exhibit positive performance againstthe benchmark portfolio. The intuition behind this approach is that themean-variance efficient frontier of informed investors will lie outside thefrontier of uninformed investors. The optimal portfolio of the informedinvestor will have a higher Sharpe (1966) performance measure than thebenchmark portfolio given the existence of a riskless asset. We know fromDybvig & Ross (1985a,b) that when a portfolio has a higher Sharpe measurethan the benchmark portfolio, that such a portfolio will also have a positiveJensen performance.

This can also be shown from equation (4). The third term captures theselectivity component of performance. When an investor has no timinginformation the first two terms will be zero. With superior selectivityinformation, the third term will be positive and the Jensen measure willcorrectly assign the investor with superior performance. This also requiresthat when an investor changes the portfolio composition in response toselectivity information that it does not induce a non-zero correlation betweenthe portfolio beta and the benchmark portfolio return. This implicitlyassumes that selectivity and timing information are independent of eachother. This also requires the assumptions in Section I. The analysis ofDybvig & Ross (1985a) and Grinblatt & Titman (1989) shows that theJensen measure will correctly measure superior selectivity ability when themanager has no timing information.

When investors also have access to timing information, Dybvig & Ross(1985a) and others have shown that the Jensen measure is biased and caneven assign superior market timers with negative performance. Bhattacharya& Pfleiderer (1983) have shown within a CAPM framework and under theassumption that the manager chooses the portfolio so as to maximize theutility of investors with constant absolute risk aversion, that the relationshipbetween the fund’s return and the market return will be quadratic. Theyshow that this can be written as:

rpt=ap+bprmt+cprmt2+ept (5)

The coefficient ap captures the selectivity ability of the fund. The coefficientcp measures both the quality of the timing information and the aggressiveness

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of the fund in response to that information. Bhattacharya & Pfleiderer(1983) show that the quality of the timing information can be identified bycombining the information in cp with that contained in the residuals. Theyalso point out that generalized least squares (GLS) estimation should beused since the residuals are heteroscedastic3 due to the presence of timinginformation. It is possible to show from this model that the selectivitymeasure will be equivalent to the Jensen measure when the manager hasno timing information. This also implies that the error terms will not beheteroscedastic.

The bias of the Jensen measure when managers have timing informationcan be illustrated from the Jensen decomposition in equation (4). Theproblem is that the OLS estimate of the fund’s beta will no longer be aconsistent estimator of biT. Grinblatt & Titman (1989) show that bp willoverestimate biT for a superior market timer under reasonable conditions.Assume that the investor’s beta response function increases monotonicallyas the timing information becomes more favourable and is symmetric aboutbiT. It is shown that the large sample estimate of beta is:

bp=biT+(r∗m/r2m)cov(bpt,rmt)

This shows that for a superior market timer, biT will be overestimated. Thedegree of bias depends upon the covariance between the portfolio beta andthe benchmark portfolio return and the mean-variance ratio of the bench-mark portfolio.

Ignoring the impact of any selectivity information, the Jensen measurecan be negative for a superior timer when the first term in equation (4)dominates the second term. Grinblatt & Titman (1989) propose a newperformance measure which attempts to overcome the timing biases in theJensen measure called the positive period weighting measure. Howeverempirical evidence in Grinblatt & Titman (1994), Cumby & Glen (1990)and Draper & Fletcher (1995) report almost identical inferences betweenthe Jensen and positive period weighting measures. This suggests that thetiming biases of the Jensen measure are of little empirical significance.

Although the evidence suggests that the timing biases in the Jensenmeasure are insignificant and it can in principle identify superior selectivityability, Ashton (1990) questions the power of current statistical tests. Ashtonargues that current tests have very little power to detect superior performanceability. This suggests that it is perhaps not surprising to observe thatmost performance studies find little evidence of abnormal performance bymanaged funds. The problem stems from the large amount of noise inreturn data and the high variability of the Jensen performance measurerelative to the estimated Jensen measure. This is an important consideration

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because it implies that even if all the theoretical conditions hold, there isstill the issue of the power of the tests.

THE IMPACT OF BENCHMARK PORTFOLIO INEFFICIENCY

The previous sections have examined the ability of the Jensen measure atcorrectly interpreting the performance of uninformed and informed in-vestors. One of the key assumptions is the unconditional mean-varianceefficiency of the benchmark portfoio. Roll (1978), Dybvig & Ross (1985b)and Green (1986) have analysed extensively the relationship between themean-variance inefficiency of the benchmark portfolio and Jensen per-formance when all investors have homogenous expectations and within thecapital asset pricing model (CAPM) framework. Roll (1978) argues that apositive (or negative) Jensen measure simply reflects the inefficiency of thebenchmark portfolio and tells us nothing about the performance of the fund.If the benchmark portfolio is ex post efficient then the Jensen performance ofthe funds, gross of expenses will be exactly zero. Similarly if the benchmarkportfolio is ex ante efficient, then the Jensen performance can be non-zerobut will be statistically insignificant and disappear in large samples whenthe return distribution is stationary. As a consequence, positive or negativeJensen performance signifies the inefficiency of the benchmark portfolio.

The simplest way to consider this is that when m is inefficient, E(eit) willno longer necessarily equal zero for each asset i in equation (2). This impliesthat the intercept in equation (1) can be positive or negative for anuninformed investor. Roll also argues that there is no way to identify theappropriate benchmark portfolio within a CAPM framework as the marketportfolio of all risky assets is unobservable. What is the correct benchmarkportfolio to use as a proxy for the market portfolio? For each benchmarkportfolio, there will be a different measure of systematic risk and Jensenmeasure for any given fund. This implies that the Jensen measure can besensitive to the choice of the benchmark portfolio.

Dybvig & Ross (1985b) and Green (1986) extend in further detail therelationship between Jensen performance and the mean-variance inefficiencyof the benchmark portfolio when all investors have identical expectations.Dybvig & Ross consider the situation where the fund manager is beingevaluated on his/her ability to pick a portfolio on the upper segment of theefficient frontier. They show that the usefulness of Jensen measure dependsupon the existence of a riskless asset. When a riskless asset exists, superiorportfolios will register positive Jensen performance against the benchmarkportfolio. However, inefficient portfolios can also show positive performance.The main result is that negative performance rules out the possibility ofsuperior ability. In the absence of a riskless asset, efficient and inefficientportfolios can either exhibit positive or negative performance. Dybvig &Ross point out that the sign of the Jensen measure does provide information

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to an investor holding the benchmark portfolio to improve their mean-variance position by making a marginal investment into (or out of) thefund.4

When we consider the situation where some investors possess superiorinformation, the Jensen measure will provide ambiguous inferences aboutfund performance. This is true even although Dybvig & Ross (1985a) haveshown that an informed investor with only selectivity ability will registerpositive performance against a benchmark portfolio regardless of its effi-ciency. We are unable to distinguish between benchmark inefficiencies andsuperior information in positive Jensen performance when the benchmarkis unconditionally mean-variance inefficient.

Grinblatt & Titman (1989) argue that an appropriate benchmark portfoliois one that is ex ante mean-variance efficient relative to the unconditionalefficient frontier of the set of assets considered tradable by investors. Thisis because it is the optimal portfolio of risky assets for an uninformedinvestor to hold. This implies that we do not require to observe the marketportfolio in performance measurement tests. Models such as the CAPM andarbitrage pricing theory (APT)5 may simply suggest candidate benchmarkportfolios that may be efficient.

Although Grinblatt & Titman have countered Roll’s critique of an ap-propriate benchmark portfolio, we still need to identify a portfolio which isex ante mean-variance efficient relative to the universe of tradable assets.This is by no means a straightforward task. A number of authors haverejected the mean-variance efficiency of various market proxies and APTbenchmark portfolios within a US and UK context. A related concern insearching for an efficient benchmark portfolio is, are some of these portfoliosactually feasible alternative strategies for investors. For example, could aninvestor actually construct a portfolio which is equivalent to the mimickingportfolios within an APT framework. Ideally we would want to identify aportfolio that is efficient and actually feasible for investors to hold.

Given that in many cases it will be difficult to use an efficient benchmarkportfolio, how sensitive is the Jensen measure to different inefficient proxies?Green (1986) shows that when benchmark portfolios are close in mean-variance space and highly correlated with each other then they shouldproduce similar Jensen performance measures for a given fund. Howeverthis will not be uniform across all funds, especially for funds with extremebetas and high standard deviations. Roll (1978) presents an example wheretwo inefficient proxies assign each fund the opposite Jensen performance toone another. Dybvig & Ross (1985b) show that this will be false when ariskless asset exists.

Existing empirical evidence on the sensitivity of Jensen performance todifferent benchmark portfolios is that the choice of the benchmark portfoliois important. Lehmann & Modest (1987), Elton, Gruber, Das & Hlvaka(1993) present results that the choice of the benchmark portfolio does havea major impact on the inferences of US mutual fund performance. Draper

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& Fletcher (1995) show for a sample of 101 UK unit trusts that thebenchmark portfolio does affect inferences about trust performance.

CONSTRAINTS ON INVESTMENTS

A second critical assumption of the Jensen measure in assigning uninformedinvestors with zero performance and informed investors with positive per-formance is that the N risky assets trade in frictionless markets. This impliesthat investors face no binding constraints except the usual budget constrainti.e. that the portfolio weights sum to one. Best & Grauer (1990) andGrauer (1991) demonstrate that when investors face binding constraints oninvestments then the Jensen measure can be non-zero. This is because allsecurities may no longer be held in the optimal portfolio. This implies thatE(eit) may no longer equal zero in equation (2) when binding constraintsare imposed. This can lead to a non-zero intercept in equation (1).

Using the notation of Best & Grauer (1990), consider an investor choosinga mean-variance efficient portfolio subject to the following linear constraints:

Ax is less than or equal to b

where X is (N∗1) vector of investment weights, A is a (m∗N) constraintmatrix where m is the number of possible constraints faced by the investor,b is an m vector consisting of the coefficients on the right side of the equationof the mth constraint. Assume that k of the m constraints are active for theinvestor and the first constraint is that the portfolio weights sum to one. Ai

is a (k∗N) matrix of the k constraints where the k rows are the correspondingrows of A and bi is a k vector associated with the constraints.

Best & Grauer show (in the absence of a riskless asset) that the followingrelationship holds for individual securities;

E(Ri)=rzp+[E(Rp)−;k

j=1(kj/t)bj]bi+;

k

j=2(kj/t)aj (6)

where E(Ri) and E(Rp) are the expected returns on asset i and the optimalportfolio p, rzp is the return on the zero-beta portfolio associated with p, kj

is the multiplier of the jth constraint j=1,......k, bj is the jth coefficient inbi, aj is the jth row in the Ai matrix, bi is the beta of security i and t can beviewed as the risk tolerance of the investor.

Best & Grauer point out that expected returns will plot on a hyperplanespanned by bi and the gradients of the k constraints with weights kj and t.This will also be true of sample data. In this situation positive or negativeJensen performance is a function of the third term. An example of this isthat it is well known that securities which cannot be sold short in the optimal

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portfolio will plot below the SML (see Ross, 1977). This implies that anyasset or portfolio which faces binding investment constraints will exhibitnon-zero Jensen performance. Additionally the optimal portfolio itself canalso have positive or negative performance unless bj=0 for j=2,....k.

The work of Best & Grauer (1990) and Grauer (1991) highlights theimportance of assuming that investors face no binding investment constraintsand that the risky assets trade in frictionless markets. Many investors inpractice will face such constraints e.g. institutional investors often facerestrictions on short selling and upper bounds on the proportion investedin any one security or portfolio. It is possible that although investors facesuch restrictions in practice, that the restrictions will not be binding on theoptimal portfolio. This occurs when the optimal portfolio chosen subject tothe restrictions still lies on the unconstrained efficient frontier. The likelihoodof this occurring is small given the evidence in Best & Grauer (1991, 1992)and Green & Hollifield (1992). It is still difficult to assess the empiricalsignificance of such binding constraints on interpreting Jensen performanceof a fund and is usually ignored in performance studies but could be afruitful area of future research.

CONCLUSIONS

This paper has examined the conditions under which the Jensen measurewill correctly assign informed investors with superior performance. It wouldappear that the Jensen measure can, at least in principle, identify superiorperformance in a mean-variance world. The critical assumptions are that ariskless asset exists, no binding constraints on investments, i.i.d. excessreturn distribution, unconditional mean-variance efficiency of the bench-mark portfolio and informed investors only possess selectivity information.

A number of performance studies assume that these assumptions holdand attribute superior Jensen performance to superior information. Howeveras discussed in the text, if some of the assumptions are invalid then theinterpretation of positive Jensen performance is ambiguous. Some of theassumptions e.g. the timing biases will have little empirical consequence ifthey fail to hold, but other assumptions can have a major empirical sig-nificance especially the mean-variance inefficiency of the benchmark port-folio. This is confirmed by a recent study by Grinblatt & Titman (1994)who compare the sensitivity of fund performance to the choice of thebenchmark portfolio and various performance measures. They find thatfor a given benchmark the different performance measures yield similarinferences. However the choice of the benchmark can have considerableimpact. This suggests that if we are able to identify an efficient benchmarkportfolio, then the Jensen measure should provide reasonably accurateinferences about fund performance. However there is still the concern aboutthe power of the tests as raised by Ashton (1990).

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N

1. Although equation (1) uses a single portfolio benchmark, the Jensen measure can alsobe estimated relative to a multiple portfolio benchmark (see Connor & Korajcyzk, 1986).

2. A mean-variance efficient portfolio is a portfolio that has the smallest variance for a givenlevel of expected return. It is assumed that investors choose portfolios on the uppersection of the efficient frontier.

3. Similar analysis is found in Adamti, Bhattacharya, Pfleiderer & Ross (1986). Othermodels for measuring selectivity and timing ability have been proposed by Hendriksson& Merton (1981).

4. This result has been extended by Jobson & Korkie (1984) to develop a measure whichallows us to identify the best fund to add to the benchmark portfolio.

5. Grinblatt & Titman note that APT benchmarks will be relevant if there exists somecombination of the factor portfolios, which mimic the unobserved factors driving securityreturns, that lies on the unconditional efficient frontier.

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