Evaluating Mutual Fund Performance

S.P. Kothari
Sloan School of Management
Massachusetts Institute of Technology
50 Memorial Drive, Cambridge, MA 02142
E-mail Kothari@MIT.edu
617 253-0994

and

Jerold B. Warner
William E. Simon Graduate School of Business Administration
University of Rochester, Rochester, NY 14627
E-mail Warner@ssb.rochester.edu
716 275-2678

First draft: August 1997

We thank Peter Wysocki for excellent research assistance. We are grateful to the Research
Foundation of the Institute of Chartered Financial Analysts and the Association for Investment
Management and Research, the Bradley Policy Research Center at the Simon School and the John
M. Olin Foundation for financial support.
 2

Abstract

We study standard mutual fund performance measures, using simulation procedures

combined with random and random-stratified samples of NYSE and AMEX securities. We track
simulated fund portfolios over time. These portfoliosÕ performance is ordinary, and well-
specified performance measures should not indicate abnormal performance. Our main result,
however, is that the performance measures are badly misspecified. Regardless of the
performance measure, there are indications of abnormal fund performance, including market-
timing ability, when none exists.
 Evaluating Mutual Fund Performance

1. Introduction

This paper studies empirical properties of performance measures for mutual funds (i.e.,
managed equity portfolios). The portfolio performance evaluation literature is extensive, but
highly controversial. Performance measures based on the Sharpe/Lintner Capital Asset Pricing
Model (CAPM) have a long history and are still used (e.g., Malkiel, 1995, and Ferson and
Schadt, 1996). At the theoretical level, however, there have been strong objections to CAPM-
based measures (e.g., Roll 1977, 1978, Admati and Ross, 1985, and Dybvig and Ross,1985a, b).
For example, the use of a security market line to measure performance can be ÒambiguousÓ (Roll,
1978, p. 1052). Inference about superior performance using this approach is sometimes regarded
as ÒhopelessÓ (Admati and Ross, 1985, p. 16) and Òin general anything is possibleÓ because
performance of a manager with superior information can plot Òbelow or above the security
market line and inside or outside of the mean-variance efficient frontier, and any combination of
these is possibleÓ (Dybvig and Ross, 1985a, p. 383).
At the empirical level, asset pricing tests have identified non-beta factors, namely size
(e.g., Banz, 1981) and book-to-market ratio (e.g., Rosenberg, Reid, and Lanstein, 1985, and Fama
and French, 1992), which are relevant in explaining cross-sectional variation in average returns. In
light of such results, some recent studies take into account multiple factors in evaluating fund
performance (e.g., Carhart, 1996). Fama and French (1993, p. 54) argue that the performance of a
managed equity portfolio should be evaluated using a three-factor model including these
additional factors, and they advocate a ÒsimpleÓ and ÒstraightforwardÓ procedure for doing so.
We provide direct evidence on commonly employed performance measures. We use
simulation procedures, coupled with random and random-stratified samples of NYSE and AMEX
securities. We form simulated fund portfolios and track their performance over time, using a
variety of measures. These portfoliosÕ performance is ordinary and could be obtained by
uninformed investors. Thus, well-specified performance measures should not indicate abnormal
 2

performance. Our approach differs from mutual fund performance studies. With few exceptions
(e.g., Ferson and Schadt, 1996, p. 448), these studies typically assume the validity of a
performance measure, and apply it to observed fund returns. In contrast, we offer independent
evidence on the specification of performance measures.
Our main result is that standard performance measures are misspecified. Regardless of
the performance measure, we find a tendency to detect abnormal fund performance, including
market-timing ability, when none is present. For example, simulated mutual-fund portfolios of
randomly-selected stocks exhibit an average abnormal performance (Jensen alpha) of over 3% per
year, which is both statistically and economically significant. Our simulations indicate that the
Fama-French three-factor modelÕs performance is better than the CAPM, but the corresponding
figure is -1.2% per year, which is also significant. Regardless of the performance measure, the
degree of misspecification is particularly troubling with the CRSP value-weighted index as the
benchmark. Ironically, this index is the most commonly used in both the academic and
practitioner literature. We document several sources of misspecification and suggest how to
reduce it. Some misspecification is due to inadequacies of the assumed asset pricing model, even
for the Fama-French three-factor model. Such results have implications beyond the context of
fund performance evaluation.
We also provide evidence on the ability to detect superior performance. We do not
introduce superior performance into the samples, but our evidence indicates that each
performance measureÕs sampling variation is large when performance is ordinary. Thus, properly
specified performance measures will have low power to distinguish superior from normal
performance. Although this point has been suggested elsewhere (e.g., Dybvig and Ross, 1985a
and b, and Siegel, 1994, p. 289), we provide evidence on how both power and specification
depend on several variables. Our results are obtained without any underlying market timing
ability, or derivatives use (Jagannathan and Korajczyk, 1986). These considerations can reduce
further the informativeness of fund performance measures.
We examine whether the performance measure misspecifications are related to time-
varying expected market returns and return distributionsÕ departures from normality, in particular
 3

skewness. The literature identifies several pre-determined information variables that are
correlated with expected market returns. These include dividend yield, book-to-market ratio,
long-term Government bond yield, term premium, and default premium. We find some evidence
that Fama-French three-factor-model-based performance measures are significantly related to the
information variables, but correlated the CAPM-based performance measures are not. We find
no evidence to suggest that the portfolio returnsÕ co-skewness with the market accounts for the
observed performance-measure misspecifications.
Section 2 outlines the issues in measuring fund performance. Section 3 describes our
baseline simulation procedure, including sample construction, portfolio performance measures,
and distributional properties of the performance measures under the null hypothesis. Section 4
discusses results of baseline simulations using mutual funds of randomly-selected stocks and
presents results of simulations using stratified-random stock portfolios (i.e., style-based
portfolios). Section 5 examines whether the observed performance-measure misspecifications are
explained by time-varying expected market returns and portfolio returnsÕ co-skewness with the
market. Section 6 gives our conclusions.

2. Issues in measuring portfolio performance

We briefly outline key issues in performance evaluation. Since the paperÕs main focus is
on test specification, we emphasize issues affecting the properties of the performance
benchmarks in the absence of any abnormal performance.

2.1 Security market lines

We study the use of a security market line, which can represent the assumed asset pricing
benchmark in any model with linear factor pricing. For example, in the Sharpe-Lintner CAPM,
expected returns on assets or portfolios are a linear function of their beta with the market
portfolio. A portfolioÕs deviation from this security market line measures abnormal performance.
 4

The deviation is typically estimated by the ÒJensen alphaÓ (Jensen, 1968, 1969), which is the
intercept in a regression of portfolio excess returns against returns on the value-weighted Index.
The security market line generalizes to multifactor models such as the arbitrage pricing
theory. Asset or portfolio returns are a linear function of factor sensitivities with respect to each
nondiversifiable factor in the economy. To implement this benchmark, excess returns can be
regressed against factor returns, and the regression intercept should measure the abnormal return
on the portfolio. In the Fama-French three factor model, the factors are the value-weighted index,
and mimicking portfolios for size and book-to-market factors. These authors argue that the
intercept in this regression should be zero in the absence of any abnormal portfolio performance.
We investigate properties of the regression intercepts involving both the CAPM (the
Jensen alpha) and the Fama-French model. In both cases, we find that the estimated intercepts
can be systematically nonzero, and are highly sensitive to index choice. These results hold even
for randomly selected portfolios, which do not have unusual size or book to market
characteristics. In addition, the sampling distribution of the intercepts is non-normal, making
inference about performance more complicated than typically assumed.

2.2 Market timing

There is a large literature on market timing. If fund managers have market timing ability,
they will shift portfolios to high beta assets when market returns are expected to be high, and
vice-versa. The resulting nonstationarity in beta will systematically bias downward the Jensen
alpha (Jensen, 1968). Explicit tests for market timing ability have been derived under both single-
factor and multifactor asset pricing benchmarks. Typically, additional terms augment the
security market line to test for market timing ability.
We examine market timing tests. Since by construction our simulations involve no market
timing ability, we should not find any market timing ability. Surprisingly, there are strong
indications of timing ability. We investigate several explanations, and in particular the relation of
market timing tests to time-varying expected returns (see Ferson and Schadt, 1996).
2.3 Reward-risk ratios
 5

We document properties of reward-risk ratios. In particular, portfolio performance is

sometimes measured by itÕs Sharpe ratio, defined as the ratio of market excess returns (over the
riskless rate) to market standard deviation. In the Sharpe-Lintner CAPM, the value-weighted
market portfolio has the highest Sharpe ratio.
The Sharpe ratio underlies performance measures for dynamic asset allocation strategies
(e.g., Graham and Harvey, 1997). To evaluate such strategies, benchmark returns are a weighted
average of the riskless and value-weighted market returns having the same standard deviation as
the portfolio under study. The Sharpe ratio is also of interest to practitioners. It is reported by
Morningstar, and is the basis for current risk measurement practice such as the Morgan-Stanley
ÒM-squaredÓ measure (see the Wall Street Journal, 2/10/97). We illustrate how CAPM
departures can easily yield higher Sharpe ratios than the value-weighted index. Although some
previous literature recognizes this general point (e.g., Grinblatt and Titman, 1987, and
MacKinlay, 1995), our results illustrate the implications of excessive reliance on the value-
weighted index in formulating a mutual fund benchmark1.

3. Baseline simulation procedure

This section describes the paperÕs baseline simulation procedure. We discuss sample
construction, mutual fund performance measures using alternative expected return models, and
test statistics under the null hypothesis of no abnormal performance. We use some of the more
commonly used measures as found in the literature, e.g., Bodie, Kane, and Marcus (1996, ch. 24).
The baseline simulations use portfolios of randomly selected stocks. Later, we examine the
sensitivity of the baseline results to stratified-random stock portfolios (e.g., style-based
portfolios), but the conclusion of misspecification is unchanged.

3.1 Sample construction

1
The precise implications of our results for M-squared measures and evaluating dynamic asset allocation strategies
are the subject of a separate study we are undertaking.
 6

We construct a 50-stock mutual fund portfolio each month from January 1964 through
December 1991. We then track these 336 simulated mutual fund portfoliosÕ performance over
three-year periods (months 1 through 36) using a number of performance measures. As
discussed later, these three-year periods are overlapping.
Stock selection. The 50 stocks in each portfolio are selected randomly and without
replacement from the population of all NYSE/AMEX securities having return data on the Center
for Research in Security Prices (CRSP) monthly returns tape. Since the number of NASDAQ
stocks is generally far greater than the number of NYSE/AMEX stocks, inclusion of NASDAQ
stocks in our sampling would have resulted in simulated mutual fund portfolios dominated by
NASDAQ stocks.
Portfolio turnover. While each portfolioÕs performance is evaluated over three years, the
portfolio composition is changed at the beginning of the second and third years (i.e., beginning of
months 13 and 25) to mimic turnover in a typical mutual fund. Specifically, we assume 100%
turnover of the stocks in the mutual fund portfolio at the end of each year.
Data availability criteria. Any NYSE/AMEX security with return data available in
month 1 is eligible for inclusion in the portfolio formed at the beginning of month 1, and similarly
any security with return data available in month 13 can be included in the portfolio formed at the
beginning of month 13. Thus, we impose minimal data-availability requirements in the baseline
simulations. For example, only the securities for which return data become available starting in
months 2 through 11 (e.g., initial public offerings) are excluded from the mutual fund portfolio
formed at the beginning of month 1.
Portfolio returns. For each of the 336 mutual fund portfolios, we construct a time series
of 36 monthly returns starting in month 1. We begin with an equal-weighted portfolio, but the
portfolio is not rebalanced at the end of each month. This is consistent with the monthly returns
earned on a mutual fund that does not trade any of its stocks in one year. We assume dividends
are re-invested, which may or may not be descriptive of an actual mutual fund. However,
performance evaluation assuming dividend reinvestment is typical. Since we reconstruct the
mutual fund at the beginning of months 13 and 25, we begin the second and third years with
equal-weighted portfolios.
 7

3.2 Portfolio performance measures

We apply the following performance measures: Sharpe measure, Jensen alpha, Treynor
measure, appraisal ratio, and Fama-French three-factor model alpha. The finance profession has
used the first four performance measures for many years. The Jensen alpha, the Treynor
measure, and the appraisal ratio are all rooted in the Sharpe-Lintner CAPM, whereas the Fama-
French three-factor alpha is the equivalent of the CAPM-based Jensen alpha in a multi-factor
setting that includes size and book-to-market factors along with the market factor. To evaluate
market timing, we employ two measures: CAPM-based market-timing alpha and gamma and
Fama-French three-factor model-based timing alpha and gamma. Table 1 summarizes the
performance measures and provides a list of selected references for each. Below we briefly
discuss each measure.
[Table 1]
Sharpe measure. The Sharpe measure (see Sharpe, 1966) provides the reward to
volatility trade-off. It is the ratio of the portfolioÕs average excess return divided by the standard
deviation of returns:
Sharpe measure = (ARP - ARf)/σP (1)
where ARP = average return on a mutual fund portfolio over the sample period, Arf = average risk
free return over the sample period, and σP = the standard deviation of excess returns over the
sample period.
Jensen alpha. The Jensen alpha measure (see Jensen, 1968, 1969) is the intercept from
the Sharpe-Lintner CAPM regression of portfolio excess returns on the market portfolio excess
returns over the sample period:
RPt - Rft = αP + βP(RMt - Rft) + εPt (2)
where RPt is the mutual fund portfolio return in month t, Rft is the risk free return in month t, RMt
is the return on the market portfolio in month t, εPt is the white noise error term, and αP βP are
the regressionÕs intercept and slope (beta risk) coefficients.
Treynor measure. The Treynor measure (see Treynor, 1965) is similar to the Sharpe
measure except that it defines reward (average excess return) as a ratio of the CAPM beta risk:
 8

Treynor measure = (ARP - ARf)/βP . (3)

Appraisal ratio. The appraisal ratio is a transformation of the JensenÕs alpha (see
Treynor and Black, 1973). It is the ratio of JensenÕs alpha to the standard deviation of the
portfolioÕs non-market risk (i.e., unsystematic risk) as estimated from eq. (2):
Appraisal ratio = αP/σ(εP). (4)
Fama-French three-factor model alpha. The Fama-French three-factor model alpha (see
Fama and French, 1993) is estimated from the following expanded form of the CAPM regression:
RPt - Rft = αP + βP1(RMt - Rft) + βP2HMLt + βP2SMBt + εPt (5)
where HMLt and SMBt are the Fama-French book-to-market and size factor returns. HMLt
is the high-minus-low book-to-market portfolio return in month t and SMBt is the small-minus-
big size portfolio return in month t. We construct the book-to-market and size factors similarly
to that in Fama and French (1993) and details are available on request.
CAPM market-timing alpha and gamma. We use the Henriksson and Merton (1981)
model to measure the market-timing ability of a mutual fund manager. The quadratic regression
of Treynor and Mazuy (1966) is an alternative measure of evaluating the market-timing ability.
Both measures are CAPM based. The Henriksson-Merton market-timing measure allows for the
beta risk to be different in ex post up and down markets. Specifically, the market-timing alpha
and gamma are given by
RPt - Rft = αP + βP(RMt - Rft) + γP(RMt - Rft)*D + εPt (6)
where D is a dummy variable that equals 1 for (RMt - Rft) > 0 and zero otherwise, and αP and γP
are the market-timing alpha and gamma. Under the null hypothesis of no market timing, both αP
and γP are expected to be zero, whereas a successful market timerÕs mutual fund should exhibit
positive values of αP and γP.
Fama-French three-factor model market-timing alpha and gamma. Henriksson and
Merton (1981) argue that market-timing ability can be inferred from a multi-factor analog of the
CAPM timing alpha and gamma (p. 517). Accordingly, we define the market-timing alpha and
gamma using the Fama-French three-factor model similar to eq. (6). The only difference is that
book-to-market and size-factors are also included as independent variables, as in eq. (5) for the
Fama-French three-factor model.
 9

3.3 Distributional properties of performance measures

Our research design and data analysis yield a time series of 336 overlapping performance
measure estimates using each of the techniques described in section 3.2. Our objective is to
examine the distributional properties of the estimated performance measures. Under the null
hypothesis of no abnormal performance in the mutual fund portfolios consisting of randomly-
selected stocks, Jensen alpha and Fama-French three-factor model alpha are expected to be zero.
We test the null hypothesis that the time series mean of the Jensen alphas and Fama-French
three-factor model alphas is zero. The test statistic is:
t = (1/T) Σt αt / S.E.(α) (7)
where S.E.(α) is the standard error of the mean of the estimated alphas. If the estimated alphas
are assumed independently distributed, then the standard error is given by:
S.E.(α) = [Σt ( αt - (1/T) Σt αt )2 ]1/2 /(T - 1). (8)
Since the alphas are estimated using 36-month overlapping windows, we use a correction for
serial dependence in estimating the standard error of the mean (see Newey and West, 1987, 1994
and Andrews, 1991) in the calculation of the t-statistic in eq. (7). We also discuss the serial
dependence in the alphas estimated using various models.
Under the null hypothesis the alpha and the up-market beta in the Henriksson-Merton
market-timing regression model are zero. This holds also in the Fama-French three-factor model
analog of the Henriksson-Merton regression. The test statistic for abnormal performance (i.e.,
alpha = 0) and market-timing ability (i.e., γP from eq. (6) = 0) are similar to that in eq. (7), with
the standard error adjusted using the Newey-West correction for serial dependence.

4. Simulation results

This section reports the paperÕs main results. We present distributional properties of
regression-based mutual fund performance measures (e.g., Jensen alpha, the associated t-statistic,
and rejection frequencies) and reward-risk ratios (e.g., the Sharpe measure) for randomly- and
non-randomly selected stock portfolios. The performance measures are often misspecified. The
 10

generally significant misspecifications of the CAPM-based performance measures are reduced,

but not eliminated, using the Fama-French three-factor model. For the randomly-selected stock
portfolios, misspecification is generally severe using the CRSP value-weighted portfolio as the
market factor proxy in the CAPM or Fama-French three-factor model regressions to evaluate
performance. Since we select stocks randomly, well-specified performance measures by
construction should not exhibit evidence of market timing. However, the market-timing
performance measures are often economically and statistically significant. In non-random
samples (style portfolios) there is evidence of misspecification using both value- and equal-
weighted indexes as market factor proxies.

4.1 Regression-based performance measures: CRSP value-weighted index as the

market-factor proxy
Table 2 reports distributional properties of the time series of 336 regression-based
performance measures for randomly-selected 50-stock portfolios. The results in this subsection
are based on using the CRSP value-weighted index. Our focus on results using the value-weighted
index is motivated by its ubiquitous use in both academic and practitioner research, due perhaps
to the ÒtrueÓ value-weighted market portfolioÕs central role in the CAPM theory.
Regressions with no market timing variables. From table 2 panel A, the average of 336
Jensen alpha estimates using the CRSP value-weighted index is 27 basis points (t-statistic = 3.08)
per month, or 3.24% per year. The average alpha using the Fama-French three-factor model has
the opposite sign, -10 basis points, and reliably is negative (t-statistic = -4.68). Abnormal
performance of economically significant magnitudes for a na•ve strategy of investing in 50
randomly-selected stocks every year is quite surprising, as are the conflicting results for the two
models.
The standard error of the average Jensen alpha, with the Newey-West correction to
account for serial correlation, is 8.8 basis points per year.2 The serial correlation in the estimated

2
The Newey-West corrected standard errors reported in this study are based on five lags selected on the basis of
sample size. There are alternative lag selection procedures discussed in Andrews (1991) and Newey and West
(1987, 1994). These alternative procedures yield 50-100% larger standard errors only in the case of CAPM-based
regression alphas estimated using the CRSP value-weighted index as the market proxy. In all other cases, i.e.,
alphas from the Fama-French three-factor model using equal- or value-weighted index and Jensen alphas using the
 11

Jensen alphas is a direct consequence of our use of overlapping three-year return windows.
However, the degree of serial correlation is sensitive to the choice of the model, due perhaps to
omitted determinants of expected returns from various models. For example, untabulated results
show that, the value-weighted index CAPM Jensen alphas exhibit autocorrelations that decline
only gradually from about 0.8 at the first lag to 0.1 at lag 33. In contrast, however, the equal-
weight CAPM Jensen alphas or the three-factor model alphas exhibit only a modest degree of
positive autocorrelation. Most of these autocorrelations are not reliably different from zero, with
point estimates generally below 0.1 and several estimates are negative. The autocorrelation-
corrected standard errors are thus substantially larger than the uncorrected standard errors only in
case of the value-weighted CAPM Jensen alphas.
[Table 2]
Specifications of the tests using Jensen alpha or the three-factor model alpha can be
examined by focusing on the distributionsÕ departures from normality or rejection rates of the
null hypothesis of zero abnormal performance. We report skewness and kurtosis properties in
table 2 and rejection frequencies in table 3, which is discussed later. The distribution of Jensen
alphas is significantly positively skewed. A skewness coefficient greater than 0.23 indicates right
skewness at the 5% level of significance (see Pearson and Hartley, 1958).3 The Jensen alpha
distribution does not exhibit significant departure from normality in the tails (i.e., neither fat nor
thin tails), but its large positive mean is likely to generate excessive rejections of the null of zero
abnormal performance.
The estimated Jensen alphas range from -1.28 to 2.69% per month. The large standard
deviation and the wide range even in the absence of abnormal performance are indications that
Jensen alpha, even if properly specified, will have low power to distinguish superior from normal

equal-weighted index, all procedures to implement the Newey-West correction yield virtually identical standard error
estimates. The source of the difference between the standard errors from alternative procedures in case of the CAPM-
based alphas using the value-weighted index appears to be in the persistently high positive autocorrelation in the
estimated alphas using overlapping three-year return data. The alphas using the equal-weighted index or the alphas
from the three-factor model regardless of the choice of the index are far less serially correlated and the standard error
estimates are insensitive to the choice of the Newey-West correction procedure.
3
Since the 336 Jensen alpha estimates are not independent, caution should be exercised in drawing inferences about
the statistical significance of the observed departures from normality.
 12

performance. Abnormal performance of economically large magnitudes in a 50-stock portfolio

might not be statistically detectable.
The magnitudes of average annual abnormal performance of 3.2% and -1.2% indicated by
the CAPM and the Fama-French model are comparable in absolute magnitude to a typical mutual
fundÕs abnormal performance reported in the literature. For example, Malkiel (1995, table III)
estimates an average Jensen alpha using returns before expenses of 239 general equity funds from
1982-1991 to be -2% per year. Employing a number of arbitrage portfolio theory factor models,
Lehmann and Modest (1987) estimate abnormal performance of approximately -3 to -4% per
year using returns after expenses for 130 mutual funds from 1968 to 1982. They conclude that
either the average mutual fund significantly under-performs or that inferences about performance
are sensitive to Òthe choice of what constitutes normal performanceÓ (p. 263). Since we find that
the CAPM and three-factor models indicate abnormal performance magnitudes using random
portfolios that are similar to those reported in the literature using actual mutual fund portfolio
returns, popularly used performance measures appear incapable of distinguishing a mutual fund
managerÕs superior from ordinary performance and/or skill.
Regressions with market timing variables. Panel A shows that the CAPM-based
Henriksson and Merton (1981) test of market timing is severely misspecified. Using the value-
weighted portfolio, the average market timing alpha for the portfolio of randomly-selected 50
stocks is a whopping 63 basis points per month or 7.6% per year (t-statistic = 4.95). Even
though there is no market timing in the simulations, the estimated average market-timing gamma is
-0.22 (t-statistic = -4.19). The Fama-French three-factor-model-based tests of market timing
exhibit a moderate degree of misspecification. The average timing alpha is -7 basis points (t-
statistic = -1.70) and the average timing gamma of -0.03 is indistinguishable from zero. Greater
misspecification of the market-timing tests compared to the Jensen-alpha tests suggests that
omitted determinants of expected returns, departures from normality (e.g., skewness) and/or
changing expected rates of returns might be the contributing factors. We explore these
explanations in section 5.
 13

4.2 Regression-based performance measures: CRSP equal-weighted index as the

market-factor proxy
Although not generally to used evaluate mutual fund performance, we also report results
using the CRSP equal-weighted index as the market factor proxy. It is well known that there are
statistically significant firm-size-related deviations from the CAPM (e.g., Banz, 1981). For
example, given size effects, equal-weighted portfolios of randomly-selected stocks should show
positive Jensen alpha using the equal-weighted index. Therefore, the use of the equal-weighted
index might mitigate any size-related Jensen-alpha misspecifications. The observed
misspecifications are unlikely to be entirely related to firm size, however, because performance
measures based on the Fama-French three-factor model, that explicitly includes a size factor,
were also misspecified.
No market timing. From panel B, consistent with the expectation of lesser
misspecification, the average Jensen alpha and the Fama-French three-factor model alpha are one
basis point or less in absolute magnitude and statistically indistinguishable from zero. The
distribution is significantly right skewed and fat tailed, but the results in table 3 suggest
departures from normality are not large enough to produce test misspecification using the Jensen
alpha performance measure. Since we construct portfolios from randomly-selected stocks, not
surprisingly, Jensen alphas using the equal-weighted index are close to zero. However, the use of
an equal-weighted market factor proxy does not always yield well-specified tests, as seen below.
Market timing. The market-timing tests using both CAPM and the three-factor model
are quite misspecified. The CAPM-based average market-timing alpha is 19 basis points per
month or 2.3% per year. The Fama-French three-factor model also yields an average market
timing alpha of similar magnitude. In both cases the average alphas are statistically highly
significant. To counterbalance the estimated average positive timing alphas in the regressions, the
timing gammas are on average negative. They are -0.10 (t-statistic = -6.47) using the CAPM and
-0.08 (t-statistic = -5.66) using the three-factor model. Thus, commonly-used methods tend to
conclude that a buy-and-hold strategy exhibits negative market-timing ability.
 14

Raw performance measures. Panel C of table 2 reports average monthly returns on the
value- and equal-weighted indexes and the portfolio of randomly-selected stocks. The averages
are calculated from the time series of 336 overlapping three-year average monthly returns. The
grand mean of the 336 three-year average returns for the value-weighted index is 0.93% return per
month with a standard deviation of 0.57%. The corresponding figures for the randomly-selected
50-stock portfolios are 1.26% and 0.94%. The difference is not surprising because the value-
weighted index is dominated by larger, less risky stocks. The average return on the CRSP equal-
weighted index is 1.26% with a standard deviation of 0.94% per month. As expected, this is
comparable to the average return and standard deviation of the portfolio of randomly-selected 50
stocks. This in part explains the lack of misspecification of the performance measures using the
equal-weighted index. That is, tests with no market-timing variables are well-specified when the
sample portfolio by construction mimics the index in virtually every dimension.
4.3 Test statistics and rejection frequencies of regression-based tests of performance
Table 3 reports distributional properties of the test statistics from the 336 CAPM and
three-factor model regressions using the equal- and value-weighted indexes with and without
market timing. To focus on the tail regions of the distributions, table 3 also reports rejection
rates of the null hypothesis of zero abnormal performance or of no market timing ability. The
results in table 3 reinforce those in table 2 and the misspecification of the performance measure
can be dramatic.
From panel A, the average t-statistics are generally large in absolute magnitude when the
regressions employed the CRSP value-weighted index. For example, the average t-statistic for the
Jensen alphas is 0.43 (standard deviation = 1.42) and for the timing alphas it is 0.82 (standard
deviation = 1.34). The standard deviations of the distributions of t-statistics are considerably
greater than 1 for the Jensen alpha and the market-timing alpha using the value-weighted index.4
If the tests were well-specified, the mean (standard deviation) of the distribution of t-statistics
should be zero (one). Panel B shows that the positive means and fat-tailed distributions of t-
statistics for the Jensen alphas and CAPM timing alphas generate excessive rates of rejections of

4
Since the regressions use overlapping return data, the reported standard deviation likely understates the true
standard deviation that would be applicable for a sample of 336 independent estimates of t-statistics.
 15

the null hypothesis in favor of positive abnormal performance. The CAPM timing alpha is
significantly positive at the 5% level of significance 27.7% of the time.
[Table 3]
Panel B shows that, using the equal-weighted index, both the CAPM and the three-factor
model timing alphas indicate positive abnormal performance moderately too often (11.6% and
9.8% compared to an expected rate of 5%). The CAPM timing gamma using both equal- and
value-weighted index and the three-factor model timing gamma using the equal-weighted index
also exhibit too many rejections in favor of negative market timing.
4.4 Reward-risk ratios
The central prediction of the Sharpe-Lintner CAPM is that ex ante the value-weighted
market portfolio has the highest Sharpe ratio. Table 4 reports descriptive statistics on reward-to-
risk ratios for the value- and equal-weighted indexes and the 336 simulated portfolios of
randomly-selected stocks.
Contrary to the CAPM prediction, mutual fund the Sharpe ratios of the CRSP equal-
weighted index and the portfolio of randomly-selected stocks substantially exceed the Sharpe
ratio of the CRSP value-weighted index. The average Sharpe ratio of the value-weighted index is
only 0.10, compared to 0.14 for the equal-weighted index and 0.13 for the simulated portfolio of
randomly-selected stocks. This finding is not driven by extreme observations. Median Sharpe
ratios yield the same inference. Given well-documented inadequacies of the CAPM, these results
should not be surprising. These inadequacies make it less probable that the value-weighted index
was ex ante efficient, but that the equal-weighted index performed better than the value-weighted
index in the 28-year sample period.5
[Table 4]
The Treynor measure uses beta in the denominator of the ratio, unlike the Sharpe
measure, which uses total volatility. Since betas (which are given an equal-weight in our mutual-
fund portfolios) estimated against the value-weighted index are generally greater than those
estimated against the equal-weighted index, one expects the Treynor measure using the value-

5
The Sharpe ratio of the CRSP equal-weighted index is greater than that of the CRSP value-weighted index over a
much longer period beginning in 1926. This makes it less likely that the higher Sharpe ratio of the equal-weighted
index over the 28-year sample period examined in this study is a period-specific phenomenon.
 16

weighted index to exceed that using the equal-weighted index.6 Table 4, however, shows that the
Treynor measure for the portfolios of randomly-selected stocks using the value-weighted index
betas is 0.63 compared to 0.72 using the equal-weighted index betas. These results are consistent
with a lower Sharpe ratio of the value-weighted index than that of the equal-weighted index.
The appraisal ratios using the equal- and value-weighted indexes provide conflicting
inferences. The appraisal ratio of the random-stocks portfolio using the value-weighted index is
0.07 (t-statistic 2.47) compared to -0.02 (t-statistic -1.60) using the equal-weighted index.

4.5 Subperiods
Table 5 reports subperiod results for 1964-71, 1972-81, and 1982-91. It provides both
regression-based measures using the value- and equal-weighted market indexes and reward-risk
ratios. The subperiod results reinforce the impression of serious misspecification.
The average Jensen alpha per month using the value-weighted index for the portfolio of
randomly-selected securities ranges from 66 basis points (t-statistic = 8.77) during 1972-81 to
-16 basis points (t-statistic = -2.11) during 1982-91 (see panel A of table 5). The large positive
average Jensen alpha in the seventies and negative average alpha in the eighties is consistent with
the well-documented time-varying size-related inadequacies of the CAPM. Misspecification is
severe even for a portfolio of randomly-selected -- not extreme size -- stocks.
[Table 5]
The three-factor model alpha is indistinguishable from zero in the 1964-71 subperiod, but
it is a highly significant -15 basis points per month in the subperiods 1972-81 and 1982-91. The
timing alphas in the subperiods using the CAPM with the value-weighted index are 1.07% (t-
statistic = 5.60) and 0.46% (t-statistic = 3.93) per month. The three-factor model timing alphas
are economically and statistically significant in all three subperiods, with a positive sign in the
first subperiod.
Panel B shows that the use of the equal-weighted index eliminates the misspecification of
the Jensen alpha and the three-factor model alpha. The average Jensen alphas ranges from -7 to 3

6
This is expected because beta is a relative risk measure and the equal-weighted index is more volatile than the
value-weighted index.
 17

basis points per month and the three-factor model alphas average -3 to 4 basis points per month
in the three subperiods. These are fairly small economically and in all but one case statistically
insignificant. The CAPM and the three-factor model timing alphas, however, continue to be
significantly non-zero, but their magnitudes are muted compared to those observed using the
value-weighted index. Both the models yield timing alphas that are consistently positive in all
three subperiods. To offset the effect of positive timing alphas in the regression, the market
timing gammas are consistently negative.

4.6 Style (non-random) portfolios

Results so far show that even when portfolios have no systematically unusual
characteristics, i.e., style, performance measures are misspecified. Therefore, our priors are that
style portfoliosÕ (i.e., portfolios comprising non-randomly selected stocks) performance
measures will also be misspecified. We report results using non-randomly selected stocks in the
hope that such analysis would provide clues to the determinants of the misspecifications.
Size portfolios. Table 6 reports results for large- (panel A) and small-capitalization stock
portfolios (panel B) using the CRSP value-weighted index as the market-factor proxy. Large
(small) stocks are defined as those belonging to CRSP market-capitalization deciles 8-10 (deciles
1-3), where the decile rankings are based only on NYSE stocksÕ market capitalizations.
The Jensen alpha and Fama-French three-factor model alpha of the large-stock portfolios
are quite small, 5 and -2 basis points per month, respectively. The corresponding alphas of the
small-stock portfolios are statistically and economically significantly non-zero, however.
Consistent with the size effect, the Jensen alpha of the small-stock mutual fund portfolios is 50
basis points (t-statistic = 3.23) per month. Interestingly, the Fama-French three-factor alpha is
-17 basis points per month (t-statistic = 3.21) or 2% per year. The three-factor timing alphas are
significantly negative for both large- and small-capitalization stocks, and for small-firm stocks it
is a whopping 1.10% per month using the CAPM.
[Table 6]
Book-to-market portfolios. From table 7, the performance measures also exhibit
misspecification when applied to the low (panel A) and high (panel B) book-to-market stock
 18

portfolios. The lowest 30% of the stocks ranked according to their book-to-market ratios are
defined as low book-to-market or growth stocks. The corresponding highest 30% stocks are high
book-to-market or value stocks. Book-to-market ratio is calculated using financial data from
Compustat. Since financial data on Compustat is not available for every NYSE/AMEX stock,
the universe of firms from which the low and high book-to-market stocks are samples is less
comprehensive than that used elsewhere in the study. From table 7, the CAPM-based measures
are misspecified for the high book-to-market (value) portfolios, whereas the three-factor-model-
based measures are misspecified in case of the low book-to-market (growth) portfolios. For
example, the three-factor alpha and the timing alpha are significantly negative for the low book-
to-market stocks, but they are indistinguishable from zero using the high book-to-market
portfolios. Since the low book-to-market stocks are generally large market capitalization stocks,
the three-factor modelÕs misspecification is not limited to small stocks.
[Table 7]

5. Exploring causes of test misspecification: Time-varying expected returns and

co-skewness

In this section we perform an exploratory analysis of whether the test misspecification

documented in the previous section are explained by time-varying expected returns and/or
departures from normality, in particular, coskewness. Neither appears to substantially account
for the performance measure misspecification. There is only weak evidence that the simulated
mutual fund portfoliosÕ estimated performance measures covary with proxies for time-variation
in market expected returns, particularly the book-to-market ratio. We do not find co-skewness to
be systematically associated with the estimated performance measures.
5.1 Association of performance measures with variables proxying for the market
expected return
In our simulated mutual fund portfolios, the null hypothesis of no market timing is true
and performance measures are not expected to show market timing. However, the existing
performance measures assume stationary expected market (or factor) returns and constant factor
 19

sensitivities of the mutual fund portfolios. Both could change through time, thus potentially
inducing test misspecification. Although we do not know the exact relation, one means of
examining whether changing expected market returns induce misspecification is to test for a
relation between the performance measures and predetermined variables that are correlated with
expected market returns.
Our approach complements the emerging literature that seeks to uncover the effect of
mutual fund managersÕ market-timing ability that might be related to time-varying expected
market return as inferred from observable indicators like the dividend yield or term premium (see,
for example, Ferson and Schadt, 1996, Glosten and Jagannathan, 1994, and Chen and Knez,
1996). Ferson and Schadt (1996) infer mutual fund managersÕ market-timing ability from the
relation between the mutual fundsÕ risks and variables correlated with expected market returns.
Ferson and Schadt make the usual assumption that in the absence of market timing and time-
varying expected returns the mutual fund performance measures are well specified. Our objective
is to ascertain whether reported market-timing results for actual funds could be in part a
manifestation of the observed performance measure misspecification.
We regress the time series of 336 estimated performance measures (i.e., estimated alphas
from the CAPM and the Fama-French three-factor models without market timing, and estimated
alphas and betas from these two models with market timing) on a set of pre-determined
information variables that previous literature has shown to be correlated with time variation in
expected market returns (e.g., Fama and French, 1989, Ferson and Harvey, 1991, Breen, Glosten,
and Jagannathan, 1989, and Evans, 1994). The information variables we use are dividend yield on
the NYSE-AMEX value-weighted portfolio, book-to-market ratio of the value-weighted NYSE-
AMEX stocks (for which book value of equity data are available on COMPUSTAT), ten-year
Government bond yield, term premium measured as the difference between the ten-year bond
yield and the one-month T-bill interest rate, and default premium measured as the difference
between the junk-bond yield and the 10-year Government bond yield. We also entertained
additional information variables like the price-earnings ratio, one-month T-bill interest rate, and
default premium defined as the difference between BAA and AAA corporate bond yields.
 20

Neither individually nor collectively did they add significantly to the reported results, so we omit
those from the tabulated results.
Table 8 reports the results using the value-weighted index as the market-factor proxy
employed in estimating various performance measures. We obtain results that are qualitatively
similar to those reported below using the equal-weighted index. Since the performance measures
are estimated using returns for overlapping three-year periods, the residuals from the regressions
of performance measures on the information variables are likely to be autocorrelated. This is
confirmed by the observed low values of the Durbin-Watson statistic. We re-estimate the models
that had significant Durbin-Watson statistic by fitting a first- and second-order autoregressive
process on the errors. The Durbin-Watson statistic of the regression models using the
transformed variables is close to 2 and statistically insignificant.
Panel A contains results using performance measures for the simulated portfolios
consisting randomly-selected 50 stocks. The CAPM-based alphas with and without market
timing do not exhibit a statistically reliable evidence of covariation between the performance
measures and market expected return proxies. The CAPM market-timing gamma is positively
related to book-to-market and negatively related to default premium at the 5% significance level
and dividend yield is significant at the 10% level. The opposite signs on the coefficients on
dividend yield and book-to-market are surprising because expected market returns increase in
both these variables (e.g., Fama and French, 1989, and Kothari and Shanken, 1997). The
significant relation between the timing gammas and the information variables in the absence of
true market timing in the simulations raises questions about the interpretation of similar
associations using mutual fund return data (e.g., Ferson and Schadt, 1996). It appears that a
portion of the observed association between information variables and market-timing gammas
using mutual fund return data might be due to model misspecification. Panel A also shows that
the Fama-French three-factor model alphas with and without timing are significantly negatively
associated with book-to-market, term premium, and default premium. The timing gammas,
however, do not exhibit significant covariation with the information variables.
[Table 8]
 21

Panels B and C report results for stratified random samples of large and small firms and
panels D and E report results for the high and low book-to-market stock portfolios. These
results indicate that the small firms and high book-to-market portfoliosÕ three-factor model
alphas are reliably negatively correlated with the book-to-market ratio and default premium. The
results suggest that in high expected market return periods the three-factor model is likely to
erroneously indicate under-performance of small and high book-to-market stocks, and conversely,
above-normal performance in low expected market return periods.
Except for the large stock portfolios, the three-factor model timing gammas are reliably
correlated with dividend yield and default premium. However, the associations between
performance measures and the information variables do not appear to entirely explain the
misspecifications noted in tables 6 and 7. There we find that the three-factor model is
misspecified in the case of low, not high, book-to-market stock portfolios. Panels B through E
show that there is limited evidence of the CAPM-based performance measures, with and without
timing, being associated with the information variables.
5.2 Association of performance measures with coskewness
The observed performance measure misspecification could be related to the return
distributionsÕ departures from normality. First, departures from joint normality of the mutual
fund portfolio returns and market returns could distort the performance measuresÕ sampling
distribution under the null hypothesis (Stapleton and Subrahmanyam, 1983). Second, if there is
coskewness in portfolio returns that is priced (Kraus and Litzenberger, 1976 and Rubinstein,
1973), then the mean-variance-analysis-based performance measures examined in this study
would likely be misspecified in part because coskewness and beta are significantly positively
correlated (Kraus and Litzenberger, 1976).
To examine coskewness-related performance-measure misspecification, we regress the
time series of performance measures on portfoliosÕ coskewness estimated contemporaneously
with the three-year period used to estimate the performance measures. Following Kraus and
Litzenberger, coskewness is defined as
Coskewness = Cov [(Rm - Avg Rm)2, (Rp - Avg Rp)]/E (Rm - Avg Rm)3
 22

which is the ratio of a portfolioÕs covariance with squared market return divided by the skewness
of the market return. Untabulated results show that all the mutual fund portfolios we examine
exhibit highly significant coskewness. However, the estimated coskewness has generally little
ability to explain the variation in estimated performance measures. Thus, although we provide
little direct evidence on the price of coskewness in the market, portfolio returnsÕ coskewness
with the market does not appear to explain the misspecifications of the performance measures
examined in this study.

6. Summary and conclusions

Although there is a large literature on mutual fund performance measures, their empirical
properties in the absence of abnormal performance have received little attention. We study these
properties. From our simulations, the main message is that standard mutual fund performance
are unreliable and can result in false inferences. In particular, it is easy to detect abnormal
performance and market-timing ability when none exists.
Our results also show that the range of measured performance is quite large even when
true performance is ordinary. This provides a benchmark to gauge mutual fund performance.
Comparisons of our numerical results with those reported in actual mutual fund studies raises the
possibility that reported results are due to misspecification, rather than abnormal performance.

Finally, the results indicate that procedures based on the Fama-French 3-factor model are
somewhat better than CAPM based measures. This is not surprising, and indicates that ÒstyleÓ
analysis is useful in benchmarking fund returns. The misspecification even for Fama-French
suggests at least two possibilities. One is that size and book-to-market do not completely
describe the characteristics relevant for expected returns. The second is related to the estimation
process, and that sampling distributions of the performance measures differ from those assumed
under the null hypothesis, for example because expected returns change over time. Further
investigation of the latter possibility could be particularly fruitful in explaining why our tests
using simulated portfolios often show market timing when none is present.
 23

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 25

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 27
Table 1
Mutual fund performance measures

Performance measure Expression Selected references

Sharpe measure1 (ARP - ARf)/σP Sharpe (1966), MacKinlay (1995)

Jensen alpha2 RPt - Rft = αP + βP(RMt - Rft) + εPt Brown and Goetzmann (1995), Carhart (1997), Elton, Gruber, and
Blake (1996b), Elton, Gruber, Das, and Hlavka (1993), Ferson and
Schadt (1996), Fama (1972), Henriksson (1984), Grinblatt and
Titman (1990), Jensen (1968, 1969), Malkiel (1995)

Treynor measure (ARP - ARf)/βP Treynor (1965)

Appraisal ratio αP/σ(εP) Elton, Gruber, and Blake (1996), Elton, Gruber, and Padberg
(1976), Treynor and Black (1973)

Fama-French three-factor model RPt - Rft = αP + βP1(RMt - Rft) + Carhart (1997), Elton, Gruber, and Blake (1996a)
alpha3 βP2HMLt + βP2SMBt + εPt

Henriksson-Merton market RPt - Rft = αP + βP(RMt - Rft) + γP(RMt Henriksson and Merton (1981), Henriksson (1984), Graham and
timing model4 - Rft)*D + εPt Harvey (1996)

Market timing using the Fama- RPt - Rft = αP + βP1(RMt - Rft) + Henriksson and Merton (1981)
French three-factor model βP2HMLt + βP2SMBt + γP(RMt -
Rft)*D + εPt

1
ARP = average return over the sample period, Arf = average risk free return over the sample period, and σP = the standard deviation of
excess returns over the sample period.
2
RPt is portfolio return in month t, Rft is the risk free return in month t, RMt is the return on the market portfolio in month t, εPt is the
white noise error term, and αP βP are the regressionÕs intercept and slope (beta risk) coefficients.
 28
3
HMLt and SMBt are the Fama-French book-to-market and size factor returns; HMLt is the high-minus-low book-to-market portfolio
return in month t and SMBt is the small-minus-big size portfolio return in month t.
4
D is a dummy variable that equals 1 for (RMt - Rft) > 0 and zero otherwise, and αP and γP are the market-timing alpha and gamma.
 29
Table 2
Distributional properties of 336 regression-based mutual fund performance measures of
portfolios of randomly-selected securities

Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is
constructed. Its performance is tracked for a three-year period (months 1 through 36). The portfolio composition is changed
100% in months 13 and 25. The 50 stocks are selected randomly and without replacement from all NYSE/AMEX stocks with
non-missing return data in month 1, and this procedure is repeated in months 13 and 25 using stocks available in those
months. For each of the 336 portfolios, a time-series of monthly returns from month 1 through 36 is constructed. Portfolios
returns are equal-weighted at the beginning of months 1, 13, and 25, but they are not rebalanced in the intervening periods.
Returns are inclusive of dividends.
All the performance measures are as defined in table 1. Performance measures in panel A (B) are estimated using the CRSP
value-weighted (equal-weighted) index return as the market-factor proxy.
Standard errors, S.E., are calculated by applying the Newey-West (1987) correction for serial dependence up to five lags.
T-statistics are ratios of the performance measuresÕ mean values to the standard errors, S.E.
Descriptive statistics in panel C are for a sample of 336 three-year average returns on the simulated mutual fund portfolios and
CRSP equal- and value-weighted indexes.

Performance measure Mean S.E.Std. t-stat

Min Median Max Skewnes Kurtosis
Dev. s
Panel A: Portfolios of 50 randomly-selected securities, CRSP value-weighted index as market factor

Jensen α in % 0.27 0.088 3.08 0.72 -1.28 0.24 2.69 0.32 2.72

FF 3-factor α in % -0.10 0.022 -4.68 0.35 -1.57 -0.12 1.01 0.14 4.24

CAPM timing α in % 0.63 0.127 4.95 1.07 -2.09 0.60 3.76 0.07 2.51

CAPM timing γ -0.22 0.053 -4.19 0.47 -1.47 -0.30 1.39 0.26 3.18

FF 3-factor timing α in % -0.07 0.043 -1.70 0.59 -2.10 -0.09 2.13 0.22 3.82

FF 3-factor timing γ -0.03 0.019 -1.48 0.29 -0.95 -0.03 1.14 -0.01 3.68

Panel B: Portfolios of 50 randomly-selected securities, CRSP equal-weighted index as market factor

Jensen α in % -0.01 0.019 -0.74 0.31 -0.97 -0.05 1.12 0.47 3.83

FF 3-factor α in % 0.00 0.020 0.00 0.34 -1.10 -0.03 1.03 0.40 3.64

CAPM timing α in % 0.19 0.030 6.17 0.50 -1.25 0.15 1.75 0.27 2.91

CAPM timing γ -0.10 0.015 -6.47 0.21 -0.84 -0.10 0.49 -0.26 3.63

FF 3-factor timing α in % 0.16 0.031 5.00 0.53 -1.64 0.15 1.92 0.04 3.16

FF 3-factor timing γ -0.08 0.014 -5.66 0.21 -0.71 -0.08 0.48 -0.19 3.39

Panel C: Descriptive statistics on returns

Random stocks portfolio 1.24 0.122 10.14 0.98 -1.49 1.28 3.87 -0.34 2.79
return %
CRSP v-wt return % 0.93 0.073 12.74 0.57 -0.98 0.95 2.40 -0.53 3.58
 30
CRSP eq-wt return % 1.26 0.122 10.29 0.94 -1.66 1.27 3.21 -0.54 2.79
 31
Table 2 (ContÕd)

The 95th and 99th percentiles of skewness coefficients for a sample of 300 are 0.230 and 0.329, and for
samples of 350 they are 0.213 and 0.305.

Selected percentiles for the kurtosis coefficient are:

Sample 1% 5% 95% 99%
300 2.46 2.59 3.47 3.79
350 2.50 2.62 3.44 3.72
 32
Table 3
Test statistics and rejection frequencies for the regression-based mutual fund performance measures:
Randomly-selected stock portfolios
Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is constructed. Its performance is tracked for a three-year
period (months 1 through 36). The portfolio composition is changed 100% in months 13 and 25. The 50 stocks are selected randomly and without replacement from all
NYSE/AMEX stocks with non-missing return data in month 1, and this procedure is repeated in months 13 and 25 using stocks available in those months. For each of the
336 portfolios, a time-series of monthly returns from month 1 through 36 is constructed. Portfolios returns are equal-weighted at the beginning of months 1, 13, and 25,
but they are not rebalanced in the intervening periods. Returns are inclusive of dividends.
All the performance measures regressions are as described in table 1. Performance measures in panel A (B) are estimated using the CRSP value-weighted (equal-weighted)
index return as the market-factor proxy.
Distributional properties of the test statistics are for samples of 336 t-statistics from the performance measure regressions described in table 1.
Rejection frequencies are based on one-sided tests of the null hypothesis of zero value of the performance measure. The table values report the pecentage of times out of 336
the null hypothesis is rejected at the specified level of significance.

Distributional properties of test-statistics Rejection frequencies

Performance measure Mean Std. Min Median Max Skewness Kurtosis <0.5% <2.5% <5% >95% >97.5% >99.5%
Dev.
Panel A: Portfolios of 50 randomly-selected securities, CRSP value-weighted index as market factor

Jensen α 0.43 1.42 -2.97 0.43 4.30 0.01 2.36 0.6 3.9 6.9 21.7 15.5 4.8

FF 3-factor α -0.36 1.02 -3.19 -0.39 3.10 0.13 3.18 1.2 4.5 8.9 2.1 1.8 0.3

CAPM timing α 0.82 1.34 -2.20 0.79 4.02 0.05 2.27 0.0 0.3 2.1 27.7 21.4 7.1

CAPM timing γ -0.71 1.26 -4.01 -0.75 2.91 -0.07 2.57 4.5 15.5 22.9 2.1 0.6 0.3

FF 3-factor timing α -0.16 1.06 -3.22 -0.19 2.96 0.20 2.97 0.6 2.7 5.7 5.1 3.0 0.3

FF 3-factor timing γ -0.09 1.13 -3.12 -0.13 3.23 0.05 2.85 0.6 5.1 8.3 5.4 3.0 1.2
 33
Table 3 (ContÕd)
Panel B: Portfolios of 50 randomly-selected securities, CRSP equal-weighted index as market factor

Jensen α -0.10 1.01 -2.59 -0.18 3.07 0.20 2.90 0.0 2.1 5.7 4.5 2.1 0.3

FF 3-factor α -0.04 1.05 -2.90 -0.09 3.33 0.22 2.99 0.3 2.1 4.5 5.7 2.4 0.9

CAPM timing α 0.38 1.07 -2.12 0.35 3.64 0.23 2.83 0.0 0.6 1.5 11.6 5.4 2.1

CAPM timing γ -0.58 1.16 -4.75 -0.58 2.44 -0.13 3.24 3.0 9.5 17.3 2.1 1.5 0.0

FF 3-factor timing α 0.32 1.09 -2.74 0.28 3.79 0.15 2.99 0.3 0.9 2.1 9.8 4.8 2.1

FF 3-factor timing γ -0.47 1.16 -4.37 -0.49 2.56 -0.26 3.22 3.0 9.5 13.7 2.4 1.2 0.0
 34
Table 4
Distributional properties of reward-risk ratios of portfolios of randomly-selected securities
Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is
constructed. Its performance is tracked for a three-year period (months 1 through 36). The portfolio composition is changed
100% in months 13 and 25. The 50 stocks are selected randomly and without replacement from all NYSE/AMEX stocks with
non-missing return data in month 1, and this procedure is repeated in months 13 and 25 using stocks available in those
months. For each of the 336 portfolios, a time-series of monthly returns from month 1 through 36 is constructed. Portfolios
returns are equal-weighted at the beginning of months 1, 13, and 25, but they are not rebalanced in the intervening periods.
Returns are inclusive of dividends.
All the reward-risk ratios are as described in table 1.
Standard errors, S.E., are calculated by applying the Newey-West (1987) correction for serial dependence up to five lags.
T-statistics are ratios of the performance measuresÕ mean values to the standard errors, S.E.
Descriptive statistics in panels A and C are for samples of 336 three-year average returns on the CRSP equal- and value-
weighted indexes.

Performance measure Mean S.E. t-stat

Std.d Min Median Max Skewnes Kurtosis
ev. s
Panel A: CRSP value-weighted index

Sharpe ratio 0.10 0.020 5.05 0.12 -0.33 0.12 0.43 -0.43 3.66

Panel B: Portfolios of 50 randomly-selected securities, CRSP value-weighted index as market factor

Sharpe ratio 0.13 0.020 6.50 0.16 -0.30 0.13 0.55 -0.11 2.70
Treynor measure 0.63 0.099 6.32 0.81 -1.65 0.70 3.13 -0.26 3.01
Appraisal ratio 0.07 0.030 2.47 0.24 -0.51 0.07 0.73 0.01 2.36

Panel C: CRSP equal-weighted index

Sharpe ratio 0.14 0.021 6.71 0.16 -0.32 0.14 0.50 -0.23 2.45

Panel D: Portfolios of 50 randomly-selected securities, CRSP equal-weighted index as market factor

Treynor measure 0.72 0.122 5.87 0.99 -2.12 0.76 3.16 -0.33 3.09
Appraisal ratio -0.02 0.010 -1.60 0.18 -0.45 -0.03 0.52 0.20 2.88
 35
Table 5
Distributional properties of mutual fund performance measures: Sub-period analysis using
portfolios of randomly-selected securities

Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is
constructed. Its performance is tracked for a three-year period (months 1 through 36). The portfolio composition is changed
100% in months 13 and 25. The 50 stocks are selected randomly and without replacement from all NYSE/AMEX stocks with
non-missing return data in month 1, and this procedure is repeated in months 13 and 25 using stocks available in those
months. For each of the 336 portfolios, a time-series of monthly returns from month 1 through 36 is constructed. Portfolios
returns are equal-weighted at the beginning of months 1, 13, and 25, but they are not rebalanced in the intervening periods.
Returns are inclusive of dividends.
Results for subperiod 1964-71 are based on samples of 96 three-year performance measures. Results for subperiods 1972-81
and 1982-91 are each based on samples of 120 performance measures.
All the regression-based performance measures and reward-risk ratio performance measures are as defined in table 1.
Performance measures in panel A (B) are estimated using the CRSP value-weighted (equal-weighted) index return as the
market-factor proxy.
Standard errors, S.E., are calculated by applying the Newey-West (1987) correction for serial dependence up to five lags.
T-statistics are ratios of the performance measuresÕ mean values to the standard errors, S.E.
Descriptive statistics in panel C are for samples of 96 (subperiod 1964-71), 120 (subperiod 1972-81), and 120 (subperiod 1982-
91) three-year average returns on the simulated mutual fund portfolios and CRSP equal- and value-weighted indexes.

Performance measure 1964-71 1972-81 1982-91

Mean S.E. t-stat Mean S.E. t-stat Mean S.E. t-stat

Panel A: Portfolios of 50 randomly-selected securities, CRSP value-weighted index as market factor

Jensen α in % 0.33 0.182 1.81 0.66 0.075 8.77 -0.16 0.076 -2.11

FF 3-factor α in % 0.01 0.042 0.19 -0.15 0.035 -4.14 -0.15 0.024 -6.29

CAPM timing α in % 0.29 0.235 1.23 1.07 0.191 5.60 0.46 0.117 3.93

CAPM timing γ 0.01 0.057 0.18 -0.22 0.100 -2.22 -0.41 0.057 -7.14

FF 3-factor timing α in % 0.28 0.054 5.18 -0.33 0.053 6.14 -0.10 0.035 2.93

FF 3-factor timing γ -0.17 0.025 7.04 0.09 0.023 3.67 -0.03 0.024 1.12

Treynor measure 0.37 0.183 2.00 0.82 0.144 5.69 0.64 0.111 5.77
Appraisal ratio 0.10 0.056 1.77 0.21 0.025 8.52 -0.09 0.029 -2.93
 36
Table 5 (contÕd)
Panel B: Portfolios of 50 randomly-selected securities, CRSP equal-weighted index as market factor

Jensen α in % 0.03 0.043 0.77 -0.07 0.027 -2.59 0.00 0.022 0.18

FF 3-factor α in % 0.04 0.041 1.00 -0.03 0.029 -0.86 -0.01 0.027 -0.30

CAPM timing α in % 0.21 0.047 4.10 0.10 0.047 2.06 0.25 0.050 5.00

CAPM timing γ -0.08 0.014 -5.50 -0.06 0.015 -4.20 -0.15 0.031 -4.74

FF 3-factor timing α in % 0.19 0.053 3.64 0.07 0.056 1.19 0.21 0.041 5.23

FF 3-factor timing γ -0.07 0.016 4.23 -0.04 0.017 2.24 -0.13 0.023 5.67

Treynor measure 0.51 0.251 2.02 0.82 0.144 5.69 0.64 0.105 6.09
Appraisal ratio -0.00 0.023 -0.09 -0.04 0.016 -2.75 0.00 0.013 0.00

Panel C: Descriptive statistics on returns and Sharpe ratios of indexes and random stock portfolio

VW index return in % 0.53 0.088 6.00 0.90 0.110 8.16 1.28 0.100 12.82
EW index return in % 0.88 0.229 3.84 1.68 0.175 9.58 1.13 0.103 10.99
Random stocks portfolio 0.93 0.243 3.81 1.61 0.174 9.25 1.11 0.100 11.12
return in %
VW index Sharpe ratio 0.04 0.024 1.79 0.06 0.019 3.16 0.19 0.016 11.81
EW index Sharpe ratio 0.10 0.041 2.37 0.17 0.027 6.44 0.14 0.025 5.68
Random stocks portfolio 0.10 0.041 2.31 0.16 0.025 6.32 0.13 0.023 5.70
Sharpe ratio
 37
Table 6
Distributional properties of mutual fund performance measures of non-randomly sampled
portfolios: large and small market capitalization securities

Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is
constructed. Its performance is tracked for a three-year period (months 1 through 36). The portfolio composition is changed
100% in months 13 and 25. In panel A (B) the 50 large (small) stocks are selected randomly and without replacement from all
NYSE/AMEX stocks whose market capitalization falls among the largest (smallest) three deciles of stocks ranked each year on
January 1 according to the equity market capitalization of all the NYSE stocks. From this universe of large and small stocks
each year, any firm with non-missing return data in month 1 is eligible for inclusion. This procedure is repeated in months 13
and 25 using large and small stocks available in those months. For each of the 336 portfolios, a time-series of monthly returns
from month 1 through 36 is constructed. Portfolios returns are equal-weighted at the beginning of months 1, 13, and 25, but
they are not rebalanced in the intervening periods. Returns are inclusive of dividends.
All the distributional properties are based on samples of 336 performance measures.
All the regression-based performance measures and reward-risk ratio performance measures are as defined in table 1. All
performance measures are estimated using the CRSP value-weighted index return as the market-factor proxy.
Standard errors, S.E., are calculated by applying the Newey-West (1987) correction for serial dependence up to five lags.
T-statistics are ratios of the performance measuresÕ mean values to the standard errors, S.E.

Performance measure Mean S.E.

t-stat Std. Min Median Max Skewnes Kurtosis
Dev. s
Panel A: Large capitalization securities, CRSP value-weighted index as market factor

Jensen α in % 0.05 0.016 2.88 0.20 -0.56 0.04 0.69 0.13 3.14

FF 3-factor α in % -0.02 0.013 -1.38 0.19 -0.62 -0.02 0.51 -0.16 3.00

CAPM timing α in % 0.01 0.032 0.19 0.36 -1.02 0.01 1.10 0.17 3.27

CAPM timing γ 0.01 0.017 0.65 0.19 -0.65 0.02 0.64 -0.56 4.33

FF 3-factor timing α in % -0.08 0.028 -2.68 0.36 -1.48 -0.08 0.98 0.12 3.78

FF 3-factor timing γ 0.02 0.016 1.25 0.19 -0.76 0.04 0.54 -0.74 4.71

Return in % 0.99 0.078 12.67 0.63 -1.35 1.03 2.36 -0.62 3.90
Sharpe ratio 0.11 0.016 6.75 0.13 -0.37 0.12 0.48 -0.27 3.87
Treynor measure 0.45 0.071 6.39 0.58 -1.73 0.50 1.84 -0.60 3.94
Appraisal ratio 0.04 0.013 2.69 0.17 -0.53 0.04 0.54 -0.03 3.24
 38
Table 6 (ContÕd)

Panel B: Small capitalization securities, CRSP value-weighted index as market factor

Jensen α in % 0.50 0.154 3.23 1.21 -1.79 0.53 3.79 0.20 2.19

FF 3-factor α in % -0.17 0.053 -3.21 0.54 -1.76 -0.15 1.17 -0.28 2.82

CAPM timing α in % 1.10 0.207 5.31 1.69 -2.81 1.06 4.91 -0.03 2.12

CAPM timing γ -0.37 0.095 -3.92 0.80 -2.49 -0.46 3.10 0.48 4.29

FF 3-factor timing α in % -0.16 0.102 -1.52 0.99 -3.15 -0.15 2.06 -0.40 2.92

FF 3-factor timing γ -0.02 0.044 -0.39 0.46 -1.20 -0.04 1.71 0.44 3.53

Return in % 1.47 0.184 7.97 1.45 -2.11 1.51 5.32 -0.06 2.37
Sharpe ratio 0.13 0.023 5.48 0.19 -0.30 0.13 0.57 -0.08 2.28
Treynor measure 0.90 0.201 4.46 2.09 -2.60 0.88 30.9 -8.93 128.7
Appraisal ratio 0.08 0.031 2.52 0.24 -0.50 0.10 0.65 2.30 7.46
 39
Table 7
Distributional properties of mutual fund performance measures of non-randomly sampled
portfolios: low and high book-to-market securities

Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is
constructed. Its performance is tracked for a three-year period (months 1 through 36). The portfolio composition is changed
100% in months 13 and 25. In panel A (B) the 50 low (high) book-to-market stocks are selected randomly and without
replacement from all NYSE/AMEX stocks whose market capitalization falls among the lowest (highest) three deciles of stocks
ranked each year on January 1 according to the book-to-market ratios of all the NYSE/AMEX stocks. Book-to-market ratios
are defined as the book value of common equity at the beginning of each year, as reported on the COMPUSTAT, divided by
the market capitalization of equity at the beginning of the year. From this universe of low and high book-to-market stocks each
year, any firm with non-missing return data in month 1 is eligible for inclusion. This procedure is repeated in months 13 and
25 using large and small stocks available in those months. For each of the 336 portfolios, a time-series of monthly returns
from month 1 through 36 is constructed. Portfolios returns are equal-weighted at the beginning of months 1, 13, and 25, but
they are not rebalanced in the intervening periods. Returns are inclusive of dividends.
All the distributional properties are based on samples of 336 performance measures.
All the regression-based performance measures and reward-risk ratio performance measures are as defined in table 1. All
performance measures are estimated using the CRSP value-weighted index return as the market-factor proxy.
Standard errors, S.E., are calculated by applying the Newey-West (1987) correction for serial dependence up to five lags.
T-statistics are ratios of the performance measuresÕ mean values to the standard errors, S.E.

Performance measure Mean S.E. Std.t-stat

Min Median Max Skewnes Kurtosis
Dev. s
Panel A: Low book-to-market securities, CRSP value-weighted index as market factor

Jensen α in % -0.05 0.069 0.67 0.59 -1.72 -0.08 1.26 0.06 2.32

FF 3-factor α in % -0.10 0.022 4.53 0.32 -1.57 -0.11 0.90 -0.19 4.25

CAPM timing α in % 0.29 0.096 2.97 0.84 -1.72 0.24 2.68 0.08 2.72

CAPM timing γ -0.19 0.032 6.05 0.32 -0.89 -0.23 0.92 0.38 3.11

FF 3-factor timing α in % -0.14 0.031 -4.58 0.49 -1.72 -0.14 1.18 -0.18 3.14

FF 3-factor timing γ 0.02 0.017 1.24 0.25 -0.66 0.02 1.09 0.40 4.05

Return in % 0.95 0.116 8.16 0.93 -2.32 1.05 3.01 -0.74 3.75
Sharpe ratio 0.09 0.018 4.86 0.14 -0.37 0.10 0.37 -0.54 3.04
Treynor measure 0.36 0.086 4.26 0.70 -2.11 0.48 1.67 -0.92 3.87
Appraisal ratio -0.01 0.030 0.24 0.01 -0.65 -0.03 0.71 0.19 2.35
 40
Table 7 (ContÕd)

Panel B: High book-to-market securities, CRSP value-weighted index as market factor

Jensen α in % 0.61 0.106 5.77 0.84 -1.72 0.68 2.58 -0.19 2.18

FF 3-factor α in % 0.03 0.048 0.60 0.46 -1.75 0.05 1.95 -0.31 4.14

CAPM timing α in % 1.00 0.157 6.32 1.28 -2.50 1.09 3.65 -0.28 2.45

CAPM timing γ -0.24 0.082 2.99 0.67 -2.26 -0.33 2.78 0.50 3.99

FF 3-factor timing α in % -0.03 0.085 -0.39 0.80 -2.65 0.11 1.58 -0.54 2.81

FF 3-factor timing γ 0.03 0.019 1.42 0.35 -0.97 -0.01 1.45 0.47 3.71

Return in % 1.56 0.129 12.07 1.03 -1.14 1.74 4.02 -0.35 2.39
Sharpe ratio 0.17 0.021 8.16 0.17 -0.25 0.19 0.56 -0.17 2.25
Treynor measure 0.99 0.128 7.73 1.07 -1.67 1.06 6.23 0.65 5.49
Appraisal ratio 0.16 0.029 5.30 0.24 -0.52 0.17 0.63 -0.15 2.38
 41
Table 8
Relation between portfolio performance measures and pre-determined state variables
Performance Measuret,t+3 = g0 + g1 Dividend Yieldt + g2 Book-to-markett + g3 Long-term Government bond yieldt + g4 Term premiumt + g6
Default premiumt + Errort,t+3
t-statistics are reported below the estimated coefficients
Performance measures are as described in table 1. All performance measures are estimated using the CRSP value-weighted index return as the market-factor proxy.
Dividend yield at the end of month t is the ratio of total dividend in the preceding 12 months on the portfolio of all the NYSE/AMEX stocks to the portfolioÕs market
capitalization at the end of month t.
Book-to-market ratio at the end of month t is the value-weighted ratio of the most recent available book-value of common equity of the NYSE/AMEX stocks to the
market capitalization of these stocks at the end of month t.
Long-term Government bond yield at the end of month t is the yield-to-maturity on 10-year Government bonds.
Term premium at the end of month t is the difference between the long-term Government bond yield and one-month Treasury bill interest rate.
Default premium at the end of month t is the difference between the yield on junk bonds and the yield on AAA-rated corporate bonds.

Sample: Each month from January 1964 through December 1991 (336 months) a 50-stock mutual fund portfolio is constructed. Its performance is tracked for a three-
year period (months 1 through 36). The portfolio composition is changed 100% in months 13 and 25. For each of the 336 portfolios, a time-series of monthly returns
from month 1 through 36 is constructed. Portfolios returns are equal-weighted at the beginning of months 1, 13, and 25, but they are not rebalanced in the intervening
periods. Returns are inclusive of dividends.
In panel A, the 50 stocks are selected randomly and without replacement from all NYSE/AMEX stocks with non-missing return data in month 1, and this procedure is
repeated in months 13 and 25 using stocks available in those months.
In panel B (C) the 50 large (small) stocks are selected randomly and without replacement from all NYSE/AMEX stocks whose market capitalization falls among the
largest (smallest) three deciles of stocks ranked each year on January 1 according to the equity market capitalization of all the NYSE stocks. From this universe of large
and small stocks each year, any firm with non-missing return data in month 1 is eligible for inclusion. This procedure is repeated in months 13 and 25 using large and
small stocks available in those months.
In panel D (E) the 50 low (high) book-to-market stocks are selected randomly and without replacement from all NYSE/AMEX stocks whose market capitalization falls
among the lowest (highest) three deciles of stocks ranked each year on January 1 according to the book-to-market ratios of all the NYSE/AMEX stocks. Book-to-market
ratios are defined as the book value of common equity at the beginning of each year, as reported on the COMPUSTAT, divided by the market capitalization of equity at
the beginning of the year. From this universe of low and high book-to-market stocks each year, any firm with non-missing return data in month 1 is eligible for
inclusion.
 42
Table 8 (ContÕd)

State variable Performance measure

Jensen alpha FF 3-factor CAPM market timing FF 3-factor market timing

model alpha
Alpha Gamma Alpha Beta

Panel A: Portfolio of randomly-selected 50 stocks

Constant 0.009 0.001 0.009 0.214 0.007 -0.32

1.98 1.08 1.34 1.13 3.68 -3.30
Dividend yield -0.175 0.055 -0.125 -16.34 -0.026 1.79
-1.12 1.35 -0.51 -1.79 -0.25 0.36
Book-to-market 0.004 -0.003 0.001 0.53 -0.006 0.14
1.04 -3.91 0.09 2.39 -4.32 1.68
LT bond yield -0.72 -0.048 0.129 -22.05 -0.033 11.18
-1.28 -0.33 0.16 -0.74 -0.10 0.73
Term premium 0.201 -0.373 -0.159 10.65 -0.397 5.08
0.57 -2.72 -0.36 0.54 -1.92 0.40
Default premium 0.344 -0.339 0.555 -87.60 -0.486 8.06
0.59 -3.26 0.655 -2.93 -1.79 1.57
 43
Table 8 (ContÕd)

Panel B: Portfolio of 50 large stocks

Constant 0.001 -0.000 -0.000 0.03 0.001 -0.09

0.72 -0.27 -0.06 0.27 0.56 -0.93
Dividend yield -0.063 -0.005 -0.015 -3.11 -0.062 2.56
-1.65 -0.15 -0.23 -1.10 -1.10 0.83
Book-to-market 0.002 -0.001 -0.000 0.13 -0.000 -0.000
2.80 -0.87 -0.12 2.02 -0.20 -0.01
LT bond yield 0.045 0.038 -0.002 5.49 0.031 4.32
0.38 0.36 -0.01 0.47 0.15 0.33
Term premium -0.118 0.093 -0.105 2.44 0.081 3.77
-1.19 1.17 -0.51 0.24 0.44 0.35
Default premium 0.186 0.157 0.542 -31.09 0.316 -10.27
1.52 2.11 1.95 -1.99 1.29 -0.64
Panel C: Portfolio of 50 small stocks

Constant 0.001 0.001 -0.001 0.38 0.007 0.49

0.21 0.28 -0.15 1.14 2.12 1.33
Dividend yield 0.287 0.222 0.645 -33.17 0.336 -0.040
1.38 1.84 2.04 -1.96 1.99 -2.73
Book-to-market -0.005 -0.007 -0.018 1.28 -0.016 0.20
-1.06 -3.18 -2.42 3.27 -4.99 1.45
LT bond yield -0.808 -0.447 0.528 -75.27 0.54 -53.02
-1.04 -1.27 0.45 -1.59 -1.09 -2.46
Term premium 0.536 -0.343 0.446 -12.95 -0.505 24.71
1.60 -0.96 0.63 -0.40 -0.89 0.99
Default premium 0.777 -0.389 0.096 -73.63 -1.434 102.39
1.27 -1.32 0.08 -1.43 -2.57 2.66
 44
Table 8 (ContÕd)

Panel D: Portfolio of 50 high book-to-market stocks

Constant 0.014 0.005 0.017 -0.16 0.008 -0.19

3.21 3.03 2.39 -0.45 2.73 -1.20
Dividend yield -0.184 -0.018 -0.277 4.12 0.017 -2.86
-1.37 -0.20 -1.19 0.35 0.13 -0.47
Book-to-market 0.005 -0.005 0.003 0.27 -0.009 0.21
1.40 -3.19 0.55 0.88 -3.03 1.29
LT bond yield -0.663 0.380 0.254 -59.36 0.411 1.33
-1.24 1.26 0.33 -1.62 0.94 0.06
Term premium 0.237 -0.444 0.532 -32.12 -0.395 1.23
0.62 -1.51 1.01 -1.13 -0.94 0.06
Default premium -0.438 -0.774 -0.656 -39.20 -1.72 72.51
-0.77 -3.40 -0.76 -0.85 -4.05 2.99
Panel E: Portfolio of 50 low book-to-market stocks

Constant 0.004 -0.000 -0.001 0.29 0.004 -0.15

1.40 -0.02 -0.17 1.94 2.52 -1.95
Dividend yield 0.009 0.063 0.210 -15.37 -0.102 6.77
0.07 1.08 1.12 -2.51 -1.62 1.92
Book-to-market 0.002 -0.001 0.001 0.24 0.001 -0.09
1.10 -0.88 0.15 1.45 1.08 -1.53
LT bond yield -1.051 -0.362 -0.56 -13.29 -0.179 -5.60
-2.45 -1.92 -0.91 -0.55 -0.83 -0.42
Term premium 0.155 0.006 0.188 -3.74 -0.001 -0.52
0.71 0.03 0.54 -0.21 -0.01 -0.03
Default premium -0.387 -0.200 -0.734 -7.67 -0.465 20.38
-0.83 -1.37 -1.53 -0.41 -2.88 1.97