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Unformatted text preview: The Performance Of Mutual Funds
In The Period 19451964
Michael C. Jensen
Harvard Business School
MJensen@hbs.edu ABSTRACT
In this paper I derive a riskadjusted measure of portfolio performance (now known as
"Jensen's Alpha") that estimates how much a manager's forecasting ability contributes to the fund's
returns. The measure is based on the theory of the pricing of capital assets by Sharpe (1964),
Lintner (1965a) and Treynor (Undated). I apply the measure to estimate the predictive ability of
115 mutual fund managers in the period 19451964—that is their ability to earn returns which are
higher than those we would expect given the level of risk of each of the portfolios. The
foundations of the model and the properties of the performance measure suggested here are
discussed in Section II.
The evidence on mutual fund performance indicates not only that these 115 mutual funds
were on average not able to predict security prices well enough to outperform a buythemarketandhold policy, but also that there is very little evidence that any individual fund was able to do
significantly better than that which we expected from mere random chance. It is also important to
note that these conclusions hold even when we measure the fund returns gross of management
expenses (that is assume their bookkeeping, research, and other expenses except brokerage
commissions were obtained free). Thus on average the funds apparently were not quite successful
enough in their trading activities to recoup even their brokerage expenses.
Keywords: Jensen's Alpha, mutual fund performance, riskadjusted returns, forecasting ability,
predictive ability. Journal of Finance, Vol. 23, No. 2 (1967) 389416. © M. C. Jensen 1967
This document is available on the
Social Science Research Network (SSRN) Electronic Library
at: http://papers.ssrn.com/ABSTRACT=244153 Created: August 2002 The Performance Of Mutual Funds
In The Period 19451964*
Michael C. Jensen
Harvard Business School
Mjensen@hbs.edu I. Introduction
A central problem in finance (and especially portfolio management) has been that
of evaluating the “performance” of portfolios of risky investments. The concept of
portfolio “performance” has at least two distinct dimensions:
1) The ability of the portfolio manager or security analyst to increase returns on
the portfolio through successful prediction of future security prices, and
2) The ability of the portfolio manager to minimize (through “efficient”
diversification) the amount of “insurable risk” born by the holders of the
portfolio.
The major difficulty encountered in attempting to evaluate the performance of a
portfolio in these two dimensions has been the lack of a thorough understanding of the
nature and measurement of “risk.” Evidence seems to indicate a predominance of risk
aversion in the capital markets, and as long as investors correctly perceive the “riskiness”
of various assets this implies that “risky” assets must on average yield higher returns than
less “risky” assets.1 Hence in evaluating the “performance” of portfolios the effects of
differential degrees of risk on the returns of those portfolios must be taken into account. 1 Assuming, of course, that investors’ expectations are on average correct. * This paper has benefited from comments and criticisms by G. Benston, E. Fama, J. Keilson, H.
Weingartner, and especially M. Scholes. Jensen 2 1967 Recent developments in the theory of the pricing of capital assets by Sharpe
(1964), Lintner (1965a) and Treynor (Undated) allow us to formulate explicit measures
of a portfolio’s performance in each of the dimensions outlined above. These measures
are derived and discussed in detail in Jensen (1967). However, we shall confine our
attention here only to the problem of evaluating a portfolio manager’s predictive
ability—that is his ability to earn returns through successful prediction of security prices
which are higher than those which we could expect given the level of riskiness of his
portfolio. The foundations of the model and the properties of the performance measure
suggested here (which is somewhat different than that proposed in Jensen (1967)) are
discussed in Section II. The model is illustrated in Section III by an application of it to
the evaluation of the performance of 115 open end mutual funds in the period 19451964.
A number of people in the past have attempted to evaluate the performance of
portfolios2 (primarily mutual funds), but almost all of these authors have relied heavily on
relative measures of performance when what we really need is an absolute measure of
performance. That is, they have relied mainly on procedures for ranking portfolios. For
example, if there are two portfolios A and B, we not only would like to know whether A
is better (in some sense) than B, but also whether A and B are good or bad relative to
some absolute standard. The measure of performance suggested below is such an
absolute measure.3 It is important to emphasize here again that the word “performance”
is used here only to refer to a fund manager’s forecasting ability. It does not refer to a
portfolio’s “efficiency” in the MarkowitzTobin sense. A measure of “efficiency” and its
relationship to certain measures of diversification and forecasting ability is derived and
discussed in detail in Jensen (1967). For purposes of brevity we confine ourselves here to
an examination of a fund manager’s forecasting ability which is of interest in and of itself
2 See for example (Cohen and Pogue, 1967; Dietz, 1966; Farrar, 1962; Friend et al., 1962; Friend and
Vickers, 1965; Horowitz, 1965; Sharpe, 1966; Treynor, 1965).
3 It is also interesting to note that the measure of performance suggested below is in many respects
quite closely related to the measure suggested by Treynor (1965). 3 Jensen 1967 (witness the widespread interest in the theory of random walks and its implications
regarding forecasting success).
In addition to the lack of an absolute measure of performance, these past studies
of portfolio performance have been plagued with problems associated with the definition
of “risk” and the need to adequately control for the varying degrees of riskiness among
portfolios. The measure suggested below takes explicit account of the effects of “risk” on
the returns of the portfolio. Finally, once we have a measure of portfolio “performance”
we also need to estimate the measure’s sampling error. That is we want to be able to
measure its “significance” in the usual statistical sense. Such a measure of significance
also is suggested below.
II. The Model
The Foundations of the Model.—As mentioned above, the measure of portfolio
performance summarized below is derived from a direct application of the theoretical
results of the capital asset pricing models derived independently by Sharpe (1964),
Lintner (1965a) and Treynor (Undated). All three models are based on the assumption
that (1) all investors are averse to risk, and are single period expected utility of terminal
wealth maximizers, (2) all investors have identical decision horizons and homogeneous
expectations regarding investment opportunities, (3) all investors are able to choose
among portfolios solely on the basis of expected returns and variance of returns, (4) all
transactions costs and taxes are zero, and (5) all assets are infinitely divisible. Given the
additional assumption that the capital market is in equilibrium, all three models yield the () ˜
following expression for the expected one period return,4 E R j , on any security (or portfolio) j: () [ ˜
˜
E R j = RF + b j E ( R M)  RF 4 Defined as the ratio of capital gains plus dividends to the initial price of the security. (1) 4 Jensen 1967 where the tildes denote random variables, and = the oneperiod risk free interest rate. RF bj = ˜˜
cov ( R j , R M )
˜
s 2 RM = the measure of risk (hereafter called systematic risk) which
the asset pricing model implies is crucial in determining the
prices of risky assets. ˜
E( R M ) = the expected oneperiod return on the “market portfolio” which consists of an
investment in each asset in the market in proportion to its fraction of the total
†
value of all assets in the market. Thus eq. (1) implies that the expected return on any asset is equal to the risk free
rate plus a risk premium given by the product of the systematic risk of the asset and the
risk premium on the market portfolio.5 The risk premium on the market portfolio is the
difference between the expected returns on the market portfolio and the risk free rate.
Equation (1) then simply tells us what any security (or portfolio) can be expected
to earn given its level of systematic risk, b j . If a portfolio manager or security analyst is
able to predict future security prices he will be able to earn higher returns than those
implied by eq. (1) and the riskiness of his portfolio. We now wish to show how (1) can be
adapted and extended to provide an estimate of the forecasting ability of any portfolio
manager. Note that (1) is stated in terms of the e xpected returns on any security or
portfolio j and the expected returns on the market portfolio. Since these expectations are
strictly unobservable we wish to show how (1) can be recast in terms of the objectively
measurable realizations of returns on any portfolio j and the market portfolio M.
In Jensen (1967) it was shown that the single period models of Sharpe, Lintner,
and Treynor can be extended to a multiperiod world in which investors are allowed to 5 Note that since s But since ( 2 ˜
( RM ) is constant for all securities the risk of any security is just ( ) ˜˜
cov R j , R M . ) ˜˜
˜
˜
cov R j , R M = s 2 ( R M) the risk of the market portfolio is just s 2 ( R M ) , and thus we are really measuring the riskiness of any security relative to the risk of the market portfolio. Hence the
systematic risk of the market portfolio, ( ) ˜˜
˜
cov R j , R M / s 2 ( R M ) , is unity, and thus the dimension of the measure of systematic risk has a convenient intuitive interpretation. 5 Jensen 1967 have heterogeneous horizon periods and in which the trading of securities takes place
continuously through time. These results indicate that we can generalize eq. (1) and
rewrite it as () [ ˜
˜
E R jt = R Ft + b j E ( RMt )  R Ft (1a) where the subscript t denotes an interval of time arbitrary with respect to length and
starting (and ending) dates.
It is also shown in Fama (1968) and Jensen (1967) that the measure of risk, b j , is
approximately equal to the coefficient bj in the “market model” given by: () ˜
˜
˜˜
R jt = E R jt + b j p t + e jt (2) j = 1, 2, ..., N where b j is a parameter which may vary from security to security and p t is an
˜
unobservable “market factor” which to some extent affects the returns on all securities,
and N is the total number of securities in the market.6 The variables p t and the e jt are
˜
˜
assumed to be independent normally distributed random variables with
(3a) E(p t ) = 0
˜ E( e jt) = 0
˜ j = 1, 2, ..., N (3b) ˜˜
cov(p t + e jt ) = 0 j = 1, 2, ..., N (3c) 6 The “market model” given in eqs. (2) and (3a)(3d) is in spirit identical to the “diagonal model”
analyzed in considerable detail by Sharpe (1963; 1967) and empirically tested by Blume (1968). The
somewhat more descriptive term “market model” was suggested by Fama (1968). The “diagonal model” is
usually stated as ˜
˜˜
R jt = a j + b j I t + u jt
I
I
where ˜ is some index of market returns, u j is a random variable uncorrelated with ˜ , and a j and b j are
˜
constants. The differences in specification between (2) and (2a) are necessary in order to avoid the over
specification (pointed out by Fama (1968)) which arises if one chooses to interpret the market index I as an
L
† average of security returns or as the returns on the market portfolio, M (cf.,† intner (1965a), Sharpe(1964)).
I
I
That is, if ˜ is some average of security returns then the assumption that u j is uncorrelated with ˜
˜
I
(equivalent to (3c)) cannot hold since ˜ contains
† ˜
uj.
† † 6 Jensen cov( e jt , e it)
˜˜ { 0
s 2 ( e j),
˜ 1967 j≠i (3d) j = 1, 2,..., N j=i It is also shown in Jensen (1967) that the linear relationships of eqs. (1a) and (2)
hold for any length time interval as long as the returns are measured as continuously
compounded rates of return. Furthermore to a close approximation the return on the
market portfolio can be expressed as7
˜
˜
˜
R Mt @ E ( R Mt) + p t (4) Since evidence given in Blume (1968) and Jensen (1967) indicates that the market
model, given by eqs. (2) and (3a) @ (3d), holds for portfolios as well as individual
securities, we can use (2) to recast (la) in terms of ex post returns.8 Substituting for ˜
˜˜
E( R Mt ) in (la) from (4) and adding b j p t + e jt to both sides of (la) we have 7 N ˜
˜
R M = Â X j R j where X j is the ratio of the total value The return on the market portfolio is given by j =1 of the j’th asset to the total value of all assets. Thus by substitution from (2) we have () ˜
˜
˜
˜
R Mt = Â X j E R jt + Â X j b j p t + Â X j e jt
j j j ˜
E( R Mt ) , and since the market factor p is
unique only up to a transformation of scale (cf. (Fama, 1968)) we can scale p such that Â X j b j = 1 and
Note that the first term on the right hand side of (3) is just j p . Furthermore by assumption, the e jt in the third term are independently
˜
2
distributed random variables with E( e jt) = 0 , and empirical evidence indicates that the s ( e j) are
˜
˜
2˜
roughly of the same order of magnitude as s (p ) (cf. (Fama, 1968; King, 1966)). Hence the variance of
the second term becomes just the last term on the right hand side of (3), given by Ê
ˆ
s 2 Á Â X j e j ˜ = Â X 2j s 2 (e j )
˜
˜
Ëj
¯
j
will be extremely small since on average X j will be equal to 1 / N 1 and N is very large. But since the
expected value of this term Ê
ˆ
Á Â X j e jt˜ is zero, and since we have shown its variance is extremely small, it
Ëj
¯ is unlikely that it will be very different from zero at any given time. Thus to a very close approximation the
returns on the market portfolio will be given by eq. (4).
8 Note that the parameters bj (in (la)) and b j , (in (2)) are not subscripted by t and are thus assumed to be stationary through time. Jensen (1967) has shown (2) to be an empirically valid description of the
behavior of the returns on the portfolios of 115 mutual funds, and Blume (1968) has found similar results
for the behavior of the returns on individual securities. 7 Jensen ˜
˜
E( R jt) + b j p t + e jt @ R Ft + b j [ R Mt  p t  R Ft ] + b j p t + e jt
˜˜
˜
˜˜ 1967 (5) ˜
But from (2) we note that the left hand side of (5) is just R jt . Hence (5) reduces to:9
˜
˜
˜
R jt = R Ft + b j [ R Mt  R Ft ] + e jt (6) Thus assuming that the asset pricing model is empirically valid,10 eq. (6) says that the
reached returns on any security or portfolio can be expressed as a linear function of its
systematic risk, the realized returns on the market portfolio, the risk free rate and a
random error, e jt , which has an expected value of zero. The term R Ft can be subtracted
˜
from both sides of eq. (6), and since its coefficient is unity the result is ˜
˜
˜
R jt  R Ft = b j [ R Mt  R Ft ] + e jt (7) The left hand side of (7) is the risk premium earned on the j’th portfolio. As long as the ˜
asset pricing model is valid this premium is equal to b j [ RMt  R Ft ] plus the random error
term e jt .
˜
The Measure of Performance.—Furthermore eq. (7) may be used directly for
empirical estimation. If we wish to estimate the systematic risk of any individual security
or of an unmanaged portfolio the constrained regression estimate of b j in eq. (7) will be
an efficient estimate11 of this systematic risk. However, we must be very careful when
applying the equation to managed portfolios. If the manager is a superior forecaster
(perhaps because of special knowledge not available to others) he will tend to
systematically select securities which realize e jt > 0 . Hence his portfolio will earn more
In addition it will be shown below that any nonstationary which might arise from attempts to
increase returns by changing the riskiness of the portfolio according to forecasts about the market factor p
lead to relatively few problems.
9 Since the error of approximation in (6) is very slight (cf. (Jensen, 1967), and note 7), we henceforth
use the equality.
10 Evidence given in Jensen (1967) suggests this is true. 11 In the statistical sense of the term. 8 Jensen 1967 than the “normal” risk premium for its level of risk. We must allow for this possibility in
estimating the systematic risk of a managed portfolio.
Allowance for such forecasting ability can be made by simply not constraining the
estimating regression to pass through the origin. That is, we allow for the possible
existence of a nonzero constant in eq. (7) by using (8) as the estimating equation. ˜
˜
˜
R jt  R Ft = a j + b j [ R Mt  RFt ] + u jt (8) ˜
The new error term u jt will now have E(u jt) = 0 , and should be serially independent.12
˜
Thus if the portfolio manager has an ability to forecast security prices, the
intercept, a j , in eq. (8) will be positive. Indeed, it represents the average incremental rate
of return on the portfolio per unit time which is due solely to the manager’s ability to
forecast future security prices. It is interesting to note that a naive random selection buy
and hold policy can be expected to yield a zero intercept. In addition if the manager is not
doing as well as a random selection buy and hold policy, a j will be negative. At first
glance it might seem difficult to do worse than a random selection policy, but such results
may very well be due to the generation of too many expenses in unsuccessful forecasting
attempts.
However, given that we observe a positive intercept in any sample of returns on a
portfolio we have the difficulty of judging whether or not this observation was due to
mere random chance or to the superior forecasting ability of the portfolio manager. Thus
in order to make inferences regarding the fund manager’s forecasting ability we need a
measure of the standard error of estimate of the performance measure. Least squares
regression theory provides an estimate of the dispersion of the sampling distribution of
the intercept a j . Furthermore, the sampling distribution of the estimate, a j , is a student t
ˆ
distribution with n j  2 degrees of freedom. These facts give us the information needed to
12 If u jt were not serially independent the manager could increase his return even more by taking
˜
account of the information contained in the serial dependence and would therefore eliminate it. 9 Jensen 1967 make inferences regarding the statistical significance of the estimated performance
measure.
It should be emphasized that in estimating a j , the measure of performance, we
are explicitly allowing for the effects of risk on return as implied by the asset pricing
model. Moreover, it should also be noted that if the model is valid, the particular nature
of general economic conditions or the particular market conditions (the behavior of p )
over the sample or evaluation period has no effect whatsoever on the measure of
performance. Thus our measure of performance can be legitimately compared across
funds of different risk levels and across differing time periods irrespective of general
economic and market conditions.
The Effects of NonStationarity of the Risk Parameter.—It was pointed out
earlier13 that by omitting the time subscript from b j (the risk parameter in eq. (8)) we
were implicitly assuming the risk level of the portfolio under consideration is stationary
through time. However, we know this need not be strictly true since the portfolio
manager can certainly change the risk level of his portfolio very easily. He can simply
switch from more risky to less risky equities (or vice versa), or he can simply change the
distribution of the assets of the portfolio between equities, bonds and cash. Indeed the
portfolio manager may consciously switch his portfolio holdings between equities, bonds
and cash in trying to outguess the movements of the market.
This consideration brings us to an important issue regarding the meaning of
“forecasting ability.” A manager’s forecasting ability may consist of an ability to forecast
the price movements of individual securities and/or an ability to forecast the general
behavior of security prices in the future (the “market factor” p in our model). Therefore
we want an evaluation model which will incorporate and reflect the ability of the 13 See note 8 above. 10 Jensen 1967 manager to forecast the market’s behavior as well as his ability to choose individual
issues.
Fortunately the model outlined above will also measure the success of these
market forecasting or “timing” activities as long as we can assume that the portfolio
manager attempts on average to maintain a given level of risk in his portfolio. More
formally as long as we can express the risk of the j’th portfolio at any time t as
˜
˜
b j = b j + e jt (9) where b j is the “target” risk level which the portfolio manager wishes to maintain on
average through time, and e jt is a normally distributed random variable (at least partially
˜ ˜
under the manager’s control) with E(e jt) = 0 . The variable e jt is the vehicle through
˜
which the manager may attempt to capitalize on any expectations he may have regarding
˜
the behavior of the market factor p in the next period. For example if the manager (correctly) perceives that there is a higher probability that p will be positive (rather than
negative) next period, he will be able to increase the returns on his portfolio by increasing
its risk,14 i.e., by making e jt positive this period. On the other hand he can reduce the
losses (and therefore increase the average returns) on the portfolio by reducing the risk
level of the portfolio (i.e., making e jt negative) when the market factor p is expected to
be negative. Thus if the manager is able to forecast market movements to some extent,
we should find a positive relationship between e jt and p t . We can state this relationship
˜
˜
formally as: ˜
˜
˜
e jt = a j p t + w jt (10) ˜
where the error term w jt is assumed to be normally distributed with E( w jt ) = 0 . The
˜
coefficient a j will be positive if the manager has any forecasting ability and zero if he 14 Perhaps by shifting resources out of bonds and into equities, or if no bonds are currently held, by
shifting into higher risk equities or by borrowing funds and investing them in equities. 11 Jensen 1967 has no forecasting ability. We can rule out a j < 0 , since as a conscious policy this would
be irrational. Moreover, we can rule out a j < 0 caused by perverse forecasting ability
since this also implies knowledge of p t and would therefore be reflected in a positive a j
˜
as long as the manager learned from past experience. Note also that eq. (10) includes no
constant term since by construction this would be included in b j in eq. (9). In addition
we note that while a j will be positive only if the manager can forecast p , its size will
˜
depend on the manager’s willingness to bet on his forecasts. His willingness to bet on his
forecasts will of course depend on his attitudes towards taking these kinds of risks and
the certainty with which he views his estimates.
Substituting from (9) into (8) the more general model appears as ˜
˜˜
˜
R jt  R Ft = a j + (b j + e jt )[ R Mt  R Ft ] + u jt (11) ˆ
Now as long as the estimated risk parameter b is an unbiased estimate of the average risk
ˆ
level b j , the estimated performance measure (a j ) will also be unbiased. Under the assumption that the forecast error w jt is uncorrelated with p t (which is certainly
˜
ˆ
reasonable), it can be shown15 that the expected value of the least squares estimator b is:
j ˆ
E( b j) = [ ˜
˜
cov ( R jt  R Ft ), ( R Mt  R Ft )
= b j  a j E( R M )
˜
s 2 ( R M) (12) Thus the estimate of the risk parameter is biased downward by an amount given by ˜
a j E( R M ) , where a j is the parameter given in eq. (10) (which describes the relationship
between e jt and p t . By the arguments given earlier a j can never be negative and will be
˜
˜
equal to zero when the manager possesses no market forecasting ability. This is important
since it† eans that if the manager is unable to forecast general market movements we
m
† 15 By substitution from (11) into the definition of the covariance and by the use of eq. (10), the
˜
˜
assumptions of the market model given in (3a)(3d), and the fact that s 2 ( RM ) @ s 2 (p ) (see note 7). 12 Jensen 1967 obtain an unbiased estimate of his ability to increase returns on the portfolio by choosing
individual securities which are “undervalued.”
However, if the manager does have an ability to forecast market movements we
have seen that a j will be positive and therefore as shown in eq. (12) the estimated risk
parameter will be biased downward. This means, of course, that the estimated
performance measure (a ) will be biased upward (since the regression line must pass
ˆ
through the point of sample means).
Hence it seems clear that if the manager can forecast market movements at all we
most certainly should see evidence of it since our techniques will tend to overstate the
magnitude of the effects of this ability. That is, the performance measure, a j , will be
positive for two reasons: (1) the extra returns actually earned on the portfolio due to the
manager’s ability, and (2) the positive bias in the estimate of a j resulting from the
negative bias in our estimate of b j .
III. The Data And Empirical Results
The Data.—The sample consists of the returns on the portfolios of 115 open end
mutual f unds f or w hich n et a sset a nd d ividend i nformation w as a vailable i n
Wiesenberger’s Investment Companies for the tenyear period 195564.16 The funds are
listed in Table 1 along with an identification number and code denoting the fund
objectives (growth, income, etc.). Annual data were gathered for the period 195564 for
all 115 funds and as many additional observations as possible were collected for these
funds in the period 194554 16 The data were obtained primarily from the 1955 and 1965 editions of Wiesenberger (1955 and 1965),
but some data not available in these editions were taken from the 194954 editions. Data on the College
Retirement Equities Fund (not listed in Wiesenberger) were obtained directly from annual reports. All per
share data were adjusted for stock splits and stock dividends to represent an equivalent share as of the end
of December1964. 13 Jensen 1967 TABLE 1
Listing Of 115 Open End Mutual Funds In The Sample
ID
Number
140
141
142
144
145
146
147
1148
2148
150
151
152
153
155
157
158
1159
2159
160
162
163
164
165
166
167
168
169
171
172
173
174
175
176
177
178
180
182
184
185
1186
2186
187
188
189
190
1191
2191
192
193
194
195 Code1
0
0
2
3
4
0
2
2
0
3
2
4
3
0
0
0
0
3
3
0
4
0
2
2
3
2
4
3
0
0
2
4
0
2
3
3
3
3
3
0
0
3
2
0
4
0
2
3
3
0
2 Fund
Aberdeen Fund
Affiliated Fund, Inc.
American Business Shares, Inc.
American Mutual Fund, Inc.
Associated Fund Trust
Atomics, Physics + Science Fund, Inc.
Axe–Houghton Fund B, Inc.
Axe–Houghton Fund A, Inc.
Axe–Houghton Stock Fund, Inc.
Blue Ridge Mutual Fund, Inc.
Boston Fund, Inc.
Broad Street Investing Corp.
Bullock Fund, Ltd.
Canadian Fund, Inc.
Century Shares Trust
The Channing Growth Fund
Channing Income Fund, Inc.
Channing Balanced Fund
Channing Common Stock Fund
Chemical Fund, Inc.
The Colonial Fund, Inc.
Colonial Growth + Energy Shares, Inc.
Commonwealth Fund–Plan C
Commonwealth Investment Co.
Commonwealth Stock Fund
Composite Fund, Inc.
Corporate Leaders Trust Fund Certificates, Series “B”
Delaware Fund, Inc.
De Vegh Mutual Fund, Inc. (No Load)
Diversified Growth Stock Fund, Inc.
Diversified Investment Fund, Inc.
Dividend Shares, Inc.
Dreyfus Fund Inc.
Eaton + Howard Balanced Fund
Eaton + Howard Stock Fund
Equity Fund, Inc.
Fidelity Fund, Inc.
Financial Industrial Fund, Inc.
Founders Mutual Fund
Franklin Custodian Funds, Inc.–Utilities Series
Franklin Custodial Funds, Inc.–Common Stock Series
Fundamental Investors, Inc.
General Investors Trust
Growth Industry Shares, Inc.
Group Securities–Common Stock Fund
Group Securities–Aerospace–Science Fund
Group Securities–Fully Administered Fund
Guardian Mutual Fund, Inc. (No Load)
Hamilton Funds, Inc.
Imperial Capital Fund, Inc.
Income Foundation Fund, Inc. 14 Jensen 1967 TABLE 1 (Continued)
ID
Number
197
198
200
201
202
203
205
206
207
208
1209
2209
210
1211
2211
1212
2212
215
216
217
218
219
220
221
222
1223
2223
224
225
226
227
231
232
233
234
235
236
239
240
241
243
244
245
246
247
249
250
251
252
253
254
255 Code1
1
3
3
2
3
1
3
2
3
3
4
0
0
1
0
1
1
2
0
3
2
4
2
0
4
0
1
1
2
2
4
3
4
3
0
1
2
2
2
3
3
2
3
2
0
0
0
3
4
0
1
0 Fund
Incorporated Income Fund
Incorporated Investors
The Investment Company of America
The Investors Mutual, Inc.
Investors Stock Fund, Inc.
Investors Selective Fund, Inc.
Investment Trust of Boston
Istel Fund, Inc.
The Johnston Mutual Fund Inc. (No–Load)
Keystone High–Grade Common Stock Fund (S–l)
Keystone Income Common Stock Fund (S–2)
Keystone Growth Common Satock Fund (S–3)
Keystone Lower–Priced Common Stock Fund (S–4)
Keystone Income Fund–(K–l)
Keystone Growth Fund (K–2)
The Keystone Bond Fund (B–3)
The Keystone Bond Fund (B–4)
Loomis–Sayles Mutual Fund, Inc. (No Load)
Massachusetts Investors Growth Stock Fund, Inc.
Massachusetts Investors Trust
Massachusetts Life Fund
Mutual Investing Foundation, MIF Fund
Mutual Investment Fund, Inc.
National Investors Corporation
National Securities Stock Series
National Securities–Growth Stock Series
National Securities–Income Series
National Securities–Dividend Series
Nation–Wide Securities Company, Inc.
New England Fund
Northeast Investors Trust (No Load)
Philadelphia Fund, Inc.
Pine Street Fund, Inc. (No Load)
Pioneer Fund, Inc.
T. Rowe Price Growth Stock Fund, Inc. (No Load)
Puritan Fund, Inc.
The George Putnam Fund of Boston
Research Investing Corp.
Scudder, Stevens + Clark Balanced Fund, Inc. (No Load)
Scudder, Stevens + Clark Common Stock Fund, Inc. (No Load)
Selected American Shares, Inc.
Shareholders’ Trust of Boston
State Street Investment Corporation (No Load)
Stein Roe + Farnham Balanced Fund, Inc. (No Load)
Stein Roe + Farnham International Fund, Inc. (No Load)
Television–Electronics Fund, Inc.
Texas Fund, Inc.
United Accumulative Fund
United Income Fund
United Science Fund
The Value Line Income Fund, Inc.
The Value Line Fund, Inc. 15 Jensen 1967 TABLE 1 (Continued)
ID
Number
256
257
259
260
1261
2261
2262
263
267
1268
2268
1000
1 Code1
4
2
3
2
3
2
2
4
4
2
2
0 Fund
Washington Mutual Investors Fund, Inc.
Wellington Fund, Inc.
Wisconsin Fund, Inc.
Composite Bond and Stock Fund, Inc.
Crown WesternDiversified Fund (D2)
Dodge + Cox Balanced Fund (No Load)
Fiduciary Mutual Investing Company, Inc.
The Knickerbocker Fund
Southwestern Investors, Inc.
Wall Street Investing Corporation
Whitehall Fund, Inc.
College Retirement Equities Fund W iesenberger c lassification a s t o f und i nvestment o bjectives:
3 = GrowthIncome, 4 = IncomeGrowth. 0 = G rowth, 1 = I ncome, 2 = B alanced, For this earlier period, 10 years of complete data were obtained for 56 of the original 115
funds.
Definitions of the Variables.— The following are the exact definitions of the
variables used in the estimation procedures: ˜
St
˜
Dt ˜
RMt
˜
NA jt = Level of the Standard and Poor Composite 500 price index17 at the end of year t.
= Estimate of dividends received on the market portfolio in year t as measured by annual observations on the four quarter moving average18 of the dividends paid by the
companies in the composite 500 Index (stated on the same scale as the level of the S&P
500 Index).
Ê˜+ ˜ˆ
= The estimated annual continuously compounded rate of return on
= loge Á S t Dt ˜
the market portfolio M for year t. Ë St1 ¯ = Per share net asset value of the j’th fund at the end of year t. ˜
ID jt
˜
CG jt = Per share “income” dividends paid by the j’th fund during year t.
= Per share “Capital gains distributions paid by the j’th fund during year t.
Ê NA jt + ID jt + CG jt ˆ
=
The annual continuously compounded rate of
˜
˜
˜
˜
˜
return on the j’th fund during year t. (Adjusted
R jt = log e Á
Á
˜
Ë
NA j,t 1
¯
for splits and stock dividends.) 19 17 Obtained from Standard and Poor (1964). Prior to March 1, 1957, the S & P index was based on only
90 securities (50 industrials, 20 rails and 20 utilities) and hence for the earlier period the index is a poorer
estimate of the returns on the market portfolio.
18 Obtained from Standard and Poor (1964). Since the use of this moving average introduces
measurement errors in the index returns it would be preferable to use an index of the actual dividends, but
such an index is not available.
19 Note that while most funds pay dividends on a quarterly basis we treat all dividends as though they
were paid as of December 31 only. This assumption of course will cause the measured returns on the fund 16 Jensen rt 1967 = Yield to maturity of a oneyear government bond at the beginning of year t (obtained from Treasury Bulletin yield curves).
= Annual continuously compounded risk free rate of return for year t.
R Ft = log e (1 + r t)
nj
= The number of yearly observations of the j’th fund. 10 £ n j £ 20 . The Empirical Results.—Table 2 presents some summary statistics of the
frequency distributions of the regression estimates of the parameters of eq. (8) for all 115
mutual funds using all sample data available for each fund in the period 194564. The
table presents the mean, median, extreme values, and mean absolute deviation of the 115
estimates of a , b , r2 , and r(ut , ut1) (the first order autocorrelation of residuals). As can be
seen in the table the average intercept was –.011 with a minimum value of –.078 and a
maximum value of .058. We defer a detailed discussion of the implications of these
estimated intercepts for a moment.
TABLE 2
Summary Of Estimated Regression Statistics For Equation (8) For
115 Mutual Funds Using All Sample Data Available In The
Period 194564. Returns Calculated Net Of All Expenses
˜ jt  RFt = a j + b j [ RMt  RFt ] + u jt
˜
˜
j=1, 2, . . . ,115
R
Extreme Values
Minimum
Maximum
0.080
0.058
0.219
1.405 Mean
Absolute
Deviation* Item Mean
Value Median
Value aj
ˆ
b .0ll
.840 .009
.848 ˆ
r2
ˆ˜˜
r(u t , ut1) * *
n .865
.077 .901
.064 0.445
0.688 0.977
0.575 .074
.211 17.0 19.0 10.0 20.0 3.12 .016
.162 115 * Defined as Â x  x1 i 1 115 ˆ2
** First order autocorrelation of residuals. The average r is .075. Since the average value of b was only .840, on average these funds tended to hold
portfolios which were less risky than the market portfolio. Thus any attempt to compare
the average returns on these funds to the returns on a market index without explicit
adjustment for differential riskiness would be biased against the funds. The average
portfolios on average to be below what they would be if dividends were considered to be reinvested when
received, but the data needed to accomplish this are not easily available. However, the resulting bias should
be quite small. In addition, the same bias is incorporated into the measured returns on the market portfolio. Jensen 17 1967 squared correlation coefficient, r 2 , was .865 and indicates in general that eq. (8) fits the
ˆ
data for most of the funds quite closely. The average first order autocorrelation of
residuals, –.077, is quite small as expected.
Our primary concern in this paper is the interpretation of the estimated intercepts.
They are presented in Table 3 along with the fund identification number and the “t”
values and sample sizes. The observations are ordered from lowest to highest on the basis
ˆ
of a . The estimates range from –.0805 to +.0582. Table 4 and Figures l4 present summary frequency distributions of these estimates (along with the distributions of the
coefficients estimated for several other time intervals which will be discussed below).
In order to obtain additional information about the forecasting success of fund
managers eq. (8) was also estimated using fund returns calculated before deduction of
fund expenses as well as after. Fund loading charges were ignored in all cases.20 Columns
1 and 2 of Table 4 and Figures 1 and 2 present the frequency distributions of the
ˆ
estimated a ’s obtained by using all sample data available for each fund. The number of observations in the estimating equation varies from 10 to 20 and the time periods are
obviously not all identical. Column 1 and Figure 1 present the frequency distribution of
the 115 intercepts estimated on the basis of fund returns calculated net of all expenses.
Column 2 of Table 4 and Figure 2 present the frequency distributions of the estimates
obtained from the fund returns calculated before deductions of management expenses (as
given by Wiesenberger (1955 and 1965)21) 20 The loading charges have been ignored since our main interest here is not to evaluate the funds from
the standpoint of the individual investor but only to evaluate the fund managers’ forecasting ability
21 Actual expense data were available only for the 10 years 195564. Therefore in estimating gross
returns for the years 194554 the expense ratio for 1955 was added (before adjustment to a continuous
base) to the returns for these earlier years. 18 Jensen 1967 TABLE 3 ˆ
Estimated Intercepts, a , And “t” Values For Individual Mutual
Funds Calculated From Equation 8 And All Sample Data
Available In The Period 194564 Using Net Returns
Fund ID
Number ˆ
t (a ) = ˆ
a
ˆ
s (a ) 1191 –.0805 –1.61 Number of
Observations
13 2211 –.0783 –1.91 14 198 –.0615 –4.82 20 222 –.0520 –4.43 20 160 –.0493 –2.41 17 146 –.0425 –1.80 11 1261 –.0424 –2.47 18 2148 –.0417 –1.89 20 184 –.0416 –4.44 20 2209 –.0412 –2.07 14 224 –.0411 –l.72 13 158 –.0410 –2.08 13 164 –.0376 –1.58 13 254 –.0372 –2.17 12 2223
194 –.0370
–.0346 –3.27
–l.27 20
13 171 –.0337 –2.57 20 220 –.0332 –2.74 20 155 –.0324 –1.61 12 263 –.0320 –1.88 20 255 –.0305 –1.10 14 210 –.0299 –1.00 13 247 –.0294 –1.35 10 1223 –.0281 –l.27 18 205 –.0278 –0.60 20 167 –.0256 –1.60 11 253 –.0249 –1.25 14 189 –.0229 –l.27 18 145 –.0224 –2.16 20 231 –.0220 –1.53 14 190 –.0213 –1.53 20 193 –.0210 –1.53 16 147 –.0207 –2.51 20 173 –.0191 –0.54 12 243
187 –.0190
–.0189 –1.82
–2.04 20
20 ˆ
a 174 –.0188 –1.75 20 2191 –.0176 –1.49 20 197 –.0157 –0.80 10 249 –.0155 –0.74 16 140 –.0155 –1.22 20 1148 –.0143 –1.02 20 182 –.0136 –1.26 20 1211 –.0134 –0.80 14 251 –.0122 –0.95 20 1159 –.0120 –0.67 11 241 –.0117 –1.04 20 216 –.0116 –0.76 20. 19 Jensen 1967 TABLE 3 (Continued) ˆ
a ˆ
a
ˆ
t (a ) =
ˆ
s (a ) 219 –.0115 –1.12 Number of
Observations
20 195 –.0111 –1.23 20 180 –.0111 –1.15 20 202 –.0111 –0.86 19 1209 –.0108 –0.79 14 153 –.0103 –0.99 20 150 –.0099 –1.14 13 2159 –.0094 –0.85 13 252 –.0093 –0.85 20 188 –.0089 –0.84 20 200 –.0088 –0.75 20 239 –.0087 –0.23 10 165
235 ~0082
–.0081 –0.52
–0.55 10
17 Fund ID
Number 259 –.0080 –0.53 20 2212 –.0080 –0.44 14 244 –.0080 –0.73 16 166 –.0080 –0.97 20 163 –.0076 –0.39 20 240 –.0073 –0.82 20 2261 –.0061 –0.66 20 185 –.0061 –0.69 20 217 –.0050 –0.91 20 236 –.0050 –0.46 20 1212 –.0037 –0.24 14 168 –.0022 –0.22 15 260 –.0017 –0.14 20 218 –.0014 –0.14 16 207 .000l 0.00 17 203 .0002 0.01 19 257 .0006 0.07 20 141 .0006 0.02 20 245
232 .0009
.0011 0.08
0.12 20
15 172 .0011 0.05 14 221 .0017 0.07 20 176 .0019 0.08 17 201 .0024 0.26 20 142 .0030 0.18 20 256 .0037 0.31 12 1000
.0040 0.30 12 208 .0044 0.40 14 1268 .0048 0.58 19 175 .0048 0.57 20 192 .0054 0.46 14 178 .0055 0.46 20 144 .0056 0.65 14 177 .0060 0.69 20 157 .0060 0.20 20 152 .0065 0.59 20 215 .0074 0.50 20 20 Jensen 1967 TABLE 3 (Continued)
Fund ID
Number
151 ˆ
a ˆ
a
ˆ
t (a ) =
ˆ
s (a ) .0108 0.82 Number of
Observations
20 226 .0108 0.85 20 246 .0112 1.06 15 2268 .0125 1.88 17 225 .0139 1.31 20 2262 .0140 1.43 15 250 .0145 1.02 15 2186 .0164 0.65 14 206
227 .0165
.0170 1.09
1.40 11
14 169 .0191 1.89 20 267 .0198 0.99 10 234 .0219 1.21 14 162 .0219 0.86 20 233 .0232 1.34 20 1186 .0582 2.03 14 TABLE 4
Frequency Distribution Of Estimated Intercepts For Equation (8)
For 115 Mutual Funds For Several Time Intervals. Fund
Returns Calculated Both Net And Gross Of Expenses Class Interval ˆ
.06 £ a < .07
ˆ
.05 £ a < .06
ˆ
.04 £ a < .05
ˆ
.03 £ a < .04
ˆ
.02 £ a < .03
ˆ < .02
.01 £ a
ˆ
.0 £ a < .01
ˆ
 .01 < a < .0
ˆ
 .02 < a £  .01
ˆ
 .03 < a £ .02
ˆ
 .04 < a £  .03
ˆ
 .05 < a £ .04
ˆ
 .06 < a £ .05
ˆ
 .07 < a £ .06
ˆ
 .08 < a £ .07
ˆ
 .09 < a £ .08 ˆ
Average a All Funds Entire
Sample Period*
Net
Gross
Gross
Returns
(1)
(2)
0
1 56 Funds
20 Years
194564
Gross
Returns All Funds
10 Years
195564
Gross
Returns (3)
0 (4)
0 1 0 0 1 0 0 0 0 0 1 1 1 3 9 2 12 12 16 8 15 23 21 13 31 22 29 17 12 21 14 6 13 12 11 5 12 9 9 2 3 8 1 1 1 1 1 1 1 1 0 0 0 1 2 0 0 1 0 0 1 .011 .004 .032 .001 * Sample sizes range from 10 to 20 annual observations among the funds. 21 Jensen 1967 ˆ
The average value of a calculated net of expenses was –.011 which indicates that on average the funds earned about 1.1% less per year (compounded continuously) than
they should have earned given their level of systematic risk. It is also clear from Figure 1
ˆ
that the distribution is skewed to the low side with 76 funds having a < 0 and only 39
ˆ
with a > 0. 35 30 25 23
22
21 FREQUENCY 20 15 12 10 12 9
8 5 3
1 1 1 1 0.09 0.08 0.07 0.06 1 0
0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ˆ
ESTIMATED INTERCEPT (a ) Figure 1 ˆ
Frequency distribution (from col. (l), Table 4) of estimated intercepts (a ) for eq. (8)
for 115 mutual funds for all years available for each fund. Fund returns calculated net of all expenses. 22 Jensen 1967 The model implies that with a random selection buy and hold policy one should
expect on average to do no worse than a = 0 . Thus it appears from the preponderance of
ˆ
negative a ’s that the funds are not able to forecast future security prices well enough to recover their research expenses, management fees and commission expenses. 35 30 29 25 21 FREQUENCY 20 16 15 14 11
10 9 9 5 2
1 1 0.06 0.05 1 1 0
0.09 0.08 0.07 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ˆ
ESTIMATED INTERCEPT (a ) Figure 2
ˆ
Frequency distribution (from col. (2), Table 4) of estimated intercepts (a ) for eq. (8)
for 115 mutual funds for all years available for each fund. Fund returns calculated
gross of all management expenses. 23 Jensen 1967 35 30 25 FREQUENCY 20 17 15 13 10 8 6
5
5 2
1 0.06 0.05 2 1 1 0
0.09 0.08 0.07 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ˆ
ESTIMATED INTERCEPT (a ) Figure 3
ˆ
Frequency distribution (from col. (3), Table 4) of estimated intercepts (a ) for eq. (8) for 56
mutual funds for which complete data were available in the period 194564. Fund returns
calculated gross of all management expenses. ˆ
In order to examine this point somewhat more closely the a ’s were also estimated on the basis of returns calculated gross of all management expenses.22 That is R jt was
˜
taken to be
Ê NA jt + CG jt + ID jt + E jt ˆ
˜
˜
˜
˜
˜
˜
R jt = log e Á
Á
˜
Ë
NA j ,t 1
¯
22 It would be desirable to use the fund returns gross of all expenses including brokerage commissions
as well as the management expenses. However, overall commission data are not yet available. 24 Jensen 1967 35 31
30 25 FREQUENCY 20 15 15 13
12 12 12 10 5 3 1 1 1 1 1 0
0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ˆ
ESTIMATED INTERCEPT (a ) Figure 4
ˆ
Frequency distribution (from col. (4), Table 4) of estimated intercepts (a ) for eq. (8) for 115 mutual funds
for the 10 years 195564. Fund returns calculated gross of al1 management expenses. where E jt , is the estimated per share dollar value of all expenses except brokerage
commissions, interest and taxes (the latter two of which are small) for the j’th fund in
year t obtained from (1955 and 1965). Now when the estimates are based on gross returns
any forecasting success of the funds (even if not sufficient to cover their expenses) should
ˆ
be revealed by positive a ’s. 25 Jensen 1967 ˆ
The results shown in Column 2 of Table 4 indicate the average a estimated from
ˆ
gross return data was –.004 or –.4% per year, with 67 funds for which a < 0 and 48 for
ˆ
which a > 0. The frequency distribution, plotted in Figure 2, is much more symmetric than the distribution obtained from the net returns. Thus it appears that on average during
this 20year period the funds were not able to increase returns enough by their trading
activities to recoup even their brokerage commissions (the only expenses which were not
added back to the fund returns).
In order to avoid the difficulties associated with nonidentical time periods and
unequal sample sizes, the measures for the 56 funds for which data were available for the
entire 20year period are summarized in Column 3 of Table 4 and Figure 3. The results
ˆ
indicate an average a of .032 with 32 funds for which a j < 0 and 24 funds for which
ˆ
ˆ
a j > 0. It is very likely that part of this apparently poorer gross performance is due to the method used in approximating the expenses for the years prior to 1955. It was noted
earlier that the expenses for these earlier years were assumed to be equal to the expenses
for 1955. But since these expense ratios were declining in the earlier period these
estimates are undoubtedly too low.
Finally in order to avoid any difficulty associated with the estimates of the
expenses before 1955, the measures were estimated for each of the 115 funds using only
ˆ
the gross return data in the 10year period 195564. The frequency distribution of the a ’s
ˆ
is given in Column 4 of Table 4 and Figure 4. The average a for this period was –.00l or
ˆ
ˆ
– .l% per year with 55 funds for which a < 0 and 60 funds for which a > 0. The reader must be careful about placing too much significance on the seemingly larger number of
ˆ
funds with a > 0. It is well known that measurement errors (even though unbiased) in any independent variable will cause the estimated regression coefficient of that variable
to be attenuated towards zero (cf. Johnson (1963, chap. 6)). Since we know that there are
undoubtedly some errors in the measurement of both the riskless rate and the estimated
ˆ
returns on the market portfolio, the coefficients b j are undoubtedly slightly downward 26 Jensen 1967 ˆ
biased. This of course results in an upward bias in the estimates of the b j since the least squares regression line must pass through the point of means.
There is one additional item which tends to bias the results slightly against the
funds. That is, the model implicitly assumes the portfolio is fully invested. But since the
mutual funds face stochastic inflows and outflows they must maintain a cash balance to
meet them. Data presented in Friend (1962, pp. 120127) indicate that on average the
funds appear to hold about 2% of their total net assets in cash. If we assume the funds had
earned the riskless rate on these assets (about 3 % per year) this would increase their
ˆ
returns (and the average a ) by about (.02) (.03) = .0006 per year. Thus the adjusted
ˆ
average a is about –.0004, and it is now getting very difficult to say that this is really different from zero. Thus, let us now give explicit consideration to these questions of
“significance.”
The “Significance” of the Estimates.—We now address ourselves to the question
regarding the statistical significance of the estimated performance measures. Table 3
presents a listing of the “t” values for the individual funds, the intercepts, and the number
of observations used in obtaining each estimate. We noted earlier that it is possible for a
fund manager to do worse than a random selection policy since it is easy to lower a
fund’s returns by unwisely spending resources in unsuccessful attempts to forecast
ˆ
security prices. The fact that the a ’s shown in Table 3 and Figure 1 are skewed to the left indicate this may well be true. Likewise an examination of the “t” values given in Table 3
and plotted in Figure 5 indicates that the t values for 14 of the funds were less than –2
and hence are all significantly negative at the 5% leve1.23 However, since we had little
doubt that it was easy to do worse than a random policy we are really interested mainly in
testing the significance of the large positive performance measures. 23 The t value for 5% level of significance (onetail) with 8 degrees of freedom (the minimum in the
sample) is 1.86 and for 18 degrees of freedom (the maximum in the sample) is 1.73. 27 Jensen 1967 40 35 32
30
30 28 25 FREQUENCY 20 15 10 10 10 5 3
1 1 0
7 6 5 4 3 2 1 0 1 2 3 4 5 6 “† VALUE” Figure 5
Frequency distribution (from col, (1), Table 5) of “t” values for estimated intercepts in eq. (8) for 115
mutual funds for all years available for each fund. Fund returns calculated net of all expenses. 7 28 Jensen 1967 41
40 35 30 28 25 FREQUENCY 21
20 15
15 10 5
5 2 2 4 3 1
0
7 6 5 2 1 0 1 2 3 4 5 6 “† VALUE” Figure 6
Frequency distribution (from col. (2), Table 5) of “t” values for estimated intercepts in eq. (8) for
115 mutual funds for all years available for each fund. Fund returns calculated gross of all expenses. 7 29 Jensen 1967 40 35 30 FREQUENCY 25 20
20 15
15 10 8
7 5 2 2 1 1 0
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 “† VALUE” Figure 7
Frequency distribution (from col. (3), Table 5) of “t” values for estimated intercepts in eq. (8) for 56
mutual funds for which complete data were available in the period 194564. Fund returns calculated
gross of all management expenses. 30 Jensen 1967 40 37
36
35 30 FREQUENCY 25 21
20 15
15 10 5 4
2 0
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 “† VALUE” Figure 8
Frequency distribution (from col. (4), Table 5) of “t” values for estimated intercepts in eq. (8) for 115
mutual funds for the 10 year period 195564. Fund returns calculated gross of all management expenses. Jensen 31 1967 An examination of Column 3 of Table 3 reveals only 3 funds which have
performance measures which are significantly positive at the 5% level. But before
concluding that these funds are superior we should remember that even if all 115 of these
funds had a true a equal to zero, we would expect (merely because of random chance) to
find 5% of them or about 5 or 6 funds yielding t values “significant” at the 5% level.
Thus, henceforth we shall concentrate on an examination of the entire frequency
distribution of the estimated t values to see whether we observe more than the expected
number of significant values. Unfortunately because of the differing degrees of freedom
among the observations plotted in Figure 5 and Figure 6 (which contains the gross
estimates), the frequency distributions are somewhat difficult to interpret.
However Figure 7 presents the frequency distribution of the t values calculated on
the basis of gross returns for the 56 funds for which 20 complete years of data were
available. The t value for the onetail 2.5 % level of significance is 2.1, and thus we
expect (.025) (56) = 1.4 observations with t values greater than 2.1. We observe just one.
Again we also observe a definite skewness towards the negative values and no evidence
of an ability to forecast security prices. It is interesting to note that if the model is valid
and if we have indeed returned all expenses to the funds, these distributions should be
symmetric about zero. However, we have not added back any of the brokerage
commissions and have used estimates of the expenses for the years 194554 which we
strongly suspect are biased low. Thus the results shown in Figure 7 are not too surprising.
As mentioned above, in order to avoid some of these difficulties and to test more
precisely whether or not the funds were on average able to forecast well enough to cover
their brokerage expenses (even if not their other expenses) the performance measures
were estimated just for the period 195564, The frequency distribution for the t values of
the intercepts of the 115 funds estimated from gross returns is given in Figure 8 and
column 4 of Table 5. All the observations have 8 degrees of freedom, and the maximum
and minimum values are respectively +2.17 and –2.84. It seems clear from the symmetry 32 Jensen 1967 of this distribution about zero and especially from the lack of any values greater than +2.2
that there is very little evidence that any of these 115 mutual funds in this l0year period
possessed substantial forecasting ability. We refrain from making a strict formal
interpretation of the statistical significance of these numbers and warn the reader to do
likewise since there is a substantial amount of evidence (cf. Fama (1965), Roll (1968))
which indicates the normality assumptions on the residuals, u jt , of (8) may not be valid.
˜
We also point out that one could also perform chisquare goodness of fit tests on the t
distributions presented, but for the same reasons mentioned above we refrain from doing
so. That is, if the residuals are not normally distributed the estimates of the parameters
will not be distributed according to the student t distribution, and therefore it doesn’t
really make sense to make formal goodness of fit tests against the “t” distribution.
TABLE 5
Frequency Distribution Of “t” Values* For Estimated Intercepts In
Equation (8) For 115 Mutual Funds For Several Time Intervals.
Fund Returns Calculated Both Net And Gross Of Expenses Class Interval ˆ
4 £ t(a ) < 5
ˆ) <4
3 £ t(a
ˆ
2 £ t (a ) < 3
ˆ
1 £ t (a ) < 2
ˆ
0 £ t (a ) < 1
ˆ
1 < t( a ) < 0
ˆ
2 < t( a ) £ 1
ˆ
3 < t(a ) £  2
ˆ
4 < t( a ) £ 3
ˆ
5 < t(a ) £  4
ˆ
* Defined as t (a j) = a j
ˆ
ˆ
s (a j ) All Funds Entire
Sample Period**
Net
Gross
Gross
Returns
(1)
(2)
0
0
0
0 56 Funds
20 Years
194564
Gross
Returns All Funds
10 Years
195564
Gross
Returns (3)
0
0 (4)
0
0 1 5 1 2 10
28 15
28 7
15 21
37 32
30 41
21 20
8 36
15 10 2 2 4 1
3 2
1 2
1 0
0 ** Sample sizes from 10 to 20 annual observations among the funds. 33 Jensen 1967 However, while the possible nonnormality of these disturbances causes problems
in attempting to perform the usual types of significance tests, it should be emphasized
that the model itself is in no way crucially dependent on this assumption. Wise (1963) has
shown that the least squares estimates of b j in (2) are unbiased and consistent if the
disturbance terms u j conform to the symmetric and finite mean members of the stable
class of distributions. Furthermore, Fama (1967) has demonstrated that the capital asset
pricing model results (eq. (1)) can still be obtained in the context of these distributions. A
complete discussion of the issues associated with this distributional problem and their
relationship to the portfolio evaluation problem is available in Jensen (1967) and will not
be repeated here. It is sufficient to reiterate the fact that the normality assumption is
necessary only in order to perform the strict tests of significance, and we warn the reader
to interpret these tests as merely suggestive until the state of stable distribution theory is
developed to the point where strict tests of significance can be legitimately performed.
It is important to note in examining the empirical results presented above that the
mutual fund industry (as represented by these 115 funds) shows very little evidence of an
ability to forecast security prices. Furthermore there is surprisingly little evidence that
indicates any individual funds in the sample might be able to forecast prices. These
results are even stronger when one realizes that the biases in the estimates24 all tend to
either exaggerate the magnitude of any forecasting ability which might exist25 or tend to
show evidence of forecasting ability where none exists.
IV. Conclusion
The evidence on mutual fund performance discussed above indicates not only that
these 115 mutual funds were on average not able to predict security prices well enough to 24 Except for the assumption of a fully invested portfolio which we have allowed for by assuming cash
earned interest at the riskless rate.
25 See Section II. Jensen 34 1967 outperform a buythemarketandhold policy, but also that there is very little evidence
that any individual fund was able to do significantly better than that which we expected
from mere random chance. It is also important to note that these conclusions hold even
when we measure the fund returns gross of management expenses (that is assume their
bookkeeping, research, and other expenses except brokerage commissions were obtained
free). Thus on average the funds apparently were not quite successful enough in their
trading activities to recoup even their brokerage expenses.
It is also important to remember that we have not considered in this paper the
question of diversification. Evidence reported elsewhere (cf. Jensen (1967)) indicates the
funds on average have done an excellent job of minimizing the “insurable” risk born by
their shareholders. Thus the results reported here should not be construed as indicating
the mutual funds are not providing a socially desirable service to investors; that question
has not been addressed here. The evidence does indicate, however, a pressing need on the
part of the funds themselves to evaluate much more closely both the costs and the
benefits of their research and trading activities in order to provide investors with
maximum possible returns for the level of risk undertaken.
(Lintner, 1965b; Mandelbrot, 1963; Markowitz, 1959)
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