Kalman J Cohen.

Some comments concerning mutual fund vs. random portfolio performance online

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JAN 2 1968





Kalman J. Cohen and Jerry A„ Pogue





Sloan School of Management
Massachusetts Institute of Technology
Cambridge^ Massachusetts 02139


JAN 2 1968



Kalman J. Cohen and Jerry A„ Pogue

7)6 . 3co -L 7


JAN 8 3 968

'^- '• I- UtiW\tiiES


Kalman J. Cohen and Jerry A„ Pogue

In the current issue of this Journal , Friend and Vickers have criti-
cized several points of the analysis of our paper on portfolio selection

which was contained in an earlier issue of this Journal .

This paper developed two multi-index models for portfolio selection and
compared their ex ante and ex post performance with other portfolio selection
models as well as with the ex post performance of randomly generated port-
folios and the actual performance of some mutual funds primarily invested in

common stocks. As a result of our analysis we raised two issues in regard

to a previous study of mutual fund performance by Friend and Vickers.

These criticisms have been labeled by Friend and Vickers as "incorrect in

The authors wish to thank Professors Richard Bower and Edwin Kuh for
helpful comments and suggestions.

'Professor of Economics and Industrial Administration^ Graduate School
of Industrial Administration^ Carnegie -Me 11 on University.

'Assistant Professor of Finance^ Sloan School of Management^ Massachu-
setts Institute of Technology.

Irwin Friend and Douglas Vickers^ "Re-Evaluation of Alternative Port-
folio Selection Models^" Journal of Business, April 1968.

Kalman J. Cohen and Jerry A. Pogue^ "An Empirical Evaluation of Alter-
native Portfolio Selection Models," Journal of Business. Vol. 40, No. 2
(April 1967), pp. 166-193.

Irwin Friend and Douglas Vickers, "Portfolio Selection and Investment
Performance," Journal of Finance, Vol. XX, No. 3 (Sept. 1965), pp. 381-415.


1 9

view of certain deficiencies' in our analysis and "gratuitous" in view

of the irrelevance of our analysis to the points in their paper which
we criticize^

Our present paper is organized into four parts. In the first section^
we will consider the specific criticisms by Friend and Vickers of our
earlier paper and indicate why we feel they are largely incorrect. In
the second section^ we shall restate our initial criticisms in somewhat
more detail than was contained in the original footnote of our previous
paper and indicate why we still feel they are relevant to the conclusions
drawn by Friend and Vickers, Finally^ in the third section we will ac-
knowledge the "two caveats" and attempt to evaluate the significance of
their conclusions about mutual fund performance in terms of them and cer-
tain other serious deficiencies in their analysis. The brief concluding
section of the paper indicates our views that both our earlier paper and
the papers of Friend and Vickers are largely irrelevant to the empirical
issue of the quality and value of mutual fund management.


It should be kept in mind that the random portfolios and mutual funds
were considered in our earlier paper primarily to provide a basis for com-
parison with the ex post performance of the ex ante efficient portfolios.

■'•Friend and Vickers ^ Journal of Business , op , cit . , p. 1,

Friend and Vickers^ Journal of Business, op . cit . , p. 1.

Cohen and Pogue^ o£. cit . , footnote 16^ p. 188.
^Friend and Vickers^ Journal of Finance , op . cit . , p. 412,

It was not our stated or implied purpose to provide a definitive test of
the performance of mutual funds versus random portfolios „ However^ this
does not preclude our observations as being used as contradictory evidence
to the rather sweeping claims made in the earlier Friend and Vickers paper^
as we shall discuss in Part II of this paper.

Friend and Vickers have challenged our use of the term "dominance."
On the basis of Figures 4 and 5 of our earlier paper we made the following
observation relating to the relative performance of the mutual fund and
random portfolio groups: "The mutual funds^ however^ are not dominated
ex post by either random portfolios or the model selected portfolios^ but
tend to be more conservative^ accepting less risk and a lower return."
As Friend and Vickers have noted^ the ex post performance of the group of
60 random portfolios and 78 mutual funds tend to have a 'first-third quad-
rant relationship. ' This relationship,, however^ is not inconsistent with
our statement that the observed mutual fund performance was not dominated
by the ex post performance of the random portfolios.

While realizing that we stdnd in danger of belaboring some rather
elementary concepts^ we have included in this present paper Figures 1 and
2 to illustrate our arguments. Figure 1 illustrates a possible ex ante
opportunity set of portfolios. The efficient frontier is spanned by effi-
cient portfolios 1 through 6. A possible ex post performance pattern is

Cohen and Pogue^ o£. cit . . p. 186.

^Only in the case of a perfect capital market where investors have
unlimited borrowing capacity at a pure rate of interest would the efficient
frontier be confined to the line extending through portfolios 1^ 2^, and 3.
This obviously is not the case in a real world situation and we are unwill-
ing to make this assumption.

shown in Figure 2. Figure 2 illustrates an ex post situation not dissimilar
to that found in Figures 4 and 5 of our earlier paper, -^ The ex post perform-
ance of the portfolios numbered 1 through 8 has tended to fall in two dis-
tinct clusters^, A and B, The performance of cluster A is not dominated ex
post by the performance of cluster B,, since the portfolios in A generally
provide a lower average return and a lower average risk than the portfolios
in B„^ Thus, some investors^ given that they base their preference for
portfolios solely on return and standard deviation of return^ would prefer
portfolios in group A^ while other investors would prefer portfolios in
in group B, Our usage of terminology in this regard is quite standard and
elementary^ and the Friend and Vickers criticism is incorrect.

The next Friend and Vickers criticism of our analysis is contained in
the statement^ "A rational investor's choice between them [the random port-
folios and the mutual funds ]^ can not be ordered solely on the basis of
the statements made to this point. The investor's preference between low
riskj low return portfolios or high risk^ high return portfolios clearly
depends on the form of nis utility function defined over the relevant ar-
guments^ and on that question Cohen and Pogue have made no statement at
all," This statement illustrates some lack of understanding of the

Cohen and Pogue 0£. cit . , p. 185 „

The portfolios within each cluster^ however^ are by no means equally

desirable ex post.

Friend and Vickers j, Journal of Business , op . cit . , p. 7.

^Added by the present authors.

-'Friend and Vickers j, Journal of Business, op . cit . , p. 7.

Figure 1

Ex Ante

Possible Ex Ante Opportunity Set

Figure 2

Ex Ante

Ex Post




Possible Ex Post Performance Pattern

Ex Post

utility concepts inherent in the assumption that the investor is risk averse
and bases his choices among portfolios solely on their expected return and
standard deviation. In effect^ we have said a great deal about the form of
the investors preference function^ namely that it is quadratic. The par-
ticular parameter values of the investors preference function completely

determine his preference for risk-return combinations. Thus^ the criticism

of Friend and Vickers in this regard is also incorrect.

Friend and Vickers have further suggested the use of a measure of rela-
tive dispersion (the "reward-to variability" ratio) as a measure of investors'
preferences for portfolios. However^ in their exuberance to assist our
efforts they have perhaps overlooked some of the assumptions inherent in
this measure if it is to be interpreted meaningfully. At best^ the R/V
ratio is an equivalent device for summarizing, in one dimension, the return-
risk relationships that we presented as two-dimensional diagrams in our
earlier paper. More seriously, however, the R/V measure in effect assumes
the existence of a perfect capital market. However, under the assumption
of an imperfect capital inarkci; the R./V ratio cannot be used to order a ra-
tional investor's preference for portfolios in either the ex ante or ex
post frame of reference. This is clearly indicated in Figures 1 and 2.

For a proof of this statement refer to H„ M. Markowitz, Portfolio
Selection ^ Cowles Foundation Monograph 16, Wiley 1959, pp. 286=288.

^The above statement by Friend and Vickers indicates that they do not
understand the utility theory implications behind the criterion used in
their own Journal of Finance paper. "We shall, therefore, assume for the
purposes of this paper that the objective of . . . investors ... is to
maximize average return, but only for a given risk." (Friend and Vickers,
Journal of Finance ^ op . cit . , p. 394) The latter was then defined as the
standard deviation of return.

-^Specifically, when borrowing becomes progressively more costly and
price-volume related transactions costs of large investors are considered.

In the ex ance case^ portfolio 7 has a higher R/V ratio than portfolio
6, but portfolio 7 would not be acceptable to a rational investor because
it is not efficient. In the ex post case a similar discussion applies to
portfolios 3 and 5,

We now consider the regression relationships of Figure 7 of our earlier
paper. These regression lines are based on the data contained in Figures
4 and 5 of our earlier paper (and similar diagrams for the other portfolio
selection models). Figure 7^ however^ as has been noted by Friend and
VickerSj tends to compress the available information by not showing the dis-
persion of the source data points in relation to the individual regression
lines. On the basis of these regression lines the mutual funds appear to
dominate the random portfolios over a wide range of ex post returns (by
virtue of the fact that the mutual fund risk-return line lies entirely
"south=erfst" of those for the two random portfolio groups). Considering
Figures 4^ 5^ and 7 together the mutual funds tend to dominate the random
portfolios below about 16 percent ex post return. Above 16 percent the
situation is unclear due to the lack of comparable data points. In retro-
spectj however^ the main observation we would like to carry away from the
mutual fund vs. random portfolio comparisons (which resulted as a by-product
of our main comparisons) is that the two groups provided different ex post
performances. The ex post performance of the group of mutual funds con-

In fact J in order to hold the portfolio with the highest ex post R/V
ratio in our example would have required an irrational ex ante decision
(i.e. selecting a non-efficient portfolio),

2lt should be noted^ however^ that the R/V ratio cannot be computed
solely from the slopes of the regression lines in Figure 7 for obvious rea-
sons, NamelVj that they do not have a common intercept at the pure rate
of interest.


sidered was not found to be dominated by the ex post performance of the
random portfolios,. This conclusion is^ of course^ conditioned by our ex-
perimental design. We have not previously^ nor will we now^ make any at-
tempt to generalize our conclusion to the performance of mutual funds gen-
erally vs„ random portfolios „ However^ we do feel that our results provide
conflicting evidence to the rather broad conclusions made by Friend and
VickerSj, and we now proceed to relate our evidence to these conclusions.


It was our opinion that we were generous, rather than "disingenuous^"
in not previously indicating our views on the so-called "two caveats"
acknowledged by Friend and Vickers. In Part III of this paper we will con-
sider these caveats and indicate why^ in terms of these and other serious
deficiencies in the Friend and Vickers analysis^ their Journal of Finance
paper has little relevance to the question of actual mutual fund vs. ran-
dom portfolio performance.

The evidence that we obtained in our research regarding the relative
performance of common stock mutual funds and random portfolios^ while ad-
mittedly limited^ was considered relevant to the issue for the following

1. We considered the actual investment performance achieved by the
mutual funds during the test period^ rather than the returns achieved by
pseudo-mutual fund portfolios which have little obvious connection to the
actual portfolios (Part III of the present paper includes a more complete
discussion of this point).

^Which included the mutual funds considered^ the method of selecting
random portfolios^ the test period^, etc.

Zo Our random portfolio returns were based on random portfolios of
individual common stocks rather than random portfolios based on indexes of
common stocks,

3. We attempted to avoid the artificial and questionable practice of
segmenting the mutual fund portfolio into two parts^ common stocks and
defensive issues^ and inferring from the former the performance of the whole,
To circumvent this problem we attempted to select basically common stock
mutual funds which held low proportions of debt issues.

Friend and Vickers have criticized our procedures on a number of
counts^ most of which we do not accept,

Firstj it is not clear that at any given time institutional investors
select their portfolios from a universe composed of many thousands of se-
curities. To the contrary^ real mutual funds (apparently as apposed to
pseudo-mutual funds) have significant information and decision-making costs
which naturally tend to reduce the number of investment alternatives con-
currently pursued to a subset-^ which is far smaller than the universe of
possible investment possibilities.

In our paper we selected our random portfolios from universes of 75
and 150 securities. It was observed (see Table in this paper} that the
group performance mean (u,^ ) for the random portfolios tended to move fur-
ther into the first quadrant in relation to the group mean for the 78 mutual
funds as the universe size was increased.

^Over timCj however^ the investment universe considered will in most
cases change significantly with respect to composition as individual se-
curities are added or dropped from consideration.

^The 75 security universe was a randomly chosen subset of the 150
security universe.


Table 1









78 Mutual Funds



f>0 Random Port, (75 sec, universe)



60 Random Port, (150 sec„ universe)



Now given the fact that distribution of the security yields tends to
be positively skewed^ and that we use telescoping universes (i.e. each
security universe is a randomly chosen subset of the next largest universe) ^
the average return u and average standard deviation 5* for the group of
random portfolios could both be expected to be non-decreasing as the number
of securities in the investment universe was increased. If this is the
casCj, then the performance of the 78 mutual funds during the test period
would continue to be preferred by a certain subset of investors^ even when
the random portfolios are selected from universes equivalent in size to
those of the mutual funds.

It is also unclear to us that our method of selecting random portfolios
is inappropriate. Given that institutional investors tend to pre-select^
either explicitly or implicitly, a security universe for detailed considera-
tion from a total universe of all investment opportunities , then we con=
sider our 'each security equally likely' criterion to be an appropriate


basis for a null test.

Friend and Vickers have also criticized the fact that our random port-
folios were held unchanged during the test period^ thus providing the
mutual funds with an additional degree of freedom. The question might be
raised as to the value of making random changes to random portfolios.
However^ another issue must be considered. If we wish to compare the per-
formance of mutual funds against comparison portfolios^ whether ex ante
efficient or random^ which are managed during the test period^ whether op-
timally in the first case or randomly in the second^ then we must take into

consideration the transactions costs involved in modifying these portfolios.

The mutual funds in managing their portfolios are forced to trade off be-
tween expected gains to be made from moving to a more efficient portfolio
with the costs involved in making the desired transition. In our opinion
the performance of randomly managed random portfolios would suffer because
of transactions costs to the extent that they would not be desirable as the
basis for a null test.

Since the evidence from our earlier paper regarding the relative per-
formance of random portfolios and mutual funds is admittedly fragmentary j,
and that from the Friend and Vickers Journal of Finance paper^ in our opinion^-

There is no requirement for micro-consistency at the level of each
mutual fund to produce macro-consistency at the level of the market. It
might also be noted that our view that institutional investors pre-select
a sub-universe would tend to support the aggregate consistency requirement^
i.e.^ mutual funds tend to invest in companies with sufficient capitaliza-
tion so that their trading actions will have minimal effect on market prices.


For large institutional investors a significant component of the trans-
actions cost is the price spread involved in buying or selling large quanti-
ties of stocks.

•^Our detailed justification for this remark is contained in Part III
of this paper.


virtually non-existent^ we shall not pursue this discussion further.

We now come to the second comment that we originally made in regard
to the Friend and Vickers Journal of Finance paper. It was focussed on a
number of statements^, which are characterized by the following excerpt
from their conclusion. "This paper^ in addition^, points up the dangers of
using past measures of return and variance as a basis for portfolio selec-
tion^ or of assuming that the procedures for portfolio selection outlined
by Markowitz provide any clues to future investment performance," In

our paper we stated that our evidence was contradictory to this strong con-


Friend and Vickers have stated that our critism is without foundation^
They further state "Some of the same points presented in connection with
our discussion of their first criticism of our paper indicate why we do not
find their analysis completely convincing with respect to the ex post per-
formance of the ex ante efficient versus random portfolios."

We reject both of these criticisms and consider the second to be ab-
solutely unjustified. While the comparison of the random portfolios of
the mutual funds was in effect a by-product analysis^ our experimental de-
sign for evaluating the relative ex post performance of the ex ante efficient
and random portfolios is completely appropriate. The random portfolios

Friend and Vickers^ Journal of Finance, op . cit . , p. 413.


Cohen and Pogue^ Journal of Business, op . cit . , p. 188^, Footnote 16,


Friend and Vickers, Journal of Business, op . cit . , p. 9.


were drawn from the same size universe as the efficient portfolios and
evaluated over the same ex post period. In addition our 'each security
equally likely' criterion would appear to be an appropriate rule for se-
lecting random portfolios^ particularly in view of the fact that market
composition was not a factor taken into consideration by the portfolios
selection models. The results of our empirical tests indicated that the
ex post performance of the efficient portfolios clearly dominates the per-
formance of the random portfolios during the 1958 to 1964 test period (see
Table 2 in this paper). Our conclusions^ which are based on the above em-
pirical evidence^ can in no way be considered as without foundation.

Table 2


Efficient Portfolios

Random Portfolios









18 oO


Friend and Vickers next state that our criticism completely misses
the mark. They state a so-called "basic theoretical proposition" that
they would not expect ex ante efficiency to be correlated highly with ex
post performance. This opinion is then supported by a further opinion

The test with which we were concerned was related to the ability of
the portfolio selection models to select portfolios which would outperform
ex post naive selection rules (as represented by the random portfolios)
applied to the same security universes.


contained in footnote 9 of their present paper. Unfortunate ly^ Friend
and Vickers have neglected to supply the results of their tests upon which
these proclamations are based.

While we consider using unweighted averages of past yields and unweighted
measures of past yield dispersions probably one of the more naive ways of
going about predicting future yield and dispersion measures^ even these
measures appear to have some predictive powers when a statistically relevant

test is performed. Fortunately^ at this point we have some empirical re-

suits to consider. Pratt, in his doctoral thesis^, tested the hypothesis

that common stocks characterized by relatively high degrees of risk, as
measured by instability of their rates of return in the past, have provided
their holders with higher rates of return, on the average and over considera-
ble periods of time, than have common stocks characterized by relatively

lower degrees of risk, measured by instability of returns. A risk factor

was computed for each common stock listed on the New York Stock Exchange

for each of 372 "base dates," at monthly intervals from January 31, 1929

to December 31, 1959. Ac each of the 372 base dates the common stocks

listed on the New York Stock Exchange were divided into portfolios based

on the values of the risk factor. The return of each portfolio was then

Friend and Vickers, Journal of Business , op . cit . , p. 10, Footnote 9.

Shannon P. Pratt, "Relationship Between Risk and Rate of Return for

Common Stocks," Unpublished Ph.D. Thesis, Indiana University, 1966.

Xhe measure of risk used in his study is the standard deviation of the
natural logarithms of the quarterly investment performance relatives for a
series of quarters immediately preceding the point in time at which it is
desired to measure the relative degree of risk associated with various stocks.


computed for one^ three^ five and seven years forward from each of the
base dates. The findings indicated that during the 30 years covered by the
study J, the average return increased with increasing risk. The findings also
supported the proposition that stocks whose returns had been unstable in
the past tended to continue to be risky in the future. This evidence stands
in disagreement with the basic proposition annunciated by Friend and Vickers.

However^ we have slighted the efforts of Friend and Vickers to some
extent. They have provided us with an illustration to support their propo-
sition. We will now consider this illustration in some detail to expose
a major fallacy in their analysis which will render a majority of their em-
pirical results of questionable value.

Farrar found that the ex ante performance of certain mutual fund port-
folios to be close to optimal (i.e. grouped near the efficient frontier).
Friend and Vickers subsequently^ in their Journal of Finance article^ re-
ported on the evaluation of mutual fund performance over subsequent test
periods using holding periods of up to 6 years beginning in 1958^, and in
subsequent tests beginning in 1959. Based on their test results'^ that in-


Online LibraryKalman J CohenSome comments concerning mutual fund vs. random portfolio performance → online text (page 1 of 2)