THE PERFORMANCE OF MUTUAL FUNDS IN THE PERIOD 1945–1964

Abstract

The ability of the portfolio manager or security analyst to increase returns on the portfolio through successful prediction of future security prices, and 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.11 Assuming, of course, that investors' expectations are on average correct. 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. Recent developments in the theory of the pricing of capital assets by Sharpe 20, Lintner 15 and Treynor 25 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 11. 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 11) 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 1945–1964. A number of people in the past have attempted to evaluate the performance of portfolios22 See for example 2, 3, 7, 8, 9, 10, 21, 24. (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.33 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 24. 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 Markowitz-Tobin sense. A measure of “efficiency” and its relationship to certain measures of diversification and forecasting ability is derived and discussed in detail in Jensen 11. 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 (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. 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.55 Note that since σ 2 ( R ~ M ) is constant for all securities the risk of any security is just cov ( R ~ j , R ~ M ). But since c o v ( R ~ M , R ~ M ) = σ 2 ( R ~ M ) the risk of the market portfolio is just σ 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, c o v ( R ~ M , R ~ M ) / σ 2 ( R ~ M ), is unity, and thus the dimension of the measure of systematic risk has a convenient intuitive interpretation. 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, β j . If a portfolio manager or security analyst is able to predict future security prices he will be able to earn higher returns that 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 expected 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. 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 β j [ R ~ M t − R F t ] plus the random error term e ~ j t . 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 β j in eq. (7) will be an efficient estimate1111 In the statistical sense of the term. 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 ~ j t > 0. Hence his portfolio will earn more 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. The new error term u ~ j t will now have E ( u ~ j t ) = 0, and should be serially independent.1212 If u ~ j t 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. Thus if the portfolio manager has an ability to forecast security prices, the intercept, α 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, α 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 α j . Furthermore, the sampling distribution of the estimate, α ˆ j , is a student t distribution with n j − 2 degrees of freedom. These facts give us the information needed to make inferences regarding the statistical significance of the estimated performance measure. It should be emphasized that in estimating α 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 π) 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 Non-Stationarity of the Risk Parameter.—It was pointed out earlier1313 See note 8 above. that by omitting the time subscript from β 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 a

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Affiliated Institutions

• University of Rochester US
• Rochester College US

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