Jensen (1968)

American Finance Association The Performance of Mutual Funds in the Period 1945-1964 Author(s): Michael C. Jensen Reviewed work(s): Source: The Journal of Finance, Vol. 23, No. 2, Papers and Proceedings of the Twenty-Sixth Annual Meeting of the American Finance Association Washington, D.C. December 28-30, 1967 (May, 1968), pp. 389-416 Published by: Wiley-Blackwell for the American Finance Association Stable URL: http://www.jstor.org/stable/2325404 . Accessed: 24/08/2012 03:07 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . Wiley-Blackwell and American Finance Association are collaborating with JSTOR to digitize, preserve and extend access to The Journal of Finance. http://www.jstor.org PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS THE PERFORMANCE OF MUTUAL FUNDS IN THE PERIOD 1945-1964 MICHAEL C. JENSEN* 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 re- turns 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 under- standing of the nature and measurement of "risk." Evidence seems to indicate a predominance of risk aversion in the capital markets, and as long as in- vestors correctly perceive the "riskiness" of various assets this implies that "risky" assets must on average yield higher returns than less "risky" assets.' 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 founda- tions 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 portfolios2 (primarily mutual funds), but almost all of these authors have * University of Rochester College of Business. This paper has benefited from comments and criticisms by G. Benston, E. Fama, J. Keilson, H. Weingartner, and especially M. Scholes. 1. Assuming, of course, that investors' expectations are on average correct. 2. See for example [2, 3, 7, 8, 9, 10, 21, 24]. 389 390 The Journal of Finance 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 "perfor- mance" 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 diversifica- tion 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 regard- ing 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 vary- ing 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 port- folio performance summarized below is derived from a direct application of the theoretical results of the capital asset pricing models derived independently by Sharpe [20], Lintner [15] and Treynor [25]. 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 invest- ment opportunities, (3) all investors are able to choose among portfolios solely on the basis of expected returns and variance of returns, (4) all trans- actions 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(Rj), on any security (or portfolio) j: E(Rj) = RF + (j3[E(Rk1) - RF] (1) where the tildes denote random variables, and 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 [24]. 4. Defined as the ratio of capital gains plus dividends to the initial price of the security. (Note, henceforth we shall use the terms asset and security interchangeably.) Performance of Mutual Funds 391 RF ~ the one-period risk free interest rate. Pij c(Rj R = the measure of risk (hereafter called systematic risk) o2(R3r) which the asset pricing model implies is crucial in determining the prices of risky assets. E(Rm) = the expected one-period 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, Pi. 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 port- folio j and the expected returns on the market portfolio. Since these expecta- tions 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 [11] 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 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(Rjt) = RFt + Pj3[E(Rkt) - RFt] (la) where the subscript t denotes an interval of time arbitrary with respect to length and starting (and ending) dates. It is also shown in [5] and [11] that the measure of risk, P3i, is approxi- mately equal to the coefficient bi in the "market model" given by: Rjt = E (Rjt) + bjTtt + eJt j - 1,)2, ... ., N (2) where bj is a parameter which may vary from security to security and ?Ct is an unobservable "market factor" which to some extent affects the returns on all 5. Note that since c2(RM) is constant for all securities the risk of any security is just cov(R1, RM). But since cov(RM, RM,) = U2(RM) the risk of the market portfolio is just Y2(RJ1), 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(RkM,R,)/o2(RM), is unity, and thus the dimension of the measure of systematic risk has a convenient intuitive interpretation. 392 The Journal of Finance securities, and N is the total number of securities in the market.6 The vari- ables ;t and the &it are assumed to be independent normally distributed random variables with E(tt) = 0 (3a) E(ejt) =? j_ 1,2, ... , N (3b) COV(it,'ijt) =? j_ 1,2, ... , N (3c) (0 j#i cov(jt, jeit) =1 j 1,2, ..., N (3d) a 2(jj), j =;i It is also shown in [11] that the linear relationships of eqs. (la) 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 approxima- tion the return on the market portfolio can be expressed as7 Rzmt -E (Rmt) + nt. (4) Since evidence given in [1, 11] indicates that the market model, given by eqs. (2) and (3a) = (3d), holds for portfolios as well as individual securities, 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 [19, 22] and empirically tested by Blume [1]. The somewhat more descriptive term "market model" was suggested by Fama [5]. The "diagonal model" is usually stated as Rjt = aj + bjIt + ujt (2a) where I is some index of market returns, iij is a random variable uncorrelated with I, and a; and bi are constants. The differences in specification between (2) and (2a) are necessary in order to avoid the overspecification (pointed out by Fama [5]) which arises if one chooses to interpret the market index I as an average of security returns or as the returns on the market portfolio, M (cf., [15, 201). That is, if I is some average of security returns then the assumption that uj is uncorrelated with I (equivalent to (3c)) cannot hold since I contains iuj. N 7. The return on the market portfolio is given by RM = Z XJRJ where Xi is the ratio of J=1 the total value of the j'th asset to the total value of all assets. Thus by substitution from (2) we have RMt = XjE(Rjt) + Xjbjt + E X,t Note that the first term on the right hand side of (3) is just E(Rkmt), and since the market factor n is unique only up to a transformation of scale (cf. [5]) we can scale x such that z Xjbj = 1 and the second term becomes just 3x. Furthermore by assumption, the ejt in the third term are independently distributed random variables with E (jt) = 0, and empirical evidence indicates that the G2(e) are roughly of the same order of magnitude as 02(x) (cf. [1, 13]). Hence the variance of the last term on the right hand side of (3), given by 2 will be extremely small since on average Xi will be equal to 1/N, and N is very large. But since the expected value of this term ( Xjejt) is zero, and since we have shown its variance is extremely small, it 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). Performance of Mutual Funds 393 we can use (2) to recast (la) in terms of ex post returns.8 Substituting for E(RMt) in (la) from (4) and adding P5jct + ejt to both sides of (la) we have E(Rjt) + Pj1tt + eCit - RFt+ ?j [RMt - Xt - RFt] + Pjt + ejt- (5) But from (2) we note that the left hand side of (5) is just Rjt. Hence (5) reduces to:9 -jt = RFt +? %[Rt - RFt] + ejt. (6) Thus assuming that the asset pricing model is empirically valid,10 eq. (6) says that the realized 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, ejt, which has an expected value of zero. The term RFt can be subtracted from both sides of eq. (6), and since its coefficient is unity the result is Rkjt -RFt =j [ Rmt -RFt] + 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 Pi [Rmt - RFt] plus the random error term 6t. 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 P in eq. (7) will be an efficient estimate" of this systematic risk. However, we must be very careful when applying the equation to man- aged 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 ijt > 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. Allowance for such forecasting ability can be made by simply not constrain- ing the estimating regression to pass through the origin. That is, we allow for the possible existence of a non-zero constant in eq. (7) by using (8) as the estimating equation. -jt RFt - aj + Pj3[RIt - RFt_ + ijt. (8) 8. Note that the parameters fj (in (la)) and bj (in (2)) are not subscripted by t and are thus assumed to be stationary through time. Jensen [11] has shown (2) to be an empirically valid description of the behavior of the returns on the portfolios of 115 mutual funds, and Blume [1] has found similar results for the behavior of the returns on individual securities. In addition it will be shown below that any non-stationarity which might arise from attempts to increase returns by changing the riskiness of the portfolio according to forecasts about the market factor it lead to relatively few problems. 9. Since the error of approximation in (6) is very slight (cf. [11], and note 7), we henceforth use the equality. 10. Evidence given in [113 suggests this is true. 11. In the statistical sense of the term. 394 The Journal of Finance The new error term uit will now have E(5it) - 0, and should be serially independent.'2 Thus if the portfolio manager has an ability to forecast security prices, the intercept, aj, in eq. (8) will be positive. Indeed, it represents the average in- cremental 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, aj 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 un- successful forecasting attempts. However, given that we observe a positive intercept in any sample of re- turns 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 aj. Furthermore, the sampling distribution of the estimate, a&, is a student t distribution with nj-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 aj, 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 r) 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 earlier'3 that by omitting the time subscript from Pi (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 man- ager 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 12. If 7it 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. 13. See note 8 above. Performance of Mutual Funds 395 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" a in our model). Therefore we want an evaluation model which will incorporate and reflect the ability of the 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 manag