The Performance of Mutual Funds in the Period 1945-1964

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.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. 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 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 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.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, 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 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 ~ 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.12 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 earlier13 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 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” π 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. 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 ε ~ j t and π ~ 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 means that if the manager is unable to forecast general market movements we 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 ( α ˆ ) will be biased upward (since the regression line must pass 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, α 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 α j resulting from the negative bias in our estimate of β j . The Data.—The sample consists of the returns on the portfolios of 115 open end mutual funds for which net asset and dividend information was available in Wiesenberger's Investment Companies for the ten-year period 1955–64.16 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 1955–64 for all 115 funds and as many additional observations as possible were collected for these funds in the period 1945–54. 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: 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 1945–64. The table presents the mean, median, extreme values, and mean absolute deviation of the 115 estimates of α, β, r 2 , and ρ ( u t , u t − 1 ) (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. Since the average value of β 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 squared correlation coefficient, r ˆ 2 , was .865 and indicates in general that eq. (8) fits the 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 of α ˆ . The estimates range from −.0805 to +.0582. Table 4 and Figures 1–4 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). Frequency distribution (from col. (1), Table 4) of estimated intercepts ( α ˆ ) for eq. (8) for 115 mutual funds for all years available for each fund. Fund returns net of all In order to obtain additional information the forecasting of fund eq. (8) was also estimated using fund returns of fund expenses as well as Fund were in all 1 and 2 of Table 4 and Figures 1 and 2 present the frequency distributions of the estimated obtained by using all sample data available for each fund. The number of observations in the estimating equation from 10 to and the time periods are not all 1 and 1 present the frequency distribution of the 115 intercepts estimated on the basis of fund returns net of all 2 of Table 4 and 2 present the frequency distributions of the estimates obtained from the fund returns of expenses given by Frequency distribution (from col. Table 4) of estimated intercepts ( α ˆ ) for eq. (8) for 115 mutual funds for all years available for each fund. Fund returns of all The average value of α ˆ net of expenses was −.011 which indicates that on average the funds earned less per than they should have earned given level of systematic risk. It is also clear from 1 that the distribution is to the side with funds α ˆ j and only with α ˆ j > 0. Frequency distribution (from col. Table 4) of estimated intercepts ( α ˆ ) for eq. (8) for 56 mutual funds for which complete data were available in the period 1945–64. Fund returns of all The model implies that with a random selection buy and hold policy should expect on average to do no worse than α = 0. Thus it from the of negative that the funds are not able to forecast future security prices well to and Frequency distribution (from col. Table 4) of estimated intercepts ( α ˆ ) for eq. (8) for 115 mutual funds for the 10 years Fund returns of all The results shown in 2 of Table 4 indicate the average α ˆ estimated from return data was or per with funds for which α ˆ and for which α ˆ > 0. The frequency in is more than the distribution obtained from the net Thus it that on average this period the funds were not able to increase returns by to (the only expenses which were not to the fund In order to the associated with time periods and sample the measures for the 56 funds for which data were available for the period are in 3 of Table 4 and The results indicate an average α ˆ of with funds for which α ˆ j and funds for which α ˆ j > 0. It is very that of this performance is due to the used in the expenses for the years to It was noted earlier that the expenses for these earlier years were to be equal to the expenses for since these were in the earlier period these estimates are too in order to any difficulty associated with the estimates of the expenses the measures were estimated for each of the 115 funds using only the return data in the period The frequency distribution of the α ˆ is given in 4 of Table 4 and The average α ˆ for this period was or per with funds for which α ˆ and funds for which α ˆ > 0. The must be careful too significance on the number of funds with α ˆ > 0. It is well that measurement in any will the estimated regression of that to be zero Since we know that there are some in the measurement of the rate and the estimated returns on the market the coefficients β ˆ j are downward This of results in an upward bias in the estimates of the α j since the squares regression line must pass through the point of is additional which to bias the results against the funds. That is, the model implicitly the portfolio is since the mutual funds and they must a cash to presented in indicate that on average the funds to hold of net assets in cash. If we the funds earned the rate on these assets per this would increase returns the average α ˆ ) by ( . ) ( . ) = . per Thus the average α ˆ is and it is now very difficult to that this is really different from us now give explicit consideration to these of The of the now ourselves to the regarding the statistical significance of the estimated performance Table 3 presents a of the “t” values for the individual the and the number of observations used in each We noted earlier that it is possible for a fund manager to do worse than a random selection policy since it is to a returns by in unsuccessful to forecast security prices. The that the α ˆ shown in Table 3 and 1 are to the left indicate this may well be an examination of the “t” values given in Table 3 and in indicates that the t values for of the funds were less than and are all negative at the However, since we that it was to do worse than a random policy we are really mainly in the significance of the positive performance Frequency distribution (from col. (1), Table of “t” values for estimated intercepts in eq. (8) for 115 mutual funds for all years available for each fund. Fund returns net of all examination of 3 of Table 3 only 3 funds which have performance measures which are positive at the that these funds are superior we should that if all 115 of these funds a true α equal to we would expect because of random to of or or funds t values at the we shall on an examination of the frequency distribution of the estimated t values to see whether we observe more than the expected number of because of the differing degrees of among the observations in and (which the the frequency distributions are somewhat difficult to presents the frequency distribution of the t values on the basis of returns for the 56 funds for which complete years of data were The t value for the level of significance is and we expect ( . ) ( 56 ) = 1 . 4 observations with t values than We observe we also observe a 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 all expenses to the these distributions should be However, we have not any of the and have used estimates of the expenses for the years which we are biased Thus the results shown in are not too As in order to some of these and to more whether or not the funds were on average able to forecast well to expenses if not other the performance measures were estimated for the period The frequency distribution for the t values of the intercepts of the 115 funds estimated from returns is given in and 4 of Table the observations have degrees of and the maximum and minimum values are and It seems clear from the of this distribution zero and from the lack of any values than that there is very evidence that any of these 115 mutual funds in this period forecasting ability. We from a interpretation of the statistical significance of these and the to do since there is a amount of evidence which indicates the on the residuals, , of (8) may not be We also point out that could also of on the t distributions but for the we from doing That is, if the are not the estimates of the parameters will not be to the student t and therefore it really make to make of against the “t” Frequency distribution (from col. Table of “t” values for estimated intercepts in eq. (8) for 115 mutual funds for all years available for each fund. Fund returns of all Frequency distribution (from col. Table of “t” values for estimated intercepts in eq. (8) for 56 mutual funds for which complete data were available in the period 1945–64. Fund returns of all Frequency distribution (from col. Table of “t” values for estimated intercepts in eq. (8) for 115 mutual funds for the 10 period Fund returns of all However, the possible of these problems in attempting to the usual of significance it should be emphasized that the model itself is in no on this has shown that the squares estimates of j in (2) are unbiased and if the terms u j to the and mean of the of Furthermore, has that the capital asset pricing model results can be obtained in the of these A complete discussion of the associated with this problem and relationship to the portfolio evaluation problem is available in and will not be It is to the that the is only in order to the of and we the to these as the of distribution theory is to the point where of significance can be legitimately It is important to note in the empirical results presented that the mutual fund by these 115 very evidence of an ability to forecast security prices. there is evidence that indicates any individual funds in the sample might be able to forecast prices. These results are when that the in the all tend to the magnitude of any forecasting ability which might or tend to show evidence of forecasting ability where The evidence on mutual fund performance discussed indicates not only that these 115 mutual funds were on average not able to predict security prices well to a policy, but also that there is very evidence that any individual fund was able to do better than that which we expected from mere random It is also important to note that these hold when we measure the fund returns of expenses is and other expenses were obtained Thus on average the funds were not quite successful in to It is also important to that we have not in this paper the of Evidence Jensen 11) indicates the funds on average have an of the risk born by Thus the results here should not be as the mutual funds are not a to that has not been The evidence does a need on the of the funds to evaluate more the and the of and in order to provide investors with maximum possible returns for the level of risk