The performance of Mutual funds In The period 1945-1964 Michael C. Jensen Harvard Business School MJensen(hbs. edu ABSTRACT In this paper I derive a risk-adjusted measure of portfolio performance (now known as 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 1945-1964-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 ll 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 buy-the-market and-hold 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 ote 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: Jensens Alpha, mutual fund performance, risk-adjusted returns, forecasting ability, predictive ability Journal of Finance, Vol. 23, No. 2(1967)389-416 CM. C. Jensen 1967 This document is available on the ocial Science Research Network(SSRN) Electronic Library athttp://papers.ssrn.com/abstract=244153 Created. August 02
Created: August 2002 The Performance Of Mutual Funds In The Period 1945-1964 Michael C. Jensen Harvard Business School MJensen@hbs.edu ABSTRACT In this paper I derive a risk-adjusted 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 1945-1964—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 buy-the-marketand-hold 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, risk-adjusted returns, forecasting ability, predictive ability. Journal of Finance, Vol. 23, No. 2 (1967) 389-416. © M. C. Jensen 1967 This document is available on the Social Science Research Network (SSRN) Electronic Library at: http://papers.ssrn.com/ABSTRACT=244153
The performance of Mutual funds In The period 1945-1964= Michael C. Jensen Harvard Business school Miensen@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. 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 Assuming, of course, that investors expectations are on average correct. s This paper has benefited from comments and criticisms by G. Benston, E. Fama, J. Keilson, H Weingartner, and especially M. Scholes
* This paper has benefited from comments and criticisms by G. Benston, E. Fama, J. Keilson, H. Weingartner, and especially M. Scholes. The Performance Of Mutual Funds In The Period 1945-1964* 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
Jensen 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 Ill 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 portfolios(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. 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 portfolios"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(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 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
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 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 (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)
Jensen (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"or the returns of the portfolio. Finally, once we have a measure of portfolio"performance e also need to estimate the measures 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 I。 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 trans-actions costs and taxes are zero, and (5)all assets are infinitely divisible. Given the dditional assumption that the capital market is in equilibrium, all three models yield the following expression for the expected one period return ER ) on any security(or portolIo) )=R+p[E(R)-8] Defined as the ratio of capital gains plus dividends to the initial of the securit
Jensen 3 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 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 j (R ˜ ), on any security (or portfolio) j: E j (R ˜ ) = RF + b j E M [ (R ˜ ) - RF] (1) 4 Defined as the ratio of capital gains plus dividends to the initial price of the security
Jensen where the tildes denote random variables and the one-period risk free interest rate β cov(R RM)=the measure of risk(hereafter called systematic risk) which the asset pricing model implies is crucial in determining the E(RM)=the expected one-period return on the"market portfolio"which consists of investment in each asset in the market in proportion to its fraction of the t 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. The risk premium on the market portfolio is the difference between the expected returns on the market portfolio and the risk free rate Cquation (1) then simply tells us what any security (or portfolio)can be expected to earn given its level of systematic risk, B,. If a portfolio manager or security analyst 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 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 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 s Note that since o(RA)is constant for all securities the risk of any security is just cov(R, RM) But since cov(R, RM)=0(RM)the risk of the market portfolio is just o(R), and thus we 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, RM)/o(R/), is unity, and thus the dimension of the measure of systematic risk has a convenient intuitive interpretatio
Jensen 4 1967 where the tildes denote random variables, and RF = the one-period risk free interest rate. b j = † cov j R ˜ , M ( R ˜ ) 2 s M R ˜ = the measure of risk (hereafter called systematic risk) which the asset pricing model implies is crucial in determining the prices of risky assets. E M (R ˜ ) = 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, 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 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. 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 2 s M (R ˜ ) is constant for all securities the risk of any security is just cov j R ˜ , M ( R ˜ ). But since cov j R ˜ , M ( R ˜ ) = 2 s M (R ˜ ) the risk of the market portfolio is just 2 s M (R ˜ ), 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 j R ˜ , M ( R ˜ )/ 2 s M (R ˜ ), is unity, and thus the dimension of the measure of systematic risk has a convenient intuitive interpretation
Jensen have heterogeneous horizon periods and in which the trading of securities takes place continuously through time. These results indicate that we can generalize eg.(1) and rewrite it as ER)=REI+BLE(RM)-Rr] 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, is approximately equal to the coefficient b, in the"market model" given by Rn=E(R)+b元十 j=1,2…,N (2) where bi is a parameter which may vary from security to security and S, 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 and the eare assumed to be independent normally distributed random variables with E(元1)=0 e(eit=0 =1,2,,N cov(元,+en)=0 =1,2,N 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). Thediagonal model"is usually stated Rit=aj+bil,+i where I is some index of market returns, u is a random variable uncorrelated with I, and a and b,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 average of security returns or as the returns on the market portfolio, M(cf, Lintner(1965a), Sharpe(1964) That is, if I is some average of security returns then the assumption that ui is uncorrelated with I (equivalent to(3c)cannot hold since I contains u
Jensen 5 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 jt (R ˜ ) = RFt + b j E Mt [ (R ˜ ) - RFt] (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: jt R ˜ = E jt (R ˜ ) + bjp ˜ t + e ˜ jt j = 1,2,..., N (2) 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 E(p ˜ t) = 0 (3a) E( jt e ˜ ) = 0 j =1,2,...,N (3b) t cov(p ˜ + jt e ˜ ) = 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 jt R ˜ = aj + bj t ˜ I + jt u ˜ where † ˜ I is some index of market returns, u ˜ j is a random variable uncorrelated with † ˜ I , 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 average of security returns or as the returns on the market portfolio, M (cf., Lintner (1965a), Sharpe(1964)). That is, if † ˜ I is some average of security returns then the assumption that u ˜ j is uncorrelated with † ˜ I (equivalent to (3c)) cannot hold since † ˜ I contains u ˜ j
Jensen 0j≠i cov(ei e le 3d) It is also shown in Jensen(1967) 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 approximation the return on the market portfolio can be expressed RM=E(R)+元 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(RMr)in(la)from(4)and adding B; I,+ei to both sides of(la)we have 7 The return on the market portfolio is given by Ru=>X,, where X; is the ratio of the total value of the j th asset to the total value of all assets. Thus by substitution from(2)we have R-∑X,E(R)+∑Xb元+∑x,已 Note that the first term on the right hand side of (3)is just E(RM), and since the market factor T is unique only up to a transformation of scale(cf(Fama, 1968)we can scale T such that > b,=land the second term becomes just T. Furthermore by assumption, the e, in the third term are independently distributed random variables with E(e )=0, and empirical evidence indicates that the o(e are roughly of the same order of magnitude as o(T)(cf.(Fama, 1968: King, 1966). Hence the variance of the last term on the right hand side of (3). given by 02x,e/-2Xo(e) will be extremely small since on average x will be equal to 1/N and N is very large. But since the expected value of this term 2Xjej 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 the market portfolio will be given by 8 Note that the parameters B, (in(la)and b;. (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
Jensen 6 1967 cov(e ˜ jt , it e ˜ ) { 0 j ≠ i 2 s (e ˜ j), j = i j = 1,2,...,N (3d) 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 Mt R ˜ @ E( Mt R ˜ ) + t p ˜ (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( Mt R ˜ ) in (la) from (4) and adding b j t p ˜ + jt e ˜ to both sides of (la) we have 7 The return on the market portfolio is given by M R ˜ = Xj j =1 N  j R ˜ where Xj is the ratio of the total value of the j’th asset to the total value of all assets. Thus by substitution from (2) we have Mt R ˜ = Xj E jt (R ˜ ) + j  Xj b jp ˜ t + j  Xj e ˜ jt j  Note that the first term on the right hand side of (3) is just E( Mt R ˜ ), and since the market factor p is unique only up to a transformation of scale (cf. (Fama, 1968)) we can scale p such that X jb j = 1 j  and the second term becomes just p . Furthermore by assumption, the e ˜ jt in the third term are independently distributed random variables with E( jt e ˜ ) = 0, and empirical evidence indicates that the 2 s (e ˜ j) are roughly of the same order of magnitude as 2 s (p ˜ ) (cf. (Fama, 1968; King, 1966)). Hence the variance of the last term on the right hand side of (3), given by 2 s Xj e ˜ j j Â Ê Ë Á ˆ ¯ ˜ = j 2 X j  2 s (e ˜ j) will be extremely small since on average Xj will be equal to 1/ N1 and N is very large. But since the expected value of this term Xj ejt j Â Ê Ë Á ˆ ¯ ˜ 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). 8 Note that the parameters b j (in (la)) and bj , (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
Jensen E(R)+B,元,+n=R+B,IRMm-元;-Rl+B,元+ep But from(2)we note that the left hand side of (5)is just Rir. Hence(5)reduces to: 9 RF+B IRMt-RErl Thus assuming that the asset pricing model is empirically valid, 1o 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, ej, which has an expected value of zero. The term RF can be subtracted from both sides of eq (6), and since its coefficient is unity the result is Rir-Re=BiIRM-Retl+e jr 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, IRM-Rerl plus the random error term 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, in eq (7)will be an efficient estimate 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 ystematically select securities which realize eit>0. Hence his portfolio will earn more In addition it will be shown below that any non-stationary which might arise from attempts to increase returns by changing the riskiness of the portfolio according to forecasts about the market factor Jt lead to relatively few problems Since the error of approximation in(6)is very slight(cf. (Jensen, 1967), and note 7), we hencefort use the equality 10 Evidence given in Jensen(1967)suggests this is true 11 In the statistical sense of the term
Jensen 7 1967 E( jt R ˜ ) + b j t p ˜ + jt e ˜ @ RFt + b j [ Mt R ˜ - t p ˜ -RFt] + b j t p ˜ + jt e ˜ (5) But from (2) we note that the left hand side of (5) is just jt R ˜ . Hence (5) reduces to:9 jt R ˜ = RFt + b j[ Mt R ˜ - RFt] + jt e ˜ (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 RFt can be subtracted from both sides of eq. (6), and since its coefficient is unity the result is jt R ˜ - RFt = b j[ Mt R ˜ - RFt] + jt e ˜ (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[ Mt R ˜ - RFt] 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 jt e ˜ > 0 . Hence his portfolio will earn more In addition it will be shown below that any non-stationary 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
Jensen than the""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 non-zero constant in eq (7) by using( 8)as the estimating equation Ri-Ro=a+B,IRM-Rerl+iir The new error term i t will now have E(u; )=0, and should be serially independent. 2 Thus if the portfolio manager has an ability to forecast security prices, the intercept, a, 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 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 square 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 ni-2 degrees of freedom. These facts give us the information needed to If uit were not serially independent the manager could increase his return even more by taking ccount of the information contained in the serial dependence and would therefore eliminate it
Jensen 8 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 non-zero constant in eq. (7) by using (8) as the estimating equation. jt R ˜ - RFt = a j + b j [ Mt R ˜ - RFt] + jt u ˜ (8) The new error term u ˜ jt will now have E jt (u ˜ ) = 0 , and should be serially independent.12 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 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, 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 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 aj . Furthermore, the sampling distribution of the estimate, a ˆ j , is a student t distribution with nj - 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
Jensen make inferences regarding the statistical significance of the estimated performance measure It should be emphasized that in estimating a, 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 t) 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 that by omitting the time subscript from B,(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 In ying to outguess the movements of th e 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 genera behavior of security prices in the future(the"market factor"Jt in our model). Therefore we want an evaluation model which will incorporate and reflect the ability of the See note 8 above
Jensen 9 1967 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 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 Non-Stationarity 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
