Jensen manager to forecast the markets 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 manager attempts on average to maintain a given level of risk in his portfolio. more formally as long as we can express the risk of the j'th portfolio at any time t β,=B+E where B, is the"target risk level which the portfolio manager wishes to maintain on average through time, and Ei is a normally distributed random variable(at least partially under the manager's control) with E(Ejt)=0. The variable Ei is the vehicle through which the manager may attempt to capitalize on any expectations he may have regarding the behavior of the market factor t in the next period. For example if the manager (correctly) perceives that there is a higher probability that t will be positive(rather than negative)next period, he will be able to increase the returns on his portfolio by increasing its risk, 4i.e, by making E i positive this period. On the other hand he can reduce the losses(and therefore increase the average returns)on the portfolio by reducing the risk level of the portfolio (i. e, making Er negative)when the market factor Jt is expected to be negative. Thus if the manager is able to forecast market movements to some extent ye should find a positive relationship between Ejt and ir. We can state this relationship formally as E 兀t+w where the error term w i is assumed to be normally distributed with E(wit)=0. The coefficient ai will be positive if the manager has any forecasting ability and zero if he 14 Perhaps by shifting resources out of bonds and into equities, or if no bonds are currently held, by shifting into higher risk equities or by borrowing funds and investing them in equities
Jensen 10 1967 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 manager attempts on average to maintain a given level of risk in his portfolio. More formally as long as we can express the risk of the j’th portfolio at any time t as j ˜ b = b j +e ˜ jt (9) where b j is the “target” risk level which the portfolio manager wishes to maintain on average through time, and e ˜ jt is a normally distributed random variable (at least partially under the manager’s control) with E( jt e ˜ ) = 0 . The variable e ˜ jt is the vehicle through which the manager may attempt to capitalize on any expectations he may have regarding the behavior of the market factor p ˜ in the next period. For example if the manager (correctly) perceives that there is a higher probability that p will be positive (rather than negative) next period, he will be able to increase the returns on his portfolio by increasing its risk,14 i.e., by making e jt positive this period. On the other hand he can reduce the losses (and therefore increase the average returns) on the portfolio by reducing the risk level of the portfolio (i.e., making e jt negative) when the market factor p is expected to be negative. Thus if the manager is able to forecast market movements to some extent, we should find a positive relationship between e ˜ jt and p ˜ t . We can state this relationship formally as: e ˜ jt = ajp ˜ t + w ˜ jt (10) where the error term w ˜ jt is assumed to be normally distributed with E jt (w ˜ ) = 0 . The coefficient a j will be positive if the manager has any forecasting ability and zero if he 14 Perhaps by shifting resources out of bonds and into equities, or if no bonds are currently held, by shifting into higher risk equities or by borrowing funds and investing them in equities
Jensen has no forecasting ability. We can rule out aj<0, since as a conscious policy this would be irrational. Moreover, we can rule out a <0 caused by perverse forecasting ability since this also implies knowledge of and would therefore be reflected in a positive aj as long as the manager learned from past experience. Note also that eq. (10)includes no constant term since by construction this would be included in Pi in eq(9). In addition we note that while a will be positive only if the manager can forecast 5, its size will depend on the managers willingness to bet on his forecasts. His willingness to bet on his forecasts will of course depend on his attitudes towards taking these kinds of risks and the certainty with which he views his estimates Substituting from(9)into(8)the more general model appears as Rt-RF=aj+(P,+EnIRMt-Reil+iir Now as long as the estimated risk parameter B is an unbiased estimate of the average risk levelB,the estimated performance measure (a, )will also be unbiased.Under the assumption that the forecast error w it is uncorrelated with ,(which is certainly reasonable), it can be shown that the expected value of the least squares estimator B,is E(B) (R -RF). (RM-Rer) 2(R) β1-aERn) (12) Thus the estimate of the risk parameter is biased downward by an amount given by a E(Ro), where ai is the parameter given in eq (10)(which describes the relationship between Ei and r. By the arguments given earlier a; 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 5 By substitution from (11)into the definition of the covariance and by the use of eq.(10),the assumptions of the market model given in (3a)-(3d), and the fact that o(RM)=O()(see note 7)
Jensen 11 1967 has no forecasting ability. We can rule out a j < 0 , since as a conscious policy this would be irrational. Moreover, we can rule out a j < 0 caused by perverse forecasting ability since this also implies knowledge of p ˜ t and would therefore be reflected in a positive a j as long as the manager learned from past experience. Note also that eq. (10) includes no constant term since by construction this would be included in b j in eq. (9). In addition we note that while a j will be positive only if the manager can forecast p ˜ , its size will depend on the manager’s willingness to bet on his forecasts. His willingness to bet on his forecasts will of course depend on his attitudes towards taking these kinds of risks and the certainty with which he views his estimates. Substituting from (9) into (8) the more general model appears as jt R ˜ - RFt = a j + (b j + jt e ˜ )[ Mt R ˜ - RFt] + jt u ˜ (11) Now as long as the estimated risk parameter ˆ b is an unbiased estimate of the average risk level b j , the estimated performance measure (a ˆ j) will also be unbiased. Under the assumption that the forecast error w ˜ jt is uncorrelated with pt (which is certainly reasonable), it can be shown15 that the expected value of the least squares estimator j ˆ b is: E( j ˆ b ) = cov ( jt R ˜ - RFt),( Mt [ R ˜ - RFt)] 2 s ( M R ˜ ) = b j - aj E(RM ) (12) Thus the estimate of the risk parameter is biased downward by an amount given by a j E( M R ˜ ), where a j is the parameter given in eq. (10) (which describes the relationship between † e ˜ jt and † p ˜ 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 15 By substitution from (11) into the definition of the covariance and by the use of eq. (10), the assumptions of the market model given in (3a)-(3d), and the fact that 2 s ( M R ˜ ) @ 2 s (p ˜ ) (see note 7)
Jensen obtain an unbiased estimate of his ability to increase returns on the portfolio by choosing ndividual securities which are"undervalued However, if the manager does have an ability to forecast market movements we have seen that ai 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 (a) 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, ai, will be positive for two reasons: (1)the extra returns actually earned on the portfolio due to the managers ability, and (2) the positive bias in the estimate of ai resulting from the negative bias in our estimate of B I. The Data And Empirical result 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. 6 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 but some data not available in these editions were taken from the 1949-54 editions. Data on the college Retirement Equities Fund(not listed in Wiesenberger)were obtained directly from annual reports. All per share data were adjusted for stock splits and stock dividends to represent an equivalent share as of the end of December 1964
Jensen 12 1967 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 (a 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, a 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 a j resulting from the negative bias in our estimate of b j . III. The Data And Empirical Results 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 16 The data were obtained primarily from the 1955 and 1965 editions of Wiesenberger (1955 and 1965), but some data not available in these editions were taken from the 1949-54 editions. Data on the College Retirement Equities Fund (not listed in Wiesenberger) were obtained directly from annual reports. All per share data were adjusted for stock splits and stock dividends to represent an equivalent share as of the end of December1964