Investment Horizon and the Functional Form of the Capital Asset Pricing Model

Abstract

D UE to Sharpe (1964), Lintner (1965) and Mossin (1966), the Capital Asset Pricing Model (CAPM) has been employed to estimate systematic and performance measure and to predict the risk-return relationship. The predictive ability of this model has been examined by Friend and Blume (FB) (1973), Black, and Scholes (BJS) (1972) and Blume and Friend (BF) (1970). They have concluded that the empirical results obtained from the CAPM are significantly different from the ex ante expectation of this model. The effects of investment horizon on the estimate of the systematic were first investigated by (1969). Based upon the instantaneous systematic concept, he concluded that the logarithmic linear form of the CAPM can be used to eliminate the effects of time horizon on the estimated systematic risk; in other words, the basic specification for the CAPM is a Cobb-Douglas type functional form. Levy (1972) has shown that the assumption of a holding period that is different from the true investment horizon will lead to systematic bias of the performance measure index. Recently, Cheng and Deets (CD) (1973) have shown that the logarithmic linear form of the CAPM not only implies a linear relationship but produces an instantaneous risk, dependent upon the length of observed horizon.' In addition, they proposed a new instantaneous systematic entitled the Cheng-Deets instantaneous systematic risk to substitute for the Jensen instantaneous Neither nor CD has ever investigated the effects of finite investment horizon when market equilibrium is not instantaneous. The main purposes of this paper are to derive two alternative functional forms for the CAPM, which will explicitly include the investment horizon parameter, to improve the explanatory power of the CAPM, and to reduce the bias of the estimated systematic risk. In the second section the risk-return relationship is reexamined under the assumptions that true investment horizon is either observable or not observable. In the third section both likelihood ratio and constant elasticity of substitution (CES) function methods are proposed to derive a testable generalized CAPM in accordance with the assumption that all investors have identical investment horizons.2 In the fourth section models derived in the third section are related to Merton's (1970, 1973) continuous time models and Fama and Macbeth's (1973, 1974) empirical work, which supports the linearity of CAPM. In the fifth section, a set of sample data from the New York Stock Exchange (NYSE) during 19671972 will be employed to estimate the related parameters of the nonlinear CAPM being derived in this paper. In addition, the results obtained from nonlinear CAPM will be compared with those obtained from the linear CAPM. Finally, in the last section, the results of this paper will be summarized.