Temporal Aggregation and the Continuous-Time Capital Asset Pricing Model

We examine how the empirical implications of the Capital Asset Pricing Model (CAPM) are affected by the length of the period over which returns are measured. We show that the continuous-time CAPM becomes a multifactor model when the asset pricing relation is aggregated temporally. We use Hansen's Generalized Method of Moments (GMM) approach to test the continuous-time CAPM at an unconditional level using size portfolio returns. The results indicate that the, continuous-time CAPM cannot be rejected. In contrast, the discrete-time CAPM is easily rejected by the tests. These results have a number of important implications for the interpretation of tests of the CAPM which have appeared in the literature. FEW RESULTS IN FINANCE are as familiar or as widely used as the linear relation between expected returns and market betas implied by the Capital Asset Pricing Model (CAPM). However, because both the discreteand continuous-time versions of the CAPM lead to this linear relation,1 the literature often fails to distinguish between their respective empirical implications. For example, discreteand continuous-time models are frequently used interchangeably to motivate empirical tests that use discretely observed return data.2 This paper examines how the empirical implications of the CAPM are affected by the length of the period over which returns are measured.3 We show that, if this period differs from the implicit time frame of the model, then the familiar linear CAPM relation need not hold for the observed returns. For example, the CAPM can imply a multifactor expression for expected returns when the asset pricing relation is properly aggregated. This aspect has many important implications for applied work as well as the interpretation of empirical tests of the CAPM that have appeared in the literature. We illustrate the importance of the temporal aggregation issue by deriving the * Academic Faculty of Finance, The Ohio State University. I am grateful for the comments and suggestions received from Warren Bailey, Stephen Buser, K. C. Chan, George Constantinides, Wayne Ferson, Campbell Harvey, Shmuel Kandel, Hersh Shefrin, Robert Stambaugh, Ren6 Stulz, Finance workshop participants at The Ohio State University, and participants at the 1988 Western Finance Association Meetings. I am particularly grateful for the referee's insights and suggestions. All errors are my responsibility. 'The CAPM has been derived in a discrete-time setting by Sharpe (1964), Lintner (1965), Black (1972), Constantinides (1980), and many others. Merton (1971, 1973) and Cox, Ingersoll, and Ross (1985b) (implicitly) have derived the continuous-time CAPM relation. 2For example, see Chan, Chen, and Hsieh (1985), Chen, Roll, and Ross (1986), Gibbons and Ferson (1985), Ferson, Kandel, and Stambaugh (1987), and Ferson (1988). 3Grossman, Melino, and Shiller (1987) discuss the effects of aggregation over time on the continuous-time CAPM. See also Christiano (1984) and Bergstrom (1984).