Dependence of Portfolio Returns Over Time and the CAPM: Diverse Holding Periods
This paper shows that under some plausible assumptions about the distributions of returns and the utility functions of the investors the CAPM holds in every single period even if investors have multiperiod diverse investment horizons. This hold even when portfolio returns are dependent over time. The dependence over time is due to the fact that portfolio decisions of the investors at any given period, depend on the actual returns of previous periods via the wealth effect. Hence, the selected portfolios returns may be dependent over time even if the securities themselves are independent. It is shown that despite the above mentioned dependency the investors must choose, at each period, a portfolio from the mean-variance efficient frontier of that period, hence the CAPM holds in each period. The paper, therefore, extends previous results of Levy and Samuelson , Merton  and Stapleton and Subrahmanyam  that showed the existence of the CAPM in a multiperiod setting. Merton proved his result in a continuous time model, a framework that prohibits transaction costs and hence is quite restrictive. Levy and Samuelson overcame the problem of dependency of portfolios by assuming either a quadratic utility function or implicitly assuming a myopic utility function, whereas Stapleton and Subrahmanyam assumed an exponential utility and a common investment horizon. The above assumptions are quite restrictive, and therefore the model given in this paper provides a much broader scope to the CAPM. It allows discrete revisions, any risk averse preferences, diverse holding periods, dependency over time, and a wide class of distribution functions of returns.