The CAPM and Stochastic Dominance

Abstract

The M-V and stochastic dominance paradigms represent two distinct branches of expected utility, each implying a different technique for portfolio investment selection. Each paradigm has its pros and cons. The advantage of the M-V approach is that it provides a method for determining the optimal diversification among risky assets which is necessary in establishing the Capital Asset Pricing Model (CAPM) (see Sharpe [1964]¹ and Lintner [1965]²). The main disadvantage of the M-V paradigm is that it relies on the assumption of normal distribution of returns, an assumption not needed for SD rules. The normality assumption is obviously inappropriate for assets traded in the stock market because asset prices cannot drop below zero (−100% rate of return) whereas the normal distribution is unbounded. As the equilibrium risk-return relationship implied by the CAPM has very important results, it has been developed under other frameworks which do not assume normality. Levy (1977)³ has shown that technically the CAPM holds even if all possible mixes of distributions are log-normal (bounded from below by zero). However, with discrete time models, a new problem emerges: if x and y are lognormally distributed, a portfolio z, where z = α x + (1 − α) y, will no longer distribute lognormally. Merton (1973)⁴ assumes continuous-time portfolio revisions and shows that under this assumption, the terminal wealth will be lognormally distributed and the CAPM will hold in each single instantaneous period. By employing the continuous portfolio revision, the nagging problem of additivity of lognormal distributions, which characterizes discrete models disappears. However, the disadvantage of the continuous time model is that the CAPM result breaks down if minor transaction costs, no matter how small, are incorporated.