The CAPM and Stochastic Dominance

Abstract

In this chapter we employ SD rules to prove that the CAPM is theoretically intact in a wide range of frameworks corresponding to the assumption about the distribution of returns and the length of the investment horizon. This is a surprising integration of the MV and SD paradigms, as these two 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 obvious advantage of the MV approach is that it provides a simple and elegant method for determining the optimal diversification among risky assets which is necessary in establishing the Capital Asset Pricing Model (CAPM) (see Sharpe and Lintner). The main disadvantage of the MV paradigm is that it relies on the assumption of normal distribution of returns (or Elliptic distribution family, where the normal distribution belongs to this family), an assumption not needed for the employment of the SD rules. The normality assumption is obviously inappropriate for most 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 theoretical and practical implications, it has been developed under other frameworks which do not assume normality. Levy has shown that technically the CAPM holds even if all possible mixes of distributions are log-normal (bounded from below by zero). However, this approach has also a drawback as with discrete time models, a new problem emerges: if x and y are log-normally distributed, a portfolio z, where z = α x + (l–α)y, will no longer distribute log-normally. Merton assumes continuous-time portfolio with infinite portfolio revisions and shows that under this assumption, the terminal wealth will be log-normally distributed and the CAPM will hold in each single instantaneous period. By employing the continuous portfolio revision, the nagging problem that the sum of log-normal distributions is not log-normal, which characterizes discrete models disappears. However, the disadvantage of the continuous time model is that the CAPM result breaks down if even minor transaction costs, no matter how small, are incorporated.