A Mean-Variance Capital Asset Pricing Model for Long Short Equity Hedge Fund Portfolios

Abstract

In this paper a single factor mean-variance Capital Asset Pricing model is derived for long short equity investment portfolios. This model is empirically tested using a statistical arbitrage momentum trading strategy based on Dow Jones 30 historical equity data from 1986 to 2002. This trading strategy commonly utilized in practice by hedge funds is typically classified as systematic statistical arbitrage within the long short equity style. The equilibrium two moment model developed has a desirable feature in that it is leverage invariant - a useful quality if used for long short hedge fund performance measurement where leverage may be unknown, non-constant and which may vary widely between various long short hedge fund portfolios making alpha estimation difficult. Alpha is modeled as a function of beta using the concept of entropy as interpreted by Shannon\cite{Shannon48}. The two moment equations derived constitute equilibrium mean-variance model and as such long only, short only and fractional beta long/short portfolios can be priced in the CAPM sense. The approach taken is to assume that during the life of the investment, the overall portfolio can be represented by the returns of the two sub-portfolios represented by long and short positions via separation with a corresponding correlation coefficient. It is shown that the expected portfolio return model agrees with the classical CAPM for expected portfolio return\cite{Sharpe64} in the theory of market equilibrium under conditions of risk. The same analytical technique used for the first moment model derivation is then used to derive the CAPM variance model. The validity of this variance model is tested at the 95 per cent confidence level by empirically simulating a variable fractional beta long short statistical arbitrage momentum trading strategy using Dow Jones 30 daily equity price data from 1988 to 2002. The mean-variance model is then developed further in leverage invariant equilibrium which leads to the formulation of an analytical dynamic alpha equation. As this function can have a maximum value as a function of beta, the usual calculus techniques are used to derive the analytical objective function which allows the determination of the optimal mean-variance long short beta portfolio. Minimum existence conditions are then derived which dictate the minimum pre-requisite performance expectations for the creation of a long/Short equity portfolio for domination of the long only market index in the mean-variance sense. Finally, since both the mean and variance expectation equations are derived, a stochastic differential equation is formulated and the Black-Scholes option price derived.