Generalization of the Sharpe Ratio and the Arbitrage-Free Pricing of Higher Moments

Abstract: We present an extension of the traditional Sharpe ratio to allow for the evaluation of non-normal return distributions. Combining earlier work in this area with stochastic simulation, we develop a procedure that allows for the construction of a benchmark for the evaluation of the performance of funds with a non-normal return distribution, while maintaining the operational ease of the Sharpe ratio. Similar to the latter, our procedure only requires the risk-free rate of interest rate, the distribution of the market index and an assumption about the type of return distributions to be evaluated. Unlike the Sharpe ratio, however, we are not restricted to normality but are able to handle any reasonable type of distribution. Since our benchmarking procedure is based on the no-arbitrage assumption, it also provides insight into the conditional arbitrage-free value of one distributional parameter in terms of another. We show that in case of the Johnson Su distribution the relationship between skewness and mean return is more or less flat. Skewness and median return on the other hand exhibit a strong negative relationship.