A Note on the Sharpe Ratio for a Class of Generalized Stochastic Volatility Processes

10.29973/JCSA.200712.0001

Hwai-Chung Ho；Pei-Yu Yang

Sharpe ratio ； long memory ； generalized stochastic volatility processes ； long-memory stochastic volatility models

中國統計學報

45卷4期（2007 / 12 / 01）

340 - 354

英文

The Sharpe ratio, which is defined as the ratio of the excess expected return of an investment to its standard deviation, is one of the most commonly used risk-adjusted measure for the returns of an asset or investment. It is a constant task for both researchers and practitioners to use Sharpe ratio to evaluate whether a portfolio performs better than a certain benchmark index. In order to achieve this based on sound and statistical justification, it is necessary to derive the asymptotic distribution of the Sharpe ratio statistics of the benchmark of interest. Considering the stochastic volatility model for returns, Ho (2006) showed that in spite of the fact that the returns form a stationary sequence of martingale differences, the Sharpe ratio statistics may converge to a normal distribution with a rate slower than √n when the latent volatility component exhibits long memory. This note aims to extend the work of Ho (2006) by assuming that the return series follows a generalized stochastic volatility model in which the volatility component is formed by a general functional of a linear process. We show that both the √n and non-√n asymptotic normality are possible and the normalization constants are determined by the decay rate of the coefficients of the linear process that governs the volatility behavior of the returns.

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