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• W3123235216 abstract "ABSTRACT Performance measures such as the Sharpe ratio and the information ratio are estimation subject to estimation error. Lo (2002) derives the explicit expressions for the statistical distribution of the Sharpe ratio. Bertrand and Protopopescu (2007) have extended his work to the bivariate case which corresponds to the Information ratio. In the present paper, we analyze the effects of skewness and kurtosis of portfolio and benchmark returns on the precision of the estimation of the Sharpe ratio and of the information ratio. We show that these effects are in line with what decision theory suggests about preferences of investors about skewness and kurtosis. Moreover, these effects are significant and can disturb the performance evaluation process if they are neglected. JEL classification: G11, G12, C10 Keywords: Sharpe ratio; Information ratio; Asymptotic distribution; Skewness; Kurtosis; Comparative static (ProQuest: ... denotes formulae omitted.) I. INTRODUCTION Investment management industry provides investors with performance measures based on some capital market theory. Two of the most popular performance measures are the Sharpe (1966,1994) ratio and the information ratio. The common practice in the money managers industry is to impose a limit on the volatility of the deviation of the active portfolio from the benchmark, namely on the tracking error volatility (TEV). The pioneer of this approach is Roll (1992). This setup leads naturally to the use of information ratios (IR), as a performance measure, defined as the ratios of the portfolio excess return over his benchmark to its TEV. In a recent paper, Lo (2002) derives the explicit expressions for the statistical distribution of the Sharpe ratio using the standard asymptotic theory under several sets of assumptions for the return-generating process. Bertrand and Protopopescu (2007) (hereafter, BP (2007)) have extended his work to the information ratio (IR), assuming that each return generating process is i.i.d. while allowing however for cross-correlation between the returns. In the present paper, we analyze the effects of skewness and kurtosis of portfolio and benchmark returns on the precision of the estimation of the Sharpe ratio and of the information ratio. In a first section, we recall a result from Lo (2002) and gives another proof for the derivation of the asymptotic variance of the Sharpe ratio statistics, SR, when the returns are supposed to be i.i.d. Then, we recall briefly two results from BP (2007). In a second section, we analyze the effects of skewness and kurtosis on the variance of the Sharpe ratio and of the information ratio. We show that these effects are in line with what decision theory suggests about preferences of investors about skewness and kurtosis. Moreover, these effects are significant and can disturb the performance evaluation process if they are neglected. II. ASYMPTOTIC VARIANCE OF PERFORMANCE MEASURES A. The Sharpe Ratio Let R^sub t^ denote the one-period simple return of a portfolio. Its mean and variance, µ and σ^sup 2^, are given by: µ=E(R^sub t^) and σ^sup 2^=Var(R^sub t^) Recall that the Sharpe ratio (SR) is defined as the ratio of the excess expected return (relative to the risk-free rate, r) to the standard deviation of return: ... As recall by Lo (2002), µ and σ are the population moments of the distribution of R^sub t^. As such, they are unobservable and must be estimated statistically using historical data and are, therefore, subject to estimation error. He then derives explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions for the return-generating process. In particular, Lo has established that the standard error of the Sharpe ratio (SR) in the i.i.d. case1 is given by: . …" @default.
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• W3123235216 date "2008-07-01" @default.
• W3123235216 modified "2023-09-28" @default.
• W3123235216 title "The Sensitivity of the Asymptotic Variance of Performance Measures with Respect to Skewness and Kurtosis" @default.
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