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The Statistics of Sharpe Ratios Andrew W

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The Statistics of Sharpe Ratios Andrew W The Statistics of Sharpe Ratios Andrew W. Lo The building blocks of the Sharpe ratio—expected returns and volatilities— are unknown quantities that must be estimated statistically and are, therefore, subject to estimation error. This raises the natural question: How accurately are Sharpe ratios measured? To address this question, I derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions for the return-generating process—independently and identically distributed returns, stationary returns, and with time aggregation. I show that monthly Sharpe ratios cannot be annualized by multiplying by 12 except under very special circumstances, and I derive the correct method of conversion in the general case of stationary returns. In an illustrative empirical example of mutual funds and hedge funds, I find that the annual Sharpe ratio for a hedge fund can be overstated by as much as 65 percent because of the presence of serial correlation in monthly returns, and once this serial correlation is properly taken into account, the rankings of hedge funds based on Sharpe ratios can change dramatically. ne of the most commonly cited statistics in show that confidence intervals, standard errors, financial analysis is the Sharpe ratio, the and hypothesis tests can be computed for the esti- ratio of the excess expected return of an mated Sharpe ratio in much the same way that they Oinvestment to its return volatility or standard devi- are computed for regression coefficients such as ation. Originally motivated by mean–variance portfolio alphas and betas. analysis and the Sharpe–Lintner Capital Asset Pric- The accuracy of Sharpe ratio estimators hinges ing Model, the Sharpe ratio is now used in many on the statistical properties of returns, and these different contexts, from performance attribution to properties can vary considerably among portfolios, tests of market efficiency to risk management.1 strategies, and over time. In other words, the Given the Sharpe ratio’s widespread use and the Sharpe ratio estimator’s statistical properties typi- myriad interpretations that it has acquired over the cally will depend on the investment style of the years, it is surprising that so little attention has been portfolio being evaluated. At a superficial level, the paid to its statistical properties. Because expected intuition for this claim is obvious: The performance returns and volatilities are quantities that are gen- of more volatile investment strategies is more dif- erally not observable, they must be estimated in ficult to gauge than that of less volatile strategies. some fashion. The inevitable estimation errors that Therefore, it should come as no surprise that the arise imply that the Sharpe ratio is also estimated results derived in this article imply that, for exam- with error, raising the natural question: How accu- ple, Sharpe ratios are likely to be more accurately rately are Sharpe ratios measured? estimated for mutual funds than for hedge funds. In this article, I provide an answer by deriving A less intuitive implication is that the time- the statistical distribution of the Sharpe ratio using series properties of investment strategies (e.g., standard econometric methods under several dif- mean reversion, momentum, and other forms of ferent sets of assumptions for the statistical behav- serial correlation) can have a nontrivial impact on ior of the return series on which the Sharpe ratio is the Sharpe ratio estimator itself, especially in com- based. Armed with this statistical distribution, I puting an annualized Sharpe ratio from monthly data. In particular, the results derived in this article show that the common practice of annualizing Andrew W. Lo is Harris & Harris Group Professor at Sharpe ratios by multiplying monthly estimates by the Sloan School of Management, Massachusetts Insti- 12 is correct only under very special circum- tute of Technology, Cambridge, and chief scientific officer stances and that the correct multiplier—which for AlphaSimplex Group, LLC, Cambridge. depends on the serial correlation of the portfolio’s 36 ©2002, AIMR® The Statistics of Sharpe Ratios returns—can yield Sharpe ratios that are consider- data, a more general distribution is derived in the ably smaller (in the case of positive serial correla- “Non-IID Returns” section, one that applies to tion) or larger (in the case of negative serial returns with serial correlation, time-varying condi- correlation). Therefore, Sharpe ratio estimators tional volatilities, and many other characteristics of must be computed and interpreted in the context of historical financial time series. In the “Time Aggre- the particular investment style with which a port- gation” section, I develop explicit expressions for folio’s returns have been generated. “time-aggregated’’ Sharpe ratio estimators (e.g., Let Rt denote the one-period simple return of expressions for converting monthly Sharpe ratio a portfolio or fund between dates t – 1 and t and estimates to annual estimates) and their distribu- denote by µ and σ2 its mean and variance: tions. To illustrate the practical relevance of these µ ≡ estimators, I apply them to a sample of monthly E(Rt), (1a) mutual fund and hedge fund returns and show that and serial correlation has dramatic effects on the annual σ2 ≡ Var (R ). (1b) Sharpe ratios of hedge funds, inflating Sharpe ratios t by more than 65 percent in some cases and deflating Recall that the Sharpe ratio (SR) is defined as the Sharpe ratios in other cases. ratio of the excess expected return to the standard deviation of return: IID Returns µ – Rf To derive a measure of the uncertainty surrounding SR ≡ -------------- , (2) σ the estimator SR, we need to specify the statistical where the excess expected return is usually com- properties of Rt because these properties determine puted relative to the risk-free rate, R Because µ and the uncertainty surrounding the component estima- f. 2 σ are the population moments of the distribution of tors µˆ and σˆ . Although this may seem like a theo- Rt, they are unobservable and must be estimated retical exercise best left for statisticians—not unlike using historical data. the specification of the assumptions needed to yield well-behaved estimates from a linear regression— Given a sample of historical returns (R1, R2, . ., there is often a direct connection between the invest- RT), the standard estimators for these moments are the sample mean and variance: ment management process of a portfolio and its statistical properties. For example, a change in the 1 T µˆ = --- R (3a) portfolio manager’s style from a small-cap value T ∑ t orientation to a large-cap growth orientation will t =1 typically have an impact on the portfolio’s volatility, and degree of mean reversion, and market beta. Even for T a fixed investment style, a portfolio’s characteristics 2 1 2 σˆ = --- ()R – µˆ ,(3b) T ∑ t can change over time because of fund inflows and t =1 outflows, capacity constraints (e.g., a microcap fund from which the estimator of the Sharpe ratio (SR) that is close to its market-capitalization limit), follows immediately: liquidity constraints (e.g., an emerging market or private equity fund), and changes in market condi- µˆ – R SR = --------------f .(4)tions (e.g., sudden increases or decreases in volatil- σˆ ity, shifts in central banking policy, and Using a set of techniques collectively known as extraordinary events, such as the default of Russian “large-sample’’ or “asymptotic’’ statistical theory government bonds in August 1998). Therefore, the in which the Central Limit Theorem is applied to investment style and market environment must be estimators such as µˆ and σˆ 2, the distribution of SR kept in mind when formulating the assumptions for and other nonlinear functions of µˆ and σˆ 2 can be the statistical properties of a portfolio’s returns. easily derived. Perhaps the simplest set of assumptions that we In the next section, I present the statistical dis- can specify for Rt is that they are independently and tribution of SR under the standard assumption that identically distributed. This means that the proba- returns are independently and identically distrib- bility distribution of Rt is identical to that of Rs for uted (IID). This distribution completely character- any two dates t and s and that Rt and Rs are statisti- izes the statistical behavior of SR in large samples cally independent for all t ≠ s. Although these con- and allows us to quantify the precision with which ditions are extreme and empirically implausible— SR estimates SR. But because the IID assumption is the probability distribution of the monthly return of extremely restrictive and often violated by financial the S&P 500 Index in October 1987 is likely to differ July/August 2002 37 Financial Analysts Journal from the probability distribution of the monthly is reminiscent of the expression for the variance of return of the S&P 500 in December 2000—they pro- the weighted sum of two random variables, except vide an excellent starting point for understanding that in Equation 7, there is no covariance term. This the statistical properties of Sharpe ratios. In the next is due to the fact that µˆ and σˆ 2 are asymptotically section, these assumptions will be replaced with a independent, thanks to our simplifying assump- more general set of conditions for returns. tion of IID returns. In the next sections, the IID Under the assumption that returns are IID and assumption will be replaced by a more general set have finite mean µ and variance σ2, it is well of conditions on returns, in which case, the covari- known that the estimators µˆ and σˆ 2 in Equation 3 ance between µˆ and σˆ 2 will no longer be zero and have the following normal distributions in large the corresponding expression for the asymptotic samples, or “asymptotically,” due to the Central variance of the Sharpe ratio estimator will be some- Limit Theorem:2 what more involved.
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