The Statistics Of Sharpe Ratios - Andrew Lo

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The Statistics of Sharpe RatiosAndrew W. LoThe 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: Howaccurately are Sharpe ratios measured? To address this question, I deriveexplicit expressions for the statistical distribution of the Sharpe ratio usingstandard asymptotic theory under several sets of assumptions for thereturn-generating process—independently and identically distributedreturns, stationary returns, and with time aggregation. I show thatmonthly Sharpe ratios cannot be annualized by multiplying by 12 exceptunder very special circumstances, and I derive the correct method ofconversion in the general case of stationary returns. In an illustrativeempirical example of mutual funds and hedge funds, I find that the annualSharpe ratio for a hedge fund can be overstated by as much as 65 percentbecause of the presence of serial correlation in monthly returns, and oncethis serial correlation is properly taken into account, the rankings of hedgefunds based on Sharpe ratios can change dramatically.One of the most commonly cited statistics infinancial analysis is the Sharpe ratio, theratio of the excess expected return of aninvestment to its return volatility or standard deviation. Originally motivated by mean–varianceanalysis and the Sharpe–Lintner Capital Asset Pricing Model, the Sharpe ratio is now used in manydifferent contexts, from performance attribution totests of market efficiency to risk management.1Given the Sharpe ratio’s widespread use and themyriad interpretations that it has acquired over theyears, it is surprising that so little attention has beenpaid to its statistical properties. Because expectedreturns and volatilities are quantities that are generally not observable, they must be estimated insome fashion. The inevitable estimation errors thatarise imply that the Sharpe ratio is also estimatedwith error, raising the natural question: How accurately are Sharpe ratios measured?In this article, I provide an answer by derivingthe statistical distribution of the Sharpe ratio usingstandard econometric methods under several different sets of assumptions for the statistical behavior of the return series on which the Sharpe ratio isbased. Armed with this statistical distribution, IAndrew W. Lo is Harris & Harris Group Professor atthe Sloan School of Management, Massachusetts Institute of Technology, Cambridge, and chief scientific officerfor AlphaSimplex Group, LLC, Cambridge.36show that confidence intervals, standard errors,and hypothesis tests can be computed for the estimated Sharpe ratio in much the same way that theyare computed for regression coefficients such asportfolio alphas and betas.The accuracy of Sharpe ratio estimators hingeson the statistical properties of returns, and theseproperties can vary considerably among portfolios,strategies, and over time. In other words, theSharpe ratio estimator’s statistical properties typically will depend on the investment style of theportfolio being evaluated. At a superficial level, theintuition for this claim is obvious: The performanceof more volatile investment strategies is more difficult to gauge than that of less volatile strategies.Therefore, it should come as no surprise that theresults derived in this article imply that, for example, Sharpe ratios are likely to be more accuratelyestimated for mutual funds than for hedge funds.A less intuitive implication is that the timeseries properties of investment strategies (e.g.,mean reversion, momentum, and other forms ofserial correlation) can have a nontrivial impact onthe Sharpe ratio estimator itself, especially in computing an annualized Sharpe ratio from monthlydata. In particular, the results derived in this articleshow that the common practice of annualizingSharpe ratios by multiplying monthly estimates by12 is correct only under very special circumstances and that the correct multiplier—whichdepends on the serial correlation of the portfolio’s 2002, AIMR

The Statistics of Sharpe Ratiosreturns—can yield Sharpe ratios that are considerably smaller (in the case of positive serial correlation) or larger (in the case of negative serialcorrelation). Therefore, Sharpe ratio estimatorsmust be computed and interpreted in the context ofthe particular investment style with which a portfolio’s returns have been generated.Let Rt denote the one-period simple return ofa portfolio or fund between dates t – 1 and t anddenote by µ and σ2 its mean and variance:µ E(Rt),(1a)σ2 Var(Rt).(1b)andRecall that the Sharpe ratio (SR) is defined as theratio of the excess expected return to the standarddeviation of return:µ – RfSR -------------- ,σ(2)where the excess expected return is usually computed relative to the risk-free rate, Rf. Because µ andσ are the population moments of the distribution ofRt , they are unobservable and must be estimatedusing historical data.Given a sample of historical returns (R1, R2, . . .,RT), the standard estimators for these moments arethe sample mean and variance:1 Tµ̂ --- R tT(3a)1 T22σ̂ --- ( R t – µ̂ ) ,T(3b)t 1andt 1from which the estimator of the Sharpe ratio (SR)follows immediately:µ̂ – R fSR -------------- .σ̂(4)Using a set of techniques collectively known as“large-sample’’ or “asymptotic’’ statistical theoryin which the Central Limit Theorem is applied toestimators such as µ̂ and σ̂ 2 , the distribution of SRand other nonlinear functions of µ̂ and σ̂ 2 can beeasily derived.In the next section, I present the statistical distribution of SR under the standard assumption thatreturns are independently and identically distributed (IID). This distribution completely characterizes the statistical behavior of SR in large samplesand allows us to quantify the precision with whichSR estimates SR. But because the IID assumption isextremely restrictive and often violated by financialJuly/August 2002data, a more general distribution is derived in the“Non-IID Returns” section, one that applies toreturns with serial correlation, time-varying conditional volatilities, and many other characteristics ofhistorical financial time series. In the “Time Aggregation” section, I develop explicit expressions for“time-aggregated’’ Sharpe ratio estimators (e.g.,expressions for converting monthly Sharpe ratioestimates to annual estimates) and their distributions. To illustrate the practical relevance of theseestimators, I apply them to a sample of monthlymutual fund and hedge fund returns and show thatserial correlation has dramatic effects on the annualSharpe ratios of hedge funds, inflating Sharpe ratiosby more than 65 percent in some cases and deflatingSharpe ratios in other cases.IID ReturnsTo derive a measure of the uncertainty surroundingthe estimator SR , we need to specify the statisticalproperties of Rt because these properties determinethe uncertainty surrounding the component estimators µ̂ and σ̂ 2 . Although this may seem like a theoretical exercise best left for statisticians—not unlikethe specification of the assumptions needed to yieldwell-behaved estimates from a linear regression—there is often a direct connection between the investment management process of a portfolio and itsstatistical properties. For example, a change in theportfolio manager’s style from a small-cap valueorientation to a large-cap growth orientation willtypically have an impact on the portfolio’s volatility,degree of mean reversion, and market beta. Even fora fixed investment style, a portfolio’s characteristicscan change over time because of fund inflows andoutflows, capacity constraints (e.g., a microcap fundthat is close to its market-capitalization limit),liquidity constraints (e.g., an emerging market orprivate equity fund), and changes in market conditions (e.g., sudden increases or decreases in volatility, shifts in central banking policy, andextraordinary events, such as the default of Russiangovernment bonds in August 1998). Therefore, theinvestment style and market environment must bekept in mind when formulating the assumptions forthe statistical properties of a portfolio’s returns.Perhaps the simplest set of assumptions that wecan specify for Rt is that they are independently andidentically distributed. This means that the probability distribution of Rt is identical to that of Rs forany two dates t and s and that Rt and Rs are statistically independent for all t s. Although these conditions are extreme and empirically implausible—the probability distribution of the monthly return ofthe S&P 500 Index in October 1987 is likely to differ37

Financial Analysts Journalfrom the probability distribution of the monthlyreturn of the S&P 500 in December 2000—they provide an excellent starting point for understandingthe statistical properties of Sharpe ratios. In the nextsection, these assumptions will be replaced with amore general set of conditions for returns.Under the assumption that returns are IID andhave finite mean µ and variance σ2, it is wellknown that the estimators µ̂ and σ̂ 2 in Equation 3have the following normal distributions in largesamples, or “asymptotically,” due to the CentralLimit Theorem:2aaT ( µ̂ – µ ) N ( 0, σ 2 ), T ( σ̂ 2 – σ 2 ) N ( 0, 2σ 4 ),(5)awhere denotes the fact that this relationship is anasymptotic one [i.e., as T increases without bound,the probability distributions of T ( µ̂ – µ ) andT ( σ̂ 2 – σ 2 ) approach the normal distribution, withmean zero and variances σ2 and 2σ4, respectively].These asymptotic distributions imply that the estimation error of µ̂ and σ̂ 2 can be approximated by24a σa 2σVar ( µ̂ ) ------ , Var ( σ̂ 2 ) --------- ,TT(6)is reminiscent of the expression for the variance ofthe weighted sum of two random variables, exceptthat in Equation 7, there is no covariance term. Thisis due to the fact that µ̂ and σ̂ 2 are asymptoticallyindependent, thanks to our simplifying assumption of IID returns. In the next sections, the IIDassumption will be replaced by a more general setof conditions on returns, in which case, the covariance between µ̂ and σ̂ 2 will no longer be zero andthe corresponding expression for the asymptoticvariance of the Sharpe ratio estimator will be somewhat more involved.The asymptotic variance of SR given in Equation 7 can be further simplified by evaluating thesensitivities explicitly— g/ µ 1/σ and g/ σ2 –(µ – Rf )/(2σ3)—and then combining terms to yield( µ – Rf )2V IID 1 --------------------2σ 2Therefore, standard errors (SEs) for the Sharpe ratioestimator SR can be computed asawhere indicates that these relations are based onasymptotic approximations. Note that in Equation6, the variances of both estimators approach zero asT increases, reflecting the fact that the estimationerrors become smaller as the sample size grows. Anadditional property of µ̂ and σ̂ in the special caseof IID returns is that they are statistically independent in large samples, which greatly simplifies ouranalysis of the statistical properties of functions ofthese estimators (e.g., the Sharpe ratio).Now, denote by the function g(µ, σ 2) theSharpe ratio defined in Equation 2; hence, theSharpe ratio estimator is simply g ( µ̂, σ̂ 2 ) SR.When the Sharpe ratio is expressed in this form, itis apparent that the estimation errors in µ̂ and σ̂ 2will affect g ( µ̂, σ̂ 2 ) and that the nature of theseeffects depends critically on the properties of thefunction g. Specifically, in the “IID Returns” section of Appendix A, I show that the asymptoticdistribution of the Sharpe ratio estimator isT ( SR – SR ) a N ( 0, V IID ), where the asymptotic variance is given by the following weightedaverage of the asymptotic variances of µ̂ and σ̂ 2 :22 g g V IID ------ σ 2 --------2- 2σ 4 . µ σ (7)The weights in Equation 7 are simply the squaredsensitivities of g with respect to µ and σ2, respectively: The more sensitive g is to a particular parameter, the more influential its asymptotic variancewill be in the weighted average. This relationship38(8)1 1 --- SR2 .2aSE(SR) 1 1--- SR 2 /T, 2(9)and this quantity can be estimated by substitutingSR for SR. Confidence intervals for SR can also beconstructed from Equation 9; for example, the 95percent confidence interval for SR around the estimator SR is simply1 2SR 1.96 1 --- SR /T. 2(10)Table 1 reports values of Equation 9 for various combinations of Sharpe ratios and samplesizes. Observe that for any given sample size T,larger Sharpe ratios imply larger standard errors.For example, in a sample of 60 observations, thestandard error of the Sharpe ratio estimator is 0.188when the true Sharpe ratio is 1.50 but is 0.303 whenthe true Sharpe ratio is 3.00. This implies that theperformance of investments such as hedge funds,for which high Sharpe ratios are one of the primaryobjectives, will tend to be less precisely estimated.However, as a percentage of the Sharpe ratio, thestandard error given by Equation 9 does approacha finite limit as SR increases sinceSE(SR)------------------ SR1 ( 1/2 ) SR 21--------------------------------- -----2TT SR2(11)as SR increases without bound. Therefore, theuncertainty surrounding the IID Sharpe ratio estimator will be approximately the same proportion 2002, AIMR

The Statistics of Sharpe RatiosTable 1. Asymptotic Standard Errors of Sharpe Ratio Estimators forCombinations of Sharpe Ratio and Sample SizeSample Size, 80.1052Note: Returns are assumed to be IID, which implies VIID 1 1/2SR .of the Sharpe ratio for higher Sharpe ratio investments with the same number of observations T.We can develop further intuition for the impactof estimation errors in µ̂ and σ̂ 2 on the Sharpe ratioby calculating the proportion of asymptotic variance that is attributable to µ̂ versus σ̂ 2 . From Equation 7, the fraction of VIID due to estimation errorin µ̂ versus σ̂ 2 is simply( g/ µ ) 2 σ 21------------------------------- ---------------------------------V IID1 ( 1/2 ) SR2(12a)and( g/ σ ) 2 2σ 4( 1/2 ) SR 2---------------------------------- ----------------------------------2 .VIID1 ( 1/2 ) SR(12b)For a small Sharpe ratio, such as 0.25, thisproportion—which depends only on the true Sharperatio—is 97.0 percent, indicating that most of thevariability in the Sharpe ratio estimator is a result ofvariability in µ̂ . However, for higher Sharpe ratios,the reverse is true: For SR 2.00, only 33.3 percentof the variability of SR comes from µ̂ , and for aSharpe ratio of 3.00, only 18.2 percent of the estimator error of SR is attributable to µ̂ .Non-IID ReturnsMany studies have documented various violationsof the assumption of IID returns for financial securities;3 hence, the results of the previous sectionmay be of limited practical value in certain circumstances. Fortunately, it is possible to derive similarresults under more general conditions, conditionsthat allow for serial correlation, conditional heteroskedasticity, and other forms of dependenceand heterogeneity in returns. In particular, ifJuly/August 2002returns satisfy the assumption of “stationarity,”then a version of the Central Limit Theorem stillapplies to most estimators and the correspondingasymptotic distribution can be derived. The formaldefinition of stationarity is that the joint probabilitydistribution F ( R t , R t , , R t ) of an arbitrary col12nlection of returns R t , R t , , R t does not change12nif all the dates are incremented by the same numberof periods; that is,F ( Rt1 k,Rt2 k, , R tn k) F ( Rt , Rt , , Rt )12n(13)for all k. Such a condition implies that mean µ andvariance σ2 (and all higher moments) are constantover time but otherwise allows for quite a broad setof dynamics for Rt , including serial correlation,dependence on such factors as the market portfolio,time-varying conditional volatilities, jumps, andother empirically relevant phenomena.Under the assumption of stationarity,4 a version of the Central Limit Theorem can still beapplied to the estimator SR . However, in this case,the expression for the variance of SR is somewhatmore complex because of the possibility of dependence between the components µ̂ and σ̂ 2 . In the“Non-IID Returns” section of Appendix A, I showthat the asymptotic distribution can be derived byusing a “robust’’ estimator—an estimator that iseffective under many different sets of assumptionsfor the statistical properties of returns—to estimatethe Sharpe ratio.5 In particular, I use a generalizedmethod of moments (GMM) estimator to estimate µ̂and σ̂ 2 , and the results of Hansen (1982) can be usedto obtain the following asymptotic distribution: g gaT ( SR – SR ) N ( 0, V GMM ), V GMM ------ -------- , (14) ′39

Financial Analysts Journalwhere the definitions of g/ and and a methodfor estimating them are given in the second sectionof Appendix A. Therefore, for non-IID returns, thestandard error of the Sharpe ratio can be estimated byaSE[SR] V̂ GMM /T(15)and confidence intervals for SR can be constructedin a similar fashion to Equation 10.Time AggregationIn many applications, it is necessary to convertSharpe ratio estimates from one frequency toanother. For example, a Sharpe ratio estimated frommonthly data cannot be directly compared with oneestimated from annual data; hence, one statisticmust be converted to the same frequency as the otherto yield a fair comparison. Moreover, in some cases,it is possible to derive a more precise estimator of anannual quantity by using monthly or daily data andthen performing time aggregation instead of estimating the quantity directly using annual data.6In the case of Sharpe ratios, the most commonmethod for performing such time aggregation is tomultiply the higher-frequency Sharpe ratio by thesquare root of the number of periods contained inthe lower-frequency holding period (e.g., multiply amonthly estimator by 12 to obtain an annual estimator). In this section, I show that this rule of thumbis correct only under the assumption of IID returns.For non-IID returns, an alternative procedure mustbe used, one that accounts for serial correlation inreturns in a very specific manner.IID Returns. Consider first the case of IIDreturns. Denote by Rt (q) the following q-periodreturn:R t ( q ) Rt Rt –1 R t – q 1 ,(16)where I have ignored the effects of compoundingfor computational convenience.7 Under the IIDassumption, the variance of Rt (q) is directly proportional to q; hence, the Sharpe ratio satisfies thesimple relationship:E [ Rt ( q ) ] – Rf ( q )SR ( q ) ------------------------------------------Var [ Rt ( q ) ]q ( µ – Rf ) ---------------------qσ (17)qSR.Despite the fact that the Sharpe ratio may seemto be “unitless’’ because it is the ratio of two quantities with the same units, it does depend on thetimescale with respect to which the numerator anddenominator are defined. The reason is that the40numerator increases linearly with aggregationvalue q whereas the denominator increases as thesquare root of q under IID returns; hence, the ratiowill increase as the square root of q, making a longerhorizon investment seem more attractive. Thisinterpretation is highly misleading and should notbe taken at face value. Indeed, the Sharpe ratio is nota complete summary of the risks of a multiperiodinvestment strategy and should never be used as thesole criterion for making an investment decision.8The asymptotic distribution of SR (q) followsdirectly from Equation 17 because SR (q) is proportional to SR:aT [ SR ( q ) – qSR ] N 0, VIID ( q ) ,V IID ( q ) qV IID(18) 1 2 q 1 --- SR .2 Non-IID Returns. The relationship betweenSR and SR(q) is somewhat more involved for nonIID returns because the variance of Rt (q) is not justthe sum of the variances of component returns butalso includes all the covariances. Specifically, underthe assumption that returns Rt are stationary,Var [ Rt ( q ) ] q –1 q –1 Cov ( Rt – i , Rt – j )i 0 j 0 qσ 2 2σ 2(19)q –1 ( q – k )ρk ,k 1where ρk Cov(Rt , Rt–k)/Var(Rt) is the kth-orderautocorrelation of Rt.9 This yields the followingrelationship between SR and SR(q):qSR ( q ) η ( q ) SR, η ( q ) ---- ,q 2 q –1 ( q – k )ρk(20)k 1Note that Equation 20 reduces to Equation 17 if allautocorrelations ρk are zero, as in the case of IIDreturns. However, for non-IID returns, the adjustment factor for time-aggregated Sharpe ratios isgenerally not q but a more complicated functionof the first q – 1 autocorrelations of returns.Example: First-Order AutoregressiveReturns. To develop some intuition for thepotential impact of serial correlation on the Sharperatio, consider the case in which returns follow afirst-order autoregressive process or “AR(1)”:Rt µ ρ(Rt–1 – µ) εt , –1 ρ 1,(21)where εt is IID with mean zero and variance σε2.In this case, the return in period t can be forecastedto some degree by the return in period t – 1, andthis “autoregression’’ leads to serial correlation atall lags. In particular, Equation 21 implies that the 2002, AIMR

The Statistics of Sharpe Ratioskth-order autocorrelation coefficient is simply ρk;hence, the scale factor in Equation 20 can be evaluated explicitly asη(q ) q2ρ 1–ρ q 1 ------------ 1 – -------------------- 1–ρ q ( 1 – ρ ) scale factor is 3.46 in the IID case and 2.88 when themonthly first-order autocorrelation is 20 percent.These patterns are summarized in Figure 1, inwhich η(q) is plotted as a function of q for fivevalues of ρ. The middle (ρ 0) curve correspondsto the standard scale factor q , which is the correct–1/2.(22)Table 2 presents values of η(q) for variousvalues of ρ and q; the row corresponding to ρ 0percent is the IID case in which the scale factor issimply q. Note that for each holding-period q,positive serial correlation reduces the scale factorbelow the IID value and negative serial correlationincreases it. The reason is that positive serial correlation implies that the variance of multiperiodreturns increases faster than holding-period q;hence, the variance of Rt (q) is more than q times thevariance of Rt, yielding a larger denominator in theSharpe ratio than the IID case. For returns withnegative serial correlation, the opposite is true: Thevariance of Rt (q) is less than q times the variance ofRt , yielding a smaller denominator in the Sharperatio than the IID case. For returns with significantserial correlation, this effect can be substantial. Forexample, the annual Sharpe ratio of a portfolio witha monthly first-order autocorrelation of –20 percentis 4.17 times the monthly Sharpe ratio, whereas theFigure 1. Scale Factors of Time-AggregatedSharpe Ratios When Returns Followan AR(1) Process: For –0.50, –0.25,0, 0.25, and 0.50Scale Factor, η(q)3025ρ 0.5020ρ 0.2515ρ 0ρ 0.2510ρ 0.5050050100150200250Aggregation Value, qTable 2. Scale Factors for Time-Aggregated Sharpe Ratios When ReturnsFollow an AR(1) Process for Various Aggregation Values and FirstOrder AutocorrelationsAggregation Value, 6.478.0912.0618.2923.3227.6146.9967.65July/August 200241

Financial Analysts Journalfactor when the correlation coefficient is zero. Thecurves above the middle one correspond to positivevalues of ρ, and those below the middle curvecorrespond to negative values of ρ. It is apparentthat serial correlation has a nontrivial effect on thetime aggregation of Sharpe ratios.The General Case. More generally, using theexpression for SR (q) in Equation 20, we can construct an estimator of SR(q) from estimators of thefirst q – 1 autocorrelations of Rt under the assumption of stationary returns. As in the “Non-IIDReturn” section, we can use GMM to estimate theseautocorrelations as well as their asymptotic jointdistribution, which can then be used to derive thefollowing limiting distribution of SR (q):T [ SR ( q ) – SR ( q ) ] a N 0, V GMM ( q ) , g gV GMM ( q ) ------ -------- , ′(23)where the definitions of g/ and and formulasfor estimating them are given in the “Time Aggregation” section of Appendix A. The standard errorof SR (q) is then given byaSE [ SR ( q ) ] V̂GMM ( q )/T(24)and confidence intervals can be constructed as inEquation 10.Using V GMM ( q ) When Returns Are IID.Although the robust estimator for SR (q) is theappropriate estimator to use when returns areserially correlated or non-IID in other ways, thereis a cost: additional estimation error induced bythe autocovariance estimator, γ̂ k, which manifestsitself in the asymptotic variance, V̂ GM M ( q ), ofSR (q). To develop a sense for the impact of estimation error on V̂ GM M ( q ), consider the robustestimator when returns are, in fact, IID. In thatcase, γk 0 for all k 0 but because the robustestimator is a function of estimators γ̂ k , the estimation errors of the autocovariance estimators willhave an impact on V̂ GM M ( q ) . In particular, in the“Using V̂ GM M ( q ) When Returns Are IID” sectionof Appendix A, I show that for IID returns, theasymptotic variance of robust estimator SR (q) isgiven byν 3 SRSR24 -------VGMM ( q ) 1 – -----------(ν–σ) 44σ 4σ3q –1 2j ( qSR ) 1 – -- ,q 2j 142(25)where ν3 E[(Rt – µ)3] and ν4 E[(Rt – µ)4] are thereturn’s third and fourth moments, respectively.Now suppose that returns are normally distributed.In that case, ν3 0 and ν4 3σ4, which implies thatV GMM ( q ) V IID ( q ) ( qSR )2q –1 j 1 – --q j 12(26) V IID ( q ).The second term on the right side of Equation 26represents the additional estimation error introduced by the estimated autocovariances in themore general estimator given in Equation A18 inAppendix A. By setting q 1 so that no time aggregation is involved in the Sharpe ratio estimator(hence, no autocovariances enter into the estimator), the expression in Equation 26 reduces to theIID case given in Equation 18.The asymptotic relative efficiency of SR (q) canbe evaluated explicitly by computing the ratio ofVGMM (q) to VIID (q) in the case of IID normal returns:–12 qj 1( 1 – j/q ) 2V GMM ( q )-,------------------------- 1 ------------------------------------------V IID ( q )1 2/SR 2(27)and Table 3 reports these ratios for various combinations of Sharpe ratios and aggregation values q.Even for small aggregation values, such as q 2,asymptotic variance VGMM (q) is significantly higherthan VIID (q)—for example, 33 percent higher for aSharpe ratio of 2.00. As the aggregation valueincreases, the asymptotic relative efficiency becomeseven worse as more estimation error is built into thetime-aggregated Sharpe ratio estimator. Even witha monthly Sharpe ratio of only 1.00, the annualized(q 12) robust Sharpe ratio estimator has an asymptotic variance that is 334 percent of VIID (q).The values in Table 3 suggest that, unlessthere is significant serial correlation in returnseries Rt , the robust Sharpe ratio estimator shouldnot be used. A useful diagnostic to check for thepresence of serial correlation is the Ljung–Box(1978) Q-statistic:q –1 ρ̂ 2kQ q –1 T ( T 2 ) ------------ ,T–k(28)k 1which is asymptotically distributed as χ 2q –1 underthe null hypothesis of no serial correlation.10 IfQq – 1 takes on a large value—for example, if itexceeds the 95 percent critical value of the χ 2q –1distribution—this signals significant serial correlation in returns and suggests that the robustSharpe ratio, SR (q), should be used instead ofqSR for estimating the Sharpe ratio of q-periodreturns. 2002, AIMR

The Statistics of Sharpe RatiosTable 3. Asymptotic Relative Efficiency of Robust Sharpe Ratio EstimatorWhen Returns Are IIDAggregation Value, 136.55Note: Asymptotic relative efficiency is given by VGMM (q)/VIID (q).An Empirical ExampleTo illustrate the potential impact of estimation errorand serial correlation in computing Sharpe ratios, Iapply the estimators described in the preceding sections to the monthly historical total returns of the 10largest (as of February 11, 2001) mutual funds fromvarious start dates through June 2000 and 12 hedgefunds from various inception

The Statistics of Sharpe Ratios July/August 2002 37 returns—can yield Sharpe ratios that are consider-ably smaller (in the case of positive serial correla-tion) or larger (in the case of negative serial correlation). Therefore, Sharpe ratio estimators must be computed and interpreted i

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