Andrew Lo - Statistics of Sharp Ratios

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CFA Institute

The Statistics of Sharpe RatiosAuthor(s): Andrew W. LoReviewed work(s):Source: Financial Analysts Journal, Vol. 58, No. 4 (Jul. - Aug., 2002), pp. 36-52Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4480405 .

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T h e Statistics o f S h a r p e R a t i o s

AndrewW. Lo

Thebuilding locks f theSharpe atio-expected eturns ndvolatilities-are unknownquantities hat must be estimated tatisticallyand are,

therefore,ubjecto estimationrror.Thisraises henatural uestion:Howaccuratelyre Sharpe atiosmeasured? oaddress his question, derive

explicit xpressionsforhestatistical istributionftheSharpeatiousing

standardasymptotic heoryunderseveralsets of assumptionsor the

return-generatingrocess-independently nd identicallydistributed

returns,stationaryreturns,and with time aggregation. show thatmonthlySharpeatios annotbeannualized y multiplying y J12exceptunder very specialcircumstances,nd I derivethe correctmethodofconversionn thegeneralcase of stationary eturns. n an illustrativeempiricalxample fmutualfunds ndhedgefunds,findthat heannual

Sharpe atiofora hedge,fundan be overstated yas muchas 65 percentbecause f thepresence f serialcorrelationn monthly eturns, ndonce

thisserial orrelations properlyakenntoaccount,herankings fhedgefundsbased nSharpe atios an changedramatically.

O ne of themost commonlycitedstatistics nfinancialanalysis is the Sharperatio, theratio of the excess expected returnof an

investmentto its returnvolatilityor standarddevi-ation. Originally motivated by mean-varianceanalysisandtheSharpe-LintnerCapitalAssetPric-ing Model, the Sharperatio is now used in manydifferentcontexts,fromperformanceattributiono

tests of market efficiency to risk management.1Given the Sharperatio's widespread use and themyriad nterpretationshat thasacquiredovertheyears, t is surprising hatsolittleattentionhasbeenpaid to its statisticalproperties.Becauseexpectedreturnsand volatilitiesare quantities hatare gen-erally not observable,they must be estimated insome fashion.Theinevitableestimationerrors hatariseimply that the Sharperatiois also estimatedwith error, aisingthenaturalquestion:How accu-ratelyare Sharperatiosmeasured?

In this article,Iprovidean answer by deriving

the statisticaldistributionof the Sharperatio usingstandardeconometricmethods under several dif-ferent sets of assumptionsfor the statisticalbehav-iorof the returnserieson which theSharperatio sbased. Armed with this statisticaldistribution,I

show that confidence intervals, standard errors,and hypothesistests can be computedforthe esti-mated Sharperatio n muchthe sameway thattheyare computed for regression coefficientssuch asportfolioalphasand betas.

Theaccuracyof Sharperatioestimatorshingeson the statistical propertiesof returns,and thesepropertiescan vary considerablyamong portfolios,

strategies, and over time. In other words, theSharperatio estimator'sstatisticalpropertiestypi-cally will depend on the investment style of theportfoliobeingevaluated.At asuperficialevel,theintuition or this claim s obvious:The performanceof more volatile investment strategies s moredif-ficult to gauge than that of less volatile strategies.Therefore, t should come as no surprise that theresultsderivedin this article mply that, for exam-ple, Sharperatios are likely to be more accuratelyestimated formutual funds thanfor hedge funds.

A less intuitive implication is that the time-

series properties of investment strategies (e.g.,mean reversion, momentum, and other forms ofserialcorrelation) an have a nontrivial mpact ontheSharperatio estimator tself,especiallyin com-puting an annualized Sharperatio from monthlydata.Inparticular,he results derivedin thisarticleshow that the common practice of annualizingSharperatiosby multiplyingmonthlyestimatesbyA/i2i s correct only under very special circum-stances and that the correct multiplier-whichdepends on the serial correlationof the portfolio's

AndrewW. Lois Harris& HarrisGroupProfessortthe SloanSchool f Management,Massachusettsnsti-tuteofTechnology,ambridge,ndchiefscientificfficerforAlphaSimplex roup,LLC,Cambridge.

36 62002, AIMR?



TheStatistics f SharpeRatios

returns-can yield Sharperatiosthatareconsider-ably smaller(in the case of positive serialcorrela-tion) or larger (in the case of negative serialcorrelation).Therefore, Sharpe ratio estimatorsmustbe computedandinterpreted n the contextofthe particular nvestmentstyle with which a port-

folio'sretums have been generated.LetRtdenote the one-periodsimple returnofa portfolioor fund between dates t - 1 and t anddenoteby g and a2 its mean and variance:

[t _ E Rt), (la)

and

cs2 Var(Rt). (lb)

Recallthat the Sharperatio (SR) s defined as theratioof the excess expected returnto the standarddeviation of return:

SR-

Rf (2)

where the excess expected returnis usually com-putedrelative othe risk-free ate,RfBecauseganda arethepopulationmomentsof thedistributionof

Rt, they are unobservableand must be estimatedusinghistoricaldata.

Givenasampleof historical eturns R1,R2,...,

RT), he standardestimators or these momentsarethe samplemeanand variance:

lT

= TERt (3a)t=1

and

T

CT= T(RD-13) , (3b)

t=1

fromwhich the estimator of the Sharperatio (SR)follows immediately:

SR = Ri Rf (4)

Using a set of techniques collectively known as"large-sample"or "asymptotic" tatisticaltheory

in which the CentralLimitTheorem s applied toestimators uchas i and &2, he distributionof SRand other nonlinearfunctionsof ,u and a2 can beeasily derived.

In the next section,I presentthe statisticaldis-tributionof SRunderthestandardassumption hatreturnsare independentlyand identicallydistrib-uted (IID).Thisdistributioncompletelycharacter-izes the statisticalbehaviorof SRin largesamplesandallows us to quantify heprecisionwith whichSR estimatesSR.ButbecausetheIIDassumption sextremelyrestrictiveand oftenviolatedbyfinancial

data,a more generaldistribution s derived in the"Non-IIDReturns" section, one that applies toreturnswith serialcorrelation,ime-varyingcondi-tionalvolatilities,andmanyothercharacteristicsfhistorical inancial imeseries.Inthe "TimeAggre-gation" section, I develop explicitexpressionsfor

"time-aggregated"Sharpe ratio estimators (e.g.,expressions for converting monthly Sharpe ratioestimatesto annual estimates)and theirdistribu-tions. To illustratethe practicalrelevanceof theseestimators,I apply them to a sample of monthlymutualfund and hedge fund returnsand showthatserialcorrelation asdramatic ffectson theannualSharpe atiosofhedgefunds, nflatingSharpe atiosbymorethan65percent nsomecasesanddeflatingSharperatios n other cases.

IIDReturns

Toderiveameasureof theuncertainty urroundingthe estimatorSR,we need to specifythe statisticalpropertiesof Rtbecausethesepropertiesdeterminethe uncertainty urrounding he componentestima-tors j and a2. Although this may seem like atheoreticalexercisebest left for statisticians-notunlike the specificationof the assumptionsneededto yieldwell-behavedestimates romalinearregres-sion-there is oftena directconnectionbetween theinvestmentmanagementprocessof a portfolioandits statisticalproperties.For example,a change intheportfoliomanager's tylefromasmall-capvalueorientationto a large-capgrowth orientationwill

typicallyhaveanimpactonthe portfolio'svolatility,degreeof mean reversion,and marketbeta.Evenfora fixedinvestmentstyle,aportfolio'scharacteristicscanchangeover time becauseof fund inflowsandoutflows,capacityconstraintse.g.,amicrocap undthat is close to its market-capitalizationlimit),liquidityconstraints e.g., an emerging marketorprivateequityfund),and changes n marketcondi-tions (e.g.,sudden increasesordecreases n volatil-ity, shifts in central banking policy, andextraordinaryvents,such as the defaultof Russiangovernmentbonds in August 1998).Therefore, he

investmentstyle and marketenvironmentmustbekept n mind whenformulatingheassumptions orthe statisticalpropertiesof a portfolio'sreturns.

Perhaps hesimplestset of assumptions hatwecanspecifyforRt s thattheyare ndependentlyandidenticallydistributed.Thismeans that the proba-bilitydistributionof Rtis identicalto thatof RS oranytwo datest ands andthatRtandRSare statisti-callyindependentfor all t s. Althoughthese con-ditionsare extreme and empirically mplausible-theprobabilitydistribution fthe monthlyreturnoftheS&P 00 Index n October1987 s likelyto differ

July/August 2002 37



FinancialAnalystsJournal

from the probability distribution of the monthly

return of the S&P500 in December 2000-they pro-

vide an excellent starting point for understanding

the statistical properties of Sharpe ratios. In the next

section, these assumptions will be replaced with a

more general set of conditions for returns.

Under the assumption that returns are IID and

have finite mean g and variance a2, it is wellknown that the estimators ,i and a2 in Equation 3

have the following normal distributions in large

samples, or "asymptotically," due to the Central

Limit Theorem:2

lT( _)a N(O,02), TT(d2 - 02) a N(O,2o4), (5)

where a denotes the fact that this relationship is an

asymptotic one [i.e., as T increases without bound,

the probability distributions of fT (Si- ,u) and

T (&2_ 0y2)approach the normal distribution, with

mean zero and variances a2 and 2o4, respectively].

These asymptotic distributions imply that the esti-

mation error of i and &2can be approximated by

a (T2 (&2)a 2ay

Var(Ti) a,Var(T2)= -y, (6)

awhere = indicates that these relations are based on

asymptotic approximations. Note that in Equation

6, the variances of both estimators approach zero as

T increases, reflecting the fact that the estimation

errorsbecome smaller as the sample size grows. An

additional property of ,i and & in the special case

of IID returns is that they are statistically indepen-

dent in large samples, which greatly simplifies our

analysis of the statistical properties of functions ofthese estimators (e.g., the Sharpe ratio).

Now, denote by the function g(., G2) the

Sharpe ratio defined in Equation 2; hence, the

Sharpe ratio estimator is simply g(,ii, &2) = SR.

When the Sharpe ratio is expressed in this form, it

is apparent that the estimation errors in ji and &2

will affect g(,, &2) and that the nature of these

effects depends critically on the properties of the

function g. Specifically, in the "IID Returns" sec-

tion of Appendix A, I show that the asymptotic

distribution of the Sharpe ratio estimator is

TT(S`- SR) a N(O, VIID), where the asymp-

totic variance is given by the following weightedaverage of the asymptotic variances of ,i and &2:

VIID =+

$2+ 264. (7)

The weights in Equation 7 are simply the squared

sensitivities of g with respect to j and a2, respec-tively: The more sensitive g is to a particular param-

eter, the more influential its asymptotic variance

will be in the weighted average. This relationship

is reminiscentof the expressionfor the varianceoftheweightedsumof two randomvariables,exceptthat nEquation7,there s no covariance erm.Thisis due to the fact that ,i and &2 areasymptoticallyindependent, thanks to our simplifying assump-tion of IID returns.In the next sections, the IIDassumptionwill be replacedby a moregeneralset

of conditionson returns, n which case,the covari-ancebetween [i and a2 will no longerbe zeroandthe corresponding expression for the asymptoticvarianceof theSharperatioestimatorwill be some-what more involved.

The asymptoticvarianceof SR given in Equa-tion 7 canbe furthersimplified by evaluatingthesensitivitiesexplicitly-ag/Dg = 1/A and ag/la2 =

- Rf)/(2&3)-and then combining terms to yield

V = 1+4(R Rf)2(8)

1+2SR.

Therefore, tandarderrors SEs) or theSharperatioestimator SR can be computed as

SE(SR) ,1 + SR2I/T, (9)

and this quantitycan be estimatedby substitutingSR for SR.Confidence ntervals for SR can also beconstructed from Equation 9; for example, the 95percentconfidence ntervalfor SRaroundthe esti-matorSRis simply

SR? 1.96x + SR / T (10)

Table 1 reportsvalues of Equation9 for vari-ous 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,thestandarderrorof theSharperatioestimator s 0.188when the trueSharperatio s 1.50but is 0.303whenthe true Sharperatiois 3.00. This implies that theperformanceof investmentssuch as hedge funds,for whichhigh Sharperatiosare one of theprimary

objectives,will tend to be less preciselyestimated.However, as a percentageof the Sharpe ratio,thestandarderrorgiven by Equation9 does approacha finitelimitas SR ncreasessince

SE(SR) 1+(1/2)SR2, 1

SR TSR _T (1

as SR increases without bound. Therefore, theuncertaintysurrounding he IIDSharperatio esti-mator will be approximately he same proportion

38 ?2002, AIMR?




Table 1. Asymptotic Standard Errors of Sharpe Ratio Estimators for

Combinations of Sharpe Ratio and Sample Size

SampleSize,T

SR 12 24 36 48 60 125 250 500

0.50 0.306 0.217 0.177 0.153 0.137 0.095 0.067 0.047

0.75 0.327 0.231 0.189 0.163 0.146 0.101 0.072 0.051

1.00 0.354 0.250 0.204 0.177 0.158 0.110 0.077 0.0551.25 0.385 0.272 0.222 0.193 0.172 0.119 0.084 0.060

1.50 0.421 0.298 0.243 0.210 0.188 0.130 0.092 0.065

1.75 0.459 0.325 0.265 0.230 0.205 0.142 0.101 0.071

2.00 0.500 0.354 0.289 0.250 0.224 0.155 0.110 0.077

2.25 0.542 0.384 0.313 0.271 0.243 0.168 0.119 0.084

2.50 0.586 0.415 0.339 0.293 0.262 0.182 0.128 0.091

2.75 0.631 0.446 0.364 0.316 0.282 0.196 0.138 0.098

3.00 0.677 0.479 0.391 0.339 0.303 0.210 0.148 0.105

Note:Returnsare assumed to be IID,which impliesVIID 1 + 1/2SR2.

of the Sharpe ratio for higher Sharpe ratio invest-ments with the same number of observations T.

We can develop further intuition for the impact

of estimation errorsin [i and (2 on the Sharpe ratio

by calculating the proportion of asymptotic vari-

ance that is attributable to i versus (2. From

Equation 7, the fraction of VIID due to estimation

error in t versus 2is simply

(_g/a_t)2_2_ 1 (12a)

VIID 1+ (1/2) SR2

and

(ag/aa)22G4 (1 /2) SR2

VIID 1+(1/2)SR2b

For a small Sharpe ratio, such as 0.25, this

proportion-which depends only on the true Sharpe

ratio-is 97.0 percent, indicating that most of the

variability in the Sharpe ratio estimator is a result of

variability in ,. However, for higher Sharpe ratios,

the reverse is true: For SR = 2.00, only 33.3 percent

of the variability of SR comes from j, and for a

Sharpe ratio of 3.00, only 18.2percent of the estima-

tor error of SR is attributable to ji.

Non-lIDReturnsMany studies have documented various violations

of the assumption of IID returns for financial secu-

rities;3 hence, the results of the previous section

may be of limited practical value in certain circum-

stances. Fortunately, it is possible to derive similar

results under more general conditions, conditions

that allow for serial correlation, conditional het-

eroskedasticity, and other forms of dependence

and heterogeneity in returns. In particular, if

returns satisfy the assumption of "stationarity,"then a version of the CentralLimitTheorem stillapplies to most estimatorsand the correspondingasymptoticdistributioncan be derived. Theformaldefinitionofstationaritys thatthe jointprobabilitydistributionF(Rt, Rt, ..., Rtn) of an arbitrary ol-lectionof returnsRtl Rt2,. ., Rtn does not changeif all the datesare ncrementedbythe samenumberof periods;thatis,

F(Rti+k t2k+k'

.RAn+k) = F(Rt,9Rt2 Rtn) (13)

for all k.Sucha conditionimpliesthat mean t and

variancec72 andall highermoments) are constantover time butotherwise allows forquiteabroadsetof dynamics for Rt, including serial correlation,dependenceon such factorsas the marketportfolio,time-varying conditional volatilities, jumps, andotherempiricallyrelevantphenomena.

Under the assumption of stationarity,4 ver-sion of the Central Limit Theorem can still beappliedto the estimatorSR. However,in this case,the expressionfor the varianceof SR is somewhatmorecomplexbecauseof the possibilityof depen-dence between the components [i and a2. In the

"Non-IIDReturns" ection of Appendix A, I showthat the asymptoticdistributioncan be derived byusing a "robust"estimator-an estimator that iseffective undermany differentsets of assumptionsfor the statisticalpropertiesof returns-to estimatethe Sharperatio.5In particular, use a generalizedmethod fmomentsGMM)estimator o estimate ,iand&2,and theresultsofHansen(1982) anbe usedto obtainthefollowingasymptoticdistribution:

,F(SR-SR) - N(O, VGMM) VGMM a , (14)

July/August 2002 39



FinancialAnalysts Journal

where the definitionsof ag/aOandX and a methodfor estimating hemaregiven in the second sectionof AppendixA. Therefore, or non-IIDreturns, hestandard rror f the Sharpe atiocanbe estimatedby

SE SR] = VGMM/T (15)

and confidence ntervalsfor SRcan be constructed

in a similarfashionto Equation10.

Time AggregationIn many applications, it is necessary to convertSharpe ratio estimates from one frequency toanother.Forexample,aSharpe atioestimated rommonthlydatacannotbe directlycomparedwithoneestimated from annual data; hence, one statisticmustbe converted othe samefrequency s theotherto yield a faircomparison.Moreover,n somecases,it is possible toderiveamorepreciseestimatorof anannualquantityby using monthlyordailydata and

then performingtime aggregation nstead of esti-matingthequantitydirectlyusingannualdata.6

In the case of Sharperatios,the most commonmethod for performing uch time aggregations tomultiplythe higher-frequency harperatioby thesquareroot of the number of periodscontained nthe lower-frequency olding period (e.g.,multiplyamonthlyestimatorby NIT2o obtainan annualesti-mator). nthissection,Ishow that his rule of thumbis correctonlyunder theassumptionof IIDreturns.Fornon-IIDreturns,an alternativeproceduremustbe used, one that accountsfor serial correlationn

returns n a very specificmanner.IIDReturns. Consider first the case of IID

returns. Denote by Rt(q) the following q-periodreturn:

Rt(q) =Rt + Rt_l + + Rt-q+l, (16)

where I have ignoredthe effects of compoundingfor computationalconvenience.7 Under the IIDassumption, he varianceofRt(q)is directlypropor-tional to q; hence, the Sharpe ratio satisfies thesimple relationship:

SR q)E[Rt(q)] Rf(q)

Var [Rt(q)]

q(g - Rf) (17)

Jq-

= IqSR.

Despitethe factthattheSharperatiomayseemto be "unitless"becauseit is the ratioof two quan-tities with the same units, it does depend on thetimescalewith respectto which the numeratoranddenominator are defined. The reason is that the

numerator increases linearly with aggregationvalue q whereas the denominator ncreases as thesquare root of qunderIIDreturns;hence, the ratiowill increaseas thesquarerootofq,makingalonger-horizon investment seem more attractive. Thisinterpretations highlymisleadingand should notbetakenatfacevalue.Indeed, heSharpe atio s not

a complete summaryof the risksof a multiperiodinvestment trategyandshould neverbeused as thesole criterion ormakinganinvestmentdecision.8

The asymptoticdistributionof SR(q)followsdirectly romEquation17becauseSR(q) s propor-tionalto SR:

,FlT"SRq) - Iq SR] N [0 VI ID(q)]

(18)

VIID(q) = qVIID = 1+ 1SRJ

Non-lID Returns. The relationshipbetween

SR and SR(q) s somewhat more involved for non-IIDreturnsbecausethevarianceof Rt(q)is not justthe sum of the variancesof componentreturnsbutalso includesall thecovariances.Specifically,undertheassumption hat returnsRtarestationary,

q-1 q-1

VarRt(q)] = E E Cov Rt-, Rt-d)z=O =O (19)

2 2q-1

= q2 + 26 E(q - k)Pk,

k=1

where Pk Cov(Rt, Rt-k)/Var(Rt) is the kth-order

autocorrelationof Rt.9This yields the followingrelationshipbetween SRandSR(q):

SR(q) = i (q)SR,rj(q)_ q9 (20)

q + 2q-l (q -k)pk

Note thatEquation20reduces to Equation17 if allautocorrelationsPkare zero, as in the case of IIDreturns.However, for non-IIDreturns,the adjust-ment factor for time-aggregatedSharpe ratios isgenerallynot {q but a morecomplicatedfunctionof the firstq- 1 autocorrelations f returns.

Example: First-Order AutoregressiveReturns. To develop some intuition for thepotential mpactof serial correlationon theSharperatio,considerthe case in which returns follow afirst-orderautoregressiveprocessor "AR(l)":

Rt =, + p(Rt-, - ) + ct, -1 < p < 1, (21)

where Et is IID with mean zero and variance42Inthiscase,thereturn n periodt can be forecastedto some degree by the returnin period t - 1, andthis "autoregression" eads to serialcorrelationatall lags. In particular,Equation21 implies thatthe

40 ?2002, AIMR?l




kth-orderautocorrelationcoefficientis simply pk;

hence, the scale factorin Equation20 can be eval-uated explicitlyas

r ~~~~~~-1/2

i (q) = q +2P(1 _-q(l_-_p) (22)

Table 2 presents values of rl(q) for variousvalues of p and q;the row corresponding o p = 0percentis the IIDcase in which the scale factorissimply Jq. Note that for each holding-periodq,

positive serial correlationreduces the scale factorbelow theIID value andnegativeserial correlationincreases t.The reason s that positiveserial corre-lation implies that the variance of multiperiodreturns increases faster than holding-period q;hence,thevarianceof Rt(q) s more than q timesthevarianceofRt,yieldinga largerdenominatorn theSharpe ratio than the IID case. For returnswithnegativeserialcorrelation,heopposite is true:Thevarianceof Rt q) s less thanqtimes thevarianceofRt, yielding a smallerdenominator n the Sharperatiothan the IID case.For returnswith significantserial correlation, hiseffect can be substantial.Forexample,the annualSharperatioofaportfoliowithamonthlyfirst-order utocorrelation f-20 percentis 4.17timesthe monthlySharperatio,whereasthe

scalefactor s 3.46 n the IID caseand 2.88 when themonthlyfirst-orderautocorrelations 20percent.

Thesepatternsare summarized n Figure 1, inwhich q q) is plotted as a function of q for fivevalues of p. The middle (p = 0) curve correspondsto the standardscalefactorJq,which is thecorrect

Figure 1. Scale Factors of Time-Aggregated

Sharpe Ratios When Returns Follow

an AR(1) Process: For p = -0.50, -0.25,

0, 0.25, and 0.50

Scale Factor,71(q)

30

25-

20-

15

0 I I

0 50 100 150 200 250

Aggregation Value, q

Table 2. Scale Factors for Time-Aggregated Sharpe Ratios When Returns

Follow an AR(1) Process for Various Aggregation Values and First-

Order Autocorrelationsp Aggregation Value, q

(%) 2 3 4 6 12 24 36 48 125 250

90 1.03 1.05 1.07 1.10 1.21 1.41 1.60 1.77 2.67 3.70

80 1.05 1.10 1.14 1.21 1.43 1.81 2.14 2.42 3.79 5.32

70 1.08 1.15 1.21 1.33 1.65 2.19 2.62 3.00 4.75 6.68

60 1.12 1.21 1.30 1.46 1.89 2.55 3.08 3.53 5.63 7.94

50 1.15 1.28 1.39 1.60 2.12 2.91 3.53 4.06 6.49 9.15

40 1.20 1.35 1.49 1.75 2.36 3.27 3.98 4.58 7.35 10.37

30 1.24 1.43 1.60 1.91 2.61 3.65 4.44 5.12 8.23 11.62

20 1.29 1.52 1.73 2.07 2.88 4.04 4.93 5.68 9.14 12.92

10 1.35 1.62 1.86 2.25 3.16 4.45 5.44 6.28 10.12 14.31

0 1.41 1.73 2.00 2.45 3.46 4.90 6.00 6.93 11.18 15.81

-10 1.49 1.85 2.16 2.66 3.80 5.39 6.61 7.64 12.35 17.47

-20 1.58 1.99 2.33 2.90 4.17 5.95 7.31 8.45 13.67 19.35

-30 1.69 2.13 2.53 3.17 4.60 6.59 8.10 9.38 15.20 21.52

-40 1.83 2.29 2.75 3.48 5.09 7.34 9.05 10.48 17.01 24.11

-50 2.00 2.45 3.02 3.84 5.69 8.26 10.21 11.84 19.26 27.31

-60 2.24 2.61 3.37 4.30 6.44 9.44 11.70 13.59 22.19 31.50

-70 2.58 2.76 3.86 4.92 7.45 11.05 13.77 16.04 26.33 37.43

-80 3.16 2.89 4.66 5.91 8.96 13.50 16.98 19.88 32.96 47.02

-90 4.47 2.97 6.47 8.09 12.06 18.29 23.32 27.61 46.99 67.65

July/August 2002 41




factorwhen the correlationcoefficient s zero. Thecurvesabove themiddleonecorrespond opositivevalues of p, and those below the middle curvecorrespondto negative values of p. It is apparentthat serialcorrelationhas a nontrivialeffecton thetimeaggregationof Sharperatios.

The General Case. Moregenerally,using theexpressionfor SR(q)in Equation20, we can con-struct an estimator of SR(q) from estimators of the

first q- 1 autocorrelations of Rt under the assump-

tion of stationary returns. As in the "Non-IIDReturn" section, we can use GMM to estimate these

autocorrelationsas well as their asymptotic jointdistribution, which can then be used to derive the

following limiting distribution of SR (q):

T[SR(q)-SR(q)] - N[O, VGMM(q)], (23)

VGMM(q) a 50ewherethedefinitions of ag/aOandX and formulasfor estimatingthem aregiven in the "TimeAggre-gation"section of AppendixA. The standarderrorof SR (q) is then given by

SEI[SR(q)] - VGMM(q)/T (24)

and confidence intervals can be constructedas inEquation 10.

Using VGMM(q)When Returns Are IID.

Although the robust estimator for SR(q) is theappropriate estimator to use when returns are

serially correlated or non-IID in other ways, there

is a cost: additional estimation error induced by

the autocovariance estimator, 'k'which manifests

itself in the asymptotic variance, VGMM(q), of

SR (q). To develop a sense for the impact of esti-

mation error on VGMM(q), consider the robustestimator when returns are, in fact, IID. In that

case, 7k = 0 for all k > 0 but because the robust

estimator is a functionofestimators 'k,the estima-

tion errors of the autocovariance estimators will

have an impact on VGMM(q). In particular, in the

"Using VGMM(q) When Returns Are IID" section

of Appendix A, I show that for IID returns, the

asymptotic variance of robust estimator SRR(q)s

given by

V ~(9) 1 V3SR+ (V SR2]

(25)

+-lVS) -

where v3 E[(Rt- p)3] and V4 E[(Rt- g)4] are thereturn's third and fourth moments, respectively.

Now suppose thatreturns arenormally distributed.In that case, V3= 0 and V4= 3c74,which implies that

VGMM(q)= VIID(q) + (JqSR) 1 ) (26)

2 VIID (q)-

The second term on the right side of Equation 26represents the additional estimation error intro-duced by the estimated autocovariances in themore general estimator given in Equation A18 inAppendix A. By setting q = 1so that no time aggre-gation is involved in the Sharpe ratio estimator(hence, no autocovariances enter into the estima-tor), the expression in Equation 26 reduces to theIID case given in Equation 18.

The asymptotic relative efficiency of SR (q) can

be evaluated explicitly by computing the ratio ofVGMM(q) o VIID(q)n the case of IID normal returns:

GMM(q)= 1+ j= ) (27)

VIID(q) 1 + 2/SR2

and Table 3 reports these ratios for various combi-nations of Sharpe ratios and aggregation values q.Even for small aggregation values, such as q = 2,asymptotic variance VGMMq) is significantly higherthan VIID(q)-for example, 33 percent higher for aSharpe ratio of 2.00. As the aggregation valueincreases, the asymptotic relative efficiency becomes

even 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 asymp-totic 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 p2

Qq-i = T(T+2)XTk, (28)k=1

which is asymptotically distributed as X2 underthe null hypothesis of no serial correlation.10 If

Qq-1 takes on a large value-for example, if itexceeds the 95 percent critical value of the X2distribution-this signals significant serial corre-lation in returns and suggests that the robustSharne ratio, SR (q), should be used instead of{qSR for estimating the Sharpe ratio of q-periodreturns.

42 ?2002, AIMR?




Table 3. Asymptotic Relative Efficiency of Robust Sharpe Ratio Estimator

When Returns Are IID

Aggregation Value, q

SR 2 3 4 6 12 24 36 48 125 250

0.50 1.06 1.12 1.19 1.34 1.78 2.67 3.56 4.45 10.15 19.41

0.75 1.11 1.24 1.38 1.67 2.54 4.30 6.05 7.81 19.07 37.37

1.00 1.17 1.37 1.58 2.02 3.34 6.00 8.67 11.34 28.45 56.22

1.25 1.22 1.49 1.77 2.34 4.08 7.59 11.09 14.60 37.11 73.66

1.50 1.26 1.59 1.93 2.62 4.72 8.95 13.18 17.42 44.59 88.71

1.75 1.30 1.67 2.06 2.85 5.25 10.08 14.92 19.76 50.81 101.22

2.00 1.33 1.74 2.17 3.04 5.69 11.01 16.34 21.67 55.89 111.45

2.25 1.36 1.80 2.25 3.19 6.04 11.76 17.49 23.23 60.02 119.75

2.50 1.38 1.84 2.33 3.31 6.32 12.37 18.43 24.49 63.38 126.51

2.75 1.40 1.88 2.38 3.42 6.56 12.87 19.20 25.52 66.12 132.02

3.00 1.41 1.91 2.43 3.50 6.75 13.28 19.83 26.37 68.37 136.55

Note: Asymptotic relative efficiency is given by VGMM q)/ VIID(q).

An Empirical ExampleToillustrate the potential impact of estimation error

and serial correlation in computing Sharpe ratios, I

apply the estimators described in the preceding sec-

tions to the monthly historical total returns of the 10

largest (as of February 11, 2001) mutual funds from

various start dates through June 2000 and 12 hedge

funds from various inception dates through Decem-

ber 2000. Monthly total returns for the mutual funds

were obtained from the University of Chicago's Cen-

ter for Research in Security Prices. The 12 hedge

funds were selected from the Altvest database to

yield a diverse range of annual Sharpe ratios (from1.00 to 5.00) computed in the standard way (_qSR,where SR is the Sharpe ratio estimator applied to

monthly returns), with the additional requirement

that the funds have a minimum five-year history of

returns. The names of the hedge funds have been

omitted to maintain their privacy, and I will refer to

them only by their investment styles (e.g., relative

value fund, risk arbitrage fund).11

Table 4 shows that the 10 mutual funds have

little serial correlation in returns, with p-values of

Q-statistics ranging from 13.2 percent to 80.2

percent.12 Indeed, the largest absolute level of

autocorrelation among the 10 mutual funds is the

12.4 percent first-order autocorrelation of the

Fidelity Magellan Fund. With a risk-free rate of 5/

12 percent per month, the monthly Sharpe ratios

of the 10 mutual funds range from 0.14 (Growth

Fund of America) to 0.32 (Janus Worldwide), with

robust standard errors of 0.05 and 0.11, respec-

tively. Because of the lack of serial correlation in

the monthly returns of these mutual funds, there

is little difference between the IID estimator for the

annual Sharpe ratio, IqSR (in Table4,12SR),

and

the robust estimator that accounts for serial

correlation, SR (12). For example, even in the caseof the Fidelity Magellan Fund, which has thehighest first-order autocorrelation among the 10

mutual funds, the difference between a 1qSR of

0.73 and a SR(12) of 0.66 is not substantial (and

certainly not statistically significant). Note that the

robust estimator is marginally lower than the IID

estimator, indicating the presence of positiveserial correlation in the monthly returns of theMagellan Fund. In contrast, for WashingtonMutual Investors, the IID estimate of the annual

Sharpe ratio is $qSR = 0.60 but the robust estimateis larger, SR(12) = 0.65, because of negative serial

correlation in the fund's monthly returns (recallthat negative serial correlation implies that thevariance of the sum of 12 monthly returns is lessthan 12 times the variance of monthly returns).

The robust standard errors SE3(12) with m = 3

for SR(12) for the mutual funds range from 0.17

(Janus) to 0.47 (Fidelity Growth and Income) andtake on similar values when m = 6, which indicatesthat the robust estimator is reasonably well behavedfor this dataset. The magnitudes of the standard

errors yield 95 percent confidence intervals forannual Sharpe ratios that do not contain 0 for anyof the 10 mutual funds. For example, the 95 percentconfidence interval for the Vanguard 500 Indexfund is 0.85 ? (1.96 x 0.26), which is (0.33, 1.36).These results indicate Sharpe ratios for the 10mutual funds that are statistically different from 0at the 95 percent confidence level.

The results for the 12 hedge funds are differentin several respects. The mean returns arehigher and

the standard deviations lower, implying much

July/August 2002 43




Table 4. Monthly and Annual Sharpe Ratio Estimates for a Sample of Mutual Funds and Hedge Funds

p-Value Monthly Annual

Start R a P2 P3 ofQ11i

Fund Date T (%) (%) (%) (%) (%) (%) SR SE3 jl-2SR SR(12) SE3(12) SE6(12)

Mutualfunds

Vanguard 500 Index 10/76 286 1.30 4.27 -4.0 -6.6 -4.9 64.5 0.21 0.06 0.72 0.85 0.26 0.25

Fidelity Magellan 1/67 402 1.73 6.23 12.4 -2.3 -0.4 28.6 0.21 0.06 0.73 0.66 0.20 0.21

Investment Company

of America 1/63 450 1.17 4.01 1.8 -3.2 -4.5 80.2 0.19 0.05 0.65 0.71 0.22 0.22

Janus 3/70 364 1.52 4.75 10.5 -0.0 -3.7 58.1 0.23 0.06 0.81 0.80 0.17 0.17

Fidelity Contrafund 5/67 397 1.29 4.97 7.4 -2.5 -6.8 58.2 0.18 0.05 0.61 0.67 0.23 0.23

Washington

Mutual Investors 1/63 450 1.13 4.09 -0.1 -7.2 -2.6 22.8 0.17 0.05 0.60 0.65 0.20 0.20

Janus Worldwide 1/92 102 1.81 4.36 11.4 3.4 -3.8 13.2 0.32 0.11 1.12 1.29 0.46 0.37

Fidelity Growth and

Income 1/86 174 1.54 4.13 5.1 -1.6 -8.2 60.9 0.27 0.09 0.95 1.18 0.47 0.40

American Century

Ultra 12/81 223 1.72 7.11 2.3 3.4 1.4 54.5 0.18 0.07 0.64 0.71 0.27 0.25

Growth Fund of

America 7/64 431 1.18 5.35 8.5 -2.7 -4.1 45.4 0.14 0.05 0.50 0.49 0.19 0.20

Hedge unds

Convertible/option

arbitrage 5/92 104 1.63 0.97 42.6 29.0 21.4 0.0 1.26 0.28 4.35 2.99 1.04 1.11

Relative value 12/92 97 0.66 0.21 25.9 19.2 -2.1 4.5 1.17 0.17 4.06 3.38 1.16 1.07

Mortgage-backed

securities 1/93 96 1.33 0.79 42.0 22.1 16.7 0.1 1.16 0.24 4.03 2.44 0.53 0.54

High-yield debt 6/94 79 1.30 0.87 33.7 21.8 13.1 5.2 1.02 0.27 3.54 2.25 0.74 0.72

Risk arbitrage A 7/93 90 1.06 0.69 -4.9 -10.8 6.9 30.6 0.94 0.20 3.25 3.83 0.87 0.85

Long-short equities 7/89 138 1.18 0.83 -20.2 24.6 8.7 0.1 0.92 0.06 3.19 2.32 0.35 0.37

Multistrategy A 1/95 72 1.08 0.75 48.9 23.4 3.3 0.3 0.89 0.40 3.09 2.18 1.14 1.19

Risk arbitrage B 11/94 74 0.90 0.77 -4.9 2.5 -8.3 96.1 0.63 0.14 2.17 2.47 0.79 0.77

Convertible arbitrage A 9/92 100 1.38 1.60 33.8 30.8 7.9 0.8 0.60 0.18 2.08 1.43 0.44 0.45

Convertible arbitrage B 7/94 78 0.78 0.62 32.4 9.7 -4.5 23.4 0.60 0.18 2.06 1.67 0.68 0.62

Multistrategy B 6/89 139 1.34 1.63 49.0 24.6 10.6 0.0 0.57 0.16 1.96 1.17 0.25 0.25

Fund of funds 10/94 75 1.68 2.29 29.7 21.1 0.9 23.4 0.56 0.19 1.93 1.39 0.67 0.70

Note:For the mutual fund sample, monthly total returns from various start dates through June 2000;for the hedge fund sample, various

start dates through December 2000. The term denotes the kthautocorrelation coefficient, and Q1l denotes the Ljung-Box Q-statistic,

which is asymptotically Xl under the null hypothesis of no serial correlation. SR denotes the usual Sharpe ratio estimator, (,i - Rf)/ 6,which is based on monthly data; Rfis assumed to be 5/12 percent per month; and SR (12) denotes the annual Sharpe ratio estimator

that takes into account serial correlation in monthly returns. All standard errors are based on GMM estimators using the Newey-West

(1982) procedure with truncation lag m = 3 for entries in the SE3and SE3 12) columns and m = 6 for entries in the SE6 12) column.

higherSharperatio estimates or hedge funds thanfor mutual funds. The monthly Sharperatio esti-

mates,SR,range rom0.56("Fundoffunds") o1.26("Convertible/option rbitrage"),n contrast o therange of 0.14 to 0.32 for the 10mutual funds.How-ever, the serialcorrelationn hedge fund returns salso muchhigher.Forexample, hefirst-order uto-correlation oefficientrangesfrom -20.2 percentto49.0percentamongthe 12hedge funds,whereas hehighest first-orderautocorrelation s 12.4 percentamongthe 10mutual funds. Thep-valuesprovideamore complete summaryof the presenceof serial

correlation:All but 5 of the 12 hedge funds havep-values ess than5percent,andseveralare ess than

1percent.Theimpact of serialcorrelationon the annual

Sharperatiosofhedge fundsis dramatic.When theIIDestimator, /Y2SR,s used for theannualSharperatio, the "Convertible/optionarbitrage" und hasa Sharpe ratio estimate of 4.35, but when serialcorrelation is properly taken into account bySR 12), he estimatedropsto2.99, mplyingthattheIIDestimatoroverstates heannualSharperatioby45 percent.The annualSharperatioestimate orthe

44 ?2002, AIMR?



TheStatisticsof SharpeRatios

"Mortgage-backedecurities"unddropsfrom4.03to2.44 when serialcorrelations taken ntoaccount,implying an overstatement f 65percent.However,theannual Sharperatioestimatefor the "Riskarbi-trage A" fund increases rom 3.25 to 3.83 because of

negative serial correlationn its monthlyreturns.

The sharp differencesbetween the annual IIDand robust Sharpe ratioestimatesunderscore theimportanceof correctly accounting for serial cor-relation in analyzing the performance of hedgefunds.Naivelyestimatingthe annualSharperatiosby multiplying SR by Jii2 will yield the rankordering given in the SR12 column of Table 4,butonce serialcorrelation s taken ntoaccount, herankordering changes to 3, 2, 5, 7, 1, 6, 8, 4, 10, 9,12, and 11.

The robust standard errors for the annualrobustSharperatioestimatesof the 12hedge fundsrange from 0.25 to 1.16,which although arger hanthose in the mutual fund sample, neverthelessimply 95 percentconfidence intervals that gener-allydo not include 0.Forexample,evenin the caseof the"MultistrategyB" und,which has thelowestrobust Sharperatioestimate (1.17), ts 95 percentconfidence intervalis 1.17? 1.96x 0.25,which is(0.68, 1.66). These statistically significant Sharperatiosareconsistentwithpreviousstudies thatdoc-ument thefactthathedge funds do seem to exhibitstatistically significantexcess returns.13The simi-larityof thestandarderrorsbetween the m= 3 andm=6 cases for thehedge fundsample ndicates hatthe robust estimator is also well behaved in thiscase,despite the presenceof significantserial cor-relation n monthlyreturns.

Insummary,theempiricalexamplesillustratethepotential mpactthatserialcorrelation anhaveon Sharpe ratio estimates and the importanceofproperly accountingfor departures rom the stan-dard IIDframework.RobustSharperatio estima-tors contain significant additional informationabout the risk-rewardtrade-offsforactive invest-mentproducts,such ashedge funds;more detailedanalysis of the risks and rewards of hedge fundinvestments is performedin Getmansky, Lo, andMakarov 2002)and Lo(2001).

ConclusionAlthough the Sharperatiohas become part of thecanonof modem financialanalysis, tsapplicationstypically do not account for the fact that it is anestimatedquantity, ubject oestimationerrors hatcan be substantial n some cases. The resultspre-sented in this articleprovideone way to gauge theaccuracyof theseestimators,and it shouldcome asno surprisethatthe statisticalpropertiesof Sharperatios depend intimately on the statisticalproper-ties of the returnseries on which they are based.This suggests that a more sophisticatedapproachto interpretingSharperatios is calledfor, one thatincorporates information about the investmentstyle that generates the returns and the marketenvironment n which thosereturnsaregenerated.For example, hedge funds have very differentreturn characteristics rom the characteristicsofmutual funds; hence, the comparisonof Sharpe

ratios between these two investmentvehicles can-not be performed naively. In light of the recentinterest n alternative nvestmentsby institutionalinvestors-investors that are accustomed to stan-dardized performanceattributionmeasures suchas the annualized Sharperatio-there is an evengreaterneed todevelopstatistics hatareconsistentwith a portfolio's nvestmentstyle.

Theempiricalexampleunderscores he practi-cal relevance of proper statistical inference forSharpe atioestimators.gnoring he mpactof serialcorrelation n hedgefund returnscanyield annual-ized Sharperatiosthatareoverstatedby more than

65percent,understatedSharperatios n thecase ofnegativelyseriallycorrelated eturns,andinconsis-tentrankingsacrosshedge funds of different tylesand objectives.By using the appropriate tatisticaldistribution orquantifyingheperformance feachreturnhistory,theSharperatio canprovidea morecompleteunderstanding f the risksandrewardsofa broadarrayof investmentopportunities.

I thankNicholasChan,ArnoutEikeboom,imHolub,Chris akob, aurelKenner, rankLinet, onMarkman,VictorNiederhoffer, an O'Reilly,Bill Sharpe,and

JonathanTayloror helpful ommentsnddiscussion.ResearchupportromAlphaSimplex roup s grate-fullyacknowledged.

July/August 2002 45




Appendix A. Asymptotic Distributions of Sharpe Ratio EstimatorsThe first section of this appendix presents results for IID returns, and the second section presentscorrespondingresults for non-IIDreturns. Results for time-aggregatedSharperatios are reportedin thethird section,and in the final section, the asymptoticvariance,VGMM(q),f the time-aggregatedrobustestimator,SR(q), s derived for the specialcase of IIDreturns.

Throughout heappendix,thefollowingconventions are maintained: 1)allvectors are columnvectors

unless otherwiseindicated; 2)vectorsandmatrixesarealways typesetin boldface(i.e.,Xandg arescalarsand Xand ,uare vectors ormatrixes).

IIDReturnsTo derive an expression for the asymptoticdistributionof SR, we must firstobtain the asymptoticjointdistribution of i and ca2.Denote by 0 the column vector (Si &2), and let 0 denote the correspondingcolumn vectorof populationvalues (g cr2)'. IfreturnsareIID, t is awell-knownconsequenceof theCentralLimitTheorem hatthe asymptoticdistributionof 0 is given by (see White):

TlT(0 0) a N(O,V0), V0 - 04 (Al)

where the notation a indicatesthat thisis anasymptoticapproximation.Because heSharperatioestimatorSRcan be writtenasafunctiong(O) of 0, itsasymptoticdistribution ollowsdirectly romTaylor's heoremor the so-called delta method(see,forexample,White):

fT [g(^) -g(0)] a N(O,Vg),Vg_ @ V , (A2)

In the caseof the Sharperatio,g(.) is given by Equation2;hence,

50 F Rf)/ I, (A3)

which yields the following asymptoticdistribution or SR:

a N(O,('ifR2 1IT(SR-SR) a N(, VIID), VIID =+ =1 +2 SR2. (A4)

Non-lID ReturnsDenote by Xtthe vector of period-t returns and lags ( Rt Rti ... Rtq+i )' and let (Xt)be a stochastic process

thatsatisfies the followingconditions:

Hl: tXt: t E (-oo, oo)} is stationary and ergodic;

H2:00e 0, 0 is an open subset of 9ik;

H3:VOE 0, p ., 0) and 'pO., 0) areBorelmeasurableandypoX, 1 s continuous on 0 forall X;

H4: po s first-momentcontinuous at 00;E[ypo(X,)] xists,is finite,andis of full rank.

H5: Let 'Pt - qp(Xt,0o)

and

46 ?C2002,AIMR??



The Statisticsof SharpeRatios

and assume

(i): E[ypopy] xists and is finite,

(ii): vj converges in mean square to 0, and

00

(iii): , E(vV.j)l12 is finite,

j-0whichimpliesE[kpXt, Ho)]= 0.

T

H6: Let 0 solve T (Xt, 0) = 0.

t=1

Then, Hansen shows that

T/T(0 0o) a N(O,V6),V -H 1YH 1 (A5)

where

HT=m i E ( (A6)

T TX _im E L 0I Xt 00*(XS 00) (A7)

L=t=1s=l

andeo(Rt, 0) denotes the derivative of(Rt, () with respect to 0 14Specifically, letp(Rt, 0) denote the

following vector function:

p(RtS ) - (Al)(Rt - ,w2 -o20

The GMM estimator of 0Edenoted by 0, is given implicitly by the solution to

1T

I wm(t,) = ?m (A9)tj=1

I~~~~~~~~

which yields the standard estimators pi and (2 given in Equation 3. For the moment conditions in EquationA8, H is given by:

H(-lm)E 1 21 0 (A14)

Therefore, Vo = X and the asymptotic distribution of the Sharpe ratio estimator follows from the delta

method as in the first section:

FT(SRSR) -NOVGMM), VGMM ao X(AL

where aglaO is given in Equation A3. An estimator for aglaO may be obtained by substituting 6 into

Equation A3, and an estimator for Y.may be obtained by using Newey and West's (1987) procedure:

m+l

July-/Auus 00 , 47,ep( -I) - 13)




and m is the truncation ag, which must satisfy the conditionmIT -- ooas T increases without bound toensureconsistency.An estimator or VSRcan then be constructedas

ag(O) ag(O)VGMM ao

,ao (A15)

Time AggregationLet 0 [=u cy2 yi ... Yq-l]'denote the vectorof parameters o be estimated,where Yks the kth-orderautocovarianceof Rt,and define the followingmoment conditions:

(pi(Xt, ) = Rt-

P2(Xt,0) = (Rt_ )2_62

P3(Xt,O) = (Rt- )(Rt_l- )-y

(p4(Xt, 0) = (Rt -g)(Rt-2

- 9) -72 (A16)

(Pq+1(XtH) = (Rt- )(Rt-q+1 - ) -Yq-1

spXt ) - Pl P2 TP3 .(Pq +1

where Xt [Rt Rt_1 ... Rt-q+l . The GMM estimator 0 is defined by EquationA9, which yields the

standardestimators 'i and 62 in Equation3 as well as the standardestimators or theautocovariances:

1T

yk = T I (Rt-i)(Rt-k-!)* (A17)t=k+l

Theestimator or the Sharperatio thenfollows directly:

qSR(q) = 1i(q)SR,r1(q)=- ,I (A18)

q+ 2 _q- (q-k)Ik

where

YkPk =2-

Asinthe firsttwo sections ofthisappendix, heasymptoticdistributionof SR(q)canbeobtainedbyapplyingthe deltamethod tog(0) where thefunctiong(-) is now given by Equation20.Recall romEquationA5 thatthe asymptoticdistributionof the GMMestimator0 is given by

,FT(6- 0) a N(O,V,,), V>aH- 1H 1, (A19)

1 T T

T,= im

TY

(Xt,O)Yp(xs,) (A20)

For the momentconditionsin EquationA16,H is

-1 0 0...0

H =lim 1T 2(RL-RE) 01O ...

(AOH =l1ElTXE 2ji -Re-Rz1 0) -1 ... 0 | =-I(A1

2JiRt Rt-q+i 0 ?. 0 4

48 ?2002, AIMR?



TheStatistics of SharpeRatios

hence, V0 =o . Theasymptoticdistributionof SR(q)thenfollows fromthe deltamethod:

fT[SR(q)-SR(q)] - N[O>VGMM(q)] VGMM(q) -3z., (A22)

where the components of ag/aO are

ag q (A23)

a,lg E_( - k)Pk

q SR~~3g q2SR 3/2'

a62 2c72[q+ 2 q1(q - k)Pk]3

ag q(q - k) SRk=1 1 (A25)

and

3g= ag agag ag (A26)

Substituting0 intoEquationA26, estimatingI according o EquationA12, andformingthe matrixproductag(O) /a 0ag(O) /aO' yields an estimator for the asymptotic variance of SR (q).

Using VGMM(q)When Returns Are IID

For IIDreturns, t is possible toevaluateI in EquationA20 explicitly as

2 v3 0 0 ... 0

V3 V4-a4 0 0 ... 0

o4 0&'...00

0 0 (6oY . ?O2 (A27)

O O

0 0 0o. 4

where V3 E[(Rt- j)3], V4 E [(Rt- p)4], and Y is partitioned into a block-diagonal matrix with a (2 x 2)

matrix E1 and a diagonal (q - 1) x (q - 1) matrix ;2 = cs41along its diagonal. Because Yk= 0 for all k > 0,

ag/aOsimplifies to

a,u '(A28)aS4t (Y'

ag IqSRa62 262 8 ~~~~~~~~~~~~~~~

a(y 2&y

a3g_q(q-k)SR k 1 (A30)

and

= ja1a32 arY1 -Yq-1) = [a bi, (A31)

July/August 002 49




where ag/aO is also partitioned to conform to the partitioned matrix X in Equation A27. Therefore, the

asymptotic variance of the robust estimator SR (q)is given by

VGMM(q)ag ag

= at1a' +b2b2

= qR

- + (v4 -4)SR4j + (FqsR)2X(1 -. (A32)

If Rt is normally distributed, then v3 = 0 and V4= 3?4; hence,

VGMM(q)= q + (3&4 _r4) SRL4+ (FqSR)2 1 -

= +2SR +qSR) q (A33)

= VIID(q) + (fqSR) 21 -IID(q).

j=1



TheStatistics of SharpeRatios

Notes

1. See Sharpe (1994) for an excellent review of its many appli-

cations, as well as some new extensions.

2. The Central Limit Theorem is a remarkable mathematical

discovery on which much of modem statistical inference is

based. It shows that under certain conditions, the probabil-

ity distribution of a properly normalized sum of random

variables must converge to the standard normal distribu-

tion, regardless of how each of the random variables in the

sum is distributed. Therefore, using the normal distribution

for calculating significance levels and confidence intervals

is often an excellent approximation, even if normality does

not hold for the particularrandom variables in question. See

White (1984) for a rigorous exposition of the role of the

Central Limit Theorem in modern econometrics.

3. See, for example, Lo and MacKinlay (1999) and their cita-

tions.

4. Additional regularity conditions are required; see Appen-

dix A, Hansen (1982), and White for further discussion.

5. The term "robust" is meant to convey the ability of an

estimator to perform well under various sets of assump-

tions. Another commonly used term for such estimators is

"nonparametric," which indicates that an estimator is notbased on any parametric assumption, such as normally

distributed retums. See Randles and Wolfe (1979) for fur-

ther discussion of nonparametric estimators and Hansen

for the generalized method of moments estimator.

6. See, for example, Campbell, Lo, and MacKinlay (1997,

Ch. 9), Lo and MacKinlay (Ch. 4), Merton (1980), and

Shiller and Perron (1985).

7. The exact expression is, of course,

q-1

Rt(q)=-n (I1Rt_j,1.j=O

For most (but not all) applications, Equation 16 is an excel-

lent approximation. Alternatively, if Rt is defined to be the

continuously compounded return [i.e., Rt = log (Pt/Pt 1),

where Pt is the price or net asset value at time t], thenEquation 16 is exact.

8. See Bodie (1995) and the ensuing debate regarding risks in

the long run for further evidence of the inadequacy of the

Sharpe ratio-or any other single statistic-for delineatingthe risk-reward profile of a dynamic investment policy.

9. The kth-order autocorrelation of a time series Rt is definedas the correlation coefficient between Rt and Rt-k, which is

simply the covariance between Rt and Rt-k divided by thesquare root of the product of the variances of Rt and Rt k.Butbecause the variances of Rtand Rt-kare the same underour assumption of stationarity, the denominator of theautocorrelation is simply the variance of Rt.

10. See, for example, Harvey (1981, Ch. 6.2).

11. These are the investment styles reported in the Altvest

database; no attempt was made to verify or to classify thehedge funds independently.

12. The p-value of a statistic is defined as the smallest level of

significance for which the null hypothesis can be rejectedbased on the statistic's value. In particular, the p-value of16.0 percent for the Q-statistic of Washington MutualInvestors in Table 4 implies that the null hypothesis of Ino

serial correlation can be rejected only at the 16.0 percent

significance level; at any lower level of significance-say,5 percent-the null hypothesis cannot be rejected. There-

fore, smaller p-values indicate stronger evidence againstthe null hypothesis and larger p-values indicate strongerevidence in favor of the null. Researchers often reportp-values instead of test statistics because p-values are easierto interpret. To interpret a test statistic, one must compareit with the critical values of the appropriate distribution.This comparison is performed in computing the p-value.For further discussion of p-values and their interpretation,see, for example, Bickel and Doksum (1977, Ch. 5.2.B).

13. See, for example, Ackermann, McEnally, and Ravenscraft(1999), Brown, Goetzmann, and Ibbotson (1999), Brown,Goetzmann, and Park (2001), Fung and Hsieh (1997a, 1997b,2000), and Liang (1999, 2000, 2001).

14. See Magnus and Neudecker (1988) for the specific defini-

tions and conventions of vector and matrix derivatives ofvector functions.

References

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