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©2002, AIMR®

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

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.

except

ne of the most commonly cited statistics in

financial analysis is the Sharpe ratio, the

ratio of the excess expected return of an

investment to its return volatility or standard devi-

ation. Originally motivated by mean–variance

analysis and the Sharpe–Lintner Capital Asset Pric-

ing Model, the Sharpe ratio is now used in many

different contexts, from performance attribution to

tests of market efficiency to risk management.1

Given the Sharpe ratio’s widespread use and the

myriad interpretations that it has acquired over the

years, it is surprising that so little attention has been

paid to its statistical properties. Because expected

returns and volatilities are quantities that are gen-

erally not observable, they must be estimated in

some fashion. The inevitable estimation errors that

arise imply that the Sharpe ratio is also estimated

with error, raising the natural question: How accu-

rately are Sharpe ratios measured?

In this article, I provide an answer by deriving

the statistical distribution of the Sharpe ratio using

standard econometric methods under several dif-

ferent sets of assumptions for the statistical behav-

ior of the return series on which the Sharpe ratio is

based. Armed with this statistical distribution, I

show that confidence intervals, standard errors,

and hypothesis tests can be computed for the esti-

mated Sharpe ratio in much the same way that they

are computed for regression coefficients such as

portfolio alphas and betas.

The accuracy of Sharpe ratio estimators hinges

on the statistical properties of returns, and these

properties can vary considerably among portfolios,

strategies, and over time. In other words, the

Sharpe ratio estimator’s statistical properties typi-

cally will depend on the investment style of the

portfolio being evaluated. At a superficial level, the

intuition for this claim is obvious: The performance

of more volatile investment strategies is more dif-

ficult to gauge than that of less volatile strategies.

Therefore, it should come as no surprise that the

results derived in this article imply that, for exam-

ple, Sharpe ratios are likely to be more accurately

estimated for mutual funds than for hedge funds.

A less intuitive implication is that the time-

series properties of investment strategies (e.g.,

mean reversion, momentum, and other forms of

serial correlation) can have a nontrivial impact on

the Sharpe ratio estimator itself, especially in com-

puting an annualized Sharpe ratio from monthly

data. In particular, the results derived in this article

show that the common practice of annualizing

returns—can yield Sharpe ratios that are consider-

ably smaller (in the case of positive serial correla-

Using a set of techniques collectively known as

“large-sample’’ or “asymptotic’’ statistical theory

in which the Central Limit Theorem is applied to

estimators such as and

and other nonlinear functions of

in large samples

data, a more general distribution is derived in the

“Non-IID Returns” section, one that applies to

returns with serial correlation, time-varying condi-

tional volatilities, and many other characteristics of

historical financial time series. In the “Time Aggre-

gation” section, I develop explicit expressions for

“time-aggregated’’ Sharpe ratio estimators (e.g.,

expressions for converting monthly Sharpe ratio

estimates to annual estimates) and their distribu-

tions. To illustrate the practical relevance of these

estimators, I apply them to a sample of monthly

mutual fund and hedge fund returns and show that

serial correlation has dramatic effects on the annual

Sharpe ratios of hedge funds, inflating Sharpe ratios

by more than 65 percent in some cases and deflating

Sharpe ratios in other cases.

IID Returns

To derive a measure of the uncertainty surrounding

the estimator. we need to specify the statistical

properties of Rt because these properties determine

the uncertainty surrounding the component estima-

tors and. Although this may seem like a theo-

retical exercise best left for statisticians—not unlike

the specification of the assumptions needed to yield

well-behaved estimates from a linear regression—

there is often a direct connection between the invest-

ment management process of a portfolio and its

statistical properties. For example, a change in the

portfolio manager’s style from a small-cap value

orientation to a large-cap growth orientation will

typically have an impact on the portfolio’s volatility,

degree of mean reversion, and market beta. Even for

a fixed investment style, a portfolio’s characteristics

can change over time because of fund inflows and

outflows, capacity constraints (e.g. a microcap fund

that is close to its market-capitalization limit),

liquidity constraints (e.g. an emerging market or

private equity fund), and changes in market condi-

tions (e.g. sudden increases or decreases in volatil-

ity, shifts in central banking policy, and

extraordinary events, such as the default of Russian

government bonds in August 1998). Therefore, the

investment style and market environment must be

kept in mind when formulating the assumptions for

the statistical properties of a portfolio’s returns.

Perhaps the simplest set of assumptions that we