Autocorrelation adjusted Sharpe ratio: Definition & Examples

An autocorrelation adjusted Sharpe ratio converts a one-period Sharpe ratio to a longer horizon while including covariance between returns at different dates. It matters when comparing strategies whose monthly returns have different serial dependence: multiplying both monthly ratios by

\sqrt{12}

can distort their annual-scale comparison.

Here, “adjusted” means Lo's time-aggregation correction for a sum of excess returns. It does not mean that the estimate has been corrected for every source of backtest bias or that its statistical significance is known. Lo (2002), Time Aggregation, equations 19–20.

Count the covariance pairs

Let $x_t=R_t-R_{f,t}$ be monthly excess return, subtracting the matching monthly cash return observation by observation. Assume covariance stationarity: a constant mean $\mu$ , a finite constant variance $\sigma^2>0$ , and lag- $k$ covariance $\gamma_k$ that depends on the lag rather than the calendar date. Define

\rho_k=\frac{\gamma_k}{\sigma^2}, \qquad SR_1=\frac{\mu}{\sigma}, \qquad X_q=\sum_{j=1}^{q}x_j.

The autocorrelation $\rho_k$ measures correlation between observations $k$ periods apart. The variance of the sum is an exact population identity:

\begin{aligned} \text{Var}(X_q) &=\sum_{i=1}^{q}\sum_{j=1}^{q}\text{Cov}(x_i,x_j)\\ &=q\sigma^2+2\sum_{k=1}^{q-1}(q-k)\gamma_k\\ &=q\sigma^2D_q, \qquad D_q=1+2\sum_{k=1}^{q-1}\left(1-\frac{k}{q}\right)\rho_k. \end{aligned}

There are $q-k$ distinct pairs $k$ months apart. Each appears twice in the double sum, once in each order. For a twelve-month sum, that means eleven lag-one pairs, ten lag-two pairs, down to one lag-eleven pair. The weights are consequences of this counting, not arbitrary choices about which lags seem important.

Since $\mathbb{E}[X_q]=q\mu$ , the horizon Sharpe ratio is

SR_q=\frac{q\mu}{\sqrt{\text{Var}(X_q)}} =\frac{\sqrt{q}\,SR_1}{\sqrt{D_q}}, \qquad D_q>0.

No normal-distribution assumption is needed for this identity. Zero autocovariance at every relevant lag gives $D_q=1$ and recovers square-root scaling; independence is sufficient but stronger than necessary. A zero horizon variance leaves the ratio undefined.

For the covariance expansion behind this result, Expectation, Variance and Covariance is available with a free signed-in account.

A monthly 0.10 becomes 0.288, not 0.346

Take a hypothetical stationary population with $SR_1=0.10$ and $\rho_k=0.2^k$ . This correlation pattern is consistent with a stationary AR(1) process; it is stipulated here, not estimated from a short return history. For $q=12$ ,

D_{12}=1+2\sum_{k=1}^{11}\left(1-\frac{k}{12}\right)0.2^k \approx1.447917.

The sum's variance is about $17.375\sigma^2$ , consisting of $12\sigma^2$ from individual months plus $5.375\sigma^2$ from covariance. Therefore,

\begin{aligned} SR_{12}^{\text{naive}}&=\sqrt{12}(0.10)\approx0.346410,\\ SR_{12}&=\frac{\sqrt{12}(0.10)}{\sqrt{1.447917}} \approx0.287885. \end{aligned}

Hypothetical stationary monthly population with Sharpe 0.10 and lag-k correlation 0.2 to the power k. Twelve-month sum variance is 17.375 monthly variance units, comprising 12 from variances and 5.375 from covariance. The adjusted horizon Sharpe is 0.288 versus naive 0.346.

Original analytical population example. Both comparisons hold the monthly mean and variance fixed; the horizon is an additive twelve-month excess-return sum.

Positive covariance makes the sum riskier than the zero-covariance calculation assumes. Negative covariance can offset risk: with the same monthly Sharpe and $\rho_k=(-0.2)^k$ , $D_{12}\approx0.689815$ and $SR_{12}\approx0.417085$ .

The direction depends on the weighted sum of all relevant autocorrelations, not just

\rho_1

. For positive monthly Sharpe,

D_q>1

reduces the horizon ratio and

0<D_q<1

increases it. With negative monthly Sharpe, the same denominator adjustment moves the signed ratio in the opposite numerical direction: “positive dependence always lowers Sharpe” is too imprecise.

A horizon correction is not a confidence interval

In data,

\mu

\sigma

and the autocorrelations are unknown. Plugging their estimates into the identity gives a horizon point estimate. Estimating eleven autocorrelations adds uncertainty; an apparent correction may itself be noisy. Fitting an AR(1) can reduce the number of parameters, but makes the geometric correlation pattern a modelling assumption. Lo (2002), The General Case and equations 23–26.

Heteroscedasticity and autocorrelation consistent (HAC) inference addresses a related, different question: how uncertain is an estimated mean or Sharpe ratio? For a stationary return series with summable autocovariances, the long-run variance is

\Omega=\gamma_0+2\sum_{k=1}^{\infty}\gamma_k.

Under suitable central-limit conditions, the variance of the estimated mean is approximately

\Omega/T

, where

T

is the history's sample size. Newey–West estimates long-run covariance using tapered sample autocovariances and a bandwidth. Inference for a Sharpe ratio also needs the joint estimation uncertainty of its mean and variance; for the adjusted horizon ratio, the estimated lag covariances enter too. It is not enough to attach a mean's HAC standard error to the final ratio. Datta and Du (2012), sections 2 and 3.2.

Keep horizon $q$ separate from sample size $T$ and HAC bandwidth. The finite-horizon weights

1-k/q

arise from the target sum. A HAC bandwidth controls estimation of a long-run object. Similar triangular weights do not make those objectives interchangeable. In this example,

D_{12}\approx1.447917

, while the infinite-horizon AR(1) variance ratio is

(1+0.2)/(1-0.2)=1.5

The membership lesson Serial Correlation: Durbin–Watson, Breusch–Godfrey, and Newey–West develops HAC inference and why it does not repair a misspecified model; the linked page provides a public preview.

What the adjustment leaves unresolved

Compounding changes the target. A sum of simple excess returns is generally different from a portfolio's compounded return minus compounded cash return. Log returns add through time, but a Sharpe ratio defined on log returns is a different statistic. The formula above does not automatically deliver a compounded annual simple-return Sharpe. Sharpe's original discussion explicitly distinguishes summation, compounding and serial correlation. Sharpe (1994), Time Dependence.

Changing regimes change the inputs. A constant lag covariance is inappropriate when a strategy or return process changes materially. The general double-sum covariance identity still holds, but the stationary reduction to one

\gamma_k

per lag does not. An autocorrelation correction cannot make the history stationary.

Observed dependence still needs an explanation. Overlapping return windows or stale valuations can create dependence. Establish what each observation represents before interpreting the correction. The resulting ratio describes the recorded series; it does not independently recover unobserved economic returns, remove omitted costs, or adjust for selecting the best of many strategies.

Practice

Two consecutive monthly excess returns have the same population mean and variance, monthly Sharpe

0.20

, and correlation

-0.25

. What is the Sharpe ratio of their sum, and why is

\sqrt{2}(0.20)

incorrect?

Answer. For two months,

D_2=1+2(1-1/2)(-0.25)=0.75

. Hence

SR_2=\sqrt{2}(0.20)/\sqrt{0.75}\approx0.326599

, versus

0.282843

from square-root scaling. The negative covariance lowers the sum's variance to

1.5\sigma^2

rather than

2\sigma^2

. No compounding is being performed.

Sources

Andrew W. Lo (2002), The Statistics of Sharpe Ratios, Financial Analysts Journal 58(4), 36–52. Time Aggregation, equations 19–26, and Appendix A: stationary aggregation and estimation uncertainty.
William F. Sharpe (1994), The Sharpe Ratio — author-hosted reprint. The Ratio and Time Dependence: differential returns, the additive convention and compounding caveats.
Deepa Dhume Datta and Wenxin Du (2012), Nonparametric HAC Estimation for Time Series Data with Missing Observations, Federal Reserve IFDP 1060. Sections 2 and 3.2: Newey–West weights, long-run variance and the role of bandwidth; this article uses the complete, equally spaced case.