Robust Sharpe Ratio
The Sharpe ratio (SR) was developed by Nobel laureate William F. Sharpe and is used to help investors understand the return of an investment compared to its risk [1]. The ratio corresponds to the average return earned in excess of the risk-free rate per unit of risk. Because of its simplicity and deep financial meaning, the Sharpe ratio is one of the most reported indicators of performance.
However, the SR is an ex-post metric and can vary significantly between time periods. A SR measured today could be very different than the same SR measured in one month's time. To mitigate the backward-looking nature and sample dependence of the SR, we implement a technique to estimate the confidence interval around the SR. This approach makes it possible to determine the statistical significance of a portfolio's SR, providing assurance that the realized risk-adjusted return is not a one-off occurrence and is likely to persist out-of-sample.
In the next sections, we explain the techniques behind the evaluation of the confidence interval around Sharpe ratios. Intuitively, the performance of a volatile asset is more difficult to measure with precision than that of a less volatile strategy. Consequently, Sharpe ratios of portfolios with low volatility are more reliable than those of portfolios with high volatility. However, volatility is not the only parameter affecting the estimation of the Sharpe ratio. Perhaps less intuitively, time-series properties such as autocorrelations can have a nontrivial impact on the Sharpe ratio estimates. We show how certain techniques allow the integration of autocorrelations in the evaluation of the SR confidence interval (see [2] and reference therein) to make them even more realistic.
Methodology
Consider a single instrument or a portfolio with a one-period simple return at time denoted by β. Additionally, assume
The Sharpe ratio (SR) is defined as a ratio of excess expected return to the standard deviation of the return, i.e.,
Given the unobservable nature of and , they must be estimated from the historical data using the sample moment estimators
which defines the Sharpe ratio sample estimator
IID Gaussian returns
To derive a measure of the uncertainty surrounding the estimator SR, we need to specify the statistical properties of returns βbecause these properties determine the uncertainty of the mean and volatility components of the SR.
Assuming that are independent and identically distributed draws from a normal distribution, the central limit theorem (CLT) implies
βThe asymptotic convergence in distribution implies that the estimation error of and can be approximated by
Encoding the Sharpe ratio in a function and defining parameter vectors , and , the above equation implies
where describes a covariance matrix. Finally, using the function that represents the Sharpe ratio, and applying the delta method [3], we get
where the gradient of is given as . Consequently, this implies
Therefore, the asymptotic standard error for the Sharpe ratio estimator is given as
where the 95% confidence interval for SR around the point estimate is simply
Note that for non-normal returns, this relationship does not generally hold as the bottom-right entry in the matrix is given as . Furthermore, this requires the first and second moments of βto be finite. Additionally, for implementation purposes, we find that it is somewhat easier to manipulate uncentered sample moments as representatives for mean and volatility.
Serially correlated returns
The above confidence interval can be improved by taking into account an important parameter of the returns: the auto-correlations. Let us denote by the vector of the period- returns and lags with and assume it is stationary and ergodic.
Compared to the i.i.d. case, the delta method, together with some regularity conditions on , implies
The limiting covariance of the unknown symmetric positive-definite matrix is given by
Note that the matrix from above captures the autocorrelation effects on top of the information included within β. Alternatively, this limit can be expressed as
where
The standard method to come up with a consistent estimator of is to use heteroskedasticity and autocorrelation robust (HAC) kernel estimation (see [4]). That is, choose a real-valued kernel such that is continuous at 0, , and , together with a bandwidth parameter β. The estimate for is then given by
where
with . Finally, analogously to the previous section, given an estimate , the standard error is calculated as
Our implementation uses a Parzen kernel whereby
βThe optimal bandwidth parameter is finally chosen using the optimal fixed bandwidth procedure (alpha method) from [4].
Annualization of the Sharpe Ratio
In many applications, it is necessary to convert Sharpe ratio estimates from one frequency to another. For example, a Sharpe ratio estimated from monthly data cannot be directly compared with one estimated from annual data; hence, one statistic must be converted to the same frequency as the other to yield a fair comparison.
First, we consider a qq-period return (simplified)
IID case
The Sharpe ratio becomes
and the standard error for the i.i.d. can be recovered from
as
Non-IID case
The relationship between SR and SR(q) is more involved for non-i.i.d. returns as the variance of includes covariances on top of the sum of the variance of its components. Specifically, assuming RtRtβ are stationary,
which implies
For the detailed derivation, see [4].
References
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1 William F. Sharpe. (1966). Mutual fund performance. The Journal of Business. 39(1): 119-138. 10.1086/294846.
2 Oliver Ledoit and Michael Wolf. (2008). Robust performance hypothesis testing with the Sharpe ratio. Journal of Empirical Finance. 15(5): 850-859. 10.1016/j.jempfin.2008.03.002.
3 J. L. Doob. (1935). The Limiting Distributions of Certain Statistics. The Annals of Mathematical Statistics. 6(3): 160-169. 10.1214/aoms/1177732594.
4 Donald W. K. Andrews. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica: Journal of the Econometric Society. 59(3): 817-858. 10.2307/2938229.
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