Computes the variance of the sample Sharpe ratio.
sr_variance(snr, n, cumulants)the variance of the sample statistic.
the population Signal Noise ratio. Often one will use the population estimate instead.
the sample size that the Shapre ratio is observed on.
a vector of the third through fourth, or the third through seventh population cumulants of the random variable. More terms are needed for the higher accuracy approximation.
Steven E. Pav shabbychef@gmail.com
The sample Sharpe ratio has variance of the form $$V = \frac{1}{n}\left(1 + \frac{\zeta^2}{2}\right) +\frac{1}{n^2}\left(\frac{19\zeta^2}{8} + 2\right) -\gamma_1\zeta\left(\frac{1}{n} + \frac{5}{2n^2}\right) +\gamma_2\zeta^2\left(\frac{1}{4n} + \frac{3}{8n^2}\right) +\frac{5\gamma_3\zeta}{4n^2} +\gamma_1^2\left(\frac{7}{4n^2} - \frac{3\zeta^2}{2n^2}\right) +\frac{39\gamma_2^2\zeta^2}{32n^2} -\frac{15\gamma_1\gamma_2\zeta}{4n^2} +o\left(n^{-2}\right),$$ where \(\zeta\) is the population Signal Noise ratio, \(n\) is the sample size, \(\gamma_1\) is the population skewness, and \(\gamma_2\) is the population excess kurtosis, and \(\gamma_3\) through \(\gamma_5\) are the fifth through seventh cumulants of the error term. This form of the variance appears as Equation (4) in Bao.
See ‘The Sharpe Ratio: Statistics and Applications’, section 3.2.3.
Bao, Yong. "Estimation Risk-Adjusted Sharpe Ratio and Fund Performance Ranking Under a General Return Distribution." Journal of Financial Econometrics 7, no. 2 (2009): 152-173. tools:::Rd_expr_doi("10.1093/jjfinec/nbn022")
Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.
sr_bias.
# variance under normality:
sr_variance(1, 100, rep(0,5))
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