Computes the variance of the sample Sharpe ratio.
sr_variance(snr, n, cumulants)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.
the variance of the sample statistic.
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. 10.1093/jjfinec/nbn022
Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.
# NOT RUN {
# variance under normality:
sr_variance(1, 100, rep(0,5))
# }
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