sr_variance {SharpeR}

sr_variance .

Description

Computes the variance of the sample Sharpe ratio.

Usage

sr_variance(snr, n, cumulants)

Arguments

snr

the population Signal Noise ratio. Often one will use the population estimate instead.

n

the sample size that the Shapre ratio is observed on.

cumulants

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.

Details

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.

Value

the variance of the sample statistic.

Author(s)

Steven E. Pav [e-mail]

References

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. doi: 10.1093/jjfinec/nbn022

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

sr_bias.

Examples

# variance under normality:
sr_variance(1, 100, rep(0,5))