Influence Functions for Risk and Performance Estimators

Abstract

A new general method for computing standard errors of risk and performance estimators is developed. The method relies on the fact that the influence function of an estimator, the Gateaux derivative of the estimator functional in the direction of point mass distributions, may be used to represent the asymptotic variance of the estimator as the expected value of the squared influence function. The law of large numbers shows that the asymptotic variance of an estimator can be estimated as the time series average of the squared influence function, thereby yielding a very simple estimator standard error calculation that does not require knowledge of the asymptotic variance formula. We derive formulas for the influence functions of six risk estimators and seven performance estimators, thereby providing a convenient portfolio performance and risk management tool to easily compute standard errors for most risk and performance estimators of interest or practical importance. We conduct a simulation study to evaluate the quality of the standard errors and confidence interval error rates for the Sharpe ratio and downside Sharpe ratio estimators. Software implementations of our proposed method in the R packages RPEIF and RPESE are publicly available on CRAN.