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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.
Highlights
Returns based risk estimators such as volatility, value-at-risk, and expected shortfall, and performance estimators such as the Sharpe ratio and Sortino ratio, have important roles in asset and portfolio risk assessment and management
In order to get a sense of the accuracy of risk and performance estimator standard errors, and ensuing confidence interval error rates, computed using the formula (45), we carried out Monte Carlo simulation studies for the standard deviation (SD), Semi-Standard Deviation (SemiSD), Sharpe Ratio (SR) and Downside Sharpe Ratio (DSR) estimators
We have introduced a new general method for computing standard errors of risk and performance estimators that is simple to implement and does not require an estimator’s asymptotic variance formula
Summary
Returns based risk estimators such as volatility, value-at-risk, and expected shortfall, and performance estimators such as the Sharpe ratio and Sortino ratio, have important roles in asset and portfolio risk assessment and management. In this paper we overcome the above impediments to computing risk and performance estimator SE’s by introducing a new deterministic method of computing standard errors based on the use of the influence function (IF) borrowed from robust statistics. The new method allows one to use a risk or performance estimator’s IF formula to derive the estimator’s asymptotic variance expression, from which one can compute an estimator standard error via the usual recipe described above. We provide supplementary material, including the derivation of IF based asymptotic variance formulas for all the risk and performance estimators treated in this paper, along with references to formal mathematical statistics derivations of most of the formulas
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