The Automated Bias-Corrected and Accelerated Bootstrap Confidence Intervals for Risk Measures

Abstract

Different approaches to determining two-sided interval estimators for risk measures such as Value-at-Risk (VaR) and conditional tail expectation (CTE) when modeling loss data exist in the actuarial literature. Two contrasting methods can be distinguished: a nonparametric one not relying on distributional assumptions or a fully parametric one relying on standard asymptotic theory to apply. We complement these approaches and take advantage of currently available computer power to propose the bias-corrected and accelerated (BCA) confidence intervals for VaR and CTE. The BCA confidence intervals allow the use of a parametric model but do not require standard asymptotic theory to apply. We outline the details to determine interval estimators for these three different approaches using general computational tools as well as with analytical formulas when assuming the truncated Lognormal distribution as a parametric model for insurance loss data. An extensive simulation study is performed to assess the performance of the proposed BCA method in comparison to the two alternative methods. A real dataset of left-truncated insurance losses is employed to illustrate the implementation of the BCA-VaR and BCA-CTE interval estimators in practice when using the truncated Lognormal distribution for modeling the loss data.