Getting Extreme VaR Right: Eliminating Convexity and Approximation Biases Under Heavy-tailed, Moderately-Sized Samples (Presentation Slides)

Abstract

Across many types of risk (e.g. insurance risk, credit risk, the risk of catastrophic loss, enterprise-level portfolio risk, and operational risk) researchers and practitioners commonly estimate the size of rare but very impactful loss events using the Compound Loss Distribution (CLD) framework. The CLD combines an estimated frequency distribution, representing the number of losses that can occur during a specified time period, with an estimated severity distribution, representing the magnitudes of these losses, to obtain an estimated compound loss distribution (CLD). The CLD allows researchers to extrapolate beyond sparse extant data to ‘fill in’ the far right tail of the loss distribution to more reliably calculate selected risk metrics such as extreme Value-at-Risk (‘VaR’ is simply the value (quantile) associated with a large percentile of the distribution, such as 99.9%tile or higher). However, under real world conditions where loss data samples often are small to moderately sized and severity distributions typically are heavy-tailed, extreme VaR is a convex function of the key severity parameters, so regardless of the choice of (unbiased) estimators used to estimate these parameters, this convexity will upwardly bias the VaR estimate due to Jensen’s inequality (see Jensen, 1906). The magnitude of this bias often is very material, sometimes even multiples of the true value of VaR, and this effect holds even when the model is based on a ‘single’ rather than a ‘compound’ loss distribution (i.e. assuming just a constant number of losses and no frequency distribution). Surprisingly, this convexity-induced quantile bias has been almost entirely overlooked in the relevant literatures despite an intense focus during the past dozen years on the estimation of very large amounts of operational risk capital in the banking industry, which has used this exact CLD model to estimate extreme VaR. We discuss herein some of the reasons for this oversight, and propose a straightforward estimator that effectively neutralizes the systematic upward bias while simultaneously notably increasing the precision and robustness of the VaR estimate. We do this within a framework that identifies, isolates, and properly treats the distinct effects of each source of error on the ultimate VaR estimate: approximation error, model error, and estimation (sampling) error. Although this has not been done previously in the literature, it is absolutely necessary to achieve our ultimate goal here, which is to obtain reasonably accurate, precise, and robust estimates of extreme VaR when using either compound or single loss distributions.