In this paper we present a framework for backtesting all currently popular risk measurement methods for quantifying market risk (including value-at-risk and expected shortfall) using the functional delta method. Estimation risk can be taken explicitly into account. Based on a simulation study we provide evidence that tests for expected shortfall with acceptable low levels have a better performance than tests for value-at-risk in realistic financial sample sizes. We propose a way to determine multiplication factors, and find that the resulting regulatory capital scheme using expected shortfall compares favorably to the current Basel Accord backtesting scheme.

We propose a procedure to take model risk into account in the computation of capital reserves. This addresses the need to make the allocation of capital reserves to positions in given markets dependent on the extent to which reliable models are available. The proposed procedure can be used in combination with any of the standard risk measures, such as Value-at-Risk and expected shortfall. We assume that models are obtained by usual econometric methods, which allows us to distinguish between estimation risk and misspecification risk. We discuss an additional source of risk which we refer to as identification risk. By way of illustration, we carry out calculations for equity and FX data sets. In both markets, estimation risk and misspecification risk together explain about half of the multiplication factors employed by the Bank for International Settlements (BIS).

Joost Driessen, Pieter Klaassen, Bertrand Melenberg, The Performance of Multi-Factor Term Structure Models for Pricing and Hedging Caps and Swaptions, The Journal of Financial and Quantitative Analysis, Vol. 38, No. 3 (Sep., 2003), pp. 635-672