Calibration of VaR models with overlapping data

Abstract

Abstract Under the European Union’s Solvency II regulations, insurance firms are required to use a one-year VaR (Value at Risk) approach. This involves a one-year projection of the balance sheet and requires sufficient capital to be solvent in 99.5% of outcomes. The Solvency II Internal Model risk calibrations require annual changes in market indices/term structure for the estimation of risk distribution for each of the Internal Model risk drivers. This presents a significant challenge for calibrators in terms of: Robustness of the calibration that is relevant to the current market regimes and at the same time able to represent the historically observed worst crisis; Stability of the calibration model year on year with arrival of new information. The above points need careful consideration to avoid credibility issues with the Solvency Capital Requirement (SCR) calculation, in that the results are subject to high levels of uncertainty. For market risks, common industry practice to compensate for the limited number of historic annual data points is to use overlapping annual changes. Overlapping changes are dependent on each other, and this dependence can cause issues in estimation, statistical testing, and communication of uncertainty levels around risk calibrations. This paper discusses the issues with the use of overlapping data when producing risk calibrations for an Internal Model. A comparison of the overlapping data approach with the alternative non-overlapping data approach is presented. A comparison is made of the bias and mean squared error of the first four cumulants under four different statistical models. For some statistical models it is found that overlapping data can be used with bias corrections to obtain similarly unbiased results as non-overlapping data, but with significantly lower mean squared errors. For more complex statistical models (e.g. GARCH) it is found that published bias corrections for non-overlapping and overlapping datasets do not result in unbiased cumulant estimates and/or lead to increased variance of the process. In order to test the goodness of fit of probability distributions to the datasets, it is common to use statistical tests. Most of these tests do not function when using overlapping data, as overlapping data breach the independence assumption underlying most statistical tests. We present and test an adjustment to one of the statistical tests (the Kolmogorov Smirnov goodness-of-fit test) to allow for overlapping data. Finally, we explore the methods of converting “high”-frequency (e.g. monthly data) to “low”-frequency data (e.g. annual data). This is an alternative methodology to using overlapping data, and the approach of fitting a statistical model to monthly data and then using the monthly model aggregated over 12 time steps to model annual returns is explored. There are a number of methods available for this approach. We explore two of the widely used approaches for aggregating the time series.