Statistical pitfalls in Solvency II Value-at-Risk models

In the first part of this thesis, we compute the effect of misspecifications and misinterpretations of Solvency II Value-at-Risk models. We investigate the effect of backtesting using a rolling window without correcting for the use of the rolling window (a misinterpretation). We show that this leads to a significant increase in the probability of finding an historic extreme event in the data the model is calibrated on. If not corrected for this effect, backtesting with a rolling window can lead to false rejections of Value-at-Risk models. We illustrate this by analyzing the evaluation of the equity stress parameter for Dutch Pension Funds, which is said to be too low. We find that the rejection of the parameter in the report can be explained by the use of a rolling window. We propose a step by step approach to correctly backtest using a rolling window. To our knowledge, this is the first time the effect of this commonly made error is quantified. Next to that, we investigate the effect of a number of possible misspecifications of Value-at-Risk models. The normality assumption is at the heart of Solvency II. We compute how much the probability of finding an extreme event increases if normality is assumed, and effects like autocorrelation, clustered volatility, and heavy tailedness of distributions are neglected. Next to that, we have extended an existing analysis of the effect of scaling up the VaR of a Jump Diffusion process to a longer time horizon by the square root of time rule. By considering different time horizons, we identify three different regimes, and their implications on the use of the square root of time rule, of which two regimes do not occur in the current analysis. For the misspecifications, we have computed possible theoretical effects, and investigated the practical relevance by computing the effects for realistic parameter values. For this purpose, historic interest rate data, and calibrations of Jump Diffusion models and GARCH models on historic equity (option) data have been used. For realistic parameter values, we find increases in the probability of finding an extreme event by a factor between 2 and 4. For backtesting with a rolling window, we even find an increase in the probability of finding an extreme event by a factor 7. Therefore, the found effects are relevant, and should be considered when calibrating VaR models. In the second part of this thesis, statistical pitfalls of the one-year horizon in Solvency II are investigated. We compute long term probabilities of ruin and default for insurance companies all starting with an initial Solvency II ratio of 100%. We vary investment strategies, liability durations and cashflow patterns, and show that this highly influences levels of safety on time horizons beyond one year. In the light of the level playing field, we plead to consider longer time horizons in Solvency II beside the currently used one-year horizon, analogous to the regulations for Dutch Pension Funds. We show that for longer time horizons, probabilities of ruin and default are more sensitive for assumptions of mean reversion of risky assets, and therefore assumptions should be set with care.