Contributions to solvency risk measurement

The thesis focuses on risk measures used to calculate solvency capital requirements. It consists of three independent papers. The first paper (Chapter 2) investigates time-consistency, the relation that should hold across risk measurements of the same financial position at different time points. Sufficient conditions are provided for coherent risk measures, in order to satisfy the requirements of acceptance-, rejection- and sequential consistency. It is shown that risk measures used in practice usually do not satisfy these requirements. Hence a method is provided to systematically construct sequentially consistent risk measures. It is also emphasized that current approaches to dynamic risk measurement do not consider that risk measures at different time points have different arguments. Here we briefly discuss this new setting highlighting that the notions of time consistency presented in the literature need to be reinterpreted. The second and third papers (Chapters 3 and 4) consider respectively the risk arising from parameter and model mis-specification due to estimation from a limited amount of available data. This risk may have a substantial impact on risk measures used to quantify solvency capital requirements. We introduce a new method to quantify this impact measured as the additional capital needed to allow for randomness in the data sample used for the estimation procedure. This level of capital we call residual estimation risk. In the second paper, for parameter uncertainty we prove the effectiveness of three approaches for reducing residual estimation risk in the case of location-scale families. These are based on (a) raising the capital requirement by adjusting the risk measure, (b) Bayesian predictive distributions under probability-matching priors and (c) residual risk estimation via parametric bootstrap. Risk measures satisfying standard properties are used, for example the popular TVaR. For more general distributions only (a) and (b) are investigated and a truncated version of TVaR is used. Numerical results obtained via Monte-Carlo simulation demonstrate that the proposed methods perform well. In the third paper (Chapter 4), we compare the effectiveness of four different approaches to estimate capital requirements in the presence of model uncertainty. For a given set of candidate models the model posterior weights can be obtained via a Bayesian approach. Then we consider approaches based on: (a) worst case scenario, (b) highest model posterior, (c) averaging the capital under each model according to the model posterior weights and (d) determining the predictive distribution of the financial loss and using it to calculate the capital. It is shown that all these methods are very sensitive to the set of candidate models specified. If this has been carefully selected (for instance via expert judgement) the approach based on the highest posterior performs slightly better than the others. Alternatively, if there is poor prior information on the model set the effectiveness of all these approaches decreases substantially. In particular, the worst case approach has a very low performance. It also emerges that mis-specifiying the model by using distributions that are more heavy-tailed than the one generating the data, may reduce the capital and thus it is not a conservative approach