Non-parametric inference of risk measures

Responding to the changes in the insurance environment of the past decade, insurance regulators globally have been revamping the valuation and capital regulations. This thesis is concerned with the design and analysis of statistical inference procedures that are used to implement these new and upcoming insurance regulations, and their analysis in a more general setting toward lending further insights into their performance in practical situations. The quantitative measure of risk that is used in these new and upcoming regulations is the risk measure known as the Tail Value-at-Risk (T-VaR). In implementing these regulations, insurance companies often have to estimate the T-VaR of product portfolios from the output of a simulation of its cash flows. The distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. A technique that has proved promising in such settings is that of importance sampling. While the asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling has been conjectured, the literature has lacked an honest result. The main goal of the first part of the thesis is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Our method also establishes such a result for the natural non-parametric estimator for the Value-at-Risk, another popular risk measure, under weaker assumptions than those used in the literature. We also report on a simulation study conducted to examine the quality of these asymptotic approximations in small samples.

The Haezendonck-Goovaerts class of risk measures corresponds to a premium principle that is a multiplicative analog of the zero utility principle, and is thus of significant academic interest. From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is, in a sense, a minimal member of the class. Hence, a study of the natural non-parametric estimator for these risk measures will lend further insights into the statistical inference for the T-VaR. Analysis of the asymptotic behavior of the generalized estimator has proved elusive, largely due to the fact that, unlike the T-VaR, it lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator. In order to conduct a simulation study, we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm has clearly been noticed in the literature, and has impeded the quality of simulation results. In this part we also design and analyze an algorithm for computing these risk measures. In the process of doing we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. We also have implemented our algorithm on the R software environment, and included its source code in the Appendix.