Ph.D. Public Lecture (DSAS) - Ang Li
"Compound Sums, Their Distributions, and Actuarial Pricing"
Compound risk models are widely used in insurance companies to mathematically describe their aggregate amount of losses during certain time period. However, evaluation of the distribution of compound random variables and the computation of the relevant risk measures are non-trivial. Therefore, the main purpose of this thesis is to study the bounds and simulation methods for both univariate and multivariate compound distributions. The premium setting principles related to dependent multivariate compound distributions are studied.
In the first part of this thesis, we consider the upper and lower bounds of the tail of bivariate compound distributions. Our results extend those in the literature (eg. Willmot and Lin (1994) and Willmot et al. (2001)) for univariate compound distributions. First, we derive the exponential upper bounds when the claim size distribution is light-tailed with finite moment generating function. Second, we present generalized upper and lower bounds when the claim size distribution is heavy-tailed without a finite moment generating function. Numerical examples are provided to illustrate the tightness of these bounds.
In the second part of the thesis, we develop several novel variance reduction techniques for simulating tail probability and mean excess loss of the univariate and bivariate compound models. These techniques stem from possible combinations of existing commonly used variance reduction techniques. Their performs are evaluated in details.
In the third part of the thesis, we investigate the premium setting principles when the claim frequencies and claim severities in multiple collective risk models are correlated via a background risk. We develop a novel methodology of premium setting and numerically illustrate how model parameters influence the premiums level. Two empirical methods and a parametric fitting method are provided for pricing and corresponding performance assessments are presented.
