 #### Thesis Format

Integrated Article

#### Degree

Doctor of Philosophy

#### Program

Statistics and Actuarial Sciences

Ren, Jiandong

#### 2nd Supervisor

Zitikis, Ricardas

Joint Supervisor

#### Abstract

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.

#### Summary for Lay Audience

Compound risk models are widely used in insurance companies to mathematically describe their aggregate amount of losses during certain time period. The tail probability and tail moments of the aggregate losses are important risk measures of the insurer's operation. Therefore, accurate evaluation of them are essential in premium setting and risk management of insurance companies. However, the calculation of the tail probabilities and the tail moments of the compound random variables are non-trivial because compound variables usually do not have explicit probability distribution function - even for the one-dimensional case.

Literature on evaluation of the tail probability and tail moments for univariate compound risk models are extensive. However results for multivariate compound risk models with dependence are much less. Therefore, the main purpose of this thesis is to develop methodology to study the tail probability and tail moments of multivariate compound risk models.

In particular, in the first part of the thesis, we derive theoretical upper bounds for the tail probability of bivariate compound distributions. The results provide an analytical way to characterize tail behavior of the insurance companies' aggregate losses in two lines of businesses. Our result generalize those for univariate models in the literature.

In the second part of the thesis, we develop several novel techniques to efficiently simulate tail probability and mean excess loss of the univariate and bivariate compound models. The methodologies are developed particularly for compound variables. We show that the our proposed simulation method is much more efficient than the simple crude simulation methods; they are essentially based on combining existing variance-reduction methods.

In the third part of the thesis, we study a multivariate compound risk model where claim frequencies and claim severities are correlated via a background risk. We introduce a new premium setting methodology and provide both non-parametric and parametric methods for parameter estimation and apply them in premium setting.

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