Thesis Format
Integrated Article
Degree
Doctor of Philosophy
Program
Statistics and Actuarial Sciences
Supervisor
Ren, Jiandong
Abstract
This thesis focuses on two areas in actuarial science: compound risk models with dependence, and parameter uncertainty.
In the first part of the thesis, we study a hierarchical risk model where an accident can cause a combination of different types of claims, whose sizes could be dependent. In addition, the frequencies of accidents that cause the different combinations of claims are dependent. We first derive formulas for computing risk measures, such as the tail conditional expectation and tail variance of the aggregate losses for a portfolio of businesses. Then, we present formulas for performing the associated capital allocation to different types of claims in the portfolio. The main tool we used is the moment (or size-biased) transform of the multivariate distributions.
In the second part of the thesis, we consider an extension of the classical compound Poisson risk model with a dependence structure between the inter-claim time and the subsequent claim size. We derive formulas for asymptotic tail moments of the aggregate claims when the claim amounts are heavy tail distributed. Our results extended those in the literature. Again, the primary technique we utilized is the moment transform.
In the third part of the thesis, we investigate actuarial credibility theory when the information about the loss model or the prior distribution of its parameters is imprecise or vague. Several approaches, such as robust Bayesian method and imprecise probability, have been proposed in the literature to study such problems. We propose to represent the imprecise/partial/vague information about model parameters as fuzzy numbers and derive formulas for “fuzzy credibility premiums”. The results extend those exist in the literature.
Summary for Lay Audience
This thesis contributes to two areas of research in actuarial science: dependent compound risk models and parameter uncertainty.
Insurance companies typically write policies in multiple lines of business, with each line potentially resulting in different types of claims. To establish both fair and adequate aggregate premiums and ensure that premiums for individual risk coverage align with their contribution to the overall risk in the portfolio, it is essential to accurately evaluate the aggregate risk and allocate the total capital requirement across different risk types within the business portfolio. In Chapter 2 of this thesis, we first derive formulas for evaluating risk measures, such as tail conditional expectation and tail variance of the aggregate losses in the hierarchical multivariate risk model. Then, we provide explicit expressions and computation methods for allocating the required capital to the different types of risks.
One key assumption of a classical compound Poisson risk model is that the inter-claim times and the claim amounts are independent. This assumption can be rather restrictive in application. Therefore, in Chapter 3 of the thesis, we study an extension of the classical compound Poisson risk model by considering the dependence between the claim occurrence time and the subsequent claim size. We derive formulas for evaluating tail moments of the aggregate losses.
In Chapter 4 of this thesis, we study actuarial credibility theory, which is a quantitative method that insurance companies use to estimate future losses of a policyholder based on the loss experiences of the policyholder as well as the average loss experiences of all policyholders in the rating class. In particular, we study an actuarial credibility problem when the information about the prior distribution of the risk parameter is imprecise or vague, so that the actuary cannot specify the exact prior distribution or the moments. However, instead of considering the imprecise information about the prior distribution from robust Bayesian analysis/imprecise probability point of view, we propose to apply fuzzy set theory.
Recommended Citation
Gao, Dechen, "Studies of compound risk models with dependence and parameter uncertainty" (2024). Electronic Thesis and Dissertation Repository. 10087.
https://ir.lib.uwo.ca/etd/10087
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