Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Statistics and Actuarial Sciences

Supervisor

Sendova, Kristina P.

2nd Supervisor

Jones, Bruce L.

Co-Supervisor

Abstract

Ruin theory studies the riskiness of an insurance portfolio by investigating the evolution of an insurer's surplus. The existing models often assume stationary increments of the surplus process, which is not always appropriate to describe an actual experience. In this thesis, we consider some modifications that are inspired by a real-life insurance data set to the existing risk models and investigate how these modifications affect ruin theory results.

We first explore potential surplus modelling improvements by investigating how well the available models describe an insurance risk process. To this end, we obtain and analyze a real-life data set that is provided by an anonymous insurer. Based on our analysis, we discover that both the purchasing process and the corresponding claim process have seasonal fluctuations. Some special events, such as public holidays, also have impact on these processes. In the existing literature, the seasonality is often stressed in the claim process, while the cash inflow usually assumes simple forms. We further suggest a possible way of modelling the dependence between these two processes. A preliminary analysis of the impact of these patterns on the surplus process is also conducted. As a result, we propose a surplus process model which utilizes a non-homogeneous Poisson process for premium counts and a Cox process for claim counts that reflect the specific features of the data.

Next, we study a risk model with stochastic premium income. It is assumed that both the premium arrival process and the claim arrival process are modelled by homogeneous Poisson processes, and that the premium amounts are modelled by independent and identically distributed random variables. After reviewing various known results of this model, a simulation approach for obtaining the probability of ultimate ruin based on importance sampling is derived. We demonstrate this approach by examples where the distribution of the sampling random variable can be identified. We then give other examples where we use fast Fourier transform to obtain an approximation of the sampling random variable. The simulated results are compared with known results in the existing literature.

In the last part of the thesis, we consider a risk model where both the premium income and the claim process have seasonal fluctuations. We obtain the probability of ruin based on the simulation approach presented in Morales (2004). We also discuss the conditions that must be satisfied for this approach to work. We give both a numerical example that is based on a simulation study and an example using a real-life auto insurance data set. Various properties of this risk model are also discussed and compared with the existing literature.

Summary for Lay Audience

Insurance companies operate by charging their customers premiums, investing these premiums in the financial market, setting aside a sufficient amount of funds, and paying out claims if the customers suffer certain losses. An insurance company must hold sufficient reserve so that it can settle claims at any given time. If the insurance company is not able to pay a claim, we say the insurance company is ruined. Ruin theory is a branch of actuarial science that studies an insurer's insolvency risk. A key component of such studies is to build a model to describe how the available funds to the insurer, or the surplus, changes over time. In the existing literature, many models have been built and various conclusions have been obtained.

In this thesis, we investigate how well the existing models fit the actual practice. To this end, we obtained and analyzed a real-life insurance data set. We discovered that both the premium income and the claim payment have seasonal fluctuations. Some special events, such as major public holidays, also have impact on the cash flow of an insurer. We demonstrate how these features affect the insurance company's insolvency risk via a simulation example. We then study two different models, each featuring one of the characteristics of the data set. Specifically, we study a model under which the premium income is stochastic, and a model under which both the premium income and the claim payment have seasonal fluctuations. We show how to obtain the probability of ruin under these models via simulation. The behaviours of these models are demonstrated by numerical examples.

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