Risk Theory
RISK THEORY - LECTURE NOTES
1. INTRODUCTION
The primary subject of Risk Theory is the development and study of mathematical and statistical models to describe and predict the behaviour of insurance portfolios, which are simply financial instruments composed of a (possibly quite large) number of individual policies. For the purposes of this course, we will define a policy as a random (or stochastic) process generating a deterministic income in the form of periodic premiums, and incurring financial losses in the form of claims, which are of a random amount and occur at random times. We will deal with these processes both in the short-term, i.e., within a given fixed period of time, as well as over the long-term. In addition, we will deal only with insurance policies for which the premiums are set to fully cover the claims made during the term of the premium payment, and for which the premiums may be changed at the time of renewal of the policy. We note that there are types of insurance which do not fit this definition, such as whole-life insurance (for which level premiums are paid throughout the duration of the policy, meaning that the initial premiums are higher than necessary to recover short-term costs, the excess being stored to offset the increase in expected claims over later periods which the premiums paid during this time would be insufficient to cover on their own). However, there are many important insurance examples which do satisfy the preceding definition. The most common of such policies, and the ones on which we shall primarily focus, are liability and property insurance. Generally, then, we will want to investigate various aspects of the behaviour of our chosen class of stochastic processes and use these results to describe and predict the associated behaviour of various types of insurance policies and portfolios. For example, we will be interested in predicting the total amount of all the claims made within a given timeframe.
1. INTRODUCTION
The primary subject of Risk Theory is the development and study of mathematical and statistical models to describe and predict the behaviour of insurance portfolios, which are simply financial instruments composed of a (possibly quite large) number of individual policies. For the purposes of this course, we will define a policy as a random (or stochastic) process generating a deterministic income in the form of periodic premiums, and incurring financial losses in the form of claims, which are of a random amount and occur at random times. We will deal with these processes both in the short-term, i.e., within a given fixed period of time, as well as over the long-term. In addition, we will deal only with insurance policies for which the premiums are set to fully cover the claims made during the term of the premium payment, and for which the premiums may be changed at the time of renewal of the policy. We note that there are types of insurance which do not fit this definition, such as whole-life insurance (for which level premiums are paid throughout the duration of the policy, meaning that the initial premiums are higher than necessary to recover short-term costs, the excess being stored to offset the increase in expected claims over later periods which the premiums paid during this time would be insufficient to cover on their own). However, there are many important insurance examples which do satisfy the preceding definition. The most common of such policies, and the ones on which we shall primarily focus, are liability and property insurance. Generally, then, we will want to investigate various aspects of the behaviour of our chosen class of stochastic processes and use these results to describe and predict the associated behaviour of various types of insurance policies and portfolios. For example, we will be interested in predicting the total amount of all the claims made within a given timeframe.