Risk Theory in Insurance

elective courses

The lecture is devoted to short term pricing of insurance risk. We will discuss basic premium calculation issues, individual and collective risk models, risk sharing, and ruin probability.

Detailed syllabus:

1. Basic premium calculation issues. Portfolio of risks and the total amount of claims. The top-down approach under simplified assumptions: mutual independency of risks and normally distributed total amount of claims over a year. Overview of issues: dependent risks, non-normal distribution, long-run decision-making horizon. Fitting probability distributions to statistical data.

2. Individual risk model. Convolutions of random variables with discrete-continuous distributions. Convolutions of arithmetic distributions. Raw and central moments, skewness and kurtosis. Moment generating function, cumulant generating function. Size of the portfolio and its characteristics.

3. Collective risk model: basic distributions of the number of claims. Poisson distribution and its basic properties. Negative binomial distribution - as a result of heterogeneity in the population of risks, and as a result of (possibly) more than one claim per accident. Empirical data analysis.

4. Collective risk model: compound distributions of the aggregate amount of claims. Compound Poisson, compound binomial and compound negative binomial distributions. Moments of the compound distribution. Panjer's formula for the distribution of the aggregate amount of claims. Discretisation of the continuous distribution. Examples of more complex distributions.

5. Risk sharing. Typical methods of splitting risks. Utility theory and optimal risk sharing. Excess of loss over a constant as a random variable. Inflation effect under non-proportional risk-sharing schemes.

6. Approximations of the distribution of aggregate amount of claims. Normal and Shifted-Gamma approximations. Normal Power approximation. Compound Poisson distribution: controlling accuracy of the approximation by limiting individual loss coverage. Decomposition of the portfolio premium into individual-risk premiums.

7. Dependent risks models. Examples of simple dependencies. Distribution of the total amount of claims when risks are conditionally independent, but risk parameters of the whole portfolio change randomly in time. Premium formulae based on the model with random claim frequency and random scale parameter of the severity distribution.

8. Short overview of ruin theory. Stochastic process of insurer's surplus. Ruin probability and the adjustment coefficient R. Discrete-time model. Classical model: Poisson claim arrival process. The simplest case: exponential severity distribution. Bounds for the probability of ruin in the discrete-time case. Cramer-Lundberg asymptotic formula.

9. Ruin probability - approximations. Typical approximation methods. Pollatschek-Khinchin formula and application of the Panjer's recursion algorithm in assessing ruin probability. Controlling ruin probability by limiting individual loss coverage.

10. Premium calculation revisited. Value at Risk. Short-term horizon and the Risk Based Capital. Short-term horizon and the optimal level of premium, capital and reinsurance. Ruin probability and the optimal level of premium, capital and reinsurance.

English-language literature will be offered on demand.

Learning outcomes

Student knows:

1) basic issues of modeling risk in insurance, in particular how to calculate premiums both on the level of the whole portfolio of insurance contracts and of an individual contract,

2) basic probability models for these two versions of the basic issue, in particular probability distributions with support on non-negative real numbers, distrubutions that are partly discrete and partly continuous, operation of convolution, mixing, and compounding of distributions,

3) detailed properties of the Poisson process and the Poisson distribution, as well as most other discrete distributions used as alternative to Poisson.

4) how to approximate distributions from their moments, when only few first moments are known.

5) the discretization of continuous distributions,

6) the techniques of mitigating the right tail of the distribution, which are strictly related to the problem of stochastic orders, being presented as well.

Student can:

1) translate practical problems into (formal) probability models. He can analyze practical problems first, and then translate them into the language of probability calculus.

2) cooperate with the practitioners - his advantage are acquired technical skills that can be used to find answers to important practical questions.

3) Assume responsibility in finding common language with the practitioners.

4) translate problems from the language of practice to the language of mathematics and vice-versa.

Grades are based on a written exam.