Actuarial Methods in Asset Insurance 2400-M1IiEMAM

1. Premium calulation (top-down approach) (4 h.)
The premium for the whole portfolio of risks under a quantile formula and normality assumption. Decomposition of the whole premium for premiums for individual risks on the basis of their marginal contribution into the risk of the whole portfolio. Why sum of marginal premiums does not cover the whole premium required, and what we can do with this problem. (Otto, book, chap. 1)

2. Individual risk model (6 h.)
Partly continuous and partly discrete distributions – integration over increments of the cumulative distribution function. Expected value and higher order moments. Moment generating and cumulant generating functions. Convolution – exact calculations and numerical approximations (Otto, book, chap. 2)

3. Collective risk model –frequency distributions (6 h.)
Binomial and Poisson distributions, equivalence between independent exponential waiting times and Poisson arrival process. Theorems on families of distributions closed under the convolution operation. Negative binomial distribution as a mixed-Poisson, as a compound Poisson distribution, and as a distribution of number of successes that happened before r-th failure. Why existence of heterogeneity of population of risk is of so much practical importance. [(Otto, book, chap. 3)

4. Collective risk model –compound distributions (4 h.)
Compound Poisson, compound binomial, and compound negative binomial distributions. Severity distributions. Closeness of families of compound distributions under convolution. Special application of a compound distribution, when a random number of claims is randomly splitted into two kinds of claims. Panjer’s recursive formula. Discretizations of continuous distributions. (Otto, book, chap. 4)

5. Non-standard distributions. (2 h.)
Empirical approach to modelling of the number of claims – distributions with Poisson tail. Other examples of departures form simplest dostributions. The case when insureds are motivated to refrain of reporting selected losses. (Otto, book, chap. 5)

6. Risk sharing issues. (4 h.)
Pfroportional and non-proportional sharing rules. Utility theory and optimal risk sharing. Properties of excess of loss over a deductible as a random variable. Inflation and non-proportional risk sharing (Otto, book, chap. 6)

7. Approximations of distribution of aggregate loss amount. (2 h.)
Approximation by normal distribution and by shifted Gamma distribution. Other approximations. Premium formula for the portfolio and for individual risk. Controlling risk of the portfolio by limiting responsibility for individual loss. (Otto, book, chap. 7)

8. Reserves and the Chain-Ladder method. (2 h.)
Typology of non-life reserves. More about oustanding claims reserve. The Chain-Ladder Method. (Otto, skrypt: lecture 1)