On-line services of the University of Warsaw | USOSownia - uniwersyteckie forum USOSoweYou are not logged in | log in
course directory - help

Risk Theory in Insurance

General data

Course ID: 1000-135TRU Erasmus code / ISCED: 11.503 / (0542) Statistics
Course title: Risk Theory in Insurance Name in Polish: Teoria ryzyka w ubezpieczeniach
Department: Faculty of Mathematics, Informatics, and Mechanics
Course groups: (in Polish) Przedmioty fakultatywne na matematyce
Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores): 6.00
view allocation of credits
Language: English
Type of course:

elective courses

Short description:

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.

Full description:

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.

Bibliography:

English-language literature will be offered on demand.

Classes in period "Winter semester 2018/19" (past)

Time span: 2018-10-01 - 2019-01-25
Choosen plan division:


magnify
see course schedule
Type of class: Class, 30 hours more information
Lecture, 30 hours more information
Coordinators: Wojciech Otto
Group instructors: Jacek Micał, Wojciech Otto
Students list: (inaccessible to you)
Examination: Course - Examination
Lecture - Examination

Classes in period "Winter semester 2019/20" (future)

Time span: 2019-10-01 - 2020-01-27
Choosen plan division:


magnify
see course schedule
Type of class: Class, 30 hours more information
Lecture, 30 hours more information
Coordinators: Wojciech Otto
Group instructors: Jacek Micał, Wojciech Otto
Students list: (inaccessible to you)
Examination: Course - Examination
Lecture - Examination
Course descriptions are protected by copyright.
Copyright by University of Warsaw.