Risk Theory in Insurance
General data
Course ID:  1000135TRU  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: 
Elective courses for 2nd stage studies in Mathematics Elective courses for 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 topdown approach under simplified assumptions: mutual independency of risks and normally distributed total amount of claims over a year. Overview of issues: dependent risks, nonnormal distribution, longrun decisionmaking horizon. Fitting probability distributions to statistical data. 2. Individual risk model. Convolutions of random variables with discretecontinuous 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 nonproportional risksharing schemes. 6. Approximations of the distribution of aggregate amount of claims. Normal and ShiftedGamma approximations. Normal Power approximation. Compound Poisson distribution: controlling accuracy of the approximation by limiting individual loss coverage. Decomposition of the portfolio premium into individualrisk 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. Discretetime model. Classical model: Poisson claim arrival process. The simplest case: exponential severity distribution. Bounds for the probability of ruin in the discretetime case. CramerLundberg asymptotic formula. 9. Ruin probability  approximations. Typical approximation methods. PollatschekKhinchin 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. Shortterm horizon and the Risk Based Capital. Shortterm horizon and the optimal level of premium, capital and reinsurance. Ruin probability and the optimal level of premium, capital and reinsurance. 

Bibliography: 
Englishlanguage literature will be offered on demand. 
Classes in period "Winter semester 2018/19" (past)
Time span:  20181001  20190125 
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" (past)
Time span:  20191001  20200127 
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 
Copyright by University of Warsaw.