# Otázky k souborné zkoušce z VS 4IM Insurance Mathematics

## Questions from the exam 4IMA

Students during the exam answer 3 questions (from 1-7, 8-14, 15-21).

1. Random vector; joint, conditional and marginal distributions; characteristics (covariance matrix, correlation matrix). Linear transformations and quadratic forms of random vectors.
2. Random samples from vector distributions; sample characteristics; linear transformations.
3. Stochastic convergence; Central limit theorems; Laws of large numbers.
4. Multivariate normal distribution. Statistical inference about parameters.
5. Estimation of vector parameters. Point estimates and their properties; maximum likelihood estimates; asymptotic normality of maximum likelihood estimates.
6. Ordered sample. Characteristics of location and variability based on the ordered sample. Empirical distribution function, sample quantile function, kernel estimates of density.
7. Kolmogorov-Smirnov tests (one sample, two samples). Chi-squared goodness-of-fit test.
8. Triangle schemes and reserving methods based on them. Principle of the chain-ladder method, frequency-severity approach, Bornhuetter Ferguson, formulation via generalized linear model, Mack’s model, over-dispersed Poisson model. Prediction error, process error, estimation error.
9. Random variables and distribution models used in non-life insurance. (Frequency, severity, aggregate loss).
10. Basics of generalized linear model – definition and comparison to classical linear model, model quality assessment (statistical tests and quality measures).
11. Collective and individual model. Assumptions and properties.
12. Ruin theory – compound Poisson process, surplus process, adjustment equation, Lundberg’s upper bound.
13. Credibility theory. Comparison of classical and Bayesian estimates, credibility formula, properties of credibility coefficient, examples of models.
14. Bonus-malus system, no claim discount system, Markovian chain, Makovian property, Transition probabilities and matrix, absolute probabilities and application.
15. Creation of life tables, selection factor. Life table applications in life insurance.
16. Traditional life products. Netto-premium calculation principles – using a) px, qx, b) lx, dx c) commutation figures, d) actuarial figures. Single/regular premium, premium paid other than in whole policy period, insurance with returning the premium in case of death, insurance with non-flat sum insured, … Brutto premium, traditional approach to include expenses in the premium. Interpretation of standard expense coefficients.
17. Traditional life products. Traditional technical provision calculation – netto, brutto. Risk and saving part of the premium, sum at risk. Zillmerization – its calculation and interpretation. Calculation of changes in traditional life products – sum insured/premium reduction, surrender value, increase/decrease of premium. Profit share – basic principles and intention – sources, typical structure.
18. Flexible life products. Basic principles, differences from traditional products, pros/cons for the insurance company/clients. Calculation of technical provisions, premium. Extra premium, partial withdrawals. Investment guarantees. Typical investment funds.
19. Real cash flow models – 1st and 2nd order assumptions. Basic differences, examples. Models structure, interpretation and way of calculation of individual parts of the cash flows.
20. Cash flow models applications. Profit and loss models applications. Both for existing as well as new life policies. Tax, cost of capital approach.
21. Solvency – basic idea, own funds, required and minimal capital requirement, role of the regulator. Solvency I and II – basic principles, differences (balance sheet, risk-based vs. factor approach, legislation, …)

Bibliography:

BOLAND, P J. (2007). Statistical and probabilistic methods in actuarial science. Boca Raton Chapman & Hall/CRC.

DICKSON, David CM, et al. Actuarial mathematics for life contingent risks. Cambridge University Press, 2013..

ROSS, S M. (2014).  Introduction to probability models.   Amsterdam Elsevier.

WASSERMAN, L. (2004). All of statistics : a concise source in statistical inference. New York Springer.