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).
- Random vector; joint, conditional and marginal distributions; characteristics (covariance matrix, correlation matrix). Linear transformations and quadratic forms of random vectors.
- Random samples from vector distributions; sample characteristics; linear transformations.
- Stochastic convergence; Central limit theorems; Laws of large numbers.
- Multivariate normal distribution. Statistical inference about parameters.
- Estimation of vector parameters. Point estimates and their properties; maximum likelihood estimates; asymptotic normality of maximum likelihood estimates.
- Ordered sample. Characteristics of location and variability based on the ordered sample. Empirical distribution function, sample quantile function, kernel estimates of density.
- Kolmogorov-Smirnov tests (one sample, two samples). Chi-squared goodness-of-fit test.
- 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.
- Random variables and distribution models used in non-life insurance. (Frequency, severity, aggregate loss).
- Basics of generalized linear model – definition and comparison to classical linear model, model quality assessment (statistical tests and quality measures).
- Collective and individual model. Assumptions and properties.
- Ruin theory – compound Poisson process, surplus process, adjustment equation, Lundberg’s upper bound.
- Credibility theory. Comparison of classical and Bayesian estimates, credibility formula, properties of credibility coefficient, examples of models.
- Bonus-malus system, no claim discount system, Markovian chain, Makovian property, Transition probabilities and matrix, absolute probabilities and application.
- Creation of life tables, selection factor. Life table applications in life insurance.
- 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.
- 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.
- 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.
- 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.
- Cash flow models applications. Profit and loss models applications. Both for existing as well as new life policies. Tax, cost of capital approach.
- 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, …)
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.