Course: Mathematics for Insurance

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Course title Mathematics for Insurance
Course code KMA/PM
Organizational form of instruction Lecture
Level of course Master
Year of study not specified
Semester Winter
Number of ECTS credits 3
Language of instruction Czech
Status of course Compulsory
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • Friesl Michal, Mgr. Ph.D.
  • Tomiczková Světlana, RNDr. Ph.D.
Course content
FOR WINTER TERM OF SCHOOL YEAR 2017/2018 Introduction. Some statistical indicators in insurance. Individual and collective model. Distribution of the amount of insurance claims. Exponential, gamma, log normal, Weibull, and Pareto distributions, skewness, moment-generating function, conditional distribution. Estimates of parameters. Maximum likelihood method, moment and quantile methods, chi-square goodness-of-fit test. Example. Distribution of the number of insurance claims. Poisson, negative binomial, and mixed Poisson distributions. Generating functions of probability distribution, example. Distribution of the total amount of claims. Compound distribution and its characteristics. Compound Poisson distribution, sums of compound Poisson distributions. Compound negative-binomial distribution, the interpretation as a compound Poisson. Approximation of individual model by collective model. Deductibles and reinsurance. Proportional and fixed amount deductibles, distribution of the number and amount of claims paid by insurer. Proportional, XL, and SL reinsurance. Distribution of claims paid by cedant, estimates of parameters. Calculation and approximation of compound distributions. Panjer recurrent formula, moments. Approximations by shifted distribution, Edgeworth, normal-power, Gram-Charlier approximation. Premium principles. Premiums from long-term perspective, the safety margin. Expected value, standard deviation, variance, quantile, zero-utility, exponential priciples and their properties. Credibility theory. Homogeneous and inhomogeneous collective of risks, collective and individual premiums. American credibility theory, full and partial credibility. Bayesian credibility theory, Bayesian and linear credibility premiums. Bühlmann and Bühlmann-Straub model. Bonus-malus systems. Bonus classes, Markov chain, limit distribution. Reserves. Reserve for claims and its estimate. Run-off triangles, Chain-Ladder and separation methods. Ruin probability. Insurance claims as a random process, Cramér-Lundberg classical model, differential equations for the probability of ruin in finite and infinite horizon. Estimation of ruin probability. Lundberg adjustment coefficient and Lundberg inequality, Cramer-Lundberg approximation, approximation of adjustment coefficient. Influence of reinsurance to adjustment coefficient, proportional reinsurance. Additional information on the web page

Learning activities and teaching methods
Lecture with practical applications, Students' self-study, Self-study of literature, Textual studies, Lecture
  • Contact hours - 26 hours per semester
  • Preparation for an examination (30-60) - 40 hours per semester
  • Preparation for formative assessments (2-20) - 13 hours per semester
professional knowledge
The course assumes knowledge of probability and statistics at least at the introductory course KMA/PSA level (other broader knowledge, or practice with routine use of the apparatus would be an advantage) and uses methods from the introductory courses of mathematical analysis.
learning outcomes
To orientate onself in applications of probability, statistics and random processes in treated areas of non-life insurance, to be able to derive the results presented. In the case of knowledge of the methods of probability and statistics (their detailed exposition is not part of this course) also to apply treated approaches under different conditions.
teaching methods
Textual studies
Students' self-study
Self-study of literature
Lecture with practical applications
assessment methods
Oral exam
Written exam
Recommended literature
  • Bühlmann, Hans. Mathematical methods in risk theory. Berlin : Springer-Verlag, 1996. ISBN 3-540-61703-5.
  • Cipra, Tomáš. Pojistná matematika. 1. vydání. Praha : Ekopress, 1999. ISBN 80-86119-17-3.
  • Mandl, Petr; Mazurová, Lucie. Matematické základy neživotního pojištění. Vyd. 1. Praha : Matfyzpress, 1999. ISBN 80-85863-42-1.
  • Sundt, Bjorn. An introduction to non-life insurance mathematics. 4th ed. Karlsruhe : VVW, 1999. ISBN 3-88487-801-8.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Applied Sciences Financial Informatics and Statistics (2011) Economy 2 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2014) Economy 2 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2014) Economy 2 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2014) Pedagogy, teacher training and social care 2 Winter