Risk Theory in Insurance I

  • Credits: 5
  • Ending: Examination
  • Range: 2P + 2C
  • Semester: summer
  • Year: 1
  • Faculty of Economic Informatics

Teachers

Included in study programs

Actuarial Science

Teaching results

After completing the course Risk theory in Insurance I, it is assumed that students acquire knowledge and skills in the field of insurance risk management through their own assessment, which can be used in partial internal models of insurance companies. Thanks to the software support of the R language and the Monte Carlo simulation method, they will also be able to handle stochastic risk modeling, which they will use to create various studies for actuarial analyzes.
Knowledge
Students will gain knowledge:
1. on stochastic measurement and risk management through risk measures,
2. on statistical techniques for estimating probability distributions from available data sets,
3. on the analysis of light and heavy tails of distributions,
4. on modeling extreme values using the Excess over Threshold method,
5. on modeling two-dimensional distributions using a copula functions that can be applied in risk aggregation,
6. on the collective and on the individual risk model, within the created partial internal model they will have knowledge of the methods of determining the distribution of total damage and calculations related to risk management.
Competences
Based on the above knowledge, students are able to assess and measure risk in a partial internal model within the acquired competencies. Based on the evaluation of input parameters, they can choose a suitable solution approach to modeling the analyzed issues. Students will be competent in interpreting the results obtained from modeling in connection with the measurement and management of risks.
Skills
After completing the course, students can:
• estimate probability distributions from data files,
• implement various statistical approaches and methods,
• implement the Excess over Threshold method,
• implement risk aggregation using a copula functions,
• implement a Monte Carlo simulation method to generate values of random variables,
• determine risk measures using different solution approaches depending on the analyzed situation,
• perform various graphical interpretations and calculations,
• use computer technology and software support (R language, MS Excel),
• orientate in the given issue and apply appropriate procedures.

Indicative content

1. Introduction to risk theory. Risk measurement: a stochastic approach. Risk functions and risk measures (survival function, hazard rate function, mean excess loss function, value at risk, expected shortfall, or conditional value at risk).
2. Creation of probabilistic risk models (data visualization: histogram, boxplot, QQ plot, Cullen - Frey graph, kernel probability density estimation, parameter estimation, goodness-of-fit tests (Pearson's chi-square, Kolmogorov-Smirnov, Anderson-Darling and Cramer von Mises test), creating m-component distributions (splicing).
3. Basic knowledge from survival analysis (nonparametric estimation: for example Kaplan-Meier). Analysis of light and heavy tail of distributions.
4. Basic knowledge of the extrem value theory. Excess over Threshold (EOT) method. Determining the threshold value. Estimation of parameters of generalized Pareto distribution.
5. Estimation of value at risk, expected shortfall using the EOT method.
6. Copula functions. Sklar's theorem. Survival copula. Dependency measures. Tail dependence. Selected types of copula functions.
7. Estimation of copula function parameters and selection of a suitable copula function. Simulation of two-dimensional random variable values using the copula function.
8. Risk aggregation methods. Aggregation by addition. Risk aggregation using copula function. Determining the value at risk and expected shortfall of aggregated risks using a copula function.
9. Compound distribution of a random variable, collective risk model (CRM), methods for determining the distribution of total claims, simulation of its values using the Monte Carlo method.
10. Compound distributions: recurrent (Panjer's relation) and approximate approach (approximation by normal and shifted gamma distributions) to determine the distribution of total claims.
11. Determination of economic capital by value at risk and expected shortfall methodology in case of distribution of total claims in CRM.
12. Surplus model. Determination of risk measures value at risk, expected shortfall and economic capital in case of a distribution of the surplus.
13. Individual risk model (IMR). Determination of the distribution of total claims by the Monte Carlo method. Approximation of IMR by composing compound Poisson distributions

Support literature

1. Horáková, G., Páleš, M. & Slaninka, F.: Teória rizika v poistení. Wolters Kluwer. 2015.
2. Cipra, T. Riziko ve financích a pojišťovnictví: Basel III a Solvency II. Praha: Ekopress. 2015
3. Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern actuarial risk theory using R, Berlin:
Springer. 2008.
4. Klugman, S., A.,Panjer, H. H., & Willmot, G. E.: Loss Models (From Data to Decision).
New York: John Wiley  Sons. 2012.
5. Páleš, M.: Jazyk R pre aktuárov. Bratislava, Letra Edu. 2019.
6. Coles, S.: An Introduction to Statistical Modeling of Extreme Values.Springer. 2001
7. Moore, F., D.: Applied survival analysis using R. Springer. 2016.
8. Charpentier, A.: Computation actuarial science with R. Taylor & Francis Group. 2015.
9. Mucha, V., Páleš, P..: Teória pravdepodobnosti pre ekonómov. S podporou jazyka R.
Letra Edu. 2018.
10. Hofert, M., Kojadinovic, I., Mächler, M., & Yan,J.: Elements of copula modeling with R.
Springer. 2018.
11. Ruppert, D., Matteson S., D.: Statistics and Data Analysis for Financial Engineering with
R examples. Springer. 2015.
12. Markovich, N.: Nonparametric Analysis of Univariate Heavy-Tail Data. John Wiley  Sons.
2007.
13. Mucha, V. Applying Simulations in the Individual Risk Model Using R. In Managing and Modelling of Financial Risks. Proceedings of 9th International Scientific Conference : VŠB - Technical University of Ostrava, 2018.
14. Mucha, V., Páleš, M., Sakálová, K. Calculation of the capital requirement using the Monte Carlo simulation for non-life. In Ekonomický časopis. Bratislava : Ekonomický ústav SAV : Prognostický ústav SAV, 2016, roč. 64, č. 9.

Syllabus

1. Introduction to risk theory. Risk measurement: a stochastic approach. Risk functions and risk measures (survival function, hazard rate function, mean excess loss function, value at risk, expected shortfall, or conditional value at risk). 2. Creation of probabilistic risk models (data visualization: histogram, boxplot, QQ plot, Cullen - Frey graph, kernel probability density estimation, parameter estimation, goodness-of-fit tests (Pearson's chi-square, Kolmogorov-Smirnov, Anderson-Darling and Cramer von Mises test), creating m-component distributions (splicing). 3. Basic knowledge from survival analysis (nonparametric estimation: for example Kaplan-Meier). Analysis of light and heavy tail of distributions. 4. Basic knowledge of the extrem value theory. Excess over Threshold (EOT) method. Determining the threshold value. Estimation of parameters of generalized Pareto distribution. 5. Estimation of value at risk, expected shortfall using the EOT method. 6. Copula functions. Sklar's theorem. Survival copula. Dependency measures. Tail dependence. Selected types of copula functions. 7. Estimation of copula function parameters and selection of a suitable copula function. Simulation of two-dimensional random variable values using the copula function. 8. Risk aggregation methods. Aggregation by addition. Risk aggregation using copula function. Determining the value at risk and expected shortfall of aggregated risks using a copula function. 9. Compound distribution of a random variable, collective risk model (CRM), methods for determining the distribution of total claims, simulation of its values using the Monte Carlo method. 10. Compound distributions: recurrent (Panjer's relation) and approximate approach (approximation by normal and shifted gamma distributions) to determine the distribution of total claims. 11. Determination of economic capital by value at risk and expected shortfall methodology in case of distribution of total claims in CRM. 12. Surplus model. Determination of risk measures value at risk, expected shortfall and economic capital in case of a distribution of the surplus. 13. Individual risk model (IMR). Determination of the distribution of total claims by the Monte Carlo method. Approximation of IMR by composing compound Poisson distributions

Requirements to complete the course

30% 2 written works (using software support),
70% written exam (using software support)

Student workload

Total study load (in hours): 130 hours
26 hours - participation in lectures,
26 hours - participation in exercises,
13 hours - preparation for exercises, homeworks,
26 hours - preparation for written works,
52 hours - self-study in preparation for the exam.

Language whose command is required to complete the course

slovak

Date of approval: 10.02.2023

Date of the latest change: 15.05.2022