# Risk Theory in Insurance II

- Credits: 6
- Ending: Examination
- Range: 2P + 2C
- Semester: winter
- Year: 2
- Faculty of Economic Informatics

## Teachers

## Included in study programs

**Teaching results**

After completing the course Risk theory in Insurance II, it is assumed that students will acquire knowledge and skills in the field of insurance risk management through their transfer in the application of forms of insurance and types of reinsurance. Thanks to the software support of the R language and the Monte Carlo simulation method, they will also be able to handle stochastic modeling of the total insurance and reinsurance benefit and the measurement of the given risk reduction effect. Furthermore, they will gain knowledge and skills in estimating the probability of ruin using modeling of a compound Poisson process in a collective risk model for longer time periods.

Knowledge

Students will gain knowledge:

1. on forms of insurance and types of reinsurance in the context of their use in risk management,

2. on the modeling of the total claim amount paid by the insurer and reinsurer within the collective

risk model based on the Monte Carlo method,

3. the reinsurer's limit and the choice of appropriate optimal reinsurance protection,

4. on stochastic processes (Poisson and Wiener process) within the estimation of the probability

of ruin in a collective risk model for longer time periods.

Competences

On the basis of the above knowledge, students are able to decide, within the acquired competencies, on the selection of a suitable implementation of risk transfer to the insured or reinsurance company and evaluate it through risk measures. They will be able to comment on the setting of input parameters in the modeled studies so that this step is reflected in the required outputs. Students will be competent in the selection of a suitable solution approach, in the interpretation of the achieved results and in the evaluation of model causality..

Skills

After completing the course, students can:

• implement stochastic modeling using Monte Carlo simulations,

• also perform various graphical interpretations and calculations,

• select a suitable reinsurance protection,

• use computer technology and software support (R language, MS Excel, mathematical software),

• orientate in the ruin theory and apply appropriate procedures,

• measure risk reduction after the application of insurance and reinsurance.

**Indicative content**

1. Forms of insurance with supplementary forms of insurance (Pure indemnity insurance, First loss insurance, Insurance with an average clause, Quota insurance, With and without an excess).

2. Risk management by applying various forms of insurance in a collective risk model.

3. Modeling of the total claim amount paid by the insurer by Monte Carlo simulations and

measurement of risk reduction using risk measures.

4. Reinsurance. Types of reinsurance. Proportional reinsurance (quota, excess reinsurance with respect to the sum insured (surplus)). Non-proportional reinsurance (excess reinsurance with respect to the amount of claims (WXL / R, Excess of Loss), WXL /E or CatXL (Per-Event Excess of Loss or Catastrophe Excess of Loss).

5. Risk management by applying various reinsurance protections in a collective risk model.

6. Application of the reinsurer limit, multiple reinsurer limit in individual types of reinsurance.

7. Modeling of the total claim amount paid by the reinsurer by Monte Carlo simulations and measurement of risk reduction using risk measures.

8. Optimization in reinsurance (minimization of Value at Risk or Conditional Value at Risk, maximization of profit with constant variance, minimization of variance with constant profit, minimization of probability of ruin with constant profit).

9. Use of the extrem value theory(the Excess over Threshold method) in Non-proportional reinsurance.

10. Collective model for longer time periods. Stochastic process: Poisson process (loading process of the number of claims), compound Poisson process (surplus process) and their modeling.

11. Probability of ruin in the distant horizon: determination of the probability of ruin by Lundberg inequality and using Monte Carlo simulations.

12. Probability of ruin in finite time: determination of the probability of ruin by the Poisson cumulative distribution function, using simulations by the Monte Carlo method.

13.Use of Wiener process, shifted Brownian motion in the field of ruin theory.

**Support literature**

Odporúčaná literatúra:

1. Horáková, G., Páleš, M. & Slaninka, F.: Teória rizika v poistení. Wolters Kluwer. 2015.

2. Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern actuarial risk theory using R, Berlin: Springer. 2008.

3. Charpentier, A.: Computation actuarial science with R. Taylor & Francis Group. 2015.

4. Albrecher, H., Beirlant, L., & Teugels, J. L.: Reinsurance: Actuarial and Statistical Aspects. New York: John Wiley & Sons. 2017.

5. Coles, S.: An Introduction to Statistical Modeling of Extreme Values.Springer. 2001.

6. Dobrow, R.: Introduction to Stochastic Processes with R. John Wiley & Sons. 2016.

7. Schilling, L. R., Partzsch, L.: Brownian motion. Walter de Gruyter GmbH & Co. KG. 2012.

8. Páleš, M., Slaninka, F.: Teória rizika v poistení : riešené príklady v jazyku R a Maxima. Letra Edu, 2021.

9. Cipra, T.: Zajištění a přenos rizik v pojišťovnictví. Grada Publishing, a.s.. 2004.

10. Skřivánková, V., Hančová, M.: Náhodné procesy a ich aplikácie. UPJŠ Košice. 2018.

11. Deelstra, G., Plantin, G.: Risk theory and reinsurance. Springer. 2014.

12. 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.

13. Horáková, G., Mucha, V. Optimálne zaisťovacie reťazce. In Ekonomický časopis. Bratislava : Ústav slovenskej a svetovej ekonomiky SAV : Prognostický ústav SAV, 2005, roč. 53, č. 6.

**Syllabus**

1. Forms of insurance with supplementary forms of insurance (Pure indemnity insurance, First loss insurance, Insurance with an average clause, Quota insurance, With and without an excess). 2. Risk management by applying various forms of insurance in a collective risk model. 3. Modeling of the total claim amount paid by the insurer by Monte Carlo simulations and measurement of risk reduction using risk measures. 4. Reinsurance. Types of reinsurance. Proportional reinsurance (quota, excess reinsurance with respect to the sum insured (surplus)). Non-proportional reinsurance (excess reinsurance with respect to the amount of claims (WXL / R, Excess of Loss), WXL /E or CatXL (Per-Event Excess of Loss or Catastrophe Excess of Loss). 5. Risk management by applying various reinsurance protections in a collective risk model. 6. Application of the reinsurer limit, multiple reinsurer limit in individual types of reinsurance. 7. Modeling of the total claim amount paid by the reinsurer by Monte Carlo simulations and measurement of risk reduction using risk measures. 8. Optimization in reinsurance (minimization of Value at Risk or Conditional Value at Risk, maximization of profit with constant variance, minimization of variance with constant profit, minimization of probability of ruin with constant profit). 9. Use of the extrem value theory(the Excess over Threshold method) in Non-proportional reinsurance. 10. Collective model for longer time periods. Stochastic process: Poisson process (loading process of the number of claims), compound Poisson process (surplus process) and their modeling. 11. Probability of ruin in the distant horizon: determination of the probability of ruin by Lundberg inequality and using Monte Carlo simulations. 12. Probability of ruin in finite time: determination of the probability of ruin by the Poisson cumulative distribution function, using simulations by the Monte Carlo method. 13.Use of Wiener process, shifted Brownian motion in the field of ruin theory.

**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): 156 hours

26 hours - participation in lectures,

26 hours - participation in exercises,

26 hours - preparation for exercises, homeworks,

20 hours - preparation for written works,

58 hours - self-study in preparation for the exam.

**Language whose command is required to complete the course**

slovak

Date of approval: 11.03.2024

Date of the latest change: 15.05.2022