Prague Economic Papers Vol. 23 No. 3

# On Sums of Claims and their Applications in Analysis of Pension Funds and Insurance Products

DOI: https://doi.org/10.18267/j.pep.488

[full text (PDF)]

*Rastislav Potocký, Helmut Waldl, Milan Stehlík*

The problem that assets of a fund are not suffi cient to cover its liabilities is of extreme importance both for its members as well as for fund managers. We show that this problem can be solved via total claims distributions and give answers to the following questions: How much money will be needed in the fi rst pillar in order to satisfy the requirements of pensioners in a time horizon and which groups of working people should join also the second pillar because their benefi ts from it will be greater than those from the fi rst pillar? Though the paper concentrates primarily on the situation with Slovakian pension funds we believe that our fi ndings are more general. We show that the alternative methods should be used for calculation of extremes. We discuss the so-called barrier strategy for treating the surplus of an insurance company and bring some new results concerning it.

**Keywords:** catastrophic events, claims, first and second, Johnson estimators, light- and heavy-tailed distributions, robust approach, the 20–80 rule

**References:**

Alexander, C. (2003), “Technical Document for OpRisk2021 Module 1.” [cit. 2009-09-02], available from

www.2021solutions.com.

Alexander, C. (2005), “Assessment of Operational Risk Capital.” In Frenkel, M., Hommel, U., Rudolf,

M., eds., Risk Management, Challenge and Opportunity. 2nd revised and Enlarged Edition. Berlin:

Springer.

Bűhlmann, H. (1970), Mathematical Methods in Risk Theory. Springer-Verlag.

Brazauskas, V., Serfl ing, R. (2000), “Robust and Effi cient Estimation of the Tail Index of a Single-

Parameter Pareto Distribution.” North American Actuarial Journal, Vol. 4, No. 4, pp. 12–27.

Brazauskas, V., Serfl ing, R. (2003), “Favorable Estimators for Fitting Pareto Models: A Study Using

Goodness-of-fi t Measures with Actual Data.” Astin Bulletin, Vol. 33, No. 2, pp. 365–381.

Christoph, G. (2005), “Exact Rates of Convergence to Ruin Probabilities for Regularly Varying Random

Variables.” In Frenkel, I. et al., eds., Proceedings of the International Symposium on Stochastic

Models in Reliability, Safety, Security and Logistics. Beer Sheva, Israel, pp. 75–78.

Christoph, G. (2004), “Exact Rates of Convergence for Compound Sums of Random Variables with

Common Regularly Varying Distribution Functions.” In Antonov, V. et al., Longevity, Aging and

Degradation Models in Reliability, Public Health, Medicine and Biology. Vol. 2. St. Petersburg,

pp. 56–67.

Embrechts, P., Kluppelberg, C., Mikosch, T. (2003), Modeling Extremal Events. Applications

of Mathematics. 4th Edition. Springer.

Fabián, Z. (2001), “Induced Cores and Their Use in Robust Parametric Estimation.” Communications in

Statistics - Theory and Methods, Vol. 30, No. 3, pp. 537–556.

Fabián, Z. (2006), “Johnson Point and Johnson Variance.” In Hušková, M., Janžura, M. eds., Proc.

Prague Stochastics 2006. Prague: Matfyzpress, pp. 354–363.

Fabián, Z. (2007), “Parametric Estimates by Generalized Moment Method.” Research Report No. 1014,

Institute of Computer Sciences, AS CR.

Fabián, Z. (2008), “New Measures of Central Tendency and Variability of Continuous Distributions.”

Communications in Statistics – Theory and Methods, Vol. 37, No. 2, pp. 159–174.

Fabián, Z., Stehlík, M. (2008), “A Note on Favorable Estimation when Data Is Contaminated.”

Communications in Dependability and Quaity Management, Vol. 11, No. 4, pp. 36–43.

Gruber, H. U. (1979), An Introduction to Mathematical Risk Theory. Philadelphia, PA.: S. S. Huebner

Foundation.

Hewitt, Ch. C., Lefkowitz, B. (1979), “Methods for Fitting Distributions to Insurance Loss Data.”

Proceedings of the Casualty Actuarial Society, Vol. 66, pp. 139–160.

Hogg, R. V., Klugman, S. A. (1984), Loss Distributions. New York: Wiley.

Hommel, U., Ritter, M. (2005), “New Approaches to Managing Catastrophic Insurance Risk.”

In Frenkel, M., Hommel, U., Rudolf, M. eds., Risk Management, Challenge and Opportunity. 2nd

Revised and Enlarged Edition. Springer.

Juárez, S. F., Schucany, W. R. (2004), “Robust and Effi cient Estimation of the Generalized Pareto

Distribution.” Extremes, Vol. 7, No. 3, pp. 237–251.

Mikosch, T., Nagaev, A. (2001), “Rates in Approximations to Ruin Probabilities for Heavy-Tailed

Distributions.” Extremes, Vol. 4, No. 1, pp. 67–78.

Pacáková, V., Linda, B. (2009), “Simulations of Extreme Losses in Non-Life Insurance.” Economics

and Management, Vol. 12, No. 4, pp. 97–102.

Petrov, V. V. (1975), Sums of Independent Random Variables. Springer.

Philbrick, S. W. (1985), “Practical Guide to Single Parameter Pareto Distribution.” Proceedings of the

Casualty Actuarial Society, Vol.72, pp.44–84.

Potocký, R. (2008), “On a Dividend Strategy of Insurance Companies.“ Economics and Management,

Vol. 11, No. 4, pp. 103–110.

370 PRAGUE ECONOMIC PAPERS, 3, 2014

Potocký, R., Stehlík, M. (2007), “Stochastic Models in Insurance and Finance with Respect to Basel II.“

Journal of Applied Mathematics, Statistics and Informatics, Vol. 3, No. 2, pp. 237–245.

Potocký, R., Stehlík, M. (2009), “Statistical Analysis of Mixtures Underlying Probability of Ruin.” Acta

universitatis agriculturae et silviculturae Mendelianae Brunensis, Vol. 62, No. 6, pp. 209–214.

ISSN 1211-8516.

Sheldon, L. X. E, Willmot, G., Drekic, S. (2003), “The Classical Risk Model with a Constant Dividend

Barrier: Analysis of the Gerber-Shiu Discounted Penalty Function.” Insurance: Mathematics and

Economics, Vol. 33, No. 3, pp. 551–566.

Stehlík, M., Potocký, R., Fabián, Z., Waldl, H. (2010), “On the Favourable Estimation of Fitting Heavy

Tailed Data.” Computational Statistics, Vol. 25, No. 3, pp. 485–503.

Stehlík, M., Wagner, H. (2011), “Exact Likelihood Ratio Testing for Homogeneity of Exponential

Distribution.” Communications in Statistics - Simulation and Computation, Vol. 40, pp. 663–684.

Stehlík, M., Strelec, L. (2009), “On Normality Assumptions for Claims in Insurance.” Acta universitatis

agriculturae et silviculturae Mendelianae Brunensis, 2009, Vol. 62, No. 3, pp.141–146.

ISSN 1211-8516.

Uherek, M., Stehlík, M., Strelec, L. (2011), “On Robust Analysis of Paycheck: Case Study.” Acta

universitatis agriculturae et silviculturae Mendelianae Brunensis, Vol. 64, No. 4, pp. 371–378.

ISSN 1211-8516.

Vandewalle, B., Beirlant, J., Christmann, A., Hubert, M. (2007), “A Robust Estimator for the Tail Index

of Pareto-Type Distributions.” Computational Statistics & Data Analysis, Vol. 51, No. 12,

pp. 6252–6268.

Whitehouse, E. (2009), “Pensions during the Crisis: Impact on Retirement Income Systems and Policy

Responses.“ The Geneva Papers, Vol. 34, No. 4, pp. 536–547.

## FOR AUTHORS

## Current issue

### 5/2018

#### Articles

Michala Moravcová

**The Impact of German Macroeconomic News on Emerging European Forex Markets**

Lukasz Kurowski, Pawel Smaga

**Monetary Policy and Cyclical Systemic Risk - Friends or Foes?**

Milan Hrdý

**Valuation Standards for Commercial Banks in the Financial Theory and their Analysis**

Aysa Ipek Erdogan

**Cash Flow Sensitivities of Financial Decisions: Evidence from an Emerging Market**

Blanka Škrabic Peric

**Have More Profitable Banks a More or a Less Risky Lending Policy? Empirical Evidence from CEE Countries**

Yiming Chang, Shangmei Zhao, Haijun Yang, Jiang He, Fei Hu

**Determinants of Deposit Insurance Coverage**

Fuhmei Wang

**The Influences of Fiscal Decentralization on Economic Performance: Empirical Evidence from OECD Countries**