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ONE-YEAR PREMIUM RISK AND EMERGENCE PATTERN OF ULTIMATE LOSS BASED ON CONDITIONAL DISTRIBUTION

Published online by Cambridge University Press:  05 May 2020

Łukasz Delong  and
Marcin Szatkowski
Łukasz Delong*
Affiliation:
SGH Warsaw School of Economics, Institute of Econometrics Niepodległości 162, Warsaw 02-554, Poland E-mail: [email protected]
Marcin Szatkowski
Affiliation:
SGH Warsaw School of Economics, Institute of Econometrics Niepodległości 162, Warsaw 02-554, Poland and Risk Department, STU ERGO Hestia SA Hestii 1, Sopot 81-731, Poland E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

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We study the relation between one-year premium risk and ultimate premium risk. In practice, the one-year risk is sometimes related to the ultimate risk by using a so-called emergence pattern formula which postulates a linear relation between both risks. We define the true emergence pattern of the ultimate loss for the one-year premium risk based on a conditional distribution of the ultimate loss derived from a multivariate distribution of the claims development process. We investigate three models commonly used in claims reserving and prove that the true emergence pattern formulas are different from the linear emergence pattern formula used in practice. We show that the one-year risk, when measured by VaR, can be under and overestimated if the linear emergence pattern formula is applied. We present two modifications of the linear emergence pattern formula. These modifications allow us to go beyond the claims development models investigated in the first part and work with an arbitrary distribution of the ultimate loss.

Keywords

One-year riskultimate riskpremium riskemergence patternSolvency IIclaims development process
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 2 , May 2020 , pp. 479 - 511
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Astin Bulletin 2020

References

AISAM-ACME. (2007) AISAM-ACME study on non-life long tail liabilities: Reserve risk and risk margin assessment under Solvency II. https://www.amice-eu.org/.Google Scholar
Berlaint, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of Extremes: Theory and Applications. Chichester: John Wiley & Sons Ltd.CrossRefGoogle Scholar
Bird, C. and Cairns, M. (2011) Practical experiences of modelling one-year risk emergence. GIRO Conference and Exhibition 2011.Google Scholar
Cline, D. and Samorodnitsky, G. (1994) Subexponentiality of the product of independent random variables. Stochastic Processes and their Applications, 49, 75–98.CrossRefGoogle Scholar
England, P., Cairns, M. and Scarth, R. (2012) The 1 year view of reserving risk: The “actuary-in-the-box” vs. emergence patterns. GIRO Conference and Exhibition 2012.Google Scholar
Geluk, J. and De Vries, C. (2006) Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities. Insurance: Mathematics and Economics, 38, 39–56.Google Scholar
Gordon, R.D. (1941) Values of Mills’ ratio of area to bounding ordinate of the normal probability integral for large values of the argument. Annals of Mathematical Statistics, 12, 364–366.CrossRefGoogle Scholar
Guo, B., Feng, Q., Zhao, J. and Luo, Q. (2015) Sharp inequalities for polygamma functions. Mathematica Slovaca, 65, 103–120.CrossRefGoogle Scholar
Hertig, J. (1985) A statistical approach to IBNR-reserves in marine reinsurance. ASTIN Bulletin, 15, 171–183.CrossRefGoogle Scholar
Johnson, R.A. and Wichern, D.W. (2007) Applied Multivariate Statistical Analysis, 6th edn. Upper Saddle River, New Jork: Pearson Prentince Hall.Google Scholar
Klenke, A. and Mattner, L. (2010) Stochastic ordering of classical discrete distributions. Advances in Applied Probability, 42, 392–410.CrossRefGoogle Scholar
Lloyd’s. (2014) Solvency II Model Validation Guidance. https://www.lloyds.com/~/media/Files/The-Market/Operating-at-Lloyds/Solvency-II/2014-Guidance/Model-Validation-Guidance-April-2014-v1.pdf Google Scholar
Mikosch, T. (1999) Regular Variation, Subexponentiality and Their Applications in Probability Theory. Eindhoven: Eindhoven University of Technology.Google Scholar
Qi, F., Guo, S., Gou, B. and Chen, S. (2008) A class of k-log-convex functions and their applications to some special functions. Integral Transforms and Special Functions, 19, 195–200 CrossRefGoogle Scholar
Radtke, M., Schmidt, K.D. and Schnaus, A. (2016) Handbook on Loss Reserving. Switzerland: Springer International Publishing.CrossRefGoogle Scholar
Resnick, S. (2007) Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. New York: Springer - Verlag.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (2001) Stochastic Processes for Insurance and Finance. Chichester: John Wiley & Sons Ltd.Google Scholar
Short, M. (2013) Improved inequalities for the Poisson and Binomial distribution and upper tail quantile functions. International Scholarly Research Notices: Probability and Statistics, 2013, 1–6.Google Scholar
Steutel, F. and Van Harn, K. (2004) Infinite Divisibility of Probability Distributions on the Real Line. Marcel and Dekker.Google Scholar
Tang, Q. (2006) The subexponentiality of products revisited. Extremes, 9, 231–241.CrossRefGoogle Scholar
Taylor, G. and McGuire, G. (2016) Stochastic Loss Reserving Using Generalized Linear Models. Casualty Actuarial Society CAS Monograph Series Number 3. Arlington, Virginia: CAS.Google Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: John Wiley & Sons Ltd.Google Scholar
Wüthrich, M.V. and Merz, M. (2010) Paid-incurred chain claims reserving method. Insurance Mathematics and Economics, 46, 568–579 Google Scholar
Wüthrich, M.V. and Merz, M. (2015) Claims run-off uncertainty: The full picture. Swiss Finance Institute Research Paper No. 14-69. SSRN: https://ssrn.com/abstract=2524352 or http://dx.doi.org/10.2139/ssrn.2524352.Google Scholar