Fuzzy pricing of high excess loss layer when modeling the tail with generalized Pareto distribution

Yıl 2011, Cilt: 4 Sayı: 1, 16 - 30, 30.06.2011

Yasemin Gençtürk Tuğba Tunç Duygu İçen , Süleyman Günay

Öz

Good estimates for the tails of loss severity distributions are essentials for pricing or positioning high-excess

loss layers (in reinsurance). Extreme value theory (EVT) provides a framework to formalize the study of

behaviour in the tails of loss severity distributions. In EVT, the excess losses over a high threshold are

modelled using generalized Pareto distribution (GPD). In any data analysis, there are various layers of

uncertainty such as parameter and/or model uncertainty. These uncertainities are magnified in extreme value

analysis. The aim of this study is to obtain fuzzy price for high excess loss layer when GPD provides good

fitting to the tail of claim data. For this purpose, parameters of GPD are estimated using Buckley’s approach

Anahtar Kelimeler

Buckley’ s approach, Extreme value theory, Fuzzy parameter estimation, Fuzzy pricing for high excess of loss layer, Generalized Pareto distribution.

Kuyruk bölgesi genelleştirilmiş Pareto dağılım ile modellendiğinde hasar fazlasının bulanık fiyatlandırması

Öz

 Reasüransta, hasar şiddeti dağılımlarının kuyruk
bölgesinin doğru tahmini fiyatlandırma ve hasar fazlasının belirlenmesinde
önemlidir. Hasar şiddeti dağılımlarının kuyruk bölgesinin modellenmesinde Uç
Değer Teorisinden yararlanılmaktadır. Uç Değer Teorisinde, oldukça yüksek bir
eşik değerini aşan hasarlar Genelleştirilmiş Pareto dağılımı kullanılarak
modellenmektedir.

Herhangi bir veri analizinde, parametre ve/veya model
belirsizliği gibi çeşitli belirsizlikler söz konusudur. Uç değer analizinde, bu
belirsizlikler daha da artmaktadır. Zadeh [33] belirsizliğin üstesinden gelmek
için bulanık mantığı önermiştir. Bu çalışmada, Buckley’ in yaklaşımı kullanarak
Genelleştirilmiş Pareto Dağılımının parametrelerini tahmin edilmiş; bu
parametre tahminleri kullanılarak hasar fazlasının bulanık fiyatlandırması elde
edilmiştir.

Anahtar Kelimeler

Buckley yaklaşımı, Bulanık parametre tahmini, Genelleştirilmiş Pareto dağılımı, Hasar fazlasının bulanık fiyatlandırması, Uç değer teorisi

Kaynakça

• J. Andréz-Sanchez, 2007, Claim reserving with fuzzy regression and Taylor’s geometric seperation method, Insurance: Mathematics and Economics, 40, 145-163.
• Y.M. Babad, B. Berliner, 1994, The use of intervals of possibilities to measure and evaluate financial risk and uncertainty, 4th AFIR International Colloquium, 1, 111–140.
• A. Balkema and L. De Haan, 1974, Residual life time at great age, The Annals of Probability, 2, 792-804.
• J. Beirlant and J. Teugels, 1992, Modelling large claims in non-life insurance, Insurance: Mathematics and Economics, 11, 17-29.
• B. Berliner and N. Buehlmann, 1993, A generalization of the fuzzy zooming of cash flows, 3rd AFIR International Colloquium, 2, Roma, 431–456.
• J.J. Buckley, 2009, Fuzzy Probability and Statistics, Springer-Verlag, Berlin.
• J.D. Cummins and R.A. Derrig, 1993, Fuzzy trends in property-liability insurance claim costs, Journal of Risk and Insurance, 60(3), 429–465.
• J.D. Cummins and R.A. Derrig, 1997, Fuzzy financial pricing of property-liability insurance, North American Actuarial Journal, 1(4), 21–44.
• R.A. Derrig and K. Ostaszewski,1995, Fuzzy techniques of pattern recognition in risk and claim classification, Journal of Risk and Insurance, 62, 447–482.
• R.A. Derrig and K. Ostaszewski, 1997, Managing the tax liability of a property liability insurance company, Journal of Risk and Insurance, 64, 695–711.
• R.A. Derrig and K. Ostaszewski, 1998, Fuzzy sets methodologies in actuarial science, in Practical Applications of Fuzzy Technologies, H.J. Zimmerman(ed.), Kluwer Academic Publishers, Boston, 531-556.
• G.W. DeWit, 1982, Underwriting and uncertainty, Insurance: Mathematics and Economics, 1, 277-285.
• B. Ebanks, W. Karkowski and K. Ostaszewski, 1992, Application of measures of fuzziness to risk classification in insurance, Proceedings of the 4th International Conference on Computing and Information (ICCI 1992), Los Alamitos, Canada, 290–291.
• P. Embrechts and C. Kluppelberg, 1993, Some aspects of insurance mathematics, Theory of Probability and Its Applications, 38, 262-295.
• P. Embrechts, C. Kluppelberg and T. Mikosch, 1997, Modelling extremal events for insurance and finance, Springer-Verlag, Berlin.
• R. Fisher and L. Tippet, 1928, Limiting forms of frequency distribution of the largest or smallest member of a sample, Proceedings of the Cambridge Philosophical Society, 24 , 180-190.
• M. Gilli and E. Kéllezi, 2006, An Application extreme value theory for measuring financial risk, Computational Economics, 27, 207-228.
• P. Horgby, R. Lohse and N. Sittaro, 1997, Fuzzy underwriting: an application of fuzzy logic to medical underwriting,Journal of Actuarial Practice, 5,79-104.
• J.R.M. Hosking and J.R. Wallis, 1987, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, 29, 339-349.
• W. Karkowski and K. Ostaszewski, 1992, An analysis of possible applications of fuzzy set theory to the actuarial credibility theory, Proceedings of the North American Fuzzy Information Processing Society, Puerto Vallarta, Mexico.
• K.H. Lee, 2005, First Course on Fuzzy Theory and Applications, Springer-Verlag, Berlin.
• J. Lemaire, 1990, Fuzzy insurance, Astin Bulletin, 20(1), 34-55.
• A. Mata,2000, Parameter uncertainty for extreme value distributions, GIRO Workshop,Birmingham, United Kingdom.
• A.J. McNeil, 1997, Estimating the tails of loss severity distributions using extreme value theory, Astin Bulletin, 27(1), 117-137.
• K.M. Ostaszewski, 1993, An Investigation into Possible Applications of Fuzzy Set Methods in Actuarial Science, The Society of Actuaries.
• J. Pickands, 1975, Statistical inference using extreme order statistics, The Annals of Statistics, 3, 119-131.
• H. Rootzen and N. Tajvidi, 1995, Extreme value statistics and wind storm losses: a case study, Scandinavian Actuarial Journal, 70-94
• A.F. Shapiro, 2004, Fuzzy logic in insurance, Insurance: Mathematics and Economics, 35, 399-424.
• M.R. Simonelli, 1999, Fuzzy insurance premium principle, Studies in Fuzziness and Soft Computing, 425– 432.
• A. Terceno, J.de Andrés, C. Belvis and M.G. Barbera, 1996, Fuzzy methods incorporated to the study of personal insurances, Fuzzy Economic Review, 1(2), 105–119.
• R.J. Verrall and Y.H. Yakoubov, 1999, A fuzzy approach to grouping by policyholder age in general insurance, Journal of Actuarial Practice, 7, 181–203.
• V.R. Young, 1993, The application of fuzzy sets to group health underwriting, Transactions of the Society of Actuaries, 45, 551–590.
• L. A. Zadeh, 1965, Fuzzy sets, Information and Control, 8(3), 338–353.