Kalın kuyruklı risk modellerinde iflas olasılığı
Bu çalışmada, hasar tutarlarının kalın kuyruklu dağılım yapısına uyup uymadığı, bu dağlım yapısının koşulları incelenerek araştırılmıştır. Kalın kuyruklu dağılım yapısına sahip risklerin, risk süreci ve iflas olasılıkları üzerindeki etkisi teorik ispatlar eşliğinde açıklanmaya çalışılmıştır. Matlab yazılımında geliştirilen program sayesinde, Türkiye Trafik Sigortası veri kümesi kullanılarak, hasar büyüklüklerinin kalın kuyruklu dağılıma uyup uymadığı sorgulanmış , uyması halinde ise hangi alt sınıfa dahil olduğu araştırılmıştır. Ayrıca belirlenen dönem için risk süreci incelemesi yapılarak, çeşitli sermaye miktarı ve güvenlik yükleme faktörlerine bağlı olarak iflas olasılıkları hesaplanmıştır.
Ruin probability in heavy tailed risk models
In this study, detailed information about heavy tailed distributions and the )ts sub-classes are revieved and theconditions necessary for any distribution to belonging to one of these special classes of distributions areexplained. It is showed that how the ruin probabilities of a risk process that includes heavy tailed claim severityamounts, can be calculated with theoretical justifications. An application is provided by using compulsorytraffic insurance data in Turkey in Matlab programming language. It is investigated if the loss severities can becategorized under the heavy-tailed distribution and also the relevant sub-categorized of the distribution isdetermined,too. Moreover, ruin probabilities are calculated in terms of different initial surplus and safetyfactors by analyzing the risk process for this particular time.
___
- [1] Asmussen, S., 2000, Ruin probabilities, Advanced Series on Statistical Science and Applied Probability
Vol.2, World Scientific Publishing, Singapore, 385p.
- [2] Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987, Regular variation, Cambridge University Press,
Cambridge.
- [3] Chistyakov, V.P., 1964, A theorem on sums of independent positive random variables and its applications to
branching random processes, Theory probability Application 9, pp 640-649.
- [4] Embrechts, P., Goldie, C.M., 1982, On convolution tails, Stochastic Processes Applied 13, pp 263-278.
- [5] Embrechts, P., Klüppelberg C., Mikosch T., 2001, Modelling Extremal Events for Insurance and Fnance,
Applications of Mathematics Stochastic Modelling and Applied Probability 33 , Springer, 648p.
- [6] Goldie, C. M., Klüppelberg C., 1998, Subexponential Distributions, A practical guide to heavy tails:
statistical techniques and applications, pp 435-460.
- [7] Klugman, S.,A., Panjer H.H., Willmot G.,E., 2008, Loss models from data to decisions, Third Edition, John
Wiley and Sons, New Jersey, 726p.
- [8] Klüppelberg, C., 1988, Subexponential distributions and integrated tails, Journal of Applied Probability,Vol
25,No:1,pp 132-14.1
- [9] Mo, K.C.K., 2002, Ruin probabilities with dependent claims, Actuarial Studies, Faculty of Commerce and
Economics, University of New South Wales, 50p.
- [10] Rolski T., Schmidli H., Schmidt V., Teugels J., 1999, Stochastic processes for Insurance and Finance, Wiley
Series in Probability and Statstics, John Wiley and Sons, England, 654p.
- [11] Sigman K., 1999, Appendix: A primer on heavy-tailed distributions, Queueing Systems 33, pp 261-275