Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-30T22:28:07.563Z Has data issue: false hasContentIssue false

The fuzzy Bornhuetter–Ferguson method: an approach with fuzzy numbers

Published online by Cambridge University Press:  22 August 2016

Jochen Heberle*
Affiliation:
School of Business, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany
Anne Thomas
Affiliation:
School of Business, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany
*
*Correspondence to: Jochen Heberle, School of Business, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany. Tel: +4940428383541; E-mail: jochen.heberle@wiso.uni-hamburg.de

Abstract

This paper shows how the well-known Bornhuetter–Ferguson claims-reserving method can be extended by applying fuzzy methods. The a priori information for the ultimate claims derives from market statistics, organisational data, etc. and might contain vagueness. Likewise, the parameters of the claims development pattern can be vague or are adapted, retrospectively, due to subjective judgement. With the help of fuzzy numbers we develop new predictors for the ultimate claims. Furthermore, we quantify the uncertainty of the ultimate claims for single and aggregated accident years.

Type
Papers
Copyright
© Institute and Faculty of Actuaries 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alai, D.H., Merz, M. & Wüthrich, M.V. (2009). Mean square error of prediction in the Borhuetter-Ferguson claims reserving method. Faculty of Actuaries and Institute of Actuaries, 4, 731.Google Scholar
Baser, F. & Apaydin, A. (2010). Hybrid fuzzy least-squares regression analysis in claims reserving with geometric separation method. Insurance: Mathematics and Economics, 47(2), 113122.Google Scholar
Bornhuetter, R.L. & Ferguson, R.E. (1972). The actuary and IBNR. Proceedings of the Casualty Actuarial Society, 59, 181195.Google Scholar
Chang, Y.-H.O. (2001). Hybrid fuzzy least-squares regression analysis and its reliability measures. Fuzzy Sets and Systems, 119(2), 225246.Google Scholar
Dahms, R. (2008). A loss reserving method for incomplete claim data. Bulletin of Swiss Association of Actuaries, 127148.Google Scholar
de Andrés Sánchez, J. (2006). Calculating insurance claim reserves with fuzzy regression. Fuzzy Sets and Systems, 157(23), 30913108.Google Scholar
de Andrés Sánchez, J. (2007). Claim reserving with fuzzy regression and Taylor’s geometric separation method. Insurance: Mathematics and Economics, 40(1), 145163.Google Scholar
de Andrés Sánchez, J. (2012). Claim reserving with fuzzy regression and the two ways of ANOVA. Applied Soft Computing, 12(8), 24352441.Google Scholar
de Andrés Sánchez, J. & Terceno Gomez, A. (2003). Applications of fuzzy regression in actuarial analysis. Journal of Risk and Insurance, 70(4), 665699.Google Scholar
de Campos Ibáñez, L.M. & González Muñoz, A. (1989). A subjective approach for ranking fuzzy numbers. Fuzzy Sets and Systems, 29(2), 145153.CrossRefGoogle Scholar
Dubois, D. & Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9(6), 613626.Google Scholar
Dubois, D. & Prade, H. (1979). Fuzzy real algebra: some results. Fuzzy Sets and Systems, 2(4), 327348.Google Scholar
Dubois, D. & Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications. Mathematics in Science and Engineering. Academic Press, New York.Google Scholar
Hanss, M. (2005). Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Springer, Heidelberg.Google Scholar
Heberle, J. & Thomas, A. (2014). Combining chain-ladder claims reserving with fuzzy numbers. Insurance: Mathematics and Economics, 55, 96104.Google Scholar
Ishibuchi, H. & Nii, M. (2001). Fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks. Fuzzy Sets and Systems, 119(2), 273290.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. & Denuit, M. (2008). Modern Actuarial Risk Theory: Using R, 2nd edition. Springer, Berlin, Heidelberg.Google Scholar
Kwakernaak, H. (1978). Fuzzy random variables – I. Definitions and theorems. Information Sciences, 15(1), 129.Google Scholar
Kwakernaak, H. (1979). Fuzzy random variables – II. Algorithms and examples for the discrete case. Information Sciences, 17(3), 253278.Google Scholar
Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23(2), 213225.Google Scholar
Mack, T. (2000). Credible claims reserves: the Benktander method. ASTIN Bulletin, 30, 333347.CrossRefGoogle Scholar
Pal, N.R. & Bezdek, J.C. (1994). Measuring fuzzy uncertainty. IEEE Transactions on Fuzzy Systems, 2(2), 107118.Google Scholar
Puri, M.L. & Ralescu, D.A. (1986). Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114(2), 409422.CrossRefGoogle Scholar
Shapiro, A.F. (2004). Fuzzy logic in insurance. Insurance: Mathematics and Economics, 35(2), 399424.Google Scholar
Shapiro, A.F. (2009). Fuzzy random variables. Insurance: Mathematics and Economics, 44(2), 307314.Google Scholar
Sherman, R.E. (1984). Extrapolating, smoothing and interpolating development factors. Proceedings of the Casualty Actuarial Society, 71, 122155.Google Scholar
Tanaka, H. & Ishibuchi, H. (1992). A possibilistic regression analysis based on linear programming. In J., Kacprzyk & M., Fedrizzi, (Eds.), Fuzzy Regression Analysis (pp. 4760). Physica-Verlag, Heidelberg.Google Scholar
Verrall, R.J. (2004). A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8(3), 6789.Google Scholar
Wagenknecht, M., Hampel, R. & Schneider, V. (2001). Computational aspects of fuzzy arithmetics based on Archimedean t-norms. Fuzzy Sets and Systems, 123(1), 4962.CrossRefGoogle Scholar
Wüthrich, M.V. & Merz, M. (2008). Stochastic Claims Reserving Methods in Insurance. Wiley Finance, Chichester.Google Scholar
Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8, 338353.Google Scholar
Zadeh, L.A. (1975 a). The concept of a linguistic variable and its application to approximate reasoning – part I. Information Sciences, 8(3), 199249.CrossRefGoogle Scholar
Zadeh, L.A. (1975 b). The concept of a linguistic variable and its application to approximate reasoning – part II. Information Sciences, 8(4), 301357.CrossRefGoogle Scholar
Zadeh, L.A. (1975 c). The concept of a linguistic variable and its application to approximate reasoning – part III. Information Sciences, 9(1), 4380.Google Scholar