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A METHOD FOR CONSTRUCTING AND INTERPRETING SOME WEIGHTED PREMIUM PRINCIPLES

Published online by Cambridge University Press:  05 June 2020

Antonia Castaño-Martínez ,
Fernando López-Blazquez ,
Gema Pigueiras  and
Miguel Á. Sordo Open the ORCID record for Miguel Á. Sordo [Opens in a new window]
Antonia Castaño-Martínez
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Cádiz Cádiz, Spain E-Mail: [email protected]
Fernando López-Blazquez
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla Sevilla, Spain E-Mail: [email protected]
Gema Pigueiras
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Cádiz Cádiz, Spain E-Mail: [email protected]
Miguel Á. Sordo*
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Cádiz Cádiz, Spain E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

We present a method for constructing and interpreting weighted premium principles. The method is based on modifying the underlying risk distribution in such a way that the risk-adjusted expected value (or premium) is greater than the expected value of some conveniently chosen function of claims, which defines the insurer’s perception of the risk. Under some assumptions on the function of claims, the method produces distortion premium principles. We provide several examples under different assumptions on the claim arrival process and different functions of claims, including record claims and kth record claims.

Keywords

Weighted premium principledistortion premium principleequilibrium distributionorder statisticsrecord valueG220
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 1037 - 1064
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

Ahsanullah, M. (1995). Record Statistics. New York: Nova Science Publishers.Google Scholar
Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. (1998) Records. New York: John Wiley and Sons.CrossRefGoogle Scholar
Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. (2008) A First Course in Order Statistics. Classics inApplied Mathematics, Vol. 54. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Barmalzan, G., Payandeh Najafabadi, A.T. and Balakrishnan, N. (2016) Likelihood ratio and dispersive orders for smallest order statistics and smallest claim amounts from heterogeneous Weibull sample. Statistics and Probability Letters, 110, 1–7.CrossRefGoogle Scholar
Barmalzan, G., Payandeh Najafabadi, A.T. and Balakrishnan, N. (2017) Ordering properties of the smallest and largest claim amounts in a general scale model. Scandinavian Actuarial Journal, 2017, 105–124.CrossRefGoogle Scholar
Bowers, N., Gerber, H., Hickman, J., Jones, D. and Nesbitt, C. (1997) Actuarial Mathematics, 2nd edition. Schaumburg, IL: Society of Actuaries.Google Scholar
Castaño-Martnez, A., Pigueiras, G. and Sordo, M.A. (2019) On a family of risk measures based on largest claims. Insurance: Mathematics and Economics, 86, 92–97.Google Scholar
Chandler, K.N. (1952) The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220–228.Google Scholar
Denneberg, D. (1990) Premium calculation: Why standard deviation should be replaced by absolute deviation. ASTIN Bulletin, 20, 181–190.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks. New York: John Wiley & Sons.CrossRefGoogle Scholar
Dziubdziela, W. and Kopociński, B. (1976) Limiting properties of the k-th record values. Applicationes Mathematicae, 15, 187–190.CrossRefGoogle Scholar
Furman, E. and Zitikis, R. (2008a) Weighted premium calculation principles. Insurance: Mathematics and Economics, 42, 459–465.Google Scholar
Furman, E. and Zitikis, R. (2008b) Weighted risk capital allocations. Insurance: Mathematics and Economics, 43, 263–269.Google Scholar
Furman, E. and Zitikis, R. (2009) Weighted pricing functionals with applications to insurance: An overview. North American Actuarial Journal, 13, 483–496.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2004) Stochastic Finance: An Introduction in Discrete Time. Berlin: De Gruyter Studies in Mathematics, Walter de Gruyter.Google Scholar
Goovaerts, M.J. and Laeven, R.J.A. (2008) Actuarial risk measures for financial derivative pricing. Insurance: Mathematics and Economics, 42, 540–547.Google Scholar
Kamps, U. (1994) Reliability properties of record values from nonidentically distributed random variables. Communications in Statistics - Theory and Methods, 23, 2101–2112.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss Models. From Data to Decisions. 3rd edition. New York: John Wiley & Sons.CrossRefGoogle Scholar
Kochar, S.C. (1990) Some partial ordering results on record values. Communications in Statistics – Theory and Methods, 19, 299–306.CrossRefGoogle Scholar
Lin, X.S. and Willmot, G.E. (2000) The moments of the time of ruin, the surplus before ruin and the deficit at ruin. Insurance: Mathematics and Economics, 27, 19–44.Google Scholar
Nevzorov, V.B. (2001) Records: Mathematical Theory. Providence: American Mathematical Society.Google Scholar
Pearson, K. (1934) Tables of the Incomplete Beta Function. Cambridge University Press., Cambridge.Google Scholar
Pflug, G.C. and Römisch, W. (2007) Modeling, Measuring and Managing Risk. Singapore: World Scientific Publishing Company.CrossRefGoogle Scholar
Psarrakos, G. and Navarro, J. (2013) Generalized cumulative residual entropy and record values. Metrika, 76, 623–640.CrossRefGoogle Scholar
Psarrakos, G. and Sordo, M.A. (2019) On a family of risk measures based on proportional hazards models and tail probabilities. Insurance: Mathematics and Economics, 86, 232–240.Google Scholar
Rajesh, G. and Sunoj, S.M. (2019) Some properties of cumulative Tsallis entropy of order a. Statistical Papers, 60, 933–943.CrossRefGoogle Scholar
Rao, C.R. (1965) On discrete distributions arising out of methods of ascertainment Sankhya: The Indian Journal of Statistics, Series A, 27, 311–324.Google Scholar
Rao, M., Chen, Y., Vemuri, B.C. and Wang, F. (2004) Cumulative residual entropy: A new measure of information. IEEE Transactions on Information Theory, 50, 1220–1228.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. SpringerSeries in Statistics, New York: Springer.Google Scholar
Sordo, M.A., Castaño-Martínez, A. and Pigueiras, G. (2016) A familiy of premium principles based on mixtures of TVaRs. Insurance: Mathematics and Economics, 70, 397–405.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 71–92.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55, 95–115.CrossRefGoogle Scholar
Zhang, Y., Cai, X. and Zhao, P. (2019) Ordering properties of extreme claim amounts from heterogeneous portfolios. ASTIN Bulletin, 49, 525–554.CrossRefGoogle Scholar
Zhu, W., Tan, K.S. and Porth, L. (2016) On a class of premium calculation principles based on the multivariate weighted distribution. Available at SSRN: https://ssrn.com/abstract=2888702 or http://dx.doi.org/10.2139/ssrn.2888702CrossRefGoogle Scholar