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Calculating premium principles from the mode of a unimodal weighted distribution

Part of: Applications

Published online by Cambridge University Press:  15 May 2024

Georgios Psarrakos Open the ORCID record for Georgios Psarrakos [Opens in a new window]
Georgios Psarrakos*
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece
*
Email: gpsarr@unipi.gr
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Abstract

The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.

Keywords

Weighted premium calculation principleexponential premiumEsscher premiumKamps premiumstochastic orderingmode orderingweighted distributionunimodal distribution

MSC classification

Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , First View , pp. 1 - 13
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Belzunce, F., Martínez-Riquelme, C. and Mulero, J. (2016) An Introduction to Stochastic Orders. Amsterdam: Elsevier-Academic Press.Google Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin, 11, 52–60.CrossRefGoogle Scholar
Castaño-Martínez, A., López-Blazquez, F., Pigueiras, G. and Sordo, M. (2020) A method for constructing and interpreting some weighted premium principles. ASTIN Bulletin, 50(3), 1037–1064.CrossRefGoogle Scholar
Di Crescenzo, A. (1999) A probabilistic analogue of the mean value theorem and its applications to reliability. Journal of Applied Probability, 36, 706–719.CrossRefGoogle Scholar
Denneberg, D. (1994) Non-Additive Measure and Integral. Dordrecht: Kluwer.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2006) Actuarial Theory for Dependent Risks: Measures, Orders and Models. Hoboken, New Jersey, USA: John Wiley & Sons.Google Scholar
Furman, E. and Zitikis, R. (2008) Weighted premium calculation principles. Insurance: Mathematics and Economics, 42, 459–465.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8, distributed by Richard D. Irwin, Homewood Illinois.Google Scholar
Gerber, H. and Shiu, E.S.W. (1994) Option pricing by Esscher transforms. Transactions of the Society of Actuaries, 46, 99–191.Google Scholar
Goovaerts, M.J., de Vylder, F. and Haezendonck, J. (1984) Insurance Premiums: Theory and Applications. Amsterdam: North-Holland.Google Scholar
Gupta, R.D. and Kundu, D. (2009) A new class of weighted exponential distributions. Statistics, 43(6), 621–634.CrossRefGoogle Scholar
Kamps, U. (1998) On a class of premium principles including the Esscher premium. Scandinavian Actuarial Journal, 1, 75–80.CrossRefGoogle Scholar
Laeven, R.J.A. and Goovaerts, M.J. (2008) Premium calculation and insurance pricing. In Encyclopedia of Quantitative Risk Analysis and Assessment. Wiley & Sons.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. Chichester: John Wiley and Sons.Google Scholar
Newton, L., Bowers, Gerber H.U. Jr., Hickman, J.A., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.Google Scholar
Pearson, K. (1895) Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 186, 343–414.Google Scholar
Psarrakos, G. (2022) How a probabilistic analogue of the mean value theorem yields Stein-type covariance identities. Journal of Applied Probability, 59, 350–365.CrossRefGoogle Scholar
Ruiz, J.M. and Navarro, J. (1994) Characterization of distributions by relationships between failure rate and mean residual life. IEEE Transactions on Reliability, 43(4), 640–644.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. New York: Springer.CrossRefGoogle Scholar
Wang, S.S. (1995) Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17, 43–54.Google Scholar
Wang, S.S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26(1), 71–92.CrossRefGoogle Scholar
Wang, S.S. (2003) Equilibrium pricing transforms: New results using Bühlmann’s 1980 economic model. ASTIN Bulletin, 33(1), 57–73.CrossRefGoogle Scholar