Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-01T02:42:11.626Z Has data issue: false hasContentIssue false
  • English
  • Français

An Extension of the Gerber-Bühlmann-Jewell Conditions for Optimal Risk Sharing

Published online by Cambridge University Press:  17 April 2015

Marek Kaluszka
Marek Kaluszka*
Affiliation:
Institute of Mathematics, Technical University of Lodz, Ul. Zwirki 36, 90-924 Lodz Poland, E-mail: kaluszka@p.lodz.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide necessary and sufficient conditions for optimality of mutual contracts for risk sharing under constraints on premiums or utility functions of participants of the agreement. These conditions are an extension of those of the Borch, Gerber and Bühlmann-Jewell ones. Some applications to optimal insurance contracts, optimal dividend sharing and optimal reinsurance are given.

Keywords

InsuranceLimited insuranceReinsuranceSharing risksDividends
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 34 , Issue 1 , May 2004 , pp. 27 - 48
Copyright
Copyright © ASTIN Bulletin 2004

References

Aase, K.K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin 23, 185–211.CrossRefGoogle Scholar
Ammeter, H., Depoid, P. and De Finetti, B. (1959) L’étude matématique des assurances non viagères dans l’europe continentale occidentale. ASTIN Bulletin 1, 46–70.CrossRefGoogle Scholar
Arrow, K.J. (1963) Uncertainty and the welfare economics of medical care. The American Economic Review 53, 941–73.Google Scholar
Barile, A. and Monti, R.G. (1995) A practical guide to finite risk insurance and reinsurance. John Wiley. New York.Google Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica 30, 424–444.CrossRefGoogle Scholar
Borch, K. (1974) The Mathematical Theory of Insurance. Lexington Books, Lexington.Google Scholar
Borch, K. (1990) Economics of Insurance. Edited by Aase, K.K. and Sandmo, A., North-Holland. Amsterdam.Google Scholar
Bühlmann, H. (1984) The general economic premium principle. ASTIN Bulletin 14, 13–21.Google Scholar
Bühlmann, H. (1996) Mathematical models in risk theory. 2nd edition, Springer-Verlag, Berlin.Google Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin 10, 243–262.CrossRefGoogle Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries. Chapman & Hall, London.Google Scholar
De Finetti, B. (1940) Il problema dei pieni. Giorn. Ist. Ital. Attuari 11, 1–88.Google Scholar
Deprez, O. and Gerber, H.U. (1985) On convex principles of premium calculation. Insurance: Mathematics & Economics 4, 179–189.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Gerathewohl, K. (1980) Reinsurance Principles and Practice. Verlag Versicherungswirtschaft, Karlsruhe.Google Scholar
Gerber, H.U. and Jones, D.A. (1974) Dividend formulas in group insurance. Transactions of the Society of Actuaries 26, 77–86.Google Scholar
Gerber, H.U. (1978) Pareto-optimal risk exchanges and related decision problems. ASTIN Bulletin 9, 25–33.CrossRefGoogle Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania, Philadelphia.Google Scholar
Gerber, H.U. and Pafumi, G. (1999) Utility functions: from risk theory to finance. North American Actuarial Journal 2, 74–100.CrossRefGoogle Scholar
Gollier, Ch. (2001) The Economics of Risk and Time, The MIT Press, Cambridge.CrossRefGoogle Scholar
Goovaerts, M.J., De Vylder, F. and Haezendonck, J. (1984) Insurance Premiums: Theory and Applications. North-Holland, Amsterdam.Google Scholar
Hesselager, O. (1990) Some results on optimal reinsurance in terms of the adjustment coefficient. Scandinavian Actuarial Journal, 80–95.CrossRefGoogle Scholar
Hesselager, O. (1993) Extensions of Ohlin’s lemma with applications to optimal reinsurance structures. Insurance: Mathematics & Economics 13, 83–97.Google Scholar
Kaluszka, M. (2001) Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics & Economics 28, 61–67.Google Scholar
Kaluszka, M. (2004) Mean-variance optimal reinsurance arrangements. Scandinavian Actuarial Journal (to appear)Google Scholar
Lemaire, J. (1991) Cooperative game theory and its insurance applications. ASTIN Bulletin 21, 17–40 CrossRefGoogle Scholar
Lemaire, J. and Quairiere, J.P. (1986) Chains of reinsurance revisited. ASTIN Bulletin 16, 77–88 CrossRefGoogle Scholar
Lienhard, M. (1986) Calculation of price equilibria for utility functions of the Hara class. ASTIN Bulletin 16, 91–98.CrossRefGoogle Scholar
Mata, A.J. (2000) Pricing excess of loss reinsurance with reinstatements. ASTIN Bulletin 30, 349–368.CrossRefGoogle Scholar
Panjer, H. (editor), Boyle, P., Cox, S., Dufresne, D., Gerber, H., Mueller, H., Pedersen, H., Pliska, S., Sherris, M., Shiu, E. and Tan, K. (1998) Financial Economics With Applications to Investments, Insurance and Pensions. Schaumburg, III., The Actuarial Foundation.Google Scholar
Pesonen, M.I. (1984) Optimal reinsurances. Scandinavian Actuarial Journal, 65–90.CrossRefGoogle Scholar
Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton. New Jersey.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. J. Wiley & Sons, Chichester.CrossRefGoogle Scholar
Samson, D. (1986) Expected utility strategic decision models for general insurers. ASTIN Bulletin 16, 45–58.Google Scholar
Sundt, B. (1991) On excess of loss reinsurance with reinstatement. Bulletin of the Swiss Actuaries, 51–66.Google Scholar
Vandebroek, M. (1988) Pareto-optimal profit-sharing. ASTIN Bulletin 18, 47–56.CrossRefGoogle Scholar
Walhin, J.F. and Paris, J. (2000) The effect of excess-of-loss reinsurance with reinstatements on the cedent’s portfolio. Blätter der Deutsche Gesellschaft für Versicherungsmathematik 24, 615–627.Google Scholar
Wyler, E. (1990) Pareto optimal risk exchanges and a system of differential equations – a duality theorem. ASTIN Bulletin 20, 23–31.CrossRefGoogle Scholar
Young, V.R. (1999) Optimal insurance under Wang’s premium principle. Insurance: Mathematics & Economics 25, 109–122.Google Scholar