Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T01:35:43.114Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium in a Reinsurance Syndicate; Existence, Uniqueness and Characterization

Published online by Cambridge University Press:  29 August 2014

Knut K. Aase
Knut K. Aase*
Affiliation:
Norwegian School of Economicsand Business Administration, 5035 Bergen-Sandviken, Norway
*
Norwegian School of Economicsand Business Administration, 5035 Bergen-Sandviken, Norway
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.

This paper attempts to give an overview of the pricing of risks in a pure exchange economy, where trade takes place at time zero and where uncertainty is revealed at time one. An economic equilibrium model under uncertainty is formulated, where conditions characterizing a Pareto optimal exchange equilibrium are derived. We present two sets of sufficient conditions for the existence of an equilibrium, and demonstrate how equilibria can be characterized through several examples. Uniqueness of equilibrium is also discussed. Special attention is given to the principal components that the premiums in a reinsurance market must depend upon. We also apply the general theory to the risk exchange problem between a policyholder and an insurer, and in particular we compute market premiums of the resulting optimal contracts.

It is emphasized throughout how the formulation of a competitive equilibrium, rather than merely a general risk exchange formulation, is of particular interest in deriving a well-defined and unique set of equilibrium premiums in an insurance market. The theory is put into a framework which is fruitful for extensions beyond the one-period case.

Keywords

Reinsurance marketcompetitive equilibriumuniqueness of premiumsPareto optimalityrisk exchangeprivate insuranceCAPMrisk tolerancecomplete markets
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 23 , Issue 2 , November 1993 , pp. 185 - 211
Copyright
Copyright © International Actuarial Association 1993

References

REFERENCES

Aase, Knut K. (1993) Premiums in a Dynamic Model of a Reinsurance Market. Scandinavian Actuarial Journal (to appear).Google Scholar
Aase, Knut K. (1992) Dynamic Equilibrium and the structure of Premiums in a Reinsurance Market. The Geneva Papers on Risk and Insurance Theory 17:2, 93–136.CrossRefGoogle Scholar
Aase, Knut K. (1990) Stochastic equilibrium and premiums in insurance. In: Approche Actuarielle des Risques Financiers, 1er Colloque International AFIR, -Paris59–79.Google Scholar
Aase, Knut K. (1988) Contingent claims valuation when the security price is a combination of an Ito process and a random point process. Stochastic Processes and their Applications 28, 185–220.CrossRefGoogle Scholar
Arrow, K. J. (1953) Le rôle des valeurs boursiers pour la répartition la meilleure des risques. Colloques Internationaux du CNRS, XL, Paris41–88.Google Scholar
Arrow, K. J. (1974) Optimal insurance and generalized deductibles. Skandinavisk Aktuarietidsskrift, 1–42.Google Scholar
Black, F. (1972) Capital market equilibrium with restricted borrowing. Journal of Business 45,444–454.CrossRefGoogle Scholar
Borch, K. H. (1960) The safety loading of reinsurance premiums. Skandinavisk Aktuarietidsskrift, 163–184.Google Scholar
Borch, K. H. (1962) Equilibrium in a reinsurance market. Econometrica 30, 424–444.CrossRefGoogle Scholar
Borch, K. H. (1967) The theory of risk. Journ. of the Royal Statist. Soc, B 29, 432–452.Google Scholar
Borch, K. H. (1968) General equilibrium in the economics of uncertainty. In: Risk and Uncertainty, (Editors: Borch, Karl Henrik and Mossin, Jan), London, Macmillan.Google Scholar
Borch, K. H. (1985) A theory of insurance premiums. The Geneva Papers on Risk and Insurance 10, 192–208.CrossRefGoogle Scholar
Borch, K. H. (1990) Economics of Insurance. Advanced Textbooks in Economics Series. (Editors: Aase, Knut K. and Sandmo, Agnar), North Holland; Amsterdam, New York, Oxford, Tokyo.Google Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin 11, 52–60.CrossRefGoogle Scholar
Bühlmann, H. (1984) The general economic premium principle. ASTIN Bulletin 14, 13–21.CrossRefGoogle Scholar
Bühlmann, H. and Jewell, W. S. (1979) Optimal risk exchanges. ASTIN Bulletin 10; 3, 249.CrossRefGoogle Scholar
Chamberlain, G. (1983) A characterization of the distributions that imply mean-variance utility functions. Journal of Economic Theory 29, 185–201.CrossRefGoogle Scholar
Cramér, H. (1930) On the mathematical theory of risk. Skandia Jubilee Volume. Stockholm.Google Scholar
Duffie, D. (1991) The theory of value in security markets. In: Handbook of Mathematical Economics vol. IV, 1615–1682. (Editors: Hildenbrand, W. and Sonneschein, H.), North Holland; Amsterdam, New York, Oxford, Tokyo.Google Scholar
Duffie, D. and Zame, W. (1989) The consumption-based capital asset pricing model. Econometrica 57, 1279–1297.CrossRefGoogle Scholar
Dreze, J. H. (1987) Essays on economic decisions under uncertainly. Cambridge University Presss; Cambridge, New York.CrossRefGoogle Scholar
Du Mouchel, W. (1968) The Pareto-optimality on an n-company reinsurance treaty. Skandinavisk Aktuarietidsskrift 51, 165–170.Google Scholar
Hart, O. D. (1974) On the existence of equilibrium in a securities model. J. Econ. Theory 9, 293–311.CrossRefGoogle Scholar
Henriet, D. et Rouchet, J.C. (1991). Microeconomie de l'assurance, Economics, Paris.Google Scholar
Holmström, B. (1979) Moral hazard and observability. Bell Journal of Economics 10, 74–91.CrossRefGoogle Scholar
Karatzas, I. (1989) Optimization problems in the theory of continuous trading. SIAM J. Control and Optimization 27; 6, 1221–1269.CrossRefGoogle Scholar
Karatzas, I.Lehoczky, J.P. and Shreve, S. E. (1990) Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model. Mathematics of Operations Research 15; 1, 80–128.CrossRefGoogle Scholar
Landsberger, M. and Meilijson, I. (1990) Lotteries, insurance, and star-shaped utility functions. Journal of Economic Theory 52, 1–17.CrossRefGoogle Scholar
Lemarie, J. (1990) Borch' s Theorem: A historical survey of applications. In : Risk, Information and Insurance. Essays in the Memory of Karl H. Borch (Editor: Loubergé, H.). Kluwer Academic Publishers; Boston, Dordrecht, London.Google Scholar
Lienhard, M. (1986) Calculation of price equilibria for utility functions of the HARA class. ASTIN Bulletin 16, 91–97.CrossRefGoogle Scholar
Lintner, J. (1965) The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13–37.CrossRefGoogle Scholar
Mas-Colell, A. (1986) The price equilibrium existence problem in topological vector lattices. Econometrica 54, 1039–1054.CrossRefGoogle Scholar
Mas-Colell, A. and Zame, W. R. (1991) Equilibrium theory in infinite dimensional spaces. In: Handbook of Mathematical Economics vol. IV, 1835–1898. (Editors: Hildenbrand, W. and Sonnenschein, H.), North Holland; Amsterdam, New York, Oxford, Tokyo.Google Scholar
Merton, R. C. (1973) An intertemporal capital asset pricing model. Econometrica 41, 867–887.CrossRefGoogle Scholar
Moffet, D. (1979) The risk sharing problem. The Geneva Papers on Risk and Insurance 11, 5–13.Google Scholar
Mossin, J. (1966) Equilibrium in a capital asset market. Econometrica 34, 768–783.CrossRefGoogle Scholar
Mossin, J. (1968) Aspects of rational insurance purchasing. Journal of Political Economy 76, 553–568.CrossRefGoogle Scholar
Nielsen, L.T. (1990) Existence of Equilibrium in CAPM. Journal of Economic Theory 52, 223–231.CrossRefGoogle Scholar
Owen, J. and Rabinovitch, R. (1983) On the class of elliptical distributions and their applications to the theory of portfolio choice. J. Finance 38, 745–752.CrossRefGoogle Scholar
Raviv, A. (1979) The design of an optimal insurance policy. American Economic Review 69, 84–96.Google Scholar
Ross, S.A. (1976) The arbitrage theory of capital asset pricing. Journal of Economic Theory, 341–360.Google Scholar
Rothschild, M. and Stiglitz, J. (1976) Equilibrium in competitive insurance markets. An essay on the economics of imperfect information. Quarterly Journal of Economics 90, 629–650.CrossRefGoogle Scholar
Rubinstein, M. E. (1973) A comparative static analysis of risk premiums. Journal of Business 46, 605–615.CrossRefGoogle Scholar
Sharpe, W. F. (1964) Capital asset prices. A theory of market equilibrium under conditions of risk. Journal of Finance 19, 425–442.Google Scholar
Townsend, R. (1979) Optimal contracts and competitive markets with costly state verification. J. Econ. Theory 21, 265–293.CrossRefGoogle Scholar
Tucker, H. G. (1967) A graduate course in probability. Academic Press, New York and London.Google Scholar
Wei, K.C.J. and Lee, C.F. (1988) The generalized Stein/Rubinstein covariance formula and its application to estimate real systematic risk. Management Science 34, 1266–1270.CrossRefGoogle Scholar
Wilson, R. (1968) The theory of syndicates. Econometrica 36, 119–131.CrossRefGoogle Scholar
Wyler, E. (1990) Pareto optimal risk exchanges and a system of differential equations: A duality theorem. ASTIN Bulletin 20, 23–32.CrossRefGoogle Scholar