Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T02:05:14.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal strategies for a non-linear premium-reserve model in a competitive insurance market

Published online by Cambridge University Press:  22 September 2016

Athanasios A. Pantelous  and
Eudokia Passalidou
Athanasios A. Pantelous*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L697ZL, UK Institute for Risk and Uncertainty, University of Liverpool, Liverpool L697ZL, UK
Eudokia Passalidou
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L697ZL, UK
*
*Correspondence to: Dr Athanasios A. Pantelous, Department of Mathematical Sciences, Institute for Risk and Uncertainty, University of Liverpool, Peach Street, Liverpool L697ZL, UK. Tel: +44 151 794 5079. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The calculation of a fair premium is always a challenging topic in the real-world insurance applications. In this paper, a non-linear premium-reserve (P-R) model is presented and the premium is derived by minimising a quadratic performance criterion. The reserve is a stochastic equation, which includes an additive random non-linear function of the state, premium and not necessarily Gaussian noise, which is, however, independently distributed in time, provided only that the mean value and the covariance of the random function is 0 and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further by the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company’s reputation. The quadratic utility function concerns the present value of the reserve. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also dependent on the company’s reserve among the other parameters. Finally, a numerical application illustrates the main findings of the paper.

Keywords

Non-linear premium-reserve pricing modelStochastic optimal controlQuadratic performance criterionCompetitive insurance marketsActuarial risk
Type
Papers
Information
Annals of Actuarial Science , Volume 11 , Issue 1 , March 2017 , pp. 1 - 19
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

Ahlgrim, K.C. & D’Arcy, S.P. (2012). The effect of deflation or high inflation on the insurance industry, Casualty Actuarial Society, Canadian Institute of Actuaries and Society of Actuaries.Google Scholar
Borch, K. (1974). The Mathematical Theory of Insurance. D.C. Heath and Co, Lexington, MA.Google Scholar
Borch, K. (1990). Economics of Insurance. North-Holland, Amsterdam.Google Scholar
Clapp, J. (1985). Quantity competition in spatial markets with incomplete information. Journal of Economics, 100(2), 519–528.Google Scholar
Cretu, A.E. & Brodie, R.J. (2007). The influence of brand image and company reputation where manufacturers market to small firms: a customer value perspective. Industrial Marketing Management, 36(2), 230–240.Google Scholar
D’Arcy, S.P. (1982). A strategy for property-liability insurers in inflationary times. Proceedings of the Casualty Actuarial Society, 69, 163–186.Google Scholar
Dutang, C., Albrecher, H. & Loisel, S. (2013). Competition among non-life insurers under solvency constraints: a game-theoretic approach. European Journal of Operational Research, 231(3), 702–711.Google Scholar
Emms, P. (2007 a). Dynamic pricing of general insurance in a competitive market. ASTIN Bulletin, 37(1), 1–34.Google Scholar
Emms, P. (2007 b). Pricing general insurance with constraints. Insurance: Mathematics and Economics, 40(2), 335–355.Google Scholar
Emms, P. (2008). A stochastic demand model for optimal pricing of non-life insurance policies. Mathematical Control Theory and Finance, Springer-Verlang Berlin Heidelberg, 113–136.Google Scholar
Emms, P. (2011). Pricing general insurance in a reactive and competitive market. Journal of Computational and Applied Mathematics, 236(6), 1314–1332.Google Scholar
Emms, P. (2012). Equilibrium pricing of general insurance policies. North American Actuarial Journal, 16(3), 323–349.Google Scholar
Emms, P. & Haberman, S. (2005). Pricing general insurance using optimal control theory. ASTIN Bulletin, 35(2), 427–453.Google Scholar
Emms, P., Haberman, S. & Savoulli, I. (2007). Optimal strategies for pricing general insurance. Insurance: Mathematics and Economics, 40(1), 15–34.Google Scholar
Fanti, L., Gori, L. & Sodini, M. (2013). Complex dynamics in an OLG model of neoclassical growth with endogenous retirement age and public pensions. Nonlinear Analysis: Real World Applications, 14(1), 829–841.Google Scholar
Gerber, H.U. & Pafumi, G. (1999). Utility functions: from risk theory to finance. North American Actuarial Journal, 2(3), 74–100.Google Scholar
Huang, H.-H., Shiu, Y.-M. & Wang, C.-P. (2011). Optimal insurance contract with stochastic background wealth. Scandinavian Actuarial Journal, 2, 119–139.Google Scholar
Jacobson, D.H. (1974). A general result in stochastic optimal control of nonlinear discrete-time systems with quadratic performance criteria. Journal of Mathematical Analysis and Applications, 47, 153–161.Google Scholar
Kremer, E. (2004). Estimating the loading of the largest claims covers. Nonlinear Analysis: Real World Applications, 5(4), 711–723.Google Scholar
Kremer, E. (2006). Net premium of the drop down excess of loss cover. Nonlinear Analysis: Real World Applications, 7(3), 478–485.Google Scholar
Krivo, R. (2009). An update to D’Arcy’s: “A strategy for property-liability insurers in inflationary times”, Casualty Actuarial Society E-Forum.Google Scholar
Kushner, H.J. (1970). An Introduction to Stochastic Control Theory. John Wiley & Sons, USA.Google Scholar
Lai, L.-H. (2011). Effects of insurance premium and deductible on production. Nonlinear Analysis: Real World Applications, 12(3), 1354–1358.Google Scholar
Lee, C.-C. & Chiu, Y.-B. (2012). The impact of real income on insurance premiums: evidence from panel data. International Review of Economics & Finance, 21(1), 246–260.CrossRefGoogle Scholar
Lee, C.-C., Chiu, Y.-B. & Chang, C.-H. (2013). Insurance demand and country risks: a nonlinear panel data analysis. Journal of International Money and Finance, 36, 68–85.Google Scholar
Lorent, B. (2010). The link between insurance and banking sectors: an international cross-section analysis of life insurance demand, CEB Working Paper No. 10/040, Solvay Brussels School of Economics and Management, Brussels.Google Scholar
Lowe, S. & Warren, R. (2010). Post-recession inflation: an emerging risk for P&C insurers. Emphasis, 3, 24–29.Google Scholar
Ohlsson, E. & Johansson, B. (2010). Non-Life Insurance Pricing with Generalized Linear Models. Springer Science & Business Media, Berlin.Google Scholar
Pantelous, A.A. & Passalidou, E. (2013). Optimal premium pricing policy in a competitive insurance market environment. Annals of Actuarial Science, 7(2), 175–191.CrossRefGoogle Scholar
Pantelous, A.A. & Passalidou, E. (2015). Optimal premium pricing strategies for competitive general insurance markets. Applied Mathematics and Computation, 259, 858–874.Google Scholar
Rothschild, M. & Stiglitz, J. (1992). Equilibrium in competitive insurance markets: an essay on the economics of imperfect information, foundations of insurance economics, Huebner International Series on Risk. Insurance and Economic Security, 14, 355–375.Google Scholar
Schlesinger, H. & Doherty, N. (1985). Incomplete markets for insurance: an overview. Journal of Risk and Insurance, 52, 402–423.Google Scholar
Searle, S.R. & Willett, L.S. (2001). Matrix Algebra for Applied Economics. Wiley Series in Probability and Statistics, NY, USA.Google Scholar
Simon, C.P. & Blume, L. (1994). Mathematics for Economists. Norton, New York.Google Scholar
Taylor, G.C. (1986). Underwriting strategy in a competitive insurance environment. Insurance: Mathematics and Economics, 5(1), 59–77.Google Scholar
Taylor, G.C. (1987). Expenses and underwriting strategy in competition. Insurance: Mathematics and Economics, 6(4), 275–287.Google Scholar
Von Neumann, J. & Morgenstern, O. (1947). Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ.Google Scholar
Wu, R. & Pantelous, A.A. (2016). Potential games with aggregation in non-cooperative general insurance markets. ASTIN Bulletin (to appear).Google Scholar