Optimal Portfolios of an Insurer and a Reinsurer under Proportional Reinsurance and Power Utility Preference

Silas A.  Ihedioha; Bright O.  Osu

doi:10.4236/oalib.1102033

Silas A. Ihedioha¹, Bright O. Osu²

Abstract: This study tackled portfolio selection problem for an insurer as well as a reinsurer aiming at maximizing the probability of survival of the Insurer and the Reinsurer, to assess the impact of proportional reinsurance on the survival of insurance companies as well as to determine the condition that would warrant reinsurance according to the optimal reinsurance proportion chosen by the insurer. It was assumed the insurer’s and the reinsurer’s surplus processes were approximated by Brownian motion with drift and the insurer could purchase proportional reinsurance from the reinsurer and their risk reserves followed Brownian motion with drift. Obtained were Hamilton-Jacobi-Bellman (HJB) equations which solutions gave the optimized values of the insurer’s and the reinsurer’s optimal investments in the risky asset and the value of the discount rate that would warrant reinsurance as a ratio of their portfolio weights in the risky asset.

Keywords: Proportional Reinsurance, Optimal Investment, Insurer, Reinsurer, Hamilton-Jacobi-Bellman (HJB) Equation, Power Utility Function

Cite this paper: Ihedioha, S. and Osu, B. (2015) Optimal Portfolios of an Insurer and a Reinsurer under Proportional Reinsurance and Power Utility Preference. Open Access Library Journal, 2, 1-11. doi: 10.4236/oalib.1102033.

References

[1] Centeno, M.L. and Simoes, O. (2009) Optimal Reinsurance. Real Academia de Ciencias Serie A. Matematica, Espana, 2, 387-405.

[2] Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S S Huebner Foundation Monographs, University of Pensylvama Waters H, 1979, Excess of Loss Reinsurance Limits, 37-43.

[3] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1987) Actuarial Mathematics. Society of Actuaries, Chicago.

[4] Centeno, M.L (1986) Measuring the Effect of Reinsurance by the Adjustment Coefficient. Insurance: Mathematics and Economics, 5, 169-182.
http://dx.doi.org/10.1016/0167-6687(86)90043-0

[5] Hesselager O, 1990, Some results on Optimal Reinsurance in terms of the Adjustment Coefficient. Scandnavian Actuarial Journal.80-95.

[6] Schmidli, H. (2000) Optimal Proportional Reinsurance Policies in a Dynamic Setting. Research Report 403, Arhus University, Arhus.

[7] Hipp, C. and Vogt, M. (2003) Optimal Dynamical XL Reinsurance. ASTIN Bulletin, 33, 193-207.
http://dx.doi.org/10.2143/AST.33.2.503690

[8] Schmidli, H. (2002) On Minimizing the Ruin Probability by Investment and Reinsurance. Annals of Applied Probability, 12, 890-907.
http://dx.doi.org/10.1214/aoap/1031863173

[9] Taksar, M.I. and Markussen, C. (2003) Optimal Dynamic Reinsurance Policies for Large Insurance Portfolios. Finance and Stochastics, 7, 97-121.
http://dx.doi.org/10.1007/s007800200073

[10] Hipp, C. and Plum, M. (2000) Optimal Investment for Insurers. Insurance: Mathematics and Economics, 27, 215-228.
http://dx.doi.org/10.1016/S0167-6687(00)00049-4

[11] Browne, S. (1995) Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research, 20, 937-958.
http://dx.doi.org/10.1287/moor.20.4.937

[12] Liu, C.S. and Yang, H. (2004) Optimal Investment for an Insurer to Minimize Its Probability of Ruin. North American Actuarial Journal, 2, 11-31.
http://dx.doi.org/10.1080/10920277.2004.10596134

[13] Castillo, M.T. and Parrocha, G. (2003) Stochastic Control Theory for Optimal Investment. Working Paper, Department of Actuarial Studies, University of New South Wales, Sydney.

[14] Irgens, C. and Paulsen, J. (2004) Optimal Control of Risk Exposure, Reinsurance and Investments for Insurance Portfolios. Insurance, Mathematics and Economics, 35, 21-51.
http://dx.doi.org/10.1016/j.insmatheco.2004.04.004

[15] Paulsen, J., Kasozi, J. and Steigen, A. (2005) A Numerical Method to Find the Probability of Ultimate Ruin in the Classical Risk Model with Stochastic Return on Investments.

[16] Paulsen, J. (2008) Ruin Models with Investment Income. Probability Surveys, 5, 416-434.
http://dx.doi.org/10.1214/08-PS134

[17] Meng, H. and Zhang, X. (2010) Optimal Risk Control for the Excess-of-Loss Reinsurancepolicies. Astin Bulletin, 40, 179-197.
http://dx.doi.org/10.2143/AST.40.1.2049224

[18] Kasumo, C. (2011) Minimizing Probability of Ultimate Ruin by Proportiona Reinsurance and Investment. Msc Dissertation, University of Dar es salaam, Dar es salaam.

[19] Mata, J.A. (2000) Pricing Excess of Loss Reinsurance with Reinstatement. Astin Bulletin, 30, 349-368.
http://dx.doi.org/10.2143/AST.30.2.504640

[20] Encarta World Dictionary, 1999.

[21] Danping, L.I. (2015) Optimal Investment Problem for an Insurer and a Reinsurer under the Proportional Reinsurance. WSEAS Transactions on Mathematics, 14, 20-35.

[22] Osu, B.O. and Ihedioha, S.A. (2012) Optimal Portfolio Selection for Pension Funds with Variable Rate of Return and Transaction Costs: Finite Horizon Case. Global Journal of Pure and Applied Mathematics, 8, 275-286.

[23] Osu, B.O., Ihedioha, S.A. and Adindu-Dick, J.I. (2014) On the Survival of Insurance Company’s Investment with Consumption under Power and Exponential Utility Functions. American Journal of Applied Mathematics, 2, 8-13.
http://dx.doi.org/10.11648/j.ajam.20140201.12

[24] Wokiyi, D. (2012) Maximizing Investment Returns of an Insurance Company While Minimizing the Probability of Ruin. Master’s Thesis, University of Dares-Salam, Dar es salaam, 1-60.

[25] Nie, M. (2010) Optimal Investment Policy for Pension Funds with Transaction Costs: The Finite Horizon. Master’s Thesis, Center Graduate School, Finance, Tilburg, 1-49.