Mixed Band Control of Mutual Proportional Reinsurance

Affiliation(s)

Department of Mathematics, University of Missouri, Columbia, USA.

College of Business, City University of Hong Kong, Hong Kong, China.

Center for Transport, Trade and Financial Studies, City University of Hong Kong, Hong Kong, China.

Department of Mathematics, University of Missouri, Columbia, USA.

College of Business, City University of Hong Kong, Hong Kong, China.

Center for Transport, Trade and Financial Studies, City University of Hong Kong, Hong Kong, China.

ABSTRACT

In this paper, we investigate the optimization of mutual proportional reinsurance—a mutual reserve system that is in- tended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual in- surance and reserve banks in the US Federal Reserve, where a mutual member is both an insurer and an insured. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (*i.e*., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (*i.e.*, both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control (*a*, *A*, *B*, *b*) with *a*=0 and *a*<*A*<*B*<*b*, coupled with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost. That is, when the reserve position reaches a lower boundary of *a*=0, the reserve should immedi- ately be raised to level *A*; when the reserve reaches an upper boundary of *b*, it should immediately be reduced to a level* **B*. An interesting finding produced by the study reported in this paper is that there exists a situation such that if the up- side fixed cost is relatively large in comparison to a finite threshold, then the optimal band control is reduced to a downside only (*i.e*., dividend payment only) control in the form of (0, 0; *B*, *b*) with *a*=*A*=0. In this case, it is opti- mal for the mutual insurance firm to go bankrupt as soon as its reserve level reaches zero, rather than to jump restart by calling for additional contingent funds. This finding partially explains why many mutual insurance companies, that were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.

Cite this paper

M. Taksar, J. Liu and J. Yuan, "Mixed Band Control of Mutual Proportional Reinsurance,"*Journal of Mathematical Finance*, Vol. 3 No. 2, 2013, pp. 256-267. doi: 10.4236/jmf.2013.32025.

M. Taksar, J. Liu and J. Yuan, "Mixed Band Control of Mutual Proportional Reinsurance,"

References

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[14] A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, “Classical and Impulse Stochastic Control for the Optimization of the Dividend and Risk Policies of an Insurance Firm,” Mathematical Finance, Vol. 16, No. 1, 2006, pp. 181-202. doi:10.1111/j.1467-9965.2006.00267.x

[15] A. Cadenillas and F. Zapatero, “Optimal Central Bank Intervention in the Foreign Exchange Market,” Journal of Economic Theory, Vol. 87, No. 1, 1999, pp. 218-242. doi:10.1006/jeth.1999.2523

[16] J. M. Harrison, T. M. Sellke and A. J. Taylor, “Impulse control of Brownian Motion,” Mathematics of Operations Research, Vol. 8, No. 3,1983, pp. 454-466. doi:10.1287/moor.8.3.454

[17] J. Paulsen, “Optimal Dividend Payments and Reinvestments of Diffusion Processes with Both Fixed and Proportional Costs,” SIAM Journal on Control and Optimization, Vol. 47, No. 5, 2008, pp. 2201-2226. doi:10.1137/070691632

[18] J. Eisenberg, “Optimal Control of Capital Injections by Reinsurance and Investments,” Blatter der DGVFM, Vol. 31, No. 2, 2010, pp. 329-345. doi:10.1007/s11857-010-0124-0.

[1] G. M. Constantinides, and S. F. Richard, “Existence of Optimal Simple Policies for Discounted-cost Inventory and Cash Management in Continuous Time,” Operations Research, Vol. 26, No. 4, 1978, pp. 620-636. doi:10.1287/opre.26.4.620

[2] A. Bensoussan, R. H. Liu and S. P. Sethi, “Optimality of an (s, S) Policy with Compound Poisson and Diffusion Demands: AQuasi-variationalInequalities Approach,” SIAM Journal on Control and Optimization. Vol. 44, No. 5, 2006, pp. 1650-1676. doi:10.1137/S0363012904443737

[3] A. Cadenillas and F. Zapatero, “Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rate and Reserves,” Mathematical Finance, Vol. 10, No. 2, 2000, pp. 141-156. doi:10.1111/1467-9965.00086

[4] B. M. Hojgaard and M. Taksar, “Optimal Dynamic Portfolio Selection for a Corporation with Controllable Risk and Dividend Distribution Policy,” Quantitative Finance, Vol. 4, No. 3, 2004, pp. 256-265. doi:10.1088/1469-7688/4/3/007

[5] J. Eisenberg and H. Schmidli,“Minimizing Expected Discounted Capital Injections by Reinsurance in a Classical Risk Model,” Scandinavian Actuarial Journal, No. 3, 2011,pp. 155-176. doi:10.1080/03461231003690747

[6] A. Sulem, “A Solvable One-dimensional Model of A Diffusion Inventory System,” Mathematics of Operations Research, Vol. 11, No. 1, 1986, pp. 125-133. doi:10.1287/moor.11.1.125

[7] J. Yuan, J, “Computational Optimization of Mutual Insurance Systems: A Quasi-variational Inequality Approach,” Ph.D. Thesis, The Hong Kong Polytechnic University, 2008.

[8] M. Dawande, M. Mehrotra, V. Mookerjee and C. Srikandarajah, “An Analysis of Coordination Mechanism for the U.S. Cash Supply Chain,” Management Science, Vol. 56, No. 3, 2010, pp. 553-570. doi:10.1287/mnsc.1090.1106

[9] A. Lokka and A. M. Zervos, “Optimal Dividend and Issuance of Equity Policies in the Presence of Proportional Costs,” Insurance: Mathematics and Economics, Vol. 42, No. 3, 2008, pp. 954-961. doi:10.1016/j.insmatheco.2007.10.013

[10] A. Bensoussan and J. L. Menaldi, “Stochastic HybridControl,” Journal of Mathematical Analysis and Applications, Vol. 249, No. 1, 2000, pp. 261-288. doi:10.1006/jmaa.2000.7102

[11] M. S. Branicky, V. S. Borkar and S. K. Mitter, “A Unified Framework for Hybrid Control: Model and Optimal Control Theory,” IEEE Transactions on Automatic Control, Vol. 43, No. 1, 1998, pp. 31-45. doi:10.1109/9.654885

[12] A. Abate, A. D. Ames and S. Sastry, “Stochastic Approximations for Hybrid Systems,” Proceedings of the 24th American Control Conference, Portland, 2005, pp. 1557-1562.

[13] G. M. Constantinides, “Stochastic Cash Management with Fixed and Proportional Transaction Costs,” Management Science, Vol. 22, No. 12, 1976, pp. 1320-1331. doi:10.1287/mnsc.22.12.1320

[14] A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, “Classical and Impulse Stochastic Control for the Optimization of the Dividend and Risk Policies of an Insurance Firm,” Mathematical Finance, Vol. 16, No. 1, 2006, pp. 181-202. doi:10.1111/j.1467-9965.2006.00267.x

[15] A. Cadenillas and F. Zapatero, “Optimal Central Bank Intervention in the Foreign Exchange Market,” Journal of Economic Theory, Vol. 87, No. 1, 1999, pp. 218-242. doi:10.1006/jeth.1999.2523

[16] J. M. Harrison, T. M. Sellke and A. J. Taylor, “Impulse control of Brownian Motion,” Mathematics of Operations Research, Vol. 8, No. 3,1983, pp. 454-466. doi:10.1287/moor.8.3.454

[17] J. Paulsen, “Optimal Dividend Payments and Reinvestments of Diffusion Processes with Both Fixed and Proportional Costs,” SIAM Journal on Control and Optimization, Vol. 47, No. 5, 2008, pp. 2201-2226. doi:10.1137/070691632

[18] J. Eisenberg, “Optimal Control of Capital Injections by Reinsurance and Investments,” Blatter der DGVFM, Vol. 31, No. 2, 2010, pp. 329-345. doi:10.1007/s11857-010-0124-0.