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PORTFOLIO SELECTION BY MINIMIZING THE PRESENT VALUE OF CAPITAL INJECTION COSTS

Published online by Cambridge University Press:  15 September 2014

Ming Zhou  and
Kam C. Yuen
Ming Zhou*
Affiliation:
China Institute for Actuarial Science, Central University of Finance and Economics, 39 South College Road, Haidian, Beijing 100081, China
Kam C. Yuen
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam, Hong Kong, China E-Mail: kcyuen@hku.hk
*
E-Mail: mzhou.act@gmail.com
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Abstract

This paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

Keywords

Backward Euler methodcapital injectionHJB equationportfolio selectiontransaction costs
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 1 , January 2015 , pp. 207 - 238
Copyright
Copyright © ASTIN Bulletin 2014 

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References

Bai, L. and Guo, J. (2008) Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance: Mathematics and Economics, 42 (3), 968–975.Google Scholar
Borodin, A.N. and Salminen, P. (2002) Handbook of Brownian Motion – Facts and Formulae, 2nd ed.Basel: Birkhäuser.Google Scholar
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 (4), 937–958.Google Scholar
Cadenillas, A., Choulli, T., Taksar, M. and Zhang, L. (2006) Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm. Mathematical Finance, 16 (1), 181–202.Google Scholar
Eisenberg, J. (2010) On optimal control of capital injections by reinsurance and investments. Blätter DGVFM, 31, 329–345.Google Scholar
Eisenberg, J. and Schmidli, H. (2009) Optimal control of capital injections by reinsurance in a diffusion approximation. Blätter DGVFM, 30, 1–13.CrossRefGoogle Scholar
Fleming, W.H. and Soner, M. (2006) Controlled Markov Processes and Viscosity Solutions, 2nd ed.New York: Springer.Google Scholar
Gerber, H. and Yang, H. (2007) Absolute ruin probabilities in a jump diffusion risk model with investment. The North American Actuarial Journal, 11 (3), 159–169.Google Scholar
He, L. and Liang, Z. (2009) Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insurance: Mathematics and Economics, 44, 88–94.Google Scholar
H⊘jgaard, B. and Taksar, M. (2001) Optimal risk control for a large corporation in the presence of returns on investments. Finance and Stochastics, 5 (4), 527–547.Google Scholar
H⊘jgaard, B. and Taksar, M. (2004) Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quantitative Finance, 4 (3), 315–327.Google Scholar
Kulenko, N. and Schimidli, H. (2008) Optimal dividend strategies in a Cramér-Lundberg model with capital injections. Insurance: Mathematics and Economics, 43, 270–278.Google Scholar
Liang, Z. and Huang, J. (2011) Optimal dividend and investing control of an insurance company with higher solvency constraints. Insurance: Mathematics and Economics, 49, 501–511.Google Scholar
Liu, C. S. and Yang, H. (2004) Optimal investment for an insurer to minimize its probability of ruin. North American Actuarial Journal, 8 (2), 11–31.Google Scholar
Luo, S., Taksar, M. and Tsoi, A. (2008) On reinsurance and investment for large insurance portfolios. Insurance: Mathematics and Economics, 42, 434–444.Google Scholar
L⊘kka, A. and Zervos, M. (2008) Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance: Mathematics and Economics, 42, 954–961.Google Scholar
Paulsen, J. (2008) Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs. SIAM Journal on Control and Optimization, 47 (5), 2201–2226.Google Scholar
Promislow, S.D. and Young, V.R. (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. The North American Actuarial Journal, 9 (3), 109–128.Google Scholar
Scheer, N. and Schmidli, H. (2011) Optimal dividend strategies in a Cramer–Lundberg model with capital injections and administration costs. European Actuarial Journal, 1, 57–92.CrossRefGoogle Scholar
Schmidli, H. (2008) Stochastic Control in Insurance. London: Springer-Verlag.Google Scholar
Sethi, S.P. and Taksar, M. (2002) Optimal financing of a corporation subject to random returns. Mathematical Finance, 12 (2), 155–172.Google Scholar
Yao, D., Yang, H. and Wang, R. (2011) Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. The European Journal of Operational Research, 3, 568–576.Google Scholar
Zhang, X. and Siu, T.K. (2009) Optimal investment and reinsurance of an insurer with model uncertainty. Insurance: Mathematics and Economics, 45, 81–88.Google Scholar
Zhou, M. and Yuen, K.C. (2012) Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle. Economic Modelling, 29, 198–207.Google Scholar