# Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

Listed:
• Azcue, Pablo
• Muler, Nora

## Abstract

We consider that the surplus of an insurance company follows a Cramer-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks. [Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide. We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.

## Suggested Citation

• Azcue, Pablo & Muler, Nora, 2009. "Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 26-34, February.
• Handle: RePEc:eee:insuma:v:44:y:2009:i:1:p:26-34
as

File URL: http://www.sciencedirect.com/science/article/pii/S0167-6687(08)00124-8

As the access to this document is restricted, you may want to search for a different version of it.

## References listed on IDEAS

as
1. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
2. Gaier, Johanna & Grandits, Peter, 2002. "Ruin probabilities in the presence of regularly varying tails and optimal investment," Insurance: Mathematics and Economics, Elsevier, vol. 30(2), pages 211-217, April.
3. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, vol. 6(2), pages 227-235.
4. Vila, Jean-Luc & Zariphopoulou, Thaleia, 1997. "Optimal Consumption and Portfolio Choice with Borrowing Constraints," Journal of Economic Theory, Elsevier, vol. 77(2), pages 402-431, December.
5. Paulsen, Jostein, 1998. "Sharp conditions for certain ruin in a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 135-148, June.
6. Bayraktar, Erhan & Young, Virginia R., 2007. "Minimizing the probability of lifetime ruin under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 196-221, July.
7. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as

Cited by:

1. Li, Zhongfei & Zeng, Yan & Lai, Yongzeng, 2012. "Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 191-203.
2. Raluca Vernic, 2016. "Optimal investment with a constraint on ruin for a fuzzy discrete-time insurance risk model," Fuzzy Optimization and Decision Making, Springer, vol. 15(2), pages 195-217, June.
3. Arash Fahim & Lingjiong Zhu, 2016. "Asymptotic Analysis for Optimal Dividends in a Dual Risk Model," Papers 1601.03435, arXiv.org, revised Feb 2016.
4. Arash Fahim & Lingjiong Zhu, 2015. "Optimal Investment in a Dual Risk Model," Papers 1510.04924, arXiv.org.
5. Tatiana Belkina & Christian Hipp & Shangzhen Luo & Michael Taksar, 2011. "Optimal Constrained Investment in the Cramer-Lundberg model," Papers 1112.4007, arXiv.org.
6. Pablo Azcue & Nora Muler, 2013. "Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(2), pages 177-206, April.
7. repec:spr:compst:v:77:y:2013:i:2:p:177-206 is not listed on IDEAS
8. repec:eee:insuma:v:74:y:2017:i:c:p:7-19 is not listed on IDEAS
9. Yi, Bo & Li, Zhongfei & Viens, Frederi G. & Zeng, Yan, 2013. "Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 601-614.

### Keywords

Cramer-Lundberg process Ruin probability Insurance Portfolio optimization Borrowing constraints Hamilton-Jacobi-Bellman equation;

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:44:y:2009:i:1:p:26-34. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.