# Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

## Abstract

**:**

## 1. Introduction

## 2. The Models

## 3. HJB, Integrodifferential and Integral Equations

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

- 1.
- For the case without diffusion (i.e., when ${\sigma}_{1}^{2}={\sigma}_{2}^{2}=0$), the kernel and forcing function are given, respectively, by$$\begin{array}{c}\hfill K(u,x)=-\frac{r+\lambda \overline{F}(u-kx)}{ru+{c}^{k}},\\ \hfill \alpha \left(u\right)=\frac{{c}^{k}}{ru+{c}^{k}}\varphi \left(0\right),\end{array}$$
- 2.
- For the case with diffusion (i.e., when ${\sigma}_{1}^{2}+{\sigma}_{2}^{2}>0$), the kernel and forcing function are, respectively,$$\begin{array}{c}\hfill K(u,x)=2\frac{(2r-3{\sigma}_{2}^{2}+\lambda )kx+{c}^{k}+\lambda G(u-kx)-(r-{\sigma}_{2}^{2}+\lambda )u}{{\sigma}_{2}^{2}{u}^{2}+{k}^{2}{\sigma}_{1}^{2}},\\ \hfill \alpha \left(u\right)=\left\{\begin{array}{cc}\frac{2{c}^{k}}{{\sigma}_{2}^{2}u}\varphi \left(0\right)\hfill & \mathit{if}{\sigma}_{1}^{2}=0,\hfill \\ \frac{{\sigma}_{1}^{2}u}{{\sigma}_{2}^{2}{u}^{2}+{k}^{2}{\sigma}_{1}^{2}}{\varphi}^{\prime}\left(0\right)\hfill & \mathit{if}{\sigma}_{1}^{2}0,\hfill \end{array}\right.\end{array}$$

**Proof.**

## 4. Numerical Results

#### 4.1. Proportional Reinsurance in the Cramér–Lundberg Model

**Example**

**1.**

**Example**

**2.**

#### 4.2. Proportional Reinsurance in the Cramér–Lundberg Model under Interest Force

**Example**

**3.**

**Example**

**4.**

#### 4.3. Proportional Reinsurance in the Diffusion-Perturbed Model

**Example**

**5.**

**Example**

**6.**

#### 4.4. Proportional Reinsurance in the Perturbed Model under Interest Force

**Example**

**7.**

**Example**

**8.**

#### 4.5. Proportional Reinsurance with Investments of Black–Scholes Type

**Example**

**9.**

**Example**

**10.**

#### 4.6. Sensitivity of Ruin Probability to Volatility of Stock Prices

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CLM | Cramér–Lundberg model |

DPM | Diffusion-Perturbed model |

NPC | Net profit condition |

SDE | Stochastic differential equation |

HJB | Hamilton–Jacobi–Bellman |

IDE | Integrodifferential equation |

VIDE | Volterra integrodifferential equation |

VIE | Volterra integral equation |

VIE-2 | Volterra integral equation of the second kind |

QS | Quota-share |

## References

- Bachelier, L. The theory of speculation. Annales Scientifiques de l’École Normale Supérieure
**1900**, 17, 21–86. [Google Scholar] [CrossRef] - Liang, Z.B.; Guo, J.Y. Upper bound for ruin probabilities under optimal investment and proportional reinsurance. Appl. Stoch. Model. Bus. Ind.
**2008**, 24, 109–128. [Google Scholar] [CrossRef] - Wang, N. Optimal investment for an insurer with exponential utility preferences. Insur. Math. Econ.
**2007**, 40, 77–84. [Google Scholar] [CrossRef] - Liang, Z.B.; Guo, J.Y. Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility. J. Appl. Math. Comput.
**2011**, 36, 11–25. [Google Scholar] [CrossRef] - Kasozi, J.; Mahera, C.W.; Mayambala, F. Controlling ultimate ruin probability by quota-share reinsurance arrangements. Int. J. Appl. Math. Stat.
**2013**, 49, 1–15. [Google Scholar] - Zhou, J.; Deng, Y.; Huang, Y.; Yang, X. Optimal proportional reinsurance and investment for a constant elasticity of variance model under variance principle. Acta Math. Sci.
**2015**, 35, 303–312. [Google Scholar] [CrossRef] - Liu, C.S.; Yang, H. Optimal investment for an insurer to minimize its ruin probability. N. Am. Actuar. J.
**2004**, 8, 11–31. [Google Scholar] [CrossRef] - Hipp, C.; Plum, M. Optimal investment for insurers. Insur. Math. Econ.
**2000**, 27, 215–228. [Google Scholar] [CrossRef][Green Version] - Schmidli, H. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab.
**2002**, 12, 890–907. [Google Scholar] [CrossRef] - Liang, X.; Young, V.R. Minimizing the probability of ruin: Optimal per-loss reinsurance. Insur. Math. Econ.
**2018**, 82, 181–190. [Google Scholar] [CrossRef] - Mossin, J. Aspects of rational insurance purchasing. J. Political Econ.
**1968**, 76, 553–568. [Google Scholar] [CrossRef] - Zhu, H.; Huang, Y.; Zhou, J.; Yang, X.; Deng, C. Optimal proportional reinsurance and investment problem with constraints on risk control in a general diffusion financial market. ANZIAM J.
**2016**, 57, 352–368. [Google Scholar] - Glineur, F.; Walhin, J.F. de Finetti’s retention problem for proportional reinsurance revisited. Math. Stat.
**2006**, 3, 451–462. [Google Scholar] [CrossRef] - Jang, B.-G.; Kim, K.T. Optimal reinsurance and asset allocation under regime switching. J. Bank. Financ.
**2015**, 56, 37–47. [Google Scholar] [CrossRef] - Zhang, X.; Liang, Z. Optimal layer reinsurance on the maximization of the adjustment coefficient. Numer. Algebra Control Optim.
**2016**, 6, 21–34. [Google Scholar] [CrossRef][Green Version] - Mikosch, T. Non-Life Insurance Mathematics: An Introduction with Stochastic Processes; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Dam, D.K.; Chung, N.Q. On finite-time ruin probabilities in a risk model under quota share reinsurance. Appl. Math. Sci.
**2017**, 11, 2609–2629. [Google Scholar] [CrossRef] - Ladoucette, S.A.; Teugels, J.L. Risk Measures for a Combination of Quota-Share and Drop Down Excess-Of-Loss Rinsurance Treaties. 2004. Available online: http://www.eurandom.nl/ (accessed on 15 December 2018).
- Lampaert, I.; Walhin, J.F. On the optimality of proportional reinsurance. Scand. Actuar. J.
**2005**, 2005, 225–239. [Google Scholar] [CrossRef][Green Version] - Hipp, C. Stochastic control with application in insurance. In Stochastic Methods in Finance; Springer: Berlin/Heidelberg, Germany, 2004; pp. 127–164. [Google Scholar]
- Taylor, G.; Buchanan, R. The Management of Solvency. In Classical Insurance Solvency Theory; Cummins, J.D., Derrig, R.A., Eds.; Kluwer Academic Publishers: Boston, MA, USA, 1988; pp. 49–151. [Google Scholar]
- Dufresne, F.; Gerber, H. Risk theory for the compound Poisson process that is perturbed by diffusion. Insur. Math. Econ.
**1991**, 10, 51–59. [Google Scholar] [CrossRef] - Morales, M. On the expected discounted penalty function for a perturbed risk process driven by a subordinator. Insur. Math. Econ.
**2007**, 40, 293–301. [Google Scholar] [CrossRef] - Sarkar, J.; Sen, A. Weak convergence approach to compound Poisson risk processes perturbed by diffusion. Insur. Math. Econ.
**2005**, 36, 421–432. [Google Scholar] [CrossRef] - Li, D.; Li, D.; Young, V.R. Optimality of excess-loss reinsurance under the mean-variance criterion. Insur. Math. Econ.
**2017**, 75, 82–89. [Google Scholar] [CrossRef] - Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes in FORTRAN 77: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]
- Young, A. The application of approximate product-integration to the numerical solution of integral equations. Proc. R. Soc. Lond. Ser. A
**1954**, 224, 561–573. [Google Scholar] - Katani, R.; Shahmorad, S. The block-by-block method with Romberg quadrature for the solution of nonlinear Volterra integral equations on large intervals. Ukr. Math. J.
**2012**, 64, 1050–1063. [Google Scholar] [CrossRef] - Linz, P. Analytical and Numerical Methods for Volterra Equations; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1985. [Google Scholar]
- Baharum, N.A.; Majid, Z.A.; Senu, N. Solving Volterra integrodifferential equations via diagonally implicit multistep block method. Int. J. Math. Math. Sci.
**2018**, 2018, 7392452. [Google Scholar] [CrossRef] - Gatto, R.; Baumgartner, B. Saddlepoint approximations to the probability of ruin in finite time for the compound Poisson risk process perturbed by diffusion. Methodol. Comput. Appl. Probab.
**2016**, 18, 217–235. [Google Scholar] [CrossRef] - Gatto, R.; Mosimann, M. Four approaches to compute the probability of ruin in the compound Poisson risk process with diffusion. Math. Comput. Model.
**2012**, 55, 1169–1185. [Google Scholar] [CrossRef] - Assari, P. The thin plate spline collocation method for solving integro-differential equations arisen from the charged particle motion in oscillating magnetic fields. Eng. Comput.
**2018**, 35, 1706–1726. [Google Scholar] [CrossRef] - Assari, P.; Dehghan, M. A local Galerkin integral equation method for solving integro-differential equations arising in oscillating magnetic fields. Mediterr. J. Math.
**2018**, 15, 90. [Google Scholar] [CrossRef] - Cardone, A.; Conte, D.; D’Ambrosio, R.; Paternoster, B. Collocation methods for Volterra integral and integro-differential equations: A review. Axioms
**2018**, 7, 45. [Google Scholar] [CrossRef] - Linz, P. A method for solving nonlinear Volterra integral equations of the second kind. Math. Comput.
**1969**, 23, 595–599. [Google Scholar] [CrossRef] - Saify, S.A.A. Numerical Methods for a System of Linear Volterra Integral Equations. Master’s Thesis, University of Technology, Baghdad, Iraq, 2005. [Google Scholar]
- Kasozi, J.; Paulsen, J. Flow of dividends under a constant force of interest. Am. J. Appl. Sci.
**2005**, 2, 1389–1394. [Google Scholar] [CrossRef] - Paulsen, J.; Kasozi, J.; Steigen, A. A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments. Insur. Math. Econ.
**2005**, 36, 399–420. [Google Scholar] [CrossRef] - Kasozi, J.; Paulsen, J. Numerical Ultimate Ruin Probabilities under Interest Force. J. Math. Stat.
**2005**, 1, 246–251. [Google Scholar] [CrossRef] - Paulsen, J. Optimal dividend payouts for diffusions with solvency constraints. Financ. Stoch.
**2003**, 7, 457–473. [Google Scholar] [CrossRef] - Paulsen, J.; Gjessing, H.K. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insur. Math. Econ.
**1997**, 20, 215–223. [Google Scholar] [CrossRef] - Paulsen, J. Ruin models with investment income. Probab. Surv.
**2008**, 5, 416–434. [Google Scholar] [CrossRef] - Kasumo, C.; Kasozi, J.; Kuznetsov, D. On minimizing the ultimate ruin probability of an insurer by reinsurance. J. Appl. Math.
**2018**, 2018, 9180780. [Google Scholar] [CrossRef] - Ma, J.; Bai, L.; Liu, J. Minimizing the probability of ruin under interest force. Appl. Math. Sci.
**2008**, 17, 843–851. [Google Scholar] - Schmidli, H. Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuar. J.
**2001**, 1, 55–68. [Google Scholar] [CrossRef] - Paulsen, J. Risk theory in a stochastic economic environment. Stoch. Proc. Appl.
**1993**, 46, 327–361. [Google Scholar] [CrossRef][Green Version] - Paulsen, J.; Gjessing, H.K. Ruin Theory with stochastic return on investments. Adv. Appl. Probab.
**1997**, 29, 965–985. [Google Scholar] [CrossRef] - Kasumo, C. Minimizing the Probability of Ultimate Ruin by Proportional Reinsurance and Investment. Master’s Thesis, University of Dar es Salaam, Dar es Salaam, Tanzania, 2011. [Google Scholar]
- Schmidli, H. Stochastic Control in Insurance; Springer: London, UK, 2008. [Google Scholar]
- Huang, J.; Tang, Y.; Vázquez, L. Convergence analysis of a block-by-block method for fractional differential equations. Numer. Math. Theor. Methods Appl.
**2012**, 5, 229–241. [Google Scholar] [CrossRef] - Kasumo, C.; Kasozi, J.; Kuznetsov, D. Dividend maximization in a diffusion-perturbed classical risk process compounded by proportional and excess-of-loss reinsurance. Int. J. Appl. Math. Stat.
**2018**, 57, 68–83. [Google Scholar] - Cheng, G.; Zhao, Y. Optimal risk and dividend strategies with transaction costs and terminal value. Econ. Model.
**2016**, 54, 522–536. [Google Scholar] [CrossRef]

**Figure 1.**Ultimate ruin probabilities for the Cramér–Lundberg Model (CLM) compounded by proportional reinsurance; (

**a**) CLM with quota-share (QS) reinsurance, Exp(0.5) claims; (

**b**) CLM with QS reinsurance, Par(3,2) claims.

**Figure 2.**Ultimate ruin probabilities for the CLM compounded by proportional reinsurance and a constant force of interest; (

**a**) CLM with interest force, Exp(0.5) claims; (

**b**) CLM with interest force, Par(3,2) claims.

**Figure 3.**Ultimate ruin probabilities for the diffusion-perturbed model (DPM) compounded by proportional reinsurance; (

**a**) DPM with QS reinsurance, Exp(0.5) claims; (

**b**) DPM with QS reinsurance, Par(3,2) claims.

**Figure 4.**Ultimate ruin probabilities for the DPM compounded by proportional reinsurance and a constant force of interest; (

**a**) DPM with interest force, Exp(0.5) claims; (

**b**) DPM with interest force, Par(3,2) claims.

**Figure 5.**Ultimate ruin probabilities for the DPM compounded by proportional reinsurance and investments of Black–Scholes type; (

**a**) DPM with stochastic interest, Exp(0.5) claims; (

**b**) DPM with stochastic interest, Par(3,2) claims.

**Figure 6.**Effects of volatility of stock prices on the ultimate ruin probability in the small and large claim cases; (

**a**) Effect of volatility coefficient on $\psi \left(u\right)$, Exp(0.5) claims; (

**b**) Effect of volatility coefficient on $\psi \left(u\right)$, Par(3,2) claims.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kasumo, C. Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments. *Math. Comput. Appl.* **2019**, *24*, 21.
https://doi.org/10.3390/mca24010021

**AMA Style**

Kasumo C. Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments. *Mathematical and Computational Applications*. 2019; 24(1):21.
https://doi.org/10.3390/mca24010021

**Chicago/Turabian Style**

Kasumo, Christian. 2019. "Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments" *Mathematical and Computational Applications* 24, no. 1: 21.
https://doi.org/10.3390/mca24010021