# In the insurance business risky investments are dangerous

## Author

Listed:
• Anna Frolova

(Alfa-Bank, Masha Poryvaeva str., 9, 107078, Moscow, Russia)

• Serguei Pergamenshchikov

(Tomsk State University, Lenin str., 36, 634050 Tomsk, Russia Manuscript)

• Yuri Kabanov

() (Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon cedex, France and Central Economics and Mathematics Institute, Moscow, Russia)

## Abstract

We find an exact asymptotics of the ruin probability $\Psi (u)$ when the capital of insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a and volatility $\sigma>0$. In contrast to the classical case of non-risky investments where the ruin probability decays exponentially as the initial endowment u tends to infinity, in this model we have, if $\rho:=2a/\sigma^2>1$, that $\Psi(u)\sim Ku^{1-\rho}$ for some $K>0$. If \$\rho

## Suggested Citation

• 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.
• Handle: RePEc:spr:finsto:v:6:y:2002:i:2:p:227-235
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## Citations

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Cited by:

1. repec:eee:insuma:v:74:y:2017:i:c:p:7-19 is not listed on IDEAS
2. Paulsen, Jostein & Kasozi, Juma & Steigen, Andreas, 2005. "A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 399-420, June.
3. Tang, Qihe & Wang, Guojing & Yuen, Kam C., 2010. "Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 362-370, April.
4. David Maher, 2005. "A Note on the Ruin Problem with Risky Investments," Papers math/0506127, arXiv.org, revised Jul 2005.
5. Serguei Pergamenchtchikov & Zeitouny Omar, 2010. "Ruin probability in the presence of risky investments," Papers 1011.1329, arXiv.org.
6. Schmidli, Hanspeter, 2005. "On optimal investment and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 25-35, February.
7. Pergamenshchikov, Serguei & Zeitouny, Omar, 2006. "Ruin probability in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 267-278, February.
8. Tatiana Belkina & Christian Hipp & Shangzhen Luo & Michael Taksar, 2011. "Optimal Constrained Investment in the Cramer-Lundberg model," Papers 1112.4007, arXiv.org.
9. Brokate, M. & Klüppelberg, C. & Kostadinova, R. & Maller, R. & Seydel, R.C., 2008. "On the distribution tail of an integrated risk model: A numerical approach," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 101-106, February.
10. Emms, P. & Haberman, S., 2007. "Asymptotic and numerical analysis of the optimal investment strategy for an insurer," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 113-134, January.
11. Grandits, Peter, 2004. "A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 297-305, April.
12. Albrecher, Hansjoerg & Constantinescu, Corina & Thomann, Enrique, 2012. "Asymptotic results for renewal risk models with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3767-3789.
13. Klüppelberg, Claudia & Kostadinova, Radostina, 2008. "Integrated insurance risk models with exponential Lévy investment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 560-577, April.
14. Cai, Jun, 2004. "Ruin probabilities and penalty functions with stochastic rates of interest," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 53-78, July.
15. Xiong, Sheng & Yang, Wei-Shih, 2011. "Ruin probability in the Cramér-Lundberg model with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1125-1137, May.
16. Tatiana Belkina & Nadezhda Konyukhova & Sergey Kurochkin, 2015. "Singular Problems for Integro-Differential Equations in Dynamic Insurance Models," Papers 1511.08666, arXiv.org.
17. Peter Grandits, 2015. "An optimal consumption problem in finite time with a constraint on the ruin probability," Finance and Stochastics, Springer, vol. 19(4), pages 791-847, October.
18. Henrik Hult & Filip Lindskog, 2011. "Ruin probabilities under general investments and heavy-tailed claims," Finance and Stochastics, Springer, vol. 15(2), pages 243-265, June.
19. Kostadinova, Radostina, 2007. "Optimal investment for insurers when the stock price follows an exponential Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 250-263, September.
20. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
21. Wang, Nan, 2007. "Optimal investment for an insurer with exponential utility preference," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 77-84, January.
22. 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.
23. Nyrhinen, Harri, 2007. "Convex large deviation rate functions under mixtures of linear transformations, with an application to ruin theory," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 947-959, July.
24. repec:eee:apmaco:v:259:y:2015:i:c:p:1030-1045 is not listed on IDEAS

### Keywords

Risk process; geometric Brownian motion; ruin probabilities;

### JEL classification:

• G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
• G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors

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