In the insurance business risky investments are dangerous

Author Info

• 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

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Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 6 (2002)
Issue (Month): 2 ()
Pages: 227-235

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Handle: RePEc:spr:finsto:v:6:y:2002:i:2:p:227-235

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Related research

Keywords: Risk process; geometric Brownian motion; ruin probabilities;

Find related papers by 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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Citations

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Cited by:
1. Tatiana Belkina & Christian Hipp & Shangzhen Luo & Michael Taksar, 2011. "Optimal Constrained Investment in the Cramer-Lundberg model," Papers 1112.4007, arXiv.org.
2. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
3. 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.
4. 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.
5. Serguei Pergamenchtchikov & Zeitouny Omar, 2010. "Ruin probability in the presence of risky investments," Papers 1011.1329, arXiv.org.
6. 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.
7. Wang, Nan, 2007. "Optimal investment for an insurer with exponential utility preference," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 77-84, January.
8. 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.
9. David Maher, 2005. "A Note on the Ruin Problem with Risky Investments," Papers math/0506127, arXiv.org, revised Jul 2005.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. 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.
19. Schmidli, Hanspeter, 2005. "On optimal investment and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 25-35, February.

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