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In the insurance business risky investments are dangerous: the case of negative risk sums

Listed author(s):
  • Yuri Kabanov


    (Université de Franche-Comté
    National Research University Higher School of Economics)

  • Serguei Pergamenshchikov


    (National Research University Higher School of Economics
    Université de Rouen)

Abstract We investigate models with negative risk sums when the company invests its reserve into a risky asset whose price follows a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to one identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome, namely the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.

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Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 20 (2016)
Issue (Month): 2 (April)
Pages: 355-379

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Handle: RePEc:spr:finsto:v:20:y:2016:i:2:d:10.1007_s00780-016-0292-4
DOI: 10.1007/s00780-016-0292-4
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  1. repec:spr:compst:v:67:y:2008:i:1:p:21-42 is not listed on IDEAS
  2. Wang, Guojing & Wu, Rong, 2001. "Distributions for the risk process with a stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 329-341, October.
  3. Erhan Bayraktar & Masahiko Egami, 2008. "Optimizing venture capital investments in a jump diffusion model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 21-42, February.
  4. 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.
  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. Norberg, Ragnar, 1999. "Ruin problems with assets and liabilities of diffusion type," Stochastic Processes and their Applications, Elsevier, vol. 81(2), pages 255-269, June.
  7. 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.
  8. Harrison, J. Michael, 1977. "Ruin problems with compounding assets," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 67-79, February.
  9. Nyrhinen, Harri, 1999. "On the ruin probabilities in a general economic environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 319-330, October.
  10. Kalashnikov, Vladimir & Norberg, Ragnar, 2002. "Power tailed ruin probabilities in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 211-228, April.
  11. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
  12. 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.
  13. Albrecher, Hansjörg & Badescu, Andrei & Landriault, David, 2008. "On the dual risk model with tax payments," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1086-1094, June.
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