The Annals of Applied Probability

Asymptotic ruin probabilities and optimal investment

J. Gaier, P. Grandits, and W. Schachermayer

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Abstract

We study the infinite time ruin probability for an insurance company in the classical Cramér--Lundberg model with finite exponential moments. The additional nonclassical feature is that the company is also allowed to invest in some stock market, modeled by geometric Brownian motion. We obtain an exact analogue of the classical estimate for the ruin probability without investment, that is, an exponential inequality. The exponent is larger than the one obtained without investment, the classical Lundberg adjustment coefficient, and thus one gets a sharper bound on the ruin probability.

A surprising result is that the trading strategy yielding the optimal asymptotic decay of the ruin probability simply consists in holding a fixed quantity (which can be explicitly calculated) in the risky asset, independent of the current reserve. This result is in apparent contradiction to the common believe that "rich" companies should invest more in risky assets than "poor" ones. The reason for this seemingly paradoxical result is that the minimization of the ruin probability is an extremely conservative optimization criterion, especially for "rich" companies.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 3 (2003), 1054-1076.

Dates
First available in Project Euclid: 6 August 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1060202834

Digital Object Identifier
doi:10.1214/aoap/1060202834

Mathematical Reviews number (MathSciNet)
MR1994044

Zentralblatt MATH identifier
1046.62113

Subjects
Primary: 60G44: Martingales with continuous parameter 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 62P05: Applications to actuarial sciences and financial mathematics 60K10: Applications (reliability, demand theory, etc.)

Keywords
Ruin probability Lundberg inequality optimal investment

Citation

Gaier, J.; Grandits, P.; Schachermayer, W. Asymptotic ruin probabilities and optimal investment. Ann. Appl. Probab. 13 (2003), no. 3, 1054--1076. doi:10.1214/aoap/1060202834. https://projecteuclid.org/euclid.aoap/1060202834


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References

  • [1] ASMUSSEN, S. (2000). Ruin Probabilities. World Scientific, Singapore.
  • [2] BROWNE, S. (1995). Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20 937-957.
  • [3] DELBAEN, F. and HAEZENDONCK, J. (1987). Classical risk theory in an economic environment. Insurance Math. Econom. 6 85-116.
  • [4] EMBRECHTS, P. and VERAVERBEKE, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55-72.
  • [5] FROVOLA, A. G., KABANOV, YU. M. and PERGAMENSHCHIKOV, S. M. (2002). In the insurance business risky investments are dangerous. Finance Stochastics 6 227-235.
  • [6] GAIER, J. and GRANDITS, P. (2002). Ruin probabilities in the presence of regularly varying tails and optimal investment. Insurance Math. Econom. 30 211-217.
  • [7] GERBER, H. U. (1973). Martingales in risk theory. Mitteilungen der Schweizer Vereinigung der Versicherungsmathematiker 73 205-216.
  • [8] GERBER, H. U. (1979). An Introduction to Mathematical Risk Theory. S. S. Huebner Foundation, Univ. Pennsy lvania.
  • [9] GRANDELL, J. (1991). Aspects of Risk Theory. Springer, Berlin.
  • [10] HIPP, C. and PLUM, M. (2000). Optimal investment for insurers. Insurance Math. Econom. 27 215-228.
  • [11] HIPP, C. and PLUM, M. (2001). Optimal investment for investors with state dependent income, and for insurers. Preprint.
  • [12] HIPP, C. and SCHMIDLI, H. (2002). Asy mptoptics of ruin probabilities for controlled risk processes in the small claims case. Preprint.
  • [13] KALASHNIKOV, V. and NORBERG, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stochastic Process. Appl. 98 211-228.
  • [14] LUNDBERG, F. (1903). Approximerad framställning av sannolikhetsfunktionen. Akad. Afhandling. Almqvist och Wiksell, Uppsala.
  • [15] PAULSEN, J. (1998). Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stochastic Process. Appl. 75 135-148.
  • [16] PAULSEN, J. and GJESSING, H. K. (1997). Ruin theory with stochastic return on investments. Adv. in Appl. Probab. 29 965-985.
  • [17] PROTTER, P. (1992). Stochastic Integration and Differential Equations: A New Approach. Springer, Berlin.
  • [18] ROGERS, L. C. G. and WILLIAMS, D. (1994). Diffusions, Markov Processes and Martingales 1. Foundations, 2nd ed. Wiley, New York.