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Minimizing the ruin probability through capital injections

Published online by Cambridge University Press:  31 May 2011

Ciyu Nie ,
David C. M. Dickson  and
Shuanming Li
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Abstract

We consider an insurer who has a fixed amount of funds allocated as the initial surplus for a risk portfolio, so that the probability of ultimate ruin for this portfolio is at a known level. We consider the question of whether the insurer can reduce this ultimate ruin probability by allocating part of the initial funds to the purchase of a reinsurance contract. This reinsurance contract would restore the insurer's surplus to a positive level k every time the surplus fell between 0 and k. The insurer's objective is to choose the level k that minimizes the ultimate ruin probability. Using different examples of reinsurance premium calculation and claim size distribution we show that this objective can be achieved, often with a substantial reduction in the ultimate ruin probability from the situation when there is no reinsurance. We also show that by purchasing reinsurance the insurer can release funds for other purposes without altering its ultimate ruin probability.

Keywords

Ruin probabilityCapital injectionsLower barrier
Type
Papers
Information
Annals of Actuarial Science , Volume 5 , Issue 2 , September 2011 , pp. 195 - 209
Copyright
Copyright © Institute and Faculty of Actuaries 2011

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References

Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J. (1997). Actuarial Mathematics, 2nd edition. Itasca, IL: Society of Actuaries.Google Scholar
Centeno, M.L. (1986). Measuring the effects of reinsurance by the adjustment coefficient. Insurance: Mathematics & Economics, 5, 169–182.Google Scholar
Dickson, D.C.M. (2005). Insurance Risk and Ruin. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Dickson, D.C.M., Drekic, S. (2006). Optimal dividends under a ruin probability constraint. Annals of Actuarial Science, 1, 291–306.CrossRefGoogle Scholar
Dickson, D.C.M., Waters, H.R. (1991). Recursive calculation of survival probabilities. ASTIN Bulletin, 21, 199–221.CrossRefGoogle Scholar
Dickson, D.C.M., Waters, H.R. (1992). The probability and severity of ruin in finite and in infinite time. ASTIN Bulletin, 22, 177–190.CrossRefGoogle Scholar
Dickson, D.C.M., Waters, H.R. (2004). Some optimal dividends problems. ASTIN Bulletin, 34, 49–74.CrossRefGoogle Scholar
Eisenberg, J., Schmidli, H. (2010). Minimising expected discounted capital injections by reinsurance in a classical risk model. Scandinavian Actuarial Journal, forthcoming.Google Scholar
Gerber, H.U. (1979). An Introduction to Mathematical Risk Theory. Philadelphia, PA: S.S. Huebner Foundation.Google Scholar
Gerber, H.U., Goovaerts, M.J., Kaas, R. (1987). On the probability and severity of ruin. ASTIN Bulletin, 17, 151–163.CrossRefGoogle Scholar
Gerber, H.U., Shiu, E.S.W. (1998). On the time value of ruin. North American Actuarial Journal, 2(1), 48–78.CrossRefGoogle Scholar
Landriault, D., Willmot, G.E. (2009). On the joint distributions of the time to ruin, the surplus prior to ruin, and the deficit at ruin in the classical risk model. North American Actuarial Journal, 13(2), 252–279.CrossRefGoogle Scholar
Pafumi, G. (1998). On the time value of ruin: Discussion. North American Actuarial Journal, 2(1), 75–76.CrossRefGoogle Scholar