On the time value of ruin. With discussion and a reply by the authors. (English) Zbl 1081.60550

Summary: This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.


60K10 Applications of renewal theory (reliability, demand theory, etc.)
60G44 Martingales with continuous parameter
60K05 Renewal theory
62E10 Characterization and structure theory of statistical distributions
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
91G50 Corporate finance (dividends, real options, etc.)
Full Text: DOI


[1] Bowers N.L., Actuarial Mathematics, 2. ed. (1997)
[2] Bühlmann H., Mathematical Methods in Risk Theory (1970) · Zbl 0209.23302
[3] De Vylder F.E., Advanced Risk Theory: A Self-Contained Introduction (1996)
[4] Dickson D.C.M., Insurance: Mathematics and Economics 11 pp 191– (1992) · Zbl 0770.62090 · doi:10.1016/0167-6687(92)90026-8
[5] Dickson D.C.M., Insurance: Mathematics and Economics 14 pp 51– (1994) · Zbl 0803.62091 · doi:10.1016/0167-6687(94)00005-0
[6] Dufresne F., Insurance: Mathematics and Economics 7 pp 193– (1988) · Zbl 0674.62072 · doi:10.1016/0167-6687(88)90076-5
[7] Egídio dos Reis A.D., Insurance: Mathematics and Economics 12 pp 23– (1993) · Zbl 0777.62096 · doi:10.1016/0167-6687(93)90996-3
[8] Feller W., An Introduction to Probability Theory and Its Applications, 2. ed. (1971) · Zbl 0219.60003
[9] Gerber H.U., S.S. Huebner Foundation Monograph Series No. 8 (1979)
[10] Gerber H.U., Insurance: Mathematics and Economics 7 pp 15– (1988) · Zbl 0657.62121 · doi:10.1016/0167-6687(88)90091-1
[11] Gerber H.U., Actuarial Research Clearing House 1 pp 145– (1997)
[12] Gerber H.U., Joint Day Proceedings Volume of the XXVIIIth International ASTIN Colloquium/7th International AFIR Colloquium pp 157– (1997)
[13] Gerber H.U., Insurance: Mathematics and Economics 21 pp 129– (1997) · Zbl 0894.90047 · doi:10.1016/S0167-6687(97)00027-9
[14] Kendall D.G., Journal of the Royal Statistical Society 19 pp 207– (1957)
[15] Lundberg F., Skandinavisk Aktuarietidskrift 15 pp 137– (1932)
[16] Prabhu N.U., Annals of Mathematical Statistics 32 pp 757– (1961) · Zbl 0103.13302 · doi:10.1214/aoms/1177704970
[17] Resnick S.I., Adventures in Stochastic Processes (1992) · Zbl 0762.60002
[18] Seah E.S., Transactions of the Society of Actuaries pp 421– (1990)
[19] Seal H.L., Stochastic Theory of a Risk Business (1969) · Zbl 0196.23501
[20] Segerdahl C.-O., Skandinavisk Aktuarietidskrift 53 pp 167– (1970)
[21] Spiegel M.R., Schaum’s Outline of Theory and Problems of Laplace Transforms (1965)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.