The hurdle-race problem. (English) Zbl 1103.91353

Summary: We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into account imposed minimum levels of the future random values of the reserve. The paper ends with numerical examples illustrating the presented approximations.


91B28 Finance etc. (MSC2000)
91B82 Statistical methods; economic indices and measures
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[1] Bäuerle, N.; Müller, A., Modeling and comparing dependencies in multivariate risk portfolios, ASTIN bulletin, 28, 59-76, (1998) · Zbl 1137.91484
[2] Dhaene, J.; Goovaerts, M., Dependency of risks and stop-loss order, ASTIN bulletin, 26, 201-212, (1996)
[3] Dhaene, J.; Goovaerts, M.J., On the dependency of risks in the individual life model, Insurance: mathematics and economics, 19, 243-253, (1997) · Zbl 0931.62089
[4] Dhaene, J.; Wang, S.; Young, V.; Goovaerts, M.J., Comonotonicity and maximal stop-loss premiums, Bulletin of the swiss association of actuaries, 2000, 2, 99-113, (2000) · Zbl 1187.91099
[5] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics 31 (1), 3-33. · Zbl 1051.62107
[6] Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: applications. Insurance: Mathematics and Economics 31 (2), 133-161. · Zbl 1037.62107
[7] Embrechts, P., Mc.Neil, A., Straumann, D., 1999. Correlation and dependency in risk management. In: Proceedings of the XXXth International ASTIN Colloquium, August 22-25, 1999, pp. 227-250.
[8] Goovaerts, M.J.; Dhaene, J., Supermodular ordering and stochastic annuities, Insurance: mathematics and economics, 24, 281-290, (1999) · Zbl 0942.60008
[9] Goovaerts, M.J., Kaas, R., 2002. Some problems in actuarial finance involving sums of dependent risks. Statistica Neerlandica 56 (3), 253-269. · Zbl 1076.62558
[10] Goovaerts, M.J.; Redant, R., On the distribution of IBNR reserves, Insurance: mathematics and economics, 25, 1-9, (1999) · Zbl 0949.62087
[11] Goovaerts, M.J.; Dhaene, J.; De Schepper, A., Stochastic upper bounds for present value functions, Journal of risk and insurance theory, 67.1, 1-14, (2000)
[12] Heilmann, W.-R., On the impact of independence of risks on stop loss premiums, Insurance: mathematics and economics, 5, 197-199, (1986) · Zbl 0596.62111
[13] Kaas, R.; Dhaene, J.; Goovaerts, M.J., Upper and lower bounds for sums of random variables, Insurance: mathematics and economics, 23, 151-168, (2000) · Zbl 0989.60019
[14] Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M.J., Denuit, M., 2002. A simple geometric proof that comonotonic risks have the convex-largest sum. ASTIN Bulletin 32 (1), 71-80. · Zbl 1061.62511
[15] Müller, A., Stop-loss order for portfolios of dependent risks, Insurance: mathematics and economics, 21, 219-223, (1997) · Zbl 0894.90022
[16] Spivak, G.; Cvitanic, J., Maximising the probability of perfect hedge, The annals of applied probability, 9, 1303-1328, (1999) · Zbl 0966.91042
[17] Vyncke, D.; Goovaerts, M.; Dhaene, J., Convex upper and lower bounds for present value functions, Applied stochastic models in business and industry, 17, 2, 149-164, (2001) · Zbl 0971.91030
[18] Wang, S.; Dhaene, J., Comonotonicity, correlation order and stop-loss premiums, Insurance: mathematics and economics, 22, 235-243, (1998)
[19] Wang, S.; Young, V., Ordering risks: expected utility versus yaari’s dual theory of choice under risk, Insurance: mathematics and economics, 22, 145-162, (1998) · Zbl 0907.90102
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