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On the accumulated aggregate surplus of a life portfolio. (English) Zbl 1055.91026

Summary: The present paper considers a simple stochastic model for a multi-period (non-profit) life portfolio with two sources of uncertainty, the returns on investments and the claims. Within this model the bonus of a portfolio is determined numerically using the concept of stable reserve associated to a financial gain. The accumulated aggregate surplus of the portfolio after a finite number of insurance periods is expressed as the difference between the accumulated value on investments and the accumulated actuarial reserves and aggregate claims. It is shown that replacing the dependence between the aggregate claims of successive periods by an assumption of independence increases the riskiness in stop-loss order of the accumulated aggregate surplus. This results in higher stable reserves under the independence assumption. It is shown that the stable reserve can be determined numerically by solving an implicit expected value equation, which involves Black–Scholes formula for the stop-loss premiums on the accumulated return on investments and De Pril’s two-stage recursive formulas for the distribution of the accumulated aggregate claims. The obtained results are illustrated numerically at a portfolio of endowment insurance policies.

MSC:

91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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[1] Ammeter, H., 1957. Die Ermittlung der Risikogewinne im Versicherungswesen auf Risikotheoretischer Grundlage. Bulletin of the Swiss Association of Actuaries 2, 145-147. · Zbl 0081.36604
[2] Ammeter, H., 1960. Stop-loss cover and experience rating. Proceedings of the XVIth International Congress of Actuaries, 649-655.
[3] Ammeter, H., 1961. Risikotheoretische Grundlagen der Erfahrungstarifierung. Bulletin of the Swiss Association of Actuaries 2, 183-217. · Zbl 0101.36801
[4] Beard, R.E., Pentikäinen, T., Pesonen, E., 1984. Risk theory. In: The Stochastic Basis of Insurance, 3rd Edition. Chapman & Hall, New York.
[5] Black, R., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-659. In: Luskin, D.L. (Ed.), Portfolio Insurance: A Guide to Dynamic Hedging. Wiley, New York (1988) (reprinted). · Zbl 1092.91524
[6] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1986. Actuarial Mathematics. Society of Actuaries, Itasca.
[7] De Pril, N.L., On the exact computation of the aggregate claims distribution in the individual life model, ASTIN bulletin, 16, 109-112, (1986)
[8] De Pril, N.L., The aggregate claims distribution in the individual model with arbitrary positive claims, ASTIN bulletin, 19, 9-24, (1989)
[9] Dhaene, J.; Denuit, M., The safest dependence structure among risks, Insurance: mathematics and economics, 25, 11-21, (1999) · Zbl 1072.62651
[10] Dickson, D.C.M.; Waters, H.R., Multi-period aggregate loss distributions for a life portfolio, ASTIN bulletin, 29, 295-309, (1999)
[11] Hürlimann, W., Pseudo compound Poisson distributions in risk theory, ASTIN bulletin, 20, 57-79, (1990)
[12] Hürlimann, W., 1991. Stochastic tariffing in life insurance. In: Proceedings of the International Colloquium Life, Disability and Pensions: Tomorrow’s Challenge, Paris, Vol. 3, pp. 203-212 (French Translation: Tarification stochastique en assurance vie, Vol. 3, pp. 81-90).
[13] Hürlimann, W., 1995. On fair premium principles and Pareto-optimal risk-neutral portfolio valuation. In: Proceedings of the 25th International Congress of Actuaries, Brussels, Vol. I, pp. 189-208.
[14] Hürlimann, W., 1998. Distribution-free excess-of-loss reserves for some actuarial protection models. In: Proceedings of the 26th International Congress of Actuaries, Birmingham, Vol. 4, pp. 291-317.
[15] Kaas, R., van Heerwaarden, A.E., Goovaerts, M.J., 1994. Ordering of Actuarial Risks. CAIRE Education Series 1, Brussels. · Zbl 0683.62060
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