## Geometric Brownian motion models for assets and liabilities: from pension funding to optimal dividends. With discussion by X. Sheldon Lin, Marc Decamps and Marc Goovaerts and a reply by the authors.(English)Zbl 1084.91517

Summary: In this paper asset and liability values are modeled by geometric Brownian motions. In the first part of the paper we consider a pension plan sponsor with the funding objective that the pension asset value is to be within a band that is proportional to the pension liability value. Whenever the asset value is about to fall below the lower barrier or boundary of the band, the sponsor will provide sufficient funds to prevent this from happening. If, on the other hand, the asset value is about to exceed the upper barrier of the band, the assets are reduced by the potential overflow and returned to the sponsor. This paper calculates the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor. In particular we are interested in situations where these two expected values are equal. In the second part of the paper the refunds at the upper barrier are interpreted as the dividends paid to the shareholders of a company according to a barrier strategy. However, if the (modified) asset value ever falls to the liability value, which is the lower barrier, ”ruin” takes place, and no more dividends can be paid. We derive an explicit expression for the expected discounted dividends before ruin. From this we find an explicit expression for the proportionality constant of the upper barrier that maximizes the expected discounted dividends. If the initial asset value is the optimal upper barrier, there is a particularly simple and intriguing expression for the expected discounted dividends, which can be interpreted as the present value of a deterministic perpetuity with exponentially growing payments.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J65 Brownian motion
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### References:

 [1] Albrecher H., Computing 68 pp 289– (2002) · Zbl 1076.91521 [2] Asmussen S., Insurance: Mathematics & Economics 20 pp 1– (1997) · Zbl 1065.91529 [3] Birkhoff G., Ordinary Differential Equations (1962) · Zbl 0102.29901 [4] Borch K., The Mathematical Theory of Insurance (1974) [5] Bühlmann H., Mathematical Methods in Risk Theory (1970) · Zbl 0209.23302 [6] Cox D. R., The Theory of Stochastic Processes (1965) · Zbl 0149.12902 [7] De Finetti B., Transactions of the XVth International Congress of Actuaries 2 pp 433– (1957) [8] Gerber H. U., An Introduction to Mathematical Risk Theory (1979) · Zbl 0431.62066 [9] Gerber H. U., North American Actuarial Journal 4 (2) pp 28– (2000) · Zbl 1083.91516 [10] Gerber H. U., North American Actuarial Journal 2 (1) pp 48– (1998) · Zbl 1081.60550 [11] Gerber H. U., North American Actuarial Journal 7 (2) pp 60– (2003) · Zbl 1084.60512 [12] Gordon M. J., The Investment, Financing and Valuation of the Corporation (1962) [13] Højgaard B., Scandinavian Actuarial Journal pp 225– (2002) · Zbl 1039.91042 [14] Højgaard B., Mathematical Finance 9 pp 153– (1999) · Zbl 0999.91052 [15] Ince E. L., Ordinary Differential Equations (1926) · Zbl 0063.02971 [16] Paulsen J., Insurance: Mathematics & Economics 20 pp 215– (1997) · Zbl 0894.90048 [17] Siegl T., Insurance: Mathematics & Economics 24 pp 51– (1999) · Zbl 0944.91032 [18] Williams J. B., The Theory of Investment Value (1938)
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