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Guaranteed investment contracts: distributed and undistributed excess return. (English) Zbl 1092.91053

The paper deals with a model consisting an investment/saving plan or contract between two parties called the insurer and the customer. At date zero the customer deposits an amount \(X\) into an account \(A\), which is invested by the insurer for a period of \(T\) years. The insurer promises the customer an annual rate of return on the account \(A\) in the year \(i\) equal to \(g_{i}+\alpha(\delta_{i}-g_{i})^{+}\), where the constant \(g_{i}\) is a specified minimum rate of return guarantee in the year \(i\); \(\delta_{i}\) is the random rate of return of the specified benchmark portfolio in the year \(i\); and \(\alpha\) is the fraction of the positive excess rate of return which is credited to the customer’s account. In return for the minimum rate of return guarantee the insurer receives a fraction \(\beta\) of the excess rate of return, i.e. the return \(\beta(\delta_{i}-g_{i})^{+}\) is credited to the insurer’s account, denoted by \(C\). In addition, the model includes a surplus distribution mechanism working through the bonus account \(B\), which is managed by the insurer. The case of Gaussian returns on the benchmark portfolio and deterministic short term interest rates are considered. A closed-form solution for the value of the customer’s account is derived. The value of the bonus account is solved by Monte Carlo simulations. Corresponding values of annual minimum rate of return guarantees and the fractions of the excess return distributed to the customer’s account and the bonus account are plotted for fair contracts.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B28 Finance etc. (MSC2000)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60J65 Brownian motion
62E10 Characterization and structure theory of statistical distributions
Full Text: DOI

References:

[1] Aase K. K., Scandinavian Actuarial Journal 1 pp 26– (1994) · Zbl 0814.62067 · doi:10.1080/03461238.1994.10413928
[2] Black F., Journal of Political Economy 81 (3) pp 637– (1973) · Zbl 1092.91524 · doi:10.1086/260062
[3] Brennan M. J., Journal of Financial Economics 3 pp 195– (1976) · doi:10.1016/0304-405X(76)90003-9
[4] Brennan M. J., Journal of Business 52 pp 63– (1979) · doi:10.1086/296034
[5] Briys, E. and F. de Varenne (1997). On the Risk of Life Insurance Liabilities: Debunking Some Common Pitfalls, The Journal of Risk and Insurance, 64(4).
[6] Donselaar, J. (1999). Guaranteed Returns: Risks assured? in 9th International AFIR Colloquium, pages 195-203, Tokyo, Japan.
[7] Grosen A., The Journal of Risk and Insurance 64 pp 481– (1997) · doi:10.2307/253761
[8] Grosen A., Insurance: Mathematics and Economics 26 pp 37– (2000) · Zbl 0977.62108 · doi:10.1016/S0167-6687(99)00041-4
[9] Hansen, M. and K. R. Miltersen (2000). Minimum Rate of Return Guarantees: The Danish Case, Working Paper, Department of Accounting, Finance, and Law, Odense University, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. · Zbl 1039.91040
[10] Matsuyama, N. (1999). A Feasibility Study of the Optimal Asset Mix for Life Insurer’s General Account, 9th International AFIR Colloquium, Tokyo, Japan.
[11] Mertens, M. (1999). Common Capital Life Insurance Contracts in Germany: Interest Rate Guarantee and Bonus Collection, Working Paper, Department of Accounting, Finance, and Law, Odense University, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.
[12] Merton R. C., Bell Journal of Economics and Management Science 4 pp 141– (1973) · doi:10.2307/3003143
[13] Merton R. C. Continuous-Time Finance Basil Blackwell Inc: Padstow, Great Britain 1990
[14] Miltersen K. R., Insurance: Mathematics and Economics 25 (3) pp 307– (1999) · Zbl 1028.91566 · doi:10.1016/S0167-6687(99)00020-7
[15] Norberg, R. (1997): Bonus in Life Insurance: Priciples and Prognoses in a Stochastic Environment, Working Paper 142, Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.
[16] Norberg R., Finance and Stochastics (4) pp 373– (1999) · Zbl 0939.62108 · doi:10.1007/s007800050067
[17] Persson S.-A., The Journal of Risk and Insurance 64 (4) pp 599– (1997) · doi:10.2307/253888
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