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Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates

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  • Marcus C. Christiansen

    () (Institute of Insurance Science, University of Ulm, 89069 Ulm, Germany)

Abstract

In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims.

Suggested Citation

  • Marcus C. Christiansen, 2013. "Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates," Risks, MDPI, Open Access Journal, vol. 1(3), pages 1-20, October.
  • Handle: RePEc:gam:jrisks:v:1:y:2013:i:3:p:81-100:d:29915
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    References listed on IDEAS

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    1. Dahl, Mikkel & Moller, Thomas, 2006. "Valuation and hedging of life insurance liabilities with systematic mortality risk," Insurance: Mathematics and Economics, Elsevier, vol. 39(2), pages 193-217, October.
    2. Young, Virginia R., 2008. "Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 691-703, April.
    3. Schrager, David F., 2006. "Affine stochastic mortality," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 81-97, February.
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    5. Cairns, Andrew J.G. & Blake, David & Dowd, Kevin, 2006. "Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 36(01), pages 79-120, May.
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    8. Parker, Gary, 1994. "Two Stochastic Approaches for Discounting Actuarial Functions," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 24(02), pages 167-181, November.
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    Citations

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    Cited by:

    1. Marcus Christiansen & Andreas Niemeyer, 2015. "On the forward rate concept in multi-state life insurance," Finance and Stochastics, Springer, vol. 19(2), pages 295-327, April.

    More about this item

    Keywords

    forward interest rate; forward mortality rate; life insurance; stochastic diffusion process; Gaussian approximation;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law

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