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Valuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals

Author

Listed:
  • A. C. Belanger
  • P. A. Forsyth
  • G. Labahn

Abstract

In this paper, we give a method for computing the fair insurance fee associated with the guaranteed minimum death benefit (GMDB) clause included in many variable annuity contracts. We allow for partial withdrawals, a common feature in most GMDB contracts, and determine how this affects the GMDB fair insurance charge. Our method models the GMDB pricing problem as an impulse control problem. The resulting quasi-variational inequality is solved numerically using a fully implicit penalty method. The numerical results are obtained under both constant volatility and regime-switching models. A complete analysis of the numerical procedure is included. We show that the discrete equations are stable, monotone and consistent and hence obtain convergence to the unique, continuous viscosity solution, assuming this exists. Our results show that the addition of the partial withdrawal feature significantly increases the fair insurance charge for GMDB contracts.

Suggested Citation

  • A. C. Belanger & P. A. Forsyth & G. Labahn, 2009. "Valuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 451-496.
  • Handle: RePEc:taf:apmtfi:v:16:y:2009:i:6:p:451-496 DOI: 10.1080/13504860903075464
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    References listed on IDEAS

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    1. Jörnsten, Kurt & Ubøe, Jan, 2010. "Quantification of preferences in markets," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 453-466, July.
    2. Foley Duncan K., 1994. "A Statistical Equilibrium Theory of Markets," Journal of Economic Theory, Elsevier, vol. 62(2), pages 321-345, April.
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    Citations

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    Cited by:

    1. Seyed Amir Hejazi & Kenneth R. Jackson, 2016. "A Neural Network Approach to Efficient Valuation of Large Portfolios of Variable Annuities," Papers 1606.07831, arXiv.org.
    2. Bacinello, Anna Rita & Millossovich, Pietro & Olivieri, Annamaria & Pitacco, Ermanno, 2011. "Variable annuities: A unifying valuation approach," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 285-297.
    3. Seyed Amir Hejazi & Kenneth R. Jackson & Guojun Gan, 2017. "A Spatial Interpolation Framework for Efficient Valuation of Large Portfolios of Variable Annuities," Papers 1701.04134, arXiv.org.
    4. Jin Sun & Pavel V. Shevchenko & Man Chung Fung, 2017. "A note on the impact of management fees on the pricing of variable annuity guarantees," Papers 1705.03787, arXiv.org, revised May 2017.
    5. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A unified pricing of variable annuity guarantees under the optimal stochastic control framework," Papers 1605.00339, arXiv.org.
    6. Christophette Blanchet-Scalliet & Etienne Chevalier & Idris Kharroubi & Thomas Lim, 2015. "Max–Min Optimization Problem For Variable Annuities Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(08), pages 1-35, December.
    7. Forsyth, Peter & Vetzal, Kenneth, 2014. "An optimal stochastic control framework for determining the cost of hedging of variable annuities," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 29-53.
    8. Wang, J. & Forsyth, P.A., 2010. "Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation," Journal of Economic Dynamics and Control, Elsevier, vol. 34(2), pages 207-230, February.
    9. Goudenège, Ludovic & Molent, Andrea & Zanette, Antonino, 2016. "Pricing and hedging GLWB in the Heston and in the Black–Scholes with stochastic interest rate models," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 38-57.
    10. Hejazi, Seyed Amir & Jackson, Kenneth R., 2016. "A neural network approach to efficient valuation of large portfolios of variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 169-181.
    11. Gan, Guojun & Lin, X. Sheldon, 2015. "Valuation of large variable annuity portfolios under nested simulation: A functional data approach," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 138-150.
    12. Ludovic Goudenege & Andrea Molent & Antonino Zanette, 2015. "Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models," Papers 1509.02686, arXiv.org.
    13. Gan, Guojun, 2013. "Application of data clustering and machine learning in variable annuity valuation," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 795-801.
    14. Gao, Jin & Ulm, Eric R., 2012. "Optimal consumption and allocation in variable annuities with Guaranteed Minimum Death Benefits," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 586-598.
    15. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2016. "Pricing and Hedging GMWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models," Papers 1602.09078, arXiv.org, revised Mar 2016.
    16. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework," Risks, MDPI, Open Access Journal, vol. 4(3), pages 1-31, July.

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