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Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products

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  • Huang, Yao Tung
  • Kwok, Yue Kuen

Abstract

Guaranteed Minimum Withdrawal Benefits (GMWB) are popular riders in variable annuities with withdrawal guarantees. With withdrawals spread over the life of the annuities contract, the benefit promises to return the entire initial annuitization amount irrespective of the market performance of the underlying fund portfolio. Treating the dynamic withdrawal rate as the control variable, the earlier works on GMWB have considered the construction of a continuous singular stochastic control model and the numerical solution of the resulting pricing model. This paper presents a more detailed characterization of the pricing properties of the GMWB and performs a full mathematical analysis of the optimal dynamic withdrawal policies under the competing factors of time value of fund, optionality value provided by the guarantee and penalty charge on excessive withdrawal. When a proportional penalty charge is applied on any withdrawal amount, we can reduce the pricing formulation to an optimal stopping problem with lower and upper obstacles. We then derive the integral equations for the determination of a pair of optimal withdrawal boundaries. When a proportional penalty charge is applied on the amount that is above the contractual withdrawal rate, we manage to characterize the behavior of the optimal withdrawal boundaries that separate the domain of the pricing models into three regions: no withdrawal, continuous withdrawal at the contractual rate and an immediate withdrawal of a finite amount. Under certain limiting scenarios such as a high policy fund value, the time close to expiry, or a low value of guarantee account, we manage to obtain analytical approximate solution to the singular stochastic control model of dynamic withdrawals.

Suggested Citation

  • Huang, Yao Tung & Kwok, Yue Kuen, 2014. "Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products," Journal of Economic Dynamics and Control, Elsevier, vol. 45(C), pages 19-43.
    • Handle: RePEc:eee:dyncon:v:45:y:2014:i:c:p:19-43
    DOI: 10.1016/j.jedc.2014.05.008
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    References listed on IDEAS

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    1. Jingjiang Peng & Kwai Sun Leung & Yue Kuen Kwok, 2012. "Pricing guaranteed minimum withdrawal benefits under stochastic interest rates," Quantitative Finance, Taylor & Francis Journals, vol. 12(6), pages 933-941, October.
    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.
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    7. Milevsky, Moshe A. & Salisbury, Thomas S., 2006. "Financial valuation of guaranteed minimum withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 21-38, February.
    8. Andreas Kunz, 2012. "Robust Hedging of Withdrawal Guarantees (Extended Version)," Papers 1202.0175, arXiv.org, revised Sep 2012.
    9. Chen, Z. & Vetzal, K. & Forsyth, P.A., 2008. "The effect of modelling parameters on the value of GMWB guarantees," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 165-173, August.
    10. Min Dai & Yue Kuen Kwok & Jianping Zong, 2008. "Guaranteed Minimum Withdrawal Benefit In Variable Annuities," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 595-611, October.
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    Citations

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    Cited by:

    1. Shevchenko, Pavel V. & Luo, Xiaolin, 2017. "Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit under stochastic interest rate," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 104-117.
    2. Ludovic Goudenege & Andrea Molent & Antonino Zanette, 2019. "Pricing and hedging GMWB in the Heston and in the Black–Scholes with stochastic interest rate models," Computational Management Science, Springer, vol. 16(1), pages 217-248, February.
    3. Runhuan Feng & Jan Vecer, 2017. "Risk based capital for guaranteed minimum withdrawal benefit," Quantitative Finance, Taylor & Francis Journals, vol. 17(3), pages 471-478, March.
    4. Kirkby, J. Lars & Nguyen, Duy, 2021. "Equity-linked Guaranteed Minimum Death Benefits with dollar cost averaging," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 408-428.
    5. Yaowen Lu & Duy-Minh Dang, 2023. "A semi-Lagrangian $\epsilon$-monotone Fourier method for continuous withdrawal GMWBs under jump-diffusion with stochastic interest rate," Papers 2310.00606, arXiv.org.
    6. Yao Tung Huang & Yue Kuen Kwok, 2016. "Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 905-928, June.
    7. Pavel V. Shevchenko & Xiaolin Luo, 2016. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate," Papers 1602.03238, arXiv.org, revised Jan 2017.
    8. Xiaolin Luo & Pavel Shevchenko, 2015. "Variable Annuity with GMWB: surrender or not, that is the question," Papers 1507.08738, arXiv.org.
    9. Huansang Xu & Ruyi Liu & Marek Rutkowski, 2023. "Equity Protection Swaps: A New Type of Investment Insurance for Holders of Superannuation Accounts," Papers 2305.09472, arXiv.org, revised Apr 2024.
    10. Parsiad Azimzadeh & Peter A. Forsyth, 2015. "The existence of optimal bang-bang controls for GMxB contracts," Papers 1502.05743, arXiv.org, revised Nov 2015.

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    More about this item

    Keywords

    C61; G22; Singular stochastic control model; Guaranteed minimum withdrawal benefits; Optimal withdrawal policies; Penalty charge;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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