IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v62y2015icp5-15.html
   My bibliography  Save this article

Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization

Author

Listed:
  • Luo, Xiaolin
  • Shevchenko, Pavel V.

Abstract

In this paper we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behavior solved as an optimal stochastic control problem. This product simultaneously deals with financial risk, mortality risk and human behavior. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss–Hermite quadrature method with cubic spline. We present results for three different types of death benefit and show that, under the optimal policyholder behavior, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value — there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic installment of fixed amount, and it is cheaper than buying a separate life insurance.

Suggested Citation

  • Luo, Xiaolin & Shevchenko, Pavel V., 2015. "Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 5-15.
  • Handle: RePEc:eee:insuma:v:62:y:2015:i:c:p:5-15
    DOI: 10.1016/j.insmatheco.2015.02.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668715000177
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.insmatheco.2015.02.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Ledlie, M. C. & Corry, D. P. & Finkelstein, G. S. & Ritchie, A. J. & Su, K. & Wilson, D. C. E., 2008. "Variable Annuities," British Actuarial Journal, Cambridge University Press, vol. 14(2), pages 327-389, July.
    3. 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.
    4. Carter, Lawrence R. & Lee, Ronald D., 1992. "Modeling and forecasting US sex differentials in mortality," International Journal of Forecasting, Elsevier, vol. 8(3), pages 393-411, November.
    5. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    7. Xiaolin Luo & Pavel V. Shevchenko, 2014. "Fast and Simple Method for Pricing Exotic Options using Gauss-Hermite Quadrature on a Cubic Spline Interpolation," Papers 1408.6938, arXiv.org, revised Dec 2014.
    8. Bauer, Daniel & Kling, Alexander & Russ, Jochen, 2008. "A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities 1," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 621-651, November.
    9. Xiaolin Luo & Pavel Shevchenko, 2014. "Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy," Papers 1410.8609, arXiv.org.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Michael A. Kouritzin & Anne MacKay, 2017. "VIX-linked fees for GMWBs via Explicit Solution Simulation Methods," Papers 1708.06886, arXiv.org, revised Apr 2018.
    4. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2019. "Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate," Papers 1903.00369, arXiv.org, revised Jul 2019.
    5. Dong, Bing & Xu, Wei & Sevic, Aleksandar & Sevic, Zeljko, 2020. "Efficient willow tree method for variable annuities valuation and risk management☆," International Review of Financial Analysis, Elsevier, vol. 68(C).
    6. Kouritzin, Michael A. & MacKay, Anne, 2018. "VIX-linked fees for GMWBs via explicit solution simulation methods," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 1-17.
    7. Hanwen Zhang & Duy-Minh Dang, 2023. "A monotone numerical integration method for mean-variance portfolio optimization under jump-diffusion models," Papers 2309.05977, arXiv.org.
    8. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework," Risks, MDPI, vol. 4(3), pages 1-31, July.
    9. Daniel Doyle & Chris Groendyke, 2018. "Using Neural Networks to Price and Hedge Variable Annuity Guarantees," Risks, MDPI, vol. 7(1), pages 1-19, December.
    10. 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.
    11. Thorsten Moenig, 2021. "Efficient valuation of variable annuity portfolios with dynamic programming," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 88(4), pages 1023-1055, December.
    12. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A unified pricing of variable annuity guarantees under the optimal stochastic control framework," Papers 1605.00339, arXiv.org.
    3. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework," Risks, MDPI, vol. 4(3), pages 1-31, July.
    4. 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.
    5. Xiaolin Luo & Pavel Shevchenko, 2014. "Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy," Papers 1410.8609, arXiv.org.
    6. Xiaolin Luo & Pavel V. Shevchenko, 2015. "Valuation of capital protection options," Papers 1508.00668, arXiv.org, revised May 2017.
    7. Claudio Fontana & Francesco Rotondi, 2022. "Valuation of general GMWB annuities in a low interest rate environment," Papers 2208.10183, arXiv.org, revised Aug 2023.
    8. 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.
    9. Fontana, Claudio & Rotondi, Francesco, 2023. "Valuation of general GMWB annuities in a low interest rate environment," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 142-167.
    10. Bacinello, Anna Rita & Maggistro, Rosario & Zoccolan, Ivan, 2024. "Risk-neutral valuation of GLWB riders in variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 114(C), pages 1-14.
    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. Xiaolin Luo & Pavel Shevchenko, 2015. "Variable Annuity with GMWB: surrender or not, that is the question," Papers 1507.08738, arXiv.org.
    13. 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.
    14. Chen, Ze & Feng, Runhuan & Li, Hong & Yang, Tianyu, 2024. "Coping with longevity via hedging: Fair dynamic valuation of variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 154-169.
    15. 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.
    16. Wang, Gu & Zou, Bin, 2021. "Optimal fee structure of variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 587-601.
    17. David Blake & Marco Morales & Enrico Biffis & Yijia Lin & Andreas Milidonis, 2017. "Special Edition: Longevity 10 – The Tenth International Longevity Risk and Capital Markets Solutions Conference," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 84(S1), pages 515-532, April.
    18. Paolo Angelis & Roberto Marchis & Antonio L. Martire & Emilio Russo, 2022. "A flexible lattice framework for valuing options on assets paying discrete dividends and variable annuities embedding GMWB riders," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 415-446, June.
    19. Delong, Łukasz, 2014. "Pricing and hedging of variable annuities with state-dependent fees," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 24-33.
    20. Ignatieva, Katja & Song, Andrew & Ziveyi, Jonathan, 2016. "Pricing and hedging of guaranteed minimum benefits under regime-switching and stochastic mortality," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 286-300.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:62:y:2015:i:c:p:5-15. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.