IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v112y2023icp142-167.html
   My bibliography  Save this article

Valuation of general GMWB annuities in a low interest rate environment

Author

Listed:
  • Fontana, Claudio
  • Rotondi, Francesco

Abstract

Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:112:y:2023:i:c:p:142-167
    DOI: 10.1016/j.insmatheco.2023.07.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668723000628
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2023.07.003?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. Dhaene, Jan & Kukush, Alexander & Luciano, Elisa & Schoutens, Wim & Stassen, Ben, 2013. "On the (in-)dependence between financial and actuarial risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 522-531.
    2. 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.
    3. Ryan Donnelly & Sebastian Jaimungal & Dmitri H. Rubisov, 2014. "Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 369-382, February.
    4. 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.
    5. 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.
    6. Kang, Boda & Shen, Yang & Zhu, Dan & Ziveyi, Jonathan, 2022. "Valuation of guaranteed minimum maturity benefits under generalised regime-switching models using the Fourier Cosine method," Insurance: Mathematics and Economics, Elsevier, vol. 105(C), pages 96-127.
    7. 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.
    8. Anna Rita Bacinello & Ivan Zoccolan, 2019. "Variable annuities with a threshold fee: valuation, numerical implementation and comparative static analysis," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 21-49, June.
    9. 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.
    10. Anna Rita Bacinello & Pietro Millossovich & Alvaro Montealegre, 2016. "The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2016(5), pages 446-465, May.
    11. Parsiad Azimzadeh & Peter A. Forsyth, 2015. "The existence of optimal bang-bang controls for GMxB contracts," Papers 1502.05743, arXiv.org, revised Nov 2015.
    12. Nikolay Gudkov & Katja Ignatieva & Jonathan Ziveyi, 2019. "Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility, stochastic interest rates and stochastic mortality via the componentwise splitting method," Quantitative Finance, Taylor & Francis Journals, vol. 19(3), pages 501-518, March.
    13. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
    14. M. Costabile, 2017. "A lattice-based model to evaluate variable annuities with guaranteed minimum withdrawal benefits under a regime-switching model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2017(3), pages 231-244, March.
    15. Wei, Wei & Zhu, Dan, 2022. "Generic improvements to least squares monte carlo methods with applications to optimal stopping problems," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1132-1144.
    16. Pitacco, Ermanno, 2004. "Survival models in a dynamic context: a survey," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 279-298, October.
    17. 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.
    18. Bing Dong & Wei Xu & Yue Kuen Kwok, 2019. "Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models," Quantitative Finance, Taylor & Francis Journals, vol. 19(10), pages 1741-1761, October.
    19. Kang, Boda & Ziveyi, Jonathan, 2018. "Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 43-56.
    20. 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.
    21. 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.
    22. Vincenzo Russo & Gabriele Torri, 2019. "Calibration of one-factor and two-factor Hull–White models using swaptions," Computational Management Science, Springer, vol. 16(1), pages 275-295, February.
    23. Kirkby, J. Lars & Nguyen, Duy & Cui, Zhenyu, 2017. "A unified approach to Bermudan and barrier options under stochastic volatility models with jumps," Journal of Economic Dynamics and Control, Elsevier, vol. 80(C), pages 75-100.
    24. Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series 280, Quantitative Finance Research Centre, University of Technology, Sydney.
    25. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
    26. 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.
    27. Kirkby, J. Lars, 2023. "Hybrid equity swap, cap, and floor pricing under stochastic interest by Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(2), pages 961-978.
    28. Bernard, Carole & Hardy, Mary & Mackay, Anne, 2014. "State-Dependent Fees For Variable Annuity Guarantees," ASTIN Bulletin, Cambridge University Press, vol. 44(3), pages 559-585, September.
    29. 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.
    30. Xiaolin Luo & Pavel V. Shevchenko, 2014. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization," Papers 1411.5453, arXiv.org, revised Apr 2015.
    31. Yang, Sharon S. & Dai, Tian-Shyr, 2013. "A flexible tree for evaluating guaranteed minimum withdrawal benefits under deferred life annuity contracts with various provisions," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 231-242.
    32. 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.
    33. Anna Battauz & Francesco Rotondi, 2022. "American options and stochastic interest rates," Computational Management Science, Springer, vol. 19(4), pages 567-604, October.
    34. Jennifer Alonso-García & Oliver Wood & Jonathan Ziveyi, 2018. "Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 1049-1075, June.
    35. Delong, Łukasz, 2014. "Pricing and hedging of variable annuities with state-dependent fees," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 24-33.
    36. 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.
    37. 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.
    38. Ludovic Goudenège & Andrea Molent & Antonino Zanette, 2021. "Gaussian process regression for pricing variable annuities with stochastic volatility and interest rate," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(1), pages 57-72, June.
    39. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A unified pricing of variable annuity guarantees under the optimal stochastic control framework," Papers 1605.00339, arXiv.org.
    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. 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.

    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. Claudio Fontana & Francesco Rotondi, 2022. "Valuation of general GMWB annuities in a low interest rate environment," Papers 2208.10183, arXiv.org, revised Aug 2023.
    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. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Moenig, Thorsten, 2021. "Variable annuities: Market incompleteness and policyholder behavior," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 63-78.
    8. 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).
    9. 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.
    10. Jin Sun & Pavel V. Shevchenko & Man Chung Fung, 2018. "The Impact of Management Fees on the Pricing of Variable Annuity Guarantees," Risks, MDPI, vol. 6(3), pages 1-20, September.
    11. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A unified pricing of variable annuity guarantees under the optimal stochastic control framework," Papers 1605.00339, arXiv.org.
    12. 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.
    13. 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.
    14. Wang, Gu & Zou, Bin, 2021. "Optimal fee structure of variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 587-601.
    15. Mrad, Fatma & Hamdi, Haykel & Naoui, Kamel & Abid, Ilyes, 2023. "The GMWB guarantee embedded in Life Insurance Contracts: Fair Value Pricing Problem," Finance Research Letters, Elsevier, vol. 51(C).
    16. Zhenyu Cui & Anne MacKay & Marie-Claude Vachon, 2022. "Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation," Papers 2207.14793, arXiv.org.
    17. 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.
    18. 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.
    19. 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.
    20. Hsieh, Ming-hua & Wang, Jennifer L. & Chiu, Yu-Fen & Chen, Yen-Chih, 2018. "Valuation of variable long-term care Annuities with Guaranteed Lifetime Withdrawal Benefits: A variance reduction approach," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 246-254.

    More about this item

    Keywords

    Variable annuity; Guaranteed minimum withdrawal benefit; Dynamic withdrawal; Step-up feature; Surrender; Stochastic interest rate; Hull-White model; Mortality risk;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

    Statistics

    Access and download statistics

    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:112:y:2023:i:c:p:142-167. 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.