IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v103y2022icp1-26.html
   My bibliography  Save this article

Valuing guaranteed minimum accumulation benefits by a change of numéraire approach

Author

Listed:
  • Huang, Yiming
  • Mamon, Rogemar
  • Xiong, Heng

Abstract

Three correlated risk factors, namely, financial, mortality and lapse risks, are modelled in an integrated way to support the valuation of guaranteed minimum accumulation benefits (GMAB). The dynamics of such risk factors under the risk-adjusted measure needed for valuation are derived. A sequence of reference probability measure changes is employed to obtain a compact and implementable price representations. We demonstrate how a change of numéraire approach facilitates the valuation of a GMAB rider in a variable annuity. A numerical illustration is presented to confirm the computational efficiency and accuracy of our proposed valuation methodology. In particular, our proposed approach on average takes only 0.07% of the computing time for the Monte-Carlo (MC) simulation technique to complete the valuation calculation. Furthermore, the standard errors of our approach's results are lower than those obtained from the MC-based computations. A sensitivity analysis is also performed with respect to various parameters. When there are no renewal options in a GMAB contract, we get the special case of a guaranteed minimum maturity benefit for which a closed-form valuation solution is established.

Suggested Citation

  • Huang, Yiming & Mamon, Rogemar & Xiong, Heng, 2022. "Valuing guaranteed minimum accumulation benefits by a change of numéraire approach," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 1-26.
    • Handle: RePEc:eee:insuma:v:103:y:2022:i:c:p:1-26
    DOI: 10.1016/j.insmatheco.2021.11.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668721001682
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2021.11.002?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. Feng, Runhuan & Jing, Xiaochen, 2017. "Analytical valuation and hedging of variable annuity guaranteed lifetime withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 36-48.
    3. Zhao, Yixing & Mamon, Rogemar & Gao, Huan, 2018. "A two-decrement model for the valuation and risk measurement of a guaranteed annuity option," Econometrics and Statistics, Elsevier, vol. 8(C), pages 231-249.
    4. 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.
    5. 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.
    6. Anne MacKay & Maciej Augustyniak & Carole Bernard & Mary R. Hardy, 2017. "Risk Management of Policyholder Behavior in Equity-Linked Life Insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 84(2), pages 661-690, June.
    7. Zhenyu Cui & Runhuan Feng & Anne MacKay, 2017. "Variable Annuities with VIX-Linked Fee Structure under a Heston-Type Stochastic Volatility Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 21(3), pages 458-483, July.
    8. Xiaojing Xi & Rogemar Mamon, 2014. "Capturing the Regime-Switching and Memory Properties of Interest Rates," Computational Economics, Springer;Society for Computational Economics, vol. 44(3), pages 307-337, October.
    9. Deelstra, Griselda & Grasselli, Martino & Van Weverberg, Christopher, 2016. "The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 205-219.
    10. Luka Jalen & Rogemar S. Mamon, 2014. "Parameter Estimation in a Regime-Switching Model with Non-normal Noise," International Series in Operations Research & Management Science, in: Rogemar S. Mamon & Robert J. Elliott (ed.), Hidden Markov Models in Finance, edition 127, chapter 0, pages 241-261, Springer.
    11. 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.
    12. LUCIANO, Elisa & VIGNA, Elena, 2008. "Mortality risk via affine stochastic intensities: calibration and empirical relevance," MPRA Paper 59627, University Library of Munich, Germany.
    13. 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.
    14. Feng, Runhuan & Yi, Bingji, 2019. "Quantitative modeling of risk management strategies: Stochastic reserving and hedging of variable annuity guaranteed benefits," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 60-73.
    15. Runhuan Feng, 2014. "A Comparative Study of Risk Measures for Guaranteed Minimum Maturity Benefits by a PDE Method," North American Actuarial Journal, Taylor & Francis Journals, vol. 18(4), pages 445-461, October.
    16. Nicolini, Esteban A., 2004. "Mortality, interest rates, investment, and agricultural production in 18th century England," Explorations in Economic History, Elsevier, vol. 41(2), pages 130-155, April.
    17. Gao, Huan & Mamon, Rogemar & Liu, Xiaoming & Tenyakov, Anton, 2015. "Mortality modelling with regime-switching for the valuation of a guaranteed annuity option," Insurance: Mathematics and Economics, Elsevier, vol. 63(C), pages 108-120.
    18. 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.
    19. 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.
    20. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2013. "Valuing equity-linked death benefits in jump diffusion models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 615-623.
    21. 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.
    22. Bernard, Carole & MacKay, Anne & Muehlbeyer, Max, 2014. "Optimal surrender policy for variable annuity guarantees," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 116-128.
    23. Marcos Escobar & Mikhail Krayzler & Franz Ramsauer & David Saunders & Rudi Zagst, 2016. "Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs," Risks, MDPI, vol. 4(4), pages 1-36, November.
    24. Hyndman, Cody B. & Wenger, Menachem, 2014. "Valuation perspectives and decompositions for variable annuities with GMWB riders," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 283-290.
    25. Fung, Man Chung & Ignatieva, Katja & Sherris, Michael, 2014. "Systematic mortality risk: An analysis of guaranteed lifetime withdrawal benefits in variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 103-115.
    26. Bauer, Daniel & Kling, Alexander & Russ, Jochen, 2008. "A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities1," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 621-651, November.
    27. Apicella, Giovanna & Dacorogna, Michel M, 2016. "A General framework for modelling mortality to better estimate its relationship with interest rate risks," MPRA Paper 75788, University Library of Munich, Germany.
    28. 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. Ankush Agarwal & Christian-Oliver Ewald & Yongjie Wang, 2023. "Hedging longevity risk in defined contribution pension schemes," Computational Management Science, Springer, vol. 20(1), pages 1-34, December.

    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. Wang, Gu & Zou, Bin, 2021. "Optimal fee structure of variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 587-601.
    2. 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.
    3. Wing Fung Chong & Haoen Cui & Yuxuan Li, 2021. "Pseudo-Model-Free Hedging for Variable Annuities via Deep Reinforcement Learning," Papers 2107.03340, arXiv.org, revised Oct 2022.
    4. Feng, Runhuan & Yi, Bingji, 2019. "Quantitative modeling of risk management strategies: Stochastic reserving and hedging of variable annuity guaranteed benefits," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 60-73.
    5. Daniel Doyle & Chris Groendyke, 2018. "Using Neural Networks to Price and Hedge Variable Annuity Guarantees," Risks, MDPI, vol. 7(1), pages 1-19, December.
    6. 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.
    7. Gan Guojun & Valdez Emiliano A., 2017. "Valuation of large variable annuity portfolios: Monte Carlo simulation and synthetic datasets," Dependence Modeling, De Gruyter, vol. 5(1), pages 354-374, December.
    8. 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.
    9. Karim Barigou & Lukasz Delong, 2021. "Pricing equity-linked life insurance contracts with multiple risk factors by neural networks," Working Papers hal-02896141, HAL.
    10. Karim Barigou & Lukasz Delong, 2020. "Pricing equity-linked life insurance contracts with multiple risk factors by neural networks," Papers 2007.08804, arXiv.org, revised Nov 2021.
    11. Karim Barigou & Lukasz Delong, 2021. "Pricing equity-linked life insurance contracts with multiple risk factors by neural networks," Post-Print hal-02896141, HAL.
    12. Feng, Runhuan & Huang, Huaxiong, 2016. "Statutory financial reporting for variable annuity guaranteed death benefits: Market practice, mathematical modeling and computation," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 54-64.
    13. 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.
    14. Wenlong Hu, 2020. "Risk management of guaranteed minimum maturity benefits under stochastic mortality and regime-switching by Fourier space time-stepping framework," Papers 2006.15483, arXiv.org, revised Dec 2020.
    15. 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.
    16. Huang, H. & Milevsky, M.A. & Salisbury, T.S., 2014. "Optimal initiation of a GLWB in a variable annuity: No Arbitrage approach," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 102-111.
    17. 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.
    18. 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.
    19. 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.
    20. 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.

    More about this item

    Keywords

    Variable annuities; Investment guarantee; Stochastic model; Change of numéraire; Lapse risk;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

    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:103:y:2022:i:c:p:1-26. 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.