Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T19:14:26.478Z Has data issue: false hasContentIssue false
  • English
  • Français

RISK-BASED CAPITAL FOR VARIABLE ANNUITY UNDER STOCHASTIC INTEREST RATE

Published online by Cambridge University Press:  25 June 2020

JinDong Wang  and
Wei Xu
JinDong Wang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China, E-Mail: jdwang@pku.edu.cn
Wei Xu*
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, Canada, E-Mail: wei.xu@ryerson.ca
*
E-Mail: wei.xu@ryerson.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Interest rate is one of the main risks for the liability of the variable annuity (VA) due to its long maturity. However, most existing studies on the risk measures of the VA assume a constant interest rate. In this paper, we propose an efficient two-dimensional willow tree method to compute the liability distribution of the VA with the joint dynamics of the mutual fund and interest rate. The risk measures can then be computed by the backward induction on the tree structure. We also analyze the sensitivity and impact on the risk measures with regard to the market model parameters, contract attributes, and monetary policy changes. It illustrates that the liability of the VA is determined by the long-term interest rate whose increment leads to a decrease in the liability. The positive correlation between the interest rate and mutual fund generates a fat-tailed liability distribution. Moreover, the monetary policy change has a bigger impact on the long-term VAs than the short-term contracts.

Keywords

Variable annuityvalue at riskconditional tail expectationGMWBGMDBGMMBtwo-factor willow tree methodstochastic interest rateHull–White model
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 959 - 999
Copyright
© 2020 by Astin Bulletin. All rights reserved

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This work is supported by National Natural Science Fund of China (No. U1811462) and the Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-04686).

References

Arrow, K.J. (1964) The role of secutrities in the optimal allocation of risk-bearing. Review of Financial Studies, 31(01), 91–96.Google Scholar
Bacinello, A.R., Millossovich, P. and Montealegre, A. (2016) The valuation of gmwb variable annuities under alternative fund distributions and policyholder behaviours. Scandinavian Actuarial Journal, 79, 446–465.CrossRefGoogle Scholar
Barraquand, J. and Pudet, T. (1996) Pring of american path-dependent contingent claims. Mathematical Finance, 6, 17–51.CrossRefGoogle Scholar
Bauer, D., Kling, A. and Russ, J. (2008) A universal pricing framework for guaranteed minimum benefits in variable annuities. Astin Bulletin, 38, 621–651.CrossRefGoogle Scholar
Bell, F.C. and Miller, M.L. (2005) Life tables for the united states social security area 1900-2100. Social Security Administration Publications, p. 69.Google Scholar
Black, F. and Karasinski, P. (1991) Bond and option pricing when short rates are lognormal. Financial Analysts Journal, 47(4), 52–59.CrossRefGoogle Scholar
Costabile, M. (2017) A lattice-based model to evaluate variable annuities with guaranteed minimum withdrawal benefits under a regime-switching model. Scandinavian Actuarial Journal, 38, 231–244.Google Scholar
Cox, J., Ingersoll, J., Jonathan, E. and Ross, S. (1985) A theory of the term structure of interest rates. Econometrica: Journal of the Econometric Society, 53, 385–407.CrossRefGoogle Scholar
Cox, J.C., Ross, S.A. and Rubinstein, M. (1979) Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229–263.CrossRefGoogle Scholar
Cui, Z., Kim, J., Lian, G. and Liu, Y. (2019) Risk measures for variable annuities: A Hermite 751 series expansion approach. Journal of management science and engineering, 4, 119–141.CrossRefGoogle Scholar
Curran, M. (2001) Willow power: Optimizing derivative pricing trees. ALGO Research Quarterly 4(4), 15–24.Google Scholar
Dai, M., Kwok, Y.K. and Zong, J.P. (2008) Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance, 18(18), 595–611.CrossRefGoogle Scholar
Dong, B., Wang, J. and Xu, W.Risk metrics evaluation for variable annuity with various guaranteed benefits. Journal of Derivatives DOI: 10.3905/jod.2020.1.109.Google Scholar
Dong, B., Xu, W. and Kwok, Y.K. (2019) Willow tree algorithms for pricing guaranteed minimum withdrawal benefits under jump-diffusion and CEV models. Quantitative Finance, 19, 1741–1761.CrossRefGoogle Scholar
Drinkwater, M., Iqbal, I. and Montiminy, J.E. (2013) Variable annuity guaranteed living benefits utilization: 2011 Experience. A Joint Study Sponsored by the Society of Actuaries and LIMRA: Schamburg.Google Scholar
Drinkwater, M., Iqbal, I. and Montiminy, J.E. (2014) Variable annuity guaranteed living benefits utilization: 2012 experience. A Joint Study Sponsored by the Society of Actuaries and LIMRA: Schamburg.Google Scholar
Drinkwater, M., Iqbal, I. and Montiminy, J.E. (2015) Variable annuity guaranteed living benefits utilization: 2013 experience. Technical Report. Society of Actuaries and LIMRA.Google Scholar
Feng, R. and Huang, H. (2012) Analytical calculation of risk measures for variable annuity guaranteed benefits. Insurance: Mathematics and Economics, 51, 636–648.Google Scholar
Feng, R. and Vecer, J. (2017) Risk based capital for guaranteed minimum withdrawal benefit. Quantitative Finance, 17(3), 471–478.CrossRefGoogle Scholar
Feng, R. and Volmer, H.M. (2014) Spectral methods for the calculation of risk measures for variable annuity guaranteed benefits. ASTIN Bulletin, 44(3), 651–681.CrossRefGoogle Scholar
Hainaut, D. and MacGilchrist, R. (2010) An interest rate tree driven by a Lévy process. Journal of Derivatives, 18(2).CrossRefGoogle Scholar
Hill, I. and Holder, R. (1976) Algorithm as 99: Fitting Johnson Curves by moments. Applied Statistics, 25(2), 180–189.CrossRefGoogle Scholar
Huang, Y. and Forsyth, P.A. (2012) Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (gmwb). IMA Journal of Numerical Analysis, 32(1), 320–351.CrossRefGoogle Scholar
Huang, Y.Q., Forsyth, P.A. and Labahn, G. (2012) Iterative methods for the solution of a singular control formulation of a gmwb pricing problem. Numerische Mathematik, 122, 133–167.CrossRefGoogle Scholar
Hull, J. and White, A.Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573–592, 1990.CrossRefGoogle Scholar
Johnson, N. (1949) System of frequency curves generated by methods of translation. Biometrika, 36, 149–176.CrossRefGoogle Scholar
Kang, B. and Ziveyi, J. (2018) Optimal surrender of guaranteed minimum maturing benefits under stochastic volatility and interest rates. Insurance: Mathematics and Economics, 79, 43–56.Google Scholar
Kelani, A. and Quittard-Pinon, F. (2017) Pricing and hedging variable annuities in a Lévy market: A risk management perspective. Journal of Risk and Insurance, 84(1), 209–238.CrossRefGoogle Scholar
Kolkiewicz, A. and Liu, Y. (2012) Semi-static hedging for gmwb in variable annuities. North American Actuarial Journal, 16(1), 112–140.CrossRefGoogle Scholar
Low, R.K.Y., Alcock, J., Faff, R. and Brailsford, T. (2013) Canonical vine copulas in the context of modern portfolio management: Are they worth it? Journal of Banking and Finance, 37, 3085–3099.CrossRefGoogle Scholar
Lu, L., Xu, W. and Qian, Z. (2017) Efficient willow tree method for European-style and American-style moving average barrier options pricing. Quantitative Finance, 17, 889–906.CrossRefGoogle Scholar
Luo, X. and Shevchenko, P. (2015) Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization. Insurance: Mathematics and Economics, 62, 5–15.Google Scholar
Ma, J., Huang, S. and Xu, W. (2020) An efficient convergent willow tree method for american and exotic option pricing under stochastic volatility models. Journal of Derivatives, 27, 75–98.CrossRefGoogle Scholar
Mackay, A. (2014) Fee structure and surrender incentives in variable annuities. Ph.D. Thesis, University of Waterloo.Google Scholar
Milevsky, M.A. and Posner, S.E. (2001) The Titantic option: Valuation of the guaranteed minimum death benefit and mutual funds. Journal of Risk and Insurance, 68(1), 49–61.CrossRefGoogle Scholar
Milevsky, M.A. and Salisbury, T.S. (2006) Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38(1), 21–38.Google Scholar
Peng, J., Leung, K.S. and Kwok, Y.K. (2012) Pricing guaranteed minimum withdrawal benefits under stochastic interest rates. Quantitative Finance, 12(6), 933–941.CrossRefGoogle Scholar
Shevchenko, P.V. and Luo, X.L. (2017) Valuation of variable annuities with guaranteed minimum withdrawal benefit under stochastic interest rate. Working paper of Macquarie University.CrossRefGoogle Scholar
Ulm, E.M. (2008) Analytic solution for return of premium and rollup guaranteed minimum death benefit options under some simple mortality laws. ASTIN Bulletin, 38, 543–563.CrossRefGoogle Scholar
Ulm, E.M. (2010) The effect of policyholder transfer behavior on the value of guaranteed minimum death benefits. North American Actuarial Journal, 14, 16–37.CrossRefGoogle Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188.CrossRefGoogle Scholar
Wang, G. and Xu, W. (2018) A unified willow tree framework for one-factor short-rate models. The Journal of Derivatives, 25(3), 33–54.CrossRefGoogle Scholar
Xu, W., Hong, Z.W. and Qin, C.X. (2013) A new sampling strategy willow tree method with application to path-dependent option pricing. Quantitative Finance, 13(6), 861–872.CrossRefGoogle Scholar
Xu, W. and Yin, Y.F. (2014) Pricing american options by willow tree method under jump-diffusion process. Journal of Derivatives, 22(1), 1–9.CrossRefGoogle Scholar
Yang, S.S. and Dai, T.S. (2013) A flexible tree for evaluating guaranteed minimum withdrawal benefits under deferred life annuity contracts with various provisions. Insurance: Mathematics and Economics, 52(2), 231–242.Google Scholar
Yao, Y. and Xu, W. (2020) Willow tree algorithms for pricing exotic derivatives on discrete realized variance under time-changed Lévy process. Working Paper.Google Scholar