Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-25T15:09:17.426Z Has data issue: false hasContentIssue false
  • English
  • Français

EFFICIENT DYNAMIC HEDGING FOR LARGE VARIABLE ANNUITY PORTFOLIOS WITH MULTIPLE UNDERLYING ASSETS

Published online by Cambridge University Press:  11 August 2020

X. Sheldon Lin  and
Shuai Yang
X. Sheldon Lin
Affiliation:
Department of Statistical Sciences, University of Toronto, 100 St George Street, Toronto, ONM5S 3G3, Canada E-Mail: sheldon@utstat.toronto.edu
Shuai Yang*
Affiliation:
Department of Statistical Sciences, University of Toronto, 100 St George Street, Toronto, ONM5S 3G3, Canada E-Mail: sheldon@utstat.toronto.edu PathWise Solutions Group LLC Aon, Suite 2300, 20 Bay Street, Toronto, ONM5J 2N9, Canada E-Mails: shuai.yang@mail.utoronto.ca, alex.yang2@aon.com
*
E-Mails: shuai.yang@mail.utoronto.ca, alex.yang2@aon.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A variable annuity (VA) is an equity-linked annuity that provides investment guarantees to its policyholder and its contributions are normally invested in multiple underlying assets (e.g., mutual funds), which exposes VA liability to significant market risks. Hedging the market risks is therefore crucial in risk managing a VA portfolio as the VA guarantees are long-dated liabilities that may span decades. In order to hedge the VA liability, the issuing insurance company would need to construct a hedging portfolio consisting of the underlying assets whose positions are often determined by the liability Greeks such as partial dollar Deltas. Usually, these quantities are calculated via nested simulation approach. For insurance companies that manage large VA portfolios (e.g., 100k+ policies), calculating those quantities is extremely time-consuming or even prohibitive due to the complexity of the guarantee payoffs and the stochastic-on-stochastic nature of the nested simulation algorithm. In this paper, we extend the surrogate model-assisted nest simulation approach in Lin and Yang [(2020) Insurance: Mathematics and Economics, 91, 85–103] to efficiently calculate the total VA liability and the partial dollar Deltas for large VA portfolios with multiple underlying assets. In our proposed algorithm, the nested simulation is run using small sets of selected representative policies and representative outer loops. As a result, the computing time is substantially reduced. The computational advantage of the proposed algorithm and the importance of dynamic hedging are further illustrated through a profit and loss (P&L) analysis for a large synthetic VA portfolio. Moreover, the robustness of the performance of the proposed algorithm is tested with multiple simulation runs. Numerical results show that the proposed algorithm is able to accurately approximate different quantities of interest and the performance is robust with respect to different sets of parameter inputs. Finally, we show how our approach could be extended to potentially incorporate stochastic interest rates and estimate other Greeks such as Rho.

Keywords

Variable annuity portfolionested simulationbalanced samplingspline regressiondynamic hedging
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 913 - 957
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.)

References

Aloise, D., Deshpande, A., Hansen, P. and Popat, P. (2009) NP-hardness of Euclidean sum-of-squares clustering. Machine Learning, 75(2), 245–248.CrossRefGoogle Scholar
Arthur, D. and Vassilvitskii, S. (2006) How slow is the k-means method? Symposium on Computational Geometry, Vol. 6, pp. 1–10.Google Scholar
Bacinello, A.R., Millossovich, P., Olivieri, A. and Pitacco, E. (2011) Variable annuities: A unifying valuation approach. Insurance: Mathematics and Economics, 49(3), 285–297.Google Scholar
Bauer, D. and Ha, H. (2015) A least-squares Monte Carlo approach to the calculation of capital requirements. World Risk and Insurance Economics Congress, Munich, Germany, pp. 2–6.Google Scholar
Bauer, D., Kling, A. and Russ, J. (2008) A universal pricing framework for guaranteed minimum benefits in variable annuities. ASTIN Bulletin: The Journal of the IAA, 38(2), 621–651.CrossRefGoogle Scholar
Bauer, D., Reuss, A. and Singer, D. (2012) On the calculation of the solvency capital requirement based on nested simulations. ASTIN Bulletin: The Journal of the IAA, 42(2), 453–499.Google Scholar
Bernard, C., Hardy, M. and MacKay, A. (2014) State-dependent fees for variable annuity guarantees. ASTIN Bulletin: The Journal of the IAA, 44(3), 559–585.CrossRefGoogle Scholar
Bollen, N.P. (1998) Valuing options in regime-switching models. Journal of Derivatives, 6, 38–50.CrossRefGoogle Scholar
Boyle, P. and Hardy, M. (2003) Guaranteed annuity options. ASTIN Bulletin: The Journal of the IAA, 33(2), 125–152.CrossRefGoogle Scholar
Cathcart, M.J., Lok, H.Y., McNeil, A.J. and Morrison, S. (2015) Calculating variable annuity liability ‘Greeks’ using Monte Carlo simulation. ASTIN Bulletin: The Journal of the IAA, 45(2), 239–266.CrossRefGoogle Scholar
Chen, P. and Yang, H. (2011) Markowitz’s mean-variance asset–liability management with regime switching: A multi-period model. Applied Mathematical Finance, 18(1), 29–50.CrossRefGoogle Scholar
Cheng, X., Luo, W., Gan, G. and Li, G. (2019) Fast valuation of large portfolios of variable annuities via transfer learning. Pacific Rim International Conference on Artificial Intelligence, pp. 716–728.CrossRefGoogle Scholar
Coleman, M., Hayes, R., Lombardo, K. and Ruiz, A. (2019) Variable annuities: Market pressures push the case for model sophistication. <ldlURLLabel>Retrieved from</ldlURLLabel> https://www.willistowerswatson.com/en/insights.Retrieved+from+https://www.willistowerswatson.com/en/insights.>Google Scholar
Dai, M., Kuen Kwok, Y. and Zong, J. (2008) Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 18(4), 595–611.CrossRefGoogle Scholar
Dai, Q. and Singleton, K. (2003) Term structure dynamics in theory and reality. The Review of Financial Studies, 16(3), 631–678.CrossRefGoogle Scholar
De Boor, C. (1978) A Practical Guide to Splines. New York: Springer.CrossRefGoogle Scholar
Deville, J.-C. and Tillé, Y. (2004) Efficient balanced sampling: the cube method. Biometrika, 91(4), 893–912.CrossRefGoogle Scholar
Duong, Q.D. (2019) Application of Bayesian penalized spline regression for internal modeling in life insurance. European Actuarial Journal, 9(1), 67–107.CrossRefGoogle Scholar
Gan, G. and Lin, X.S. (2015) Valuation of large variable annuity portfolios under nested simulation: A functional data approach. Insurance: Mathematics and Economics, 62, 138–150.Google Scholar
Gan, G. and Lin, X.S. (2017) Efficient Greek calculation of variable annuity portfolios for dynamic hedging: A two-level metamodeling approach. North American Actuarial Journal, 21(2), 161–177.CrossRefGoogle Scholar
Gan, G. and Valdez, E.A. (2017) Valuation of large variable annuity portfolios: Monte Carlo simulation and synthetic datasets. Dependence Modeling, 5(1), 354–374.CrossRefGoogle Scholar
Gan, G. and Valdez, E.A. (2018) Regression modeling for the valuation of large variable annuity portfolios. North American Actuarial Journal, 22(1), 40–54.CrossRefGoogle Scholar
Glasserman, P. (2013) Monte Carlo Methods in Financial Engineering, Vol. 53. New York: Springer Science & Business Media.Please provide publisher location for ref. “Glasserman (2013).”Google Scholar
Hejazi, S.A. and Jackson, K.R. (2016) A neural network approach to efficient valuation of large portfolios of variable annuities. Insurance: Mathematics and Economics, 70, 169–181.Google Scholar
Hong, L.J., Juneja, S. and Liu, G. (2017) Kernel smoothing for nested estimation with application to portfolio risk measurement. Operations Research, 65(3), 657–673.CrossRefGoogle Scholar
Ketchen, D.J. and Shook, C.L. (1996) The application of cluster analysis in strategic management research: An analysis and critique. Strategic Management Journal, 17(6), 441–458.3.0.CO;2-G>CrossRefGoogle Scholar
Krah, A.-S., Nikolić, Z. and Korn, R. (2018) A least-squares Monte Carlo framework in proxy modeling of life insurance companies. Risks, 6(2), 62.CrossRefGoogle Scholar
Lin, X.S. and Tan, K.S. (2003) Valuation of equity-indexed annuities under stochastic interest rates. North American Actuarial Journal, 7(4), 72–91.Google Scholar
Lin, X.S., Tan, K.S. and Yang, H. (2009) Pricing annuity guarantees under a regime-switching model. North American Actuarial Journal, 13(3), 316–332.CrossRefGoogle Scholar
Lin, X.S. and Yang, S. (2020) Fast and efficient nested simulation for large variable annuity portfolios: A surrogate modeling approach. Insurance: Mathematics and Economics, 91, 85–103.Google Scholar
Lloyd, S. (1982) Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2), 129–137.CrossRefGoogle Scholar
Meyricke, R. and Sherris, M. (2014) Longevity risk, cost of capital and hedging for life insurers under solvency ii. Insurance: Mathematics and Economics, 55, 147–155.Google Scholar
Milevsky, M.A. and Posner, S.E. (2001) The titanic option: valuation of the guaranteed minimum death benefit in variable annuities and mutual funds. Journal of Risk and Insurance, 68(1), 93–128.Please provide volume number for ref. “Milevsky and Posner (2001).”Google 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
Nedyalkova, D. and Tillé, Y. (2008) Optimal sampling and estimation strategies under the linear model. Biometrika, 95(3), 521–537.CrossRefGoogle Scholar
Ng, A.C.-Y. and Li, J.S.-H. (2013) Pricing and hedging variable annuity guarantees with multiasset stochastic investment models. North American Actuarial Journal, 17(1), 41–62.CrossRefGoogle Scholar
Rebagliati, N. (2013) Strict monotonicity of sum of squares error and normalized cut in the lattice of clusterings. International Conference on Machine Learning, pp. 163–171.Google Scholar
Thorndike, R.L. (1953) Who belongs in the family? Psychometrika, 18(4), 267–276.CrossRefGoogle Scholar
Varnell, E., Kent, J., Ward, R., Osman, R. and Gilchrist, A. (2019) Insurers face challenges on management actions. <ldlURLLabel>Retrieved from</ldlURLLabel> http://ch.milliman.com/uploadedfiles/insight/life-published/pdfs/solvency-ii-presents-challenges.pdf.Retrieved+from+http://ch.milliman.com/uploadedfiles/insight/life-published/pdfs/solvency-ii-presents-challenges.pdf.>Google Scholar
Wood, S.N. (2017) Generalized Additive Models: An Introduction with R. London: Chapman and Hall/CRC.CrossRefGoogle Scholar