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A MIXED BOND AND EQUITY FUND MODEL FOR THE VALUATION OF VARIABLE ANNUITIES

Published online by Cambridge University Press:  04 November 2020

Maciej Augustyniak ,
Frédéric Godin  and
Emmanuel Hamel
Maciej Augustyniak
Affiliation:
Département de Mathématiques et de Statistique Université de Montréal Montreal (Quebec), Canada E-Mail: [email protected]
Frédéric Godin*
Affiliation:
Department of Mathematics and Statistics Concordia University Montreal (Quebec), Canada École d’Actuariat Université Laval Quebec (Quebec), Canada E-Mail: [email protected]
Emmanuel Hamel
Affiliation:
École d’Actuariat Université Laval Quebec (Quebec), Canada E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

Variable annuity (VA) policies are typically issued on mutual funds invested in both fixed income and equity asset classes. However, due to the lack of specialized models to represent the dynamics of fixed income fund returns, the literature has primarily focused on studying long-term investment guarantees on single-asset equity funds. This article develops a mixed bond and equity fund model in which the fund return is linked to movements of the yield curve. Theoretical motivation for our proposed specification is provided through an analogy with a portfolio of rolling horizon bonds. Moreover, basis risk between the portfolio return and its risk drivers is naturally incorporated into our framework. Numerical results show that the fit of our model to Canadian VA data is adequate. Finally, the valuation of VAs is illustrated and it is found that the prevailing interest rate environment can have a substantial impact on guarantee costs.

Keywords

Mutual fund modelbond fundyield curveinvestment guaranteevariable annuitiesbasis riskG32C51C63
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 1 , January 2021 , pp. 131 - 159
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

Alberts, M.E. (2020) Negative Interest Rates and the Insurance Industry. Document 220037, Canadian Institute of Actuaries.Google Scholar
Amf (2018) Ligne directrice sur les exigences de suffisance du capital. Autorité des Marchés Financiers.Google Scholar
Andersson, P. and Lagerås, A.N. (2013) Optimal bond portfolios with fixed time to maturity. Insurance: Mathematics & Economics, 53(2), 429–438.Google Scholar
Ardia, D., Mullen, K.M., Peterson, B.G. and Ulrich, J. (2016) DEoptim: Differential Evolution in R. version 2.2-4.Google Scholar
Armstrong, M.J. (2001) The reset decision for segregated fund maturity guarantees. Insurance: Mathematics and Economics, 29(2), 257–269.Google Scholar
Asai, M. and McAleer, M. (2011) Alternative asymmetric stochastic volatility models. Econometric Reviews, 30(5), 548–564.CrossRefGoogle Scholar
Augustyniak, M. and Boudreault, M. (2012) An out-of-sample analysis of investment guarantees for equity-linked products: Lessons from the financial crisis of the late 2000s. North American Actuarial Journal, 16(2), 183–206.CrossRefGoogle Scholar
Augustyniak, M. and Boudreault, M. (2017) Mitigating interest rate risk in variable annuities: An analysis of hedging effectiveness under model risk. North American Actuarial Journal, 21(4), 502–525.CrossRefGoogle 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
Bernard, C. and Moenig, T. (2019) Where less is more: Reducing variable annuity fees to benefit policyholder and insurer. Journal of Risk and Insurance, 86(3), 761–782.CrossRefGoogle Scholar
Bolder, D., Metzler, A. and Johnson, G. (2004) An empirical analysis of the Canadian term structure of zero-coupon interest rates. Staff working paper 2004-48, Bank of Canada.CrossRefGoogle Scholar
Boudreault, M. and Panneton, C.-M. (2009) Multivariate models of equity returns for investment guarantees valuation. North American Actuarial Journal, 13(1), 36–53.CrossRefGoogle Scholar
Brigo, D. and Mercurio, F. (2007) Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit. Berlin: Springer Science & Business Media.Google Scholar
Carnero, M.A., Peña, D. and Ruiz, E. (2004) Persistence and kurtosis in GARCH and stochastic volatility models. Journal of Financial Econometrics, 2(2), 319–342.CrossRefGoogle Scholar
Chen, N.-F., Roll, R. and Ross, S.A. (1986) Economic forces and the stock market. The Journal of Business, 59(3), 383–403.CrossRefGoogle Scholar
Cia (2010) Mortality Improvement Research Paper. Document 210065, Canadian Institute of Actuaries.Google Scholar
Cia (2014) Final Report: Canadian Pensioners Mortality. Document 214013, Canadian Institute of Actuaries.Google Scholar
Cia (2017) Calibration of Stochastic Risk-Free Interest Rate Models for Use in CALM Valuation. Document 217053, Canadian Institute of Actuaries.Google Scholar
Diebold, F.X. and Rudebusch, G.D. (2013) Yield Curve Modeling and Forecasting: The Dynamic Nelson-Siegel Approach. Princeton, NJ: Princeton University Press. With an introduction by Philip Hans Franses and Herman K. van Dijk.Google Scholar
Donnelly, R., Jaimungal, S. and Rubisov, D.H. (2014) Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility. Quantitative Finance, 14(2), 369–382.CrossRefGoogle Scholar
Duan, J.-C. (1995) The GARCH option pricing model. Mathematical Finance, 5(1), 13–32.CrossRefGoogle Scholar
Duan, J.-C., Gauthier, G., Simonato, J.-G. and Sasseville, C. (2006) Approximating the GJR-GARCH and EGARCH option pricing models analytically. Journal of Computational Finance, 9, 41–69.CrossRefGoogle Scholar
Ekeland, I. and Taflin, E. (2005) A theory of bond portfolios. The Annals of Applied Probability, 15(2), 1260–1305.CrossRefGoogle Scholar
Elton, E.J., Gruber, M.J. and Blake, C.R. (1995) Fundamental economic variables, expected returns, and bond fund performance. The Journal of Finance, 50(4), 1229–1256.Google Scholar
Feng, R., Jing, X. and Dhaene, J. (2017) Comonotonic approximations of risk measures for variable annuity guaranteed benefits with dynamic policyholder behavior. Journal of Computational and Applied Mathematics, 311, 272–292.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
Ghalanos, A. (2018) rugarch: Univariate GARCH models. R package version 1.4-0.Google Scholar
Ghalanos, A. and Theussl, S. (2015) Rsolnp: General Non-linear Optimization Using Augmented Lagrange Multiplier Method. R package version 1.16.Google Scholar
Guidolin, M. and Timmermann, A. (2006) An econometric model of nonlinear dynamics in the joint distribution of stock and bond returns. Journal of Applied Econometrics, 21(1), 1–22.CrossRefGoogle Scholar
Hardy, M.R. (2003) Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance. New Jersey: John Wiley & Sons.Google Scholar
Kling, A., Ruez, F. and Russ, J. (2011) The impact of stochastic volatility on pricing, hedging and hedge efficiency of withdrawal benefit guarantees in variable annuities. ASTIN Bulletin: The Journal of the IAA, 41(2), 511–545.Google Scholar
Knoller, C., Kraut, G. and Schoenmaekers, P. (2016) On the propensity to surrender a variable annuity contract: An empirical analysis of dynamic policyholder behavior. Journal of Risk and Insurance, 83(4), 979–1006.CrossRefGoogle Scholar
Ledlie, M.C., Corry, D.P., Finkelstein, G.S., Ritchie, A.J., Su, K. and Wilson, D.C.E. (2008) Variable annuities. British Actuarial Journal, 14(02), 327–389.CrossRefGoogle Scholar
Luethi, D., Erb, P. and Otziger, S. (2018) FKF: Fast Kalman Filter. R package version 0.1.5.Google Scholar
MacKay, A., Augustyniak, M., Bernard, C. and Hardy, M.R. (2017) Risk management of policyholder behavior in equity-linked life insurance. Journal of Risk and Insurance, 84(2), 661–690.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347–370.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
Ngai, A. and Sherris, M. (2011) Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives. Insurance: Mathematics and Economics, 49(1), 100–114.Google Scholar
Nystrup, P., Hansen, B.W., Larsen, H.O., Madsen, H. and Lindström, E. (2017) Dynamic allocation or diversification: A regime-based approach to multiple assets. The Journal of Portfolio Management, 44(2), 62–73.CrossRefGoogle Scholar
Rodríguez, M.J. and Ruiz, E. (2012) Revisiting several popular GARCH models with leverage effect: Differences and similarities. Journal of Financial Econometrics, 10(4), 637–668.CrossRefGoogle Scholar
Ross, S.A. (1976) The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341–360.CrossRefGoogle Scholar
Rutkowski, M. (1999) Self-financing trading strategies for sliding, rolling-horizon, and consol bonds. Mathematical Finance, 9(4), 361–385.CrossRefGoogle Scholar
Sharpe, W.F. (1988) Determining a fund’s effective asset mix. Investment Management Review, 2(6), 59–69.Google Scholar
Sharpe, W.F. (1992) Asset allocation: Management style and performance measurement. Journal of Portfolio Management, 18(2), 7–19.CrossRefGoogle Scholar
Shumway, R.H. and Stoffer, D.S. (2017) Time Series Analysis and Its Applications . Springer Texts in Statistics. Cham, Springer, fourth edition. With R examples.Google Scholar
Society of Actuaries (2011) Policyholder behavior in the tail: Variable annuity guaranteed benefits survey 2011 results. Research report, Society of Actuaries.Google Scholar
Stefanovits, D. and Wüthrich, M.V. (2014) Hedging of long term zero-coupon bonds in a market model with reinvestment risk. European Actuarial Journal, 4, 49–75.CrossRefGoogle Scholar
Sun, P. and Mo, X. (2011) Variable annuity dynamic lapse study: A data mining approach. Research report, Milliman.Google Scholar
Trottier, D.-A., Godin, F. and Hamel, E. (2018a) Local hedging of variable annuities in the presence of basis risk. ASTIN Bulletin: The Journal of the IAA, 48(2), 611–646.CrossRefGoogle Scholar
Trottier, D.-A., Godin, F. and Hamel, E. (2018b) On fund mapping regressions applied to segregated funds hedging under regime-switching dynamics. Risks, 6(3).CrossRefGoogle Scholar
Witmer, J. and Yang, J. (2016) Estimating Canada’s Effective Lower Bound. Bank of Canada Review, pp. 3–14.Google Scholar
Wüthrich, M.V. and Merz, M. (2013) Financial Modeling, Actuarial Valuation and Solvency in Insurance. Berlin: Springer.CrossRefGoogle Scholar
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