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Analytical Pricing of the Unit-Linked Endowment with Guarantees and Periodic Premiums

Published online by Cambridge University Press:  09 August 2013

Werner Hürlimann
Werner Hürlimann*
Affiliation:
FRSGlobal Switzerland AG, c/o. Regus Business Center, Seefeldstrasse 69, CH-8008 Zürich, E-Mail: [email protected]
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Abstract

We consider the unit-linked endowment with guarantee and periodic premiums, where at each premium payment date the insurance company invests a certain fraction of the premium into a risky reference portfolio. In the dual random environment of stochastic interest rates with deterministic volatilities and mortality risk, and for a fixed guarantee, simple analytical lower and upper bounds for the fair periodic premium are explicitly derived. We also consider contracts with guaranteed minimum benefits that vary over time and we obtain tight lower and upper bounds for both fair periodic premiums and guaranteed minimum benefits that increase over time. The numerical illustrations of our results reveal that the analytical bounds are very tight. Moreover, the simple, fast and very reliable analytical numerical calculations with controlled accuracy avoid time consuming Monte Carlo calculations and are almost always preferred by practitioners. Some analytical closed-form solutions for one- and two-year maturity dates are also stated.

Keywords

Unit-linked endowmentguaranteed minimum death and accumulation benefitsfair premiumconvex lower and upper bounds
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 40 , Issue 2 , November 2010 , pp. 631 - 653
Copyright
Copyright © International Actuarial Association 2010

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References

Aase, K.K. and Persson, S.-A. (1994) Pricing of unit-linked life insurance policies. Scandinavian Actuarial Journal, 1, 26–52.CrossRefGoogle Scholar
Bacinello, A. (2007) Equity linked life insurance. Encyclopedia of Quantitative Risk Assessment. J. Wiley.Google Scholar
Bacinello, A. and Ortu, F. (1993a) Pricing equity-linked life insurance with endogenous minimum guarantees. Insurance: Mathematics and Economics 12, 245–257. Corrigendum. Insurance: Mathematics and Economics 13, 303–304.Google Scholar
Bacinello, A. and Ortu, F. (1993b) Pricing guaranteed securities-linked life insurance under interest rate risk. 3rd International AFIR Colloquium, 4, 35–55. Available as http://www.actuaries.org/AFIR/colloquia/Rome/Bacinello_Ortu.pdf Google Scholar
Bacinello, A. and Ortu, F. (1994) Single and periodic premiums for guaranteed equity-linked life insurance under interest-rate risk: the lognormal + Vasicek case. In: Peccati, L., Viren, M. (Eds.). Financial Modelling: Recent Research. Physica-Verlag, Heidelberg, 1–25.Google Scholar
Bacinello, A., Biffis, E. and Millossovich, P. (2009a) Pricing life insurance contracts with early exercise features. Journal Computational Applied Mathematics 233(1), 27–35.CrossRefGoogle Scholar
Bacinello, A., Biffis, E. and Millossovich, P. (2009b) Regression-based algorithms for life insurance contracts with surrender guarantees. Appears in Quantitative Finance.Google Scholar
Boyle, P.P. and Schwartz, E.S. (1977) Equilibrium prices of guarantees under equity-linked contracts. Journal of Risk and Insurance 44, 639–680.CrossRefGoogle Scholar
Brennan, M.J. and Schwartz, E.S. (1976) The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financial Economics 3, 195–213.CrossRefGoogle Scholar
Brennan, M.J. and Schwartz, E.S. (1979a) Alternative investment strategies for the issuers of equity linked life insurance with an asset value guarantee. Journal of Business 52, 63–93.CrossRefGoogle Scholar
Brennan, M.J. and Schwartz, E.S. (1979b) Pricing and investment strategies for equity-linked life insurance. Monograph no. 7, Huebner Foundation for Insurance Education. Wharton School, University of Pennsylvania, Philadelphia.Google Scholar
Corby, F.B. (1977) Reserves for maturity guarantees under unit-linked policies. Journal of the Institute of Actuaries 104, 259–273.CrossRefGoogle Scholar
Costabile, M., Gaudenzi, M., Massabo, I. and Zanette, A. (2009) Evaluating fair premiums of equity-linked policies with surrender option in a bivariate model. Insurance: Mathematics and Economics 45(2), 286–295.Google Scholar
De Felice, M. and Moriconi, F. (2005) Market based tools for managing the life insurance company. ASTIN Bulletin 35, 79–111.CrossRefGoogle Scholar
Delbaen, F. (1990) Equity linked policies. Bulletin Association des Actuaires Belges, 33–52.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002) The concept of comonotonicity in actuarial science and finance: applications. Insurance: Mathematics and Economics 31(2), 133–161.Google Scholar
Geman, H., El Karoui, N. and Rochet, J.-C. (1995) Changes of numeraire, changes of probability measure and option pricing. Journal of Applied Probability 32, 443–458.CrossRefGoogle Scholar
Hardy, M. (2003) Investment Guarantees: Modelling and Risk Management for Equity-Linked Life Insurance. Wiley Finance 168.Google Scholar
Ho, T.S.Y. and Lee, S. (1986) Term structure movements and pricing interest rate contingent claims. Journal of Finance 41, 1011–1029.CrossRefGoogle Scholar
Hull, J. and White, A. (1990) Pricing interest rate derivative securities. The Review of Financial Studies 3, 573–592.CrossRefGoogle Scholar
Jamshidian, F. (1989) An exact bond option formula. Journal of Finance 44(1), 205–209.CrossRefGoogle Scholar
Jamshidian, F. (1991) Bond and option evaluation in the Gaussian interest rate model. Research in Finance 9, 131–170.Google Scholar
Kaas, R., Dhaene, J. and Goovaerts, M.J. (2000) Upper and lower bounds for sums of random variables. Insurance: Mathematics and Economics 27(2), 151–168.Google Scholar
Kurz, A. (1996) Pricing of equity-linked life insurance policies with an asset value guarantee and periodic premiums. 6th International AFIR Colloquium, Nurnberg, Germany. Available as http://www.actuaries.org/AFIR/colloquia/Nuernberg/Kurz.pdf Google Scholar
Mercurio, F. and Moraleda, J. (2000) An analytically tractable interest rate model with humped volatility. European Journal of Operational Research 120, 205–214.CrossRefGoogle Scholar
Mercurio, F. and Moraleda, J. (2001) A family of humped volatility models. The European Journal of Finance 7, 93–116.CrossRefGoogle Scholar
Moraleda, J. and Vorst, T. (1997) Pricing American interest rate claims with humped volatility models. Journal of Banking and Finance 21(8), 1131–1157.CrossRefGoogle Scholar
Nielsen, J.A. and Sandmann, K. (1995) Equity-linked life insurance – a model with stochastic interest rates. Insurance: Mathematics and Economics 16(3), 225–253.Google Scholar
Nielsen, J.A. and Sandmann, K. (1996) Uniqueness of the fair premium for equity-linked life insurance contracts. The GENEVA Papers on Risk and Insurance Theory 21, 65–102.CrossRefGoogle Scholar
Nielsen, J.A. and Sandmann, K. (2002) The fair premium of an equity-linked life and pension insurance. In Sandman, K., Schönbucher, P.J. (Eds.). Advances in Finance and Stochastics – Essays in Honor of Dieter Sondermann, 219–256. Springer Verlag.Google Scholar
Persson, S.-A. (1994) Pricing life insurance contracts under financial uncertainty. PhD-thesis, Norwegian School of Economics and Business Administration, Bergen.Google Scholar
Ritchken, P. and Chuang, I. (1999) Interest rate option pricing with volatility humps. Review of Derivative Research 3, 237–262.CrossRefGoogle Scholar
Rogers, L.C.G. and Shi, Z. (1995) The value of an Asian option. Journal of Applied Probability 32(4), 1077–1088.CrossRefGoogle Scholar
Vanduffel, S., Hoedemakers, T. and Dhaene, J (2005a) Comparing approximations for sums of non-independent lognormal random variables. North American Actuarial Journal 9(4), 71–82.CrossRefGoogle Scholar
Vanduffel, S., Dhaene, J. and Goovaerts, M. (2005b) On the evaluation of saving-consumption plans. Journal of Pension Economics and Finance 4(1), 17–30.CrossRefGoogle Scholar
Vanduffel, S., Chen, X., Dhaene, J., Goovaerts, M., Henrard, L. and Kaas, R. (2008a) Optimal approximations for risk measures of sums of lognormals based on conditional expectations. Journal of Computational and Applied Mathematics 221(1), 202–218.CrossRefGoogle Scholar
Vanduffel, S., Shang, Z., Henrard, L., Dhaene, J. and Valdez, E. (2008b) Analytical bounds and approximations for annuities and Asian options. Insurance: Mathematics and Economics 42(3), 1109–1117.Google Scholar
Vasicek, O.A. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.CrossRefGoogle Scholar
Yolcu, Y. (2005) One-factor interest rate models: analytical solutions and approximations. Master Thesis, Department of Financial Mathematics. The Middle East Technical University. http://www3.iam.metu.edu.tr/iam/images/9/9e/Yelizyolcuthesis.pdf Google Scholar