Abstract
Contracts paying a guaranteed minimum rate of return and a fraction of a positive excess rate, which is specified relative to a benchmark portfolio, are closely related to unit-linked life-insurance products and can be considered as alternatives to direct investment in the underlying benchmark. They contain an embedded power option, and the key issue is the tractable and realistic hedging of this option, in order to rigorously justify valuation by arbitrage arguments and prevent the guarantees from becoming uncontrollable liabilities to the issuer. We show how to determine the contract parameters conservatively and implement robust risk-management strategies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Avellaneda, Marco, Arnon Levy and Antonio Parás (1995): Pricing and Hedging Derivative Securities in Markets with Uncertain Volatilities, Applied Mathematical Finance, 2 (2): 73–88.
Bacinello, Anna Rita (2001): Fair Pricing of Life Insurance Participating Policies with a Minimum Interest Rate Guaranteed, Astin Bulletin, 31 (2): 275–297.
Bacinello, Anna Rita and Fulvio Ortu (1993): Pricing Equity-Linked Life Insurance with Endogenous Minimum Guarantee, Insurance: Mathematics and Economics, 12 (3): 245–258.
Bernard, Carole, Olivier Le Courtois and François Quittard-Pinon (2005): Market Value of Life Insurance Contracts under Stochastic Interest Rates and Default Risk, Insurance: Mathematics and Economics, 36 (3): 499–516.
Black, Fischer S. and Myron Scholes (1973): The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (3): 637–654.
Bodie, Zvi and Dwight B. Crane (1999): The Design and Production of New Retirement Savings Products, Journal of Portfolio Management, 25 (2): 77–82.
Boyle, Phelim P. and Mary R. Hardy (1997): Reserving for maturity guarantees: Two approaches, Insurance: Mathematics and Economics, 21 (2): 113–127.
Boyle, Phelim P. and Eduardo S. Schwartz (1977): Equilibrium Prices of Guarantees under Equity-Linked Contracts, Journal of Risk and Insurance, 44 (4): 639–660.
Branger, Nicole and Antje B. Mahayni (2006): Tractable Hedging-AnImplementation of Robust Hedging Strategies, Journal of Economic Dynamics and Control, 30 (11): 1937–1962.
Brennan, Michael J. and Eduardo S. Schwartz (1976): The Pricing of Equity-Linked Life Insurance Policies with an Asset Value Guarantee, Journal of Financial Economics, 3 (3): 195–213.
Brennan, Michael J. and Eduardo S. Schwartz (1979): Alternative Investment Strategies for the Issuers of Equity Linked Life Insurance Policies with an Asset Value Guarantee, Journal of Business, 52 (1): 63–93.
Briys, Eric and François de Varenne (1994): Life Insurance in a Contingent Claim Framework: Pricing and Regulatory Implications, The Geneva Papers on Risk and Insurance Theory, 19 (1): 53–72.
Briys, Eric and François de Varenne (1997): On the Risk of Insurance Liabilities: Debunking Some Common Pitfalls, The Journal of Risk and Insurance, 64 (4): 673–694.
Chen, An and Michael Suchanecki (2007): Default Risk, Bankruptcy Procedures and the Market Value of Life Insurance Liabilities, Insurance: Mathematics and Economics, 40 (2): 231–255.
Coleman, Thomas, Yohan Kim, Yuying Li and Maria-Cristina Patron (2007): Robustly Hedging Variable Annuities with Guarantees under Jump and Volatility Risks, Journal of Risk and Insurance, 74 (2): 347–376.
Dudenhausen, Antje B., Erik Schlögl and Lutz Schlögl (1998): Robustness of Gaussian Hedges under Parameter and Model Misspecification, Technical report, University of Bonn, Department of Statistics.
El Karoui, Nicole, Monique Jeanblanc-Picqué and Steven E. Shreve (1998): Robustness of the Black and Scholes Formula, Mathematical Finance, 8 (2): 93–126.
Esser, Angelika (2003): General Valuation Principles for Arbitrary Payoffs and Applications to Power Options under Stochastic Volatility Models, Financial Markets and Portfolio Management, 17 (3): 351–372.
Geman, Helyette, Nicole El Karoui and Jean-Charles Rochet (1995): Changes of Numeraire, Changes of Probability Measure and Option Pricing, Journal of Applied Probability, 32 (2): 443–458.
Grosen, Anders and Peter L. Jørgensen (2002): Life Insurance Liabilities at Market Value: An Analysis of Insovency Risk, Bonus Policy, and Regulatory Intervention Rules in a Barrier Option Framework, The Journal of Risk and Insurance, 69 (1): 63–91.
Grosen, Anders and Peter L. Jørgensen (1997): Valuation of Early Exercisable Interest Rate Guarantees, Journal of Risk and Insurance, 64 (3): 481–503.
Grosen, Anders and Peter L. Jørgensen (1999): Fair Valuation of Life Insurance Liabilities: The Impact of Interest Guarantees, Surrender Options, and Bonus Policies, Aarhus School of Business, University of Aarhus, working paper.
Harrison, J. Michael and David M. Kreps (1979): Martingales and Arbitrage in Multiperiod Securities Markets, Journal of Economic Theory, 20 (3): 381–408.
Harrison, J. Michael and Stanley R. Pliska (1981): Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stochastic Processes and their Applications, 11 (3): 215–260.
Heath, David R., Robert A. Jarrow and Aandrew Morton (1992): Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation, Econometrica, 60 (1): 77–105.
Jørgensen, Peter L. (2001): Life Insurance Contracts with Embedded Options, The Journal of Risk Finance, 3 (1): 19–30.
Lyons, Terry J. (1995): Uncertain Volatility and the Risk-free Synthesis of Derivatives, Applied Mathematical Finance, 2 (2): 117–133.
Mahayni, Antje B. and Klaus Sandmann (2006): Return Guarantees with Delayed Payment, forthcoming: German Economic Review.
Miltersen, Kristian R. and Svein-Arne Persson (1998): Guaranteed Investment Contracts: Distributed and Undistributed Excess Return, Odense University, working paper.
Miltersen, Kristian R. and Svein-Arne Persson (2003): Guarantees Investment Contracts: Distributed and Undistributed Excess Return, Scandinavian Actuarial Journal, 2003 (4): 257–279.
Musiela, Marek and Marek Rutkowski (1997): Martingale Methods in Financial Modelling, Vol. 36 of Applications of Mathematics, Springer-Verlag, New York, New York, USA.
Nielsen, Jørgen A. and Klaus Sandmann (1995): Equity-Linked Life Insurance: A Model with Stochastic Interest Rates, Insurance: Mathematics and Economics, 16 (3): 225–253.
Pelsser, Antoon (2002): Pricing and Hedging Guaranteed Annuity Options via Static Option Replication, Erasmus University of Rotterdam, working paper.
Pelsser, Antoon and David F. Schrager (2004): Pricing Rate of Return Guarantees in Regular Premium Unit Linked Insurance, Insurance, Mathematics and Economics, 35 (2): 369–398.
Pooley, David M., Peter A. Forsyth and Kenneth R. Vetzal (2003): Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility, IMA Journal of Numerical Analysis, 23 (2): 241–267.
Vanderhoof, Irwin T. and Edward I. Altman (1998): The Fair Value of Insurance Liabilities, The New York University Salomon Center Series on Financial Markets and Institutions, Kluwer Academic Publishers.
Vargiolu, Tiziano (2001): Existence, Uniqueness and Smoothness for the Black-Scholes-Barenblatt Equation, Technical report, Department of Pure and Applied Mathematics, University of Padova.
Zhang, Peter G. (1998): Exotic Options — A Guide to Second Generation Options, World Scientific.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors would like to thank Klaus Sandmann for fruitful discussions. The usual disclaimers apply. The research of the second author is supported under the Australian Research Council’s Discovery funding scheme (project number DP0559879).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mahayni, A., Schlögl, E. The Risk Management of Minimum Return Guarantees. Bus Res 1, 55–76 (2008). https://doi.org/10.1007/BF03342702
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03342702