Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-11T16:47:36.624Z Has data issue: false hasContentIssue false
  • English
  • Français

VALUATION OF HYBRID FINANCIAL AND ACTUARIAL PRODUCTS IN LIFE INSURANCE BY A NOVEL THREE-STEP METHOD

Published online by Cambridge University Press:  14 August 2020

Griselda Deelstra ,
Pierre Devolder ,
Kossi Gnameho  and
Peter Hieber Open the ORCID record for Peter Hieber [Opens in a new window]
Griselda Deelstra
Affiliation:
Department of Mathematics Université libre de Bruxelles Boulevard du Triomphe, 1050 Brussels, Belgium
Pierre Devolder
Affiliation:
Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA/LIDAM) Université Catholique de Louvain 20 Voie du Roman Pays, 1348 Louvain la Neuve, Belgium
Kossi Gnameho
Affiliation:
Department of Mathematics Université libre de Bruxelles Boulevard du Triomphe, 1050 Brussels, Belgium
Peter Hieber*
Affiliation:
Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA/LIDAM) Université Catholique de Louvain 20 Voie du Roman Pays, 1348 Louvain la Neuve, Belgium Institute of Insurance Science University of Ulm Helmholtzstr. 20, 89069 Ulm, Germany E-Mail: peter.hieber@uni-ulm.de
*
E-Mail: peter.hieber@uni-ulm.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

Keywords

Financial riskactuarial valuationcontract valuationincomplete marketsrisk decompositionhedging
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 709 - 742
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2020 by Astin Bulletin. All rights reserved

References

Ballotta, L., Deelstra, G. and Rayée, G. (2017) Multivariate FX models with jumps: Triangles, quantos and implied correlation. European Journal of Operational Research, 260 (3), 1181–1199.CrossRefGoogle Scholar
Barigou, K. and Dhaene, J. (2019) Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting. Scandinavian Actuarial Journal, 2, 163–187.CrossRefGoogle Scholar
Bielecki, T.R. and Rutkowski, M. (2004) Credit Risk: Modeling, Valuation and Hedging. Berlin, Heidelberg, Springer.CrossRefGoogle Scholar
Boyle, P.P. and Schwartz, E.S. (1977) Equilibrium prices of guarantees under equity-linked contracts. Journal of Risk and Insurance, 44(4), 639–660.CrossRefGoogle Scholar
Brennan, M.J. and Schwartz, E.S. (1979) Alternative investment strategies for the issuers of equity-linked life insurance with an asset value guarantee. Journal of Business, 52(1), 63–93.CrossRefGoogle Scholar
Chen, A., Hieber, P. and Klein, J.K. (2019) Tonuity: A novel individual-oriented retirement plan. ASTIN Bulletin, 49(1), 5–30.CrossRefGoogle Scholar
Chen, A. and Vigna, E. (2017) A unisex stochastic mortality model to comply with EU gender directive. Insurance: Mathematics & Economics, 73, 124–136.Google Scholar
Cont, R. and Tankov, P. (2003) Financial Modelling with Jump Processes. Boca Raton, London, New York, Chapman & Hall/CRC Financial Mathematics Series.CrossRefGoogle Scholar
Dahl, M. and Møller, T. (2006) Valuation and hedging of life insurance liabilities with systematic mortality risk. Insurance: Mathematics and Economics, 39 (2), 193–217.Google Scholar
Delong, Ł., Dhaene, J. and Barigou, K. (2019a) Fair valuation of insurance liability cash-flow streams in continuous time: Applications. ASTIN Bulletin, 49(2), 299–333.CrossRefGoogle Scholar
Delong, Ł., Dhaene, J. and Barigou, K. (2019b) Fair valuation of insurance liability cash-flow streams in continuous time: Theory. Insurance: Mathematics & Economics, 88, 196–208.Google Scholar
Dhaene, J., Stassen, B., Barigou, K., Linders, D. and Chen, Z. (2017) Fair valuation of insurance liabilities: Merging actuarial judgement and market-consistency. Insurance: Mathematics and Economics, 76, 14–27.Google Scholar
Eberlein, E., Papapantoleon, A. and Shiryaev, A. (2009) Esscher transform and the duality principle for multidimensional semimartingales. The Annals of Applied Probability, 19, 1944–1971.CrossRefGoogle Scholar
Engsner, H., Lindensjö, K. and Lindskog, F. (2020) The value of a liability cash flow in discrete time subject to capital requirements. Finance and Stochastics, 24(1), 125–167.CrossRefGoogle Scholar
Gerber, H.U. (1997) Life Insurance Mathematics, 3rd edition. Springer.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (1994) Option pricing by Esscher transform. Transactions of Society of Actuaries, 46, 99–191.Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1996) Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 18(3), 183–218.Google Scholar
Hirbod, A. and Gospodinov, N. (2018) Market consistent valuation with financial imperfection. Decision in Economics and Finance, 41, 65–90.Google Scholar
Ikeda, N. and Watanabe, S. (2014) Stochastic Differential Equations and Diffusion Processes. Elsevier.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008) Modern Actuarial Risk Theory: Using R, Vol. 128. Springer Science & Business Media.CrossRefGoogle Scholar
Keller, P. and Luder, T. (2004) White paper of the Swiss Solvency Test. Swiss Federal Office of Private Insurance, Switzerland.CrossRefGoogle Scholar
Kijima, M. (2006) A multivariate extension of equilibrium pricing transform: The multivariate Esscher and Wang transform for pricing financial and insurance risks. ASTIN Bulletin, 36 (1), 269–283.CrossRefGoogle Scholar
Laeven, R.J. and Goovaerts, M.J. (2008) Premium calculation and insurance pricing. In Encyclopedia of Quantitative Risk Analysis and Assessment, vol. 3, pp. 1302–1314.CrossRefGoogle Scholar
Lin, Y. and Cox, S. (2008) Securitization of catastrophe mortality risks. Insurance: Mathematics & Economics, 42(2), 628–637.Google Scholar
Luciano, E., Regis, L. and Vigna, E. (2012) Delta–gamma hedging of mortality and interest rate risk. Insurance: Mathematics and Economics, 50 (3), 402–412.Google Scholar
Luciano, E. and Vigna, E. (2008) Mortality risk via affine stochastic intensities: Calibration and empirical relevance. Belgian Actuarial Journal, 8(1), 5–16.Google Scholar
Malamud, S., Trubowitz, E. and Wüthrich, M.V. (2008) Market consistent pricing of insurance products. Astin Bulletin, 38(2), 483–526.CrossRefGoogle Scholar
Möhr, C. (2011) Market-consistent valuation of insurance liabilities by cost of capital. ASTIN Bulletin, 41(2), 315–341.Google Scholar
Møller, T. (2002) On valuation and risk management at the interface of insurance and finance. British Actuarial Journal, 8 (4), 787–827.CrossRefGoogle Scholar
Pelsser, A. and Ghalehjooghi, A.S. (2016) Time-consistent actuarial valuations. Insurance: Mathematics & Economics, 66, 97–112.Google Scholar
Pelsser, A. and Stadje, M. (2014) Time-consistent and market-consistent evaluations. Mathematical Finance, 24 (1), 25–65.CrossRefGoogle Scholar
Perlman, M.D. (1974) Jensen’s inequality for a convex vector-valued function on an infinite-dimensional space. Journal of Multivariate Analysis, 4, 52–65.CrossRefGoogle Scholar
Rotar, V.I. (2014) Actuarial Models: the Mathematics of Insurance. Boca Raton, London, New York, CRC Press.CrossRefGoogle Scholar
Schoutens, W. (2003) Lévy Processes in Finance: Pricing Financial Derivatives. Berlin, Heidelberg, John Wiley & Sons, Ltd.CrossRefGoogle Scholar
Tsai, J. and Tzeng, L. (2013) Securitization of catastrophe mortality risks. Astin Bulletin, 43(2), 97–121.CrossRefGoogle Scholar
Zeddouk, F. and Devolder, P. (2019) Pricing of longevity derivatives and cost of capital. Risks, 7 (2), 1–29.CrossRefGoogle Scholar