Hostname: page-component-5b777bbd6c-6lqsf Total loading time: 0 Render date: 2025-06-18T14:54:53.114Z Has data issue: false hasContentIssue false
  • English
  • Français

Impact of correlation between interest rates and mortality rates on the valuation of various life insurance products

Published online by Cambridge University Press:  09 September 2024

Griselda Deelstra ,
Pierre Devolder  and
Benjamin Roelants du Vivier Open the ORCID record for Benjamin Roelants du Vivier [Opens in a new window]
Griselda Deelstra
Affiliation:
Department of Mathematics, Université libre de Bruxelles, Boulevard du Triomphe, CP 210, Brussels 1050, Belgium
Pierre Devolder
Affiliation:
Institute of Statistics, Université Catholique de Louvain, Voie du Roman Pays 20, Louvain-La-Neuve 1348, Belgium
Benjamin Roelants du Vivier*
Affiliation:
Department of Mathematics, Université libre de Bruxelles, Boulevard du Triomphe, CP 210, Brussels 1050, Belgium
*
Corresponding author: Benjamin Roelants du Vivier; Email: benjamin.roelants.du.vivier@ulb.be
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we question the traditional independence assumption between mortality risk and financial risk and model the correlation between these two risks, estimating its impact on the price of different life insurance products. The interest rate and the mortality intensity are modelled as two correlated Hull and White models in an affine set-up. We introduce two building blocks, namely the zero-coupon survival bond and the mortality density, calculate them in closed form and perform an investigation about their dependence on the correlation between mortality and financial risk, both with theoretical results and numerical analysis. We study the impact of correlation also for more structured insurance products, such as pure endowment, annuity, term insurance, whole life insurance and mixed endowment. We show that in some cases, the inclusion of correlation can lead to a severe underestimation or overestimation of the best estimate. Finally, we illustrate that the results obtained using a traditional affine diffusive set-up can be generalised to affine jump diffusion by computing the price of the zero-coupon survival bond in the presence of jumps.

Keywords

Stochastic mortalitycorrelation between mortality and financial riskszero-coupon survival bondmortality densityaffine jump diffusion
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 54 , Issue 3 , September 2024 , pp. 569 - 599
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

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.)

Article purchase

Temporarily unavailable

References

Artzner, P., Eisele, K-T. and Schmidt, T. (2023) Insurance-finance arbitrage. Mathematical Finance, 1–35. https://doi.org/10.1111/mafi.12412.Google Scholar
Barro, R.J., Ursúa, J.F. and Weng, J. (2020) The coronavirus and the great influenza pandemic: Lessons from the “Spanish Flu” for the coronavirus’s potential effects on mortality and economic activity (No. w26866). National Bureau of Economic Research.CrossRefGoogle Scholar
Biffis, E. (2005) Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics and Economics, 37(3), 443–468.Google Scholar
Brahmbhatt, M. (2005) Avian Influenza: Economic and Social Impacts, Economic Document of the World Bank, Available online: http://go.worldbank.org/P6GU56G250 Google Scholar
Brigo, D. and Alfonsi, A. (2005) Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model. Finance and Stochastics, 9, 29–42.CrossRefGoogle Scholar
Brigo, D. and Mercurio, F. (2007) Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit, 2nd ed. Springer.Google Scholar
Brouwers, L., Cakici, B., Camitz, M., Tegnell, A. and Boman, M. (2009) Economic consequences to society of pandemic H1N1 influenza 2009 - preliminary results for Sweden. Euro surveillance : bulletin Europeen sur les maladies transmissibles $=$ European communicable disease bulletin, 14 37.Google ScholarPubMed
Burns, A., Van der Mensbrugghe, D. and Timmer, H. (2008) Evaluating the Economic Consequences of Avian Influenza. Washington, D.C.: World Bank Group.Google Scholar
Cairns, A., Blake, D. and Dowd, K. (2006) Pricing death: Frameworks for the valuation and securitisation of mortality risk. ASTIN Bulletin, 36, 79–120.CrossRefGoogle Scholar
Casamassima, E., Grzelak, L.A., Mulder, F.A. and Oosterlee, C.W. (2022) Pricing and hedging prepayment risk in a mortgage portfolio. International Journal of Theoretical and Applied Finance, 25(4-5), [2250016].CrossRefGoogle Scholar
Cox, J.C., Ingersoll, J.E. Jr and Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica, 53, 385–407.CrossRefGoogle Scholar
Dacorogna, M.M. and Apicella, G. (2016) A general framework for modelling mortality to better estimate its relationship with interest rate risks. SCOR Papers, 39.Google Scholar
Dacorogna, M.M. and Cadena, M. (2015) Exploring the dependence between mortality and market risks. SCOR Papers, 33.Google Scholar
Dahl, M. (2004) Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts. Insurance: Mathematics and Economics, 35(1), 113–136.Google Scholar
Deelstra, G., Devolder, P., Gnameho, K. and Hieber, P. (2020) Valuation of hybrid financial and actuarial products in life insurance: A universal 3-step method. ASTIN Bulletin, 50(3), 709–742.CrossRefGoogle Scholar
Deelstra, G., Grasselli, M. and Weverberg, C. (2016) The role of the dependence between mortality and interest rates when pricing guaranteed annuity options. Insurance: Mathematics and Economics, 71, 205–219.Google Scholar
Devolder, P. and Azizieh, C. (2013) Margrabe Option and Life Insurance with participation. Bulletin d’actuariat français, 13, 47–77.Google Scholar
Diez, F. and Korn, R. (2020) Yield curve shapes of Vasicek interest rate models, measure transformations and an application for the simulation of pension products. European Actuarial Journal, 10, 91–120.CrossRefGoogle Scholar
Duffie, D., Filipovic, D. and Schachermayer, W. (2003) Affine processes and applications in finance. The Annals of Applied Probability, 13(3), 984–1053.CrossRefGoogle Scholar
Duffie, D., Pan, J. and Singleton, K. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 68(6), 1343–1376.CrossRefGoogle Scholar
Favero, C., Gozluklu, A. and Tamoni, A. (2011) Demographic trends, the dividend-price ratio, and the predictability of long-run stock market returns. Journal of Financial and Quantitative Analysis, 46(5), 1493–1520.CrossRefGoogle Scholar
Filipovic, D. and Mayerhofer, E. (2009) Affine diffusion processes: Theory and applications. Radon Series on Computational and Applied Mathematics, 8, 1–40.Google Scholar
Grzelak, L.A. and Oosterlee, C.W. (2011) On the Heston model with stochastic interest rates. SIAM Journal on Financial Mathematics, 2(1), 255–286.CrossRefGoogle Scholar
Hull, J. and White, A. (1996) Using hull-white interest rate trees. The Journal of Derivatives, 4, 26–36.CrossRefGoogle Scholar
Jalen, L. and Mamon, R. (2009) Valuation of contingent claims with mortality and interest rate risks. Mathematical and Computer Modelling, 49, 1893–1904.CrossRefGoogle Scholar
Kou, S.G. (2002) A jump-diffusion model for option pricing. Management Science, 48(8), 1086–1101.CrossRefGoogle Scholar
Li, H., Liu, H., Tang, Q. and Yuan, Z. (2023) Pricing extreme mortality risk in the wake of the COVID-19 pandemic. Insurance: Mathematics and Economics, 108, 84–106.Google ScholarPubMed
Lin, H. and Zhang, Z. (2022) Extreme co-movements between infectious disease events and crude oil futures prices: From extreme value analysis perspective. Energy Economics, 110, 106054.CrossRefGoogle Scholar
Liu, X., Mamon, R. and Gao, H. (2014) A generalized pricing framework addressing correlated mortality and interest risks: A change of probability measure approach. Stochastics, 86(4), 594–608.CrossRefGoogle Scholar
Luciano, E. and Vigna, E. (2008) Mortality risk via affine stochastic intensities: Calibration and empirical relevance. Belgian Actuarial Bulletin, 8, 5–16.Google Scholar
Maurer, T.A. (2011) Asset pricing implications of demographic change. 24th Australasian Finance and Banking Conference 2011 Paper.CrossRefGoogle Scholar
Merton, R.C. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125–144.CrossRefGoogle Scholar
Milevsky, M. and Promislow, S. (2001) Mortality derivatives and the option to annuitise. Insurance: Mathematics and Economics, 29(3), 299–318.Google Scholar
Miltersen, K. and Persson, S.-A. (2005) Is mortality dead? Stochastic forward force of mortality rate determined by no arbitrage.Google Scholar
Nicolini, E. (2004) Mortality, interest rates, investment, and agricultural production in 18th century England. Explorations in Economic History, 41(2), 130–155.CrossRefGoogle Scholar
Polyakova, M., Kocks, G., Udalova, V. and Finkelstein, A. (2020) Initial economic damage from the COVID-19 pandemic in the United States is more widespread across ages and geographies than initial mortality impacts. Proceedings of the National Academy of Sciences, 117(45), 27934–27939.CrossRefGoogle Scholar
Singor, S., Grzelak, L.,Bragt, D. and Oosterlee, C. (2013) Pricing inflation products with stochastic volatility and stochastic interest rates. Insurance: Mathematics and Economics, 52, 286–299.Google Scholar
Wu, Y. and Liang, X. (2018) Vasicek model with mixed-exponential jumps and its applications in finance and insurance. Advances in Difference Equations, 138.CrossRefGoogle Scholar
Zeddouk, F. and Devolder, P. (2020) Mean reversion in stochastic mortality: Why and how? European Actuarial Journal, 10, 499–525.CrossRefGoogle Scholar
Supplementary material: File

Deelstra et al. supplementary material

Deelstra et al. supplementary material
Download Deelstra et al. supplementary material(File)
File 415.2 KB