A General framework for modelling mortality to better estimate its relationship with interest rate risks

Apicella, Giovanna and Dacorogna, Michel M (2016): A General framework for modelling mortality to better estimate its relationship with interest rate risks.

Abstract

The need for having a good knowledge of the degree of dependence between various risks is fundamental for understanding their real impacts and consequences, since dependence reduces the possibility to diversify the risks.

This paper expands in a more theoretical approach the methodology developed in for exploring the dependence between mortality and market risks in case of stress. In particular, we investigate, using the Feller process, the relationship between mortality and interest rate risks. These are the primary sources of risk for life (re)insurance companies. We apply the Feller process to both mortality and interest rate intensities.

Our study cover both the short and the long-term interest rates (3m and 10y) as well as the mortality indices of ten developed countries and extending over the same time horizon. Specifically, this paper deals with the stochastic modelling of mortality. We calibrate two different specifications of the Feller process (a two-parameters Feller process and a three-parameters one) to the survival probabilities of the generation of males born in 1940 in ten developed countries. Looking simultaneously at different countries gives us the possibility to find regularities that go beyond one particular case and are general enough to gain more confidence in the results. The calibration provides in most of the cases a very good fit to the data extrapolated from the mortality tables. On the basis of the principle of parsimony, we choose the two-parameters Feller process, namely the hypothesis with the fewer assumptions. These results provide the basis to study the dynamics of both risks and their dependence.

• Item Type: MPRA Paper
• Language: English
• Keywords: Mortality Model, Interest Rate Model, Dependence
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C40 - General

References

[1] D. Blake A. J. G. Cairns and K. Dowd. Pricing death: Framework for the valuation and securitization of mortality risk. Astin Bulletin, 36(1):79–120, 2006.

[2] G. Apicella and M. Dacorogna. A Comprehensive Study of Mortality Dynamics in Ten Developed Countries Using the Feller Process. in Ibit 2016 Proceedings, Paestum, 2016.

[3] K. R. Berends and Plestis T. Rosen R.J. McMenamin, R. The Sensitivity of Life Insurance Firms to Interest Rate Changes. Economic Perspectives, Available at SSRN: http: //ssrn.com/abstract=2386163, 37(2), 2013.

[4] E. Biffis. Affine processes for dynamic mortality and actuarial valuations. Insurance Mathematics and Economics, 37:443–468, 2005.

[5] C. Blackburn and M. Sherris. Consistent dynamic affine mortality models for longevity risk applications. Insurance Mathematics and Economics, 53:64–73, 2013.

[6] M. Dacorogna and M. Cadena. Exploring the dependence between mortality and market risks. SCOR Papers n33, 2015.

[7] M. Dacorogna and M. Kratz. Living in a stochastic world and managing complex risks. Available at SSRN: http://ssrn.com/abstract=2668468, 2015.

[8] M. Dahl. Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts. Insurance Mathematics and Economics, 35:113–136, 2004.

[9] D. Duffie and K.J. Singleton. Credit risk. Princeton Press, 2003.

[10] M. Grasselli G. Deelstra and C. VanWaverberg. The role of the dependence between mortality and interest rates when pricing guaranteed annuity options. SSRN Archive, 2015.

[11] L. Jalen and R. Mamon. Valuation of contingent claims with mortality and interest rate risks. Mathematical and Computer Modelling, 49:1893–1904, 2009.

[12] E. Luciano and E. Vigna. Non mean reverting affine processes for stochastic mortality. Carlo Alberto Notebook 30/06 and ICER WP 4/05, 2005.

[13] E. Luciano and E. Vigna. Mortality risk via affine stochastic intensities: calibration and empirical relevance. Belgian Actuarial Bulletin, 8(1):5–16, 2008.

[14] A. Mahayni and D. Steuten. Deferred life annuities: on the combined effects of stochastic mortality and interest rates. Review of Managerial Science, Vol. 7, Issue 1, pp 1-28, 2013.

[15] K. Ignatieva M.C. Fung and M. Sherris. Systematic mortality risk: An analysis of guaranteed lifetime withdrawal benefits in variable annuities. Insurance Mathematics and Economics, (58):103–115, 2014.

[16] M. A. Milevsky and S. D. Promislow. Mortality derivatives and the option to annuitise. Insurance Mathematics and Economics, 29:299–318, 2001.

[17] E. Pitacco. Survival models in a dynamic context: a survey. Insurance Mathematics and Economics, 29:270–298, 2004.

[18] E. Pitacco. Mortality and Longevity: a risk management perspective. In Proceedings of the 1st IAA Life Colloquium, Stockholm, 2007.

[19] M. M. Dacorogna R. Bürgi and R. Iles. Risk Aggregation, dependence structure and diversification benefit. in "Stress Testing for financial institutions", edited by Daniel Rösch and Harald Scheule, Riskbooks, Incisive Media, London, chap. 12:265–306, 2008.

[20] D. F. Schrager. Affine stochastic mortality. Insurance Mathematics and Economics, 38:81–97, 2006.

[21] S. Ortobelli S. Rachev F. J. Fabozzi V. Russo, R. Giacometti. Calibrating affine stochastic mortality models using term assurances premiums. Insurance: Mathematics and Economics, 49:53–60, 2011.

[22] R. Mamon X. Liu and H. Gao. A generalized pricing framework addressing correlated mortality and interest risks: a change of probability measure approach. Stochastics An International of Probability and Stochastic Processes, 86:4:594–608, 2014.