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ANALYZING MORTALITY BOND INDEXES VIA HIERARCHICAL FORECAST RECONCILIATION

Published online by Cambridge University Press:  03 July 2019

Han Li  and
Qihe Tang
Han Li*
Affiliation:
Department of Actuarial Studies and Business Analytics Macquarie UniversitySydney, NSW 2109, Australia E-Mail: han.li@mq.edu.au.
Qihe Tang
Affiliation:
School of Risk and Actuarial Studies UNSW SydneySydney, NSW 2052, Australia E-Mail: qihe.tang@unsw.edu.au. Department of Statistics and Actuarial Science University of IowaIowa City, IA 52242, USA E-Mail: qihe-tang@uiowa.edu.
*
E-Mail: han.li@mq.edu.au
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Abstract

In recent decades, there has been significant growth in the capital market for mortality- and longevity-linked bonds. Therefore, modeling and forecasting the mortality indexes underlying these bonds have crucial implications for risk management in life insurance companies. In this paper, we propose a hierarchical reconciliation approach to constructing probabilistic forecasts for mortality bond indexes. We apply this approach to analyzing the Swiss Re Kortis bond, which is the first “longevity trend bond” introduced in the market. We express the longevity divergence index associated with the bond’s principal reduction factor (PRF) in a hierarchical setting. We first adopt time-series models to obtain forecasts on each hierarchical level, and then apply a minimum trace reconciliation approach to ensure coherence of forecasts across all levels. Based on the reconciled probabilistic forecasts of the longevity divergence index, we estimate the probability distribution of the PRF of the Kortis bond, and compare our results with those stated in Standard and Poor’s report on pre-sale information. We also illustrate the strong performance of the approach by comparing the reconciled forecasts with unreconciled forecasts as well as those from the bottom-up approach and the optimal combination approach. Finally, we provide first insights on the interest spread of the Kortis bond throughout its risk period 2010–2016.

Keywords

Forecast reconciliationprobabilistic forecasttime-series modelsmortalitymodelingthe Kortis bond
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 823 - 846
Copyright
© Astin Bulletin 2019 

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References

Akaike, H. (1974) A new look at the statistical model identification. IEEE transactions on automatic control, 19(6), 716–723.CrossRefGoogle Scholar
Athanasopoulos, G., Ahmed, R.A. and Hyndman, R.J. (2009) Hierarchical forecasts for australian domestic tourism. International Journal of Forecasting, 25(1), 146–166.CrossRefGoogle Scholar
Bauer, D. and Kramer, F. (2016) The risk of a mortality catastrophe. Journal of Business & Economic Statistics, 34(3), 391–405.CrossRefGoogle Scholar
Bauer, D., Börger, M. and Russ, J. (2010) On the pricing of longevity-linked securities. Insurance: Mathematics and Economics, 46(1), 139–149.Google Scholar
Ben Taieb, S., Taylor, J.W. and Hyndman, R.J. (2017) Coherent probabilistic forecasts for hierarchical time series. In International Conference on Machine Learning, pp. 3348–3357.Google Scholar
Biagini, F., Rheinländer, T. and Widenmann, J. (2013) Hedging mortality claims with longevity bonds. ASTIN Bulletin: The Journal of the IAA, 43(2), 123–157.CrossRefGoogle Scholar
Blake, D., Cairns, A., Coughlan, G., Dowd, K. and MacMinn, R. (2013) The new life market. Journal of Risk and Insurance, 80(3), 501–558.CrossRefGoogle Scholar
Borges, C.E., Penya, Y.K. and Fernandez, I. (2013) Evaluating combined load forecasting in large power systems and smart grids. IEEE Transactions on Industrial Informatics, 9(3), 1570–1577.CrossRefGoogle Scholar
Braun, A. (2016) Pricing in the primary market for cat bonds: new empirical evidence. Journal of Risk and Insurance, 83(4), 811–847.CrossRefGoogle Scholar
Cairns, A.J., Blake, D., Dawson, P. and Dowd, K. (2005) Pricing the risk on longevity bonds. Life and Pensions, 1(2), 41–44.Google Scholar
Cairns, A.J., Blake, D. and Dowd, K. (2006) Pricing death: frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin: The Journal of the IAA, 36(1), 79–120.CrossRefGoogle Scholar
Capistrán, C., Constandse, C. and Ramos-Francia, M. (2010) Multi-horizon inflation forecasts using disaggregated data. Economic Modelling, 27(3), 666–677.CrossRefGoogle Scholar
Chen, H. and Cox, S.H. (2009) Modeling mortality with jumps: applications to mortality securitization. Journal of Risk and Insurance, 76(3), 727–751.CrossRefGoogle Scholar
Chen, H., MacMinn, R. and Sun, T. (2015) Multi-population mortality models: A factor copula approach. Insurance: Mathematics and Economics, 63, 135–146.Google Scholar
Chen, H., MacMinn, R.D. and Sun, T. (2017) Mortality dependence and longevity bond pricing: A dynamic factor copula mortality model with the gas structure. Journal of Risk and Insurance, 84(S1), 393–415.CrossRefGoogle Scholar
Chulia, H., Guillen, M. and Uribe, J.M. (2016) Modeling longevity risk with generalized dynamic factor models and vine-copulae. ASTIN Bulletin: The Journal of the IAA, 46(1), 165–190.CrossRefGoogle Scholar
Cowley, A. and Cummins, J.D. (2005) Securitization of life insurance assets and liabilities. Journal of Risk and Insurance, 72(2), 193–226.CrossRefGoogle Scholar
Cox, S.H., Fairchild, J.R. and Pedersen, H.W. (2000) Economic aspects of securitization of risk. ASTIN Bulletin: The Journal of the IAA, 30(1), 157–193.CrossRefGoogle Scholar
Cummins, J.D. and Trainar, P. (2009) Securitization, insurance, and reinsurance. Journal of Risk and Insurance, 76(3), 463–492.CrossRefGoogle Scholar
Dangerfield, B.J. and Morris, J.S. (1992) Top-down or bottom-up: Aggregate versus disaggregate extrapolations. International Journal of Forecasting, 8(2), 233–241.CrossRefGoogle Scholar
Deng, Y., Brockett, P.L. and MacMinn, R.D. (2012) Longevity/mortality risk modeling and securities pricing. Journal of Risk and Insurance, 79(3), 697–721.CrossRefGoogle Scholar
Engle, R. (2001) Garch 101: The use of arch/garch models in applied econometrics. Journal of economic perspectives, 15(4), 157–168.CrossRefGoogle Scholar
Gao, Q. and Hu, C. (2009) Dynamic mortality factor model with conditional heteroskedasticity. Insurance: Mathematics and Economics, 45(3), 410–423.Google Scholar
Ghalanos, A. (2019) Introduction to the rugarch package. (Version 1.4-1). http://cran.rproject.org/web/packages/rugarch.Google Scholar
Giacometti, R., Bertocchi, M., Rachev, S.T. and Fabozzi, F.J. (2012). A comparison of the Lee–Carter model and AR–ARCH model for forecasting mortality rates. Insurance: Mathematics and Economics, 50(1), 85–93.Google Scholar
Hunt, A. and Blake, D. (2015) Modelling longevity bonds: Analysing the swiss re kortis bond. Insurance: Mathematics and Economics, 63, 12–29.Google Scholar
Hyndman, R.J. and Athanasopoulos, G. (2014) Forecasting: Principles and practice. https://otexts.com/fpp2/.Google Scholar
Hyndman, R.J. and Khandakar, Y. (2008) Automatic time series forecasting: The forecast package for r. Journal of Statistical Software, 27(3), 1–22.CrossRefGoogle Scholar
Hyndman, R.J., Ahmed, R.A., Athanasopoulos, G. and Shang, H.L. (2011) Optimal combination forecasts for hierarchical time series. Computational Statistics & Data Analysis, 55(9), 2579–2589.CrossRefGoogle Scholar
Hyndman, R.J., Athanasopoulos, G., et al. (2014) Optimally reconciling forecasts in a hierarchy. Foresight: The International Journal of Applied Forecasting, 35, 42–48.Google Scholar
Hyndman, R.J., Lee, A., Wang, E., Wickramasuriya, S. and Wang, M.E. (2018) Package ‘hts’.Google Scholar
Jeon, J., Panagiotelis, A. and Petropoulos, F. (2019) Probabilistic forecast reconciliation with applications to wind power and electric load, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.05.020.CrossRefGoogle Scholar
Kahn, K.B. (1998) Revisiting top-down versus bottom-up forecasting. The Journal of Business Forecasting, 17(2), 14.Google Scholar
Lane, M. and Beckwith, R. (2011) Prague spring or Louisiana morning? Annual review for the four quarters, Q2 2010 to Q1 2011. Tech. Rep. Lane Finacial LLC.Google Scholar
Lane, M. and Beckwith, R. (2012) More return; more risk: Annual review for the four quarters, Q2 2011 to Q1 2012. Tech. Rep. Lane Finacial LLC.Google Scholar
Lane, M. and Beckwith, R. (2013) Soft markets ahead!? Annual review for the four quarters, Q2 2012 to Q1 2013. Tech. Rep. Lane Finacial LLC.Google Scholar
Lane, M. and Beckwith, R. (2014) Straw hats in winter: Annual review for the four quarters, Q2 2013 to Q1 2014. Tech. Rep. Lane Finacial LLC.Google Scholar
Lane, M. and Beckwith, R. (2015) Crawling along or coming off bottom? Annual review for the four quarters, Q2 2014 to Q1 2015. Tech. Rep. Lane Finacial LLC.Google Scholar
Lane, M. and Beckwith, R. (2016) Trace data twenty one months on - ILS trade or quote data? Annual review for the four quarters, Q2 2015 to Q1 2016. Tech. Rep. Lane Finacial LLC.Google Scholar
Lane, M. and Beckwith, R. (2017) Annual review and commentary for the four quarters, Q2 2016 to Q1 2017. Tech. Rep. Lane Finacial LLC.Google Scholar
Lee, R. and Miller, T. (2001) Evaluating the performance of the lee-carter method for forecasting mortality. Demography, 38(4), 537–549.CrossRefGoogle ScholarPubMed
Lin, T., Wang, C.-W. and Tsai, C.C.-L. (2015) Age-specific copula-ar-garch mortality models. Insurance: Mathematics and Economics, 61, 110–124.Google Scholar
Lin, Y. and Cox, S.H. (2008) Securitization of catastrophe mortality risks. Insurance: Mathematics and Economics, 42(2), 628–637.Google Scholar
Lin, Y., Liu, S. and Yu, J. (2013) Pricing mortality securities with correlated mortality indexes. Journal of Risk and Insurance, 80(4), 921–948.CrossRefGoogle Scholar
MacMinn, R., Brockett, P. and Blake, D. (2006) Longevity risk and capital markets. Journal of Risk and Insurance, 73(4), 551–557.CrossRefGoogle Scholar
Sarpong, S.A. (2013) Modeling and forecasting maternal mortality; an application of ARIMA models. International Journal of Applied Science and Technology, 3(1), 19–28.Google Scholar
Schwarzkopf, A.B., Tersine, R.J. and Morris, J.S. (1988) Top-down versus bottom-up forecasting strategies. The International Journal Of Production Research, 26(11), 1833–1843.CrossRefGoogle Scholar
Shang, H.L. and Haberman, S. (2017) Grouped multivariate and functional time series forecasting: An application to annuity pricing. Insurance: Mathematics and Economics, 75, 166–179.Google Scholar
Shang, H.L. and Hyndman, R.J. (2017) Grouped functional time series forecasting: An application to age-specific mortality rates. Journal of Computational and Graphical Statistics, 26(2), 330–343.CrossRefGoogle Scholar
Shlifer, E. and Wolff, R. (1979) Aggregation and proration in forecasting. Management Science, 25(6), 594–603.CrossRefGoogle Scholar
Standard and Poor’s (2010) Presale information: Kortis capital ltd. Tech. Rep. Standard and Poors.Google Scholar
Stone, R., Champernowne, D.G. and Meade, J.E. (1942) The precision of national income estimates. The Review of Economic Studies, 9(2), 111–125.CrossRefGoogle Scholar
Stupfler, G. and Yang, F. (2018) Analyzing and predicting cat bond premiums: A financial loss premium principle and extreme value modeling. ASTIN Bulletin: The Journal of the IAA, 48(1), 375–411.CrossRefGoogle Scholar
Syntetos, A.A., Babai, Z., Boylan, J.E., Kolassa, S. and Nikolopoulos, K. (2016) Supply chain forecasting: Theory, practice, their gap and the future. European Journal of Operational Research, 252(1), 1–26.CrossRefGoogle Scholar
Van Erven, T. and Cugliari, J. (2015) Game-theoretically optimal reconciliation of contemporaneous hierarchical time series forecasts. In Modeling and Stochastic Learning for Forecasting in High Dimensions, pp. 297–317. Cham, Switzerland: Springer.CrossRefGoogle Scholar
Wang, C.-W., Huang, H.-C. and Liu, I.-C. (2013) Mortality modeling with non-gaussian innovations and applications to the valuation of longevity swaps. Journal of Risk and Insurance, 80(3), 775–798.CrossRefGoogle Scholar
Weale, M. (1988) The reconciliation of values, volumes and prices in the national accounts. Journal of the Royal Statistical Society. Series A (Statistics in Society), 151(1), 211–221.CrossRefGoogle Scholar
Wickramasuriya, S.L., Athanasopoulos, G. and Hyndman, R.J. (2018) Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization. Journal of the American Statistical Association, 114(526), 804–819.CrossRefGoogle Scholar
Zellner, A. and Tobias, J. (2000) A note on aggregation, disaggregation and forecasting performance. Journal of Forecasting, 19(5), 457–465.3.0.CO;2-6>CrossRefGoogle Scholar