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Analyzing mortality bond indexes via hierarchical forecast reconciliation. (English) Zbl 1427.91236

Summary: 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.

MSC:

91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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[1] Akaike, H. (1974) A new look at the statistical model identification. IEEE transactions on automatic control, 19(6), 716-723. · Zbl 0314.62039
[2] Athanasopoulos, G., Ahmed, R.A. and Hyndman, R.J. (2009) Hierarchical forecasts for australian domestic tourism. International Journal of Forecasting, 25(1), 146-166.
[3] Bauer, D. and Kramer, F. (2016) The risk of a mortality catastrophe. Journal of Business & Economic Statistics, 34(3), 391-405.
[4] Bauer, D., Börger, M. and Russ, J. (2010) On the pricing of longevity-linked securities. Insurance: Mathematics and Economics, 46(1), 139-149. · Zbl 1231.91142
[5] 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.
[6] 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. · Zbl 1346.91104
[7] 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.
[8] 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.
[9] Braun, A. (2016) Pricing in the primary market for cat bonds: new empirical evidence. Journal of Risk and Insurance, 83(4), 811-847.
[10] Cairns, A.J., Blake, D., Dawson, P. and Dowd, K. (2005) Pricing the risk on longevity bonds. Life and Pensions, 1(2), 41-44.
[11] 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. · Zbl 1162.91403
[12] Capistrán, C., Constandse, C. and Ramos-Francia, M. (2010) Multi-horizon inflation forecasts using disaggregated data. Economic Modelling, 27(3), 666-677.
[13] Chen, H. and Cox, S.H. (2009) Modeling mortality with jumps: applications to mortality securitization. Journal of Risk and Insurance, 76(3), 727-751.
[14] Chen, H., MacMinn, R. and Sun, T. (2015) Multi-population mortality models: A factor copula approach. Insurance: Mathematics and Economics, 63, 135-146. · Zbl 1348.91131
[15] 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.
[16] 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. · Zbl 1390.91174
[17] Cowley, A. and Cummins, J.D. (2005) Securitization of life insurance assets and liabilities. Journal of Risk and Insurance, 72(2), 193-226.
[18] 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.
[19] Cummins, J.D. and Trainar, P. (2009) Securitization, insurance, and reinsurance. Journal of Risk and Insurance, 76(3), 463-492.
[20] 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.
[21] 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.
[22] Engle, R. (2001) Garch 101: The use of arch/garch models in applied econometrics. Journal of economic perspectives, 15(4), 157-168.
[23] Gao, Q. and Hu, C. (2009) Dynamic mortality factor model with conditional heteroskedasticity. Insurance: Mathematics and Economics, 45(3), 410-423. · Zbl 1231.91187
[24] Ghalanos, A. (2019) Introduction to the rugarch package. (Version 1.4-1). http://cran.rproject.org/web/packages/rugarch.
[25] 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. · Zbl 1235.91089
[26] Hunt, A. and Blake, D. (2015) Modelling longevity bonds: Analysing the swiss re kortis bond. Insurance: Mathematics and Economics, 63, 12-29. · Zbl 1348.91150
[27] Hyndman, R.J. and Athanasopoulos, G. (2014) Forecasting: Principles and practice. https://otexts.com/fpp2/.
[28] Hyndman, R.J. and Khandakar, Y. (2008) Automatic time series forecasting: The forecast package for r. Journal of Statistical Software, 27(3), 1-22.
[29] 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. · Zbl 1464.62095
[30] Hyndman, R.J., Athanasopoulos, G., et al. (2014) Optimally reconciling forecasts in a hierarchy. Foresight: The International Journal of Applied Forecasting, 35, 42-48.
[31] Hyndman, R.J., Lee, A., Wang, E., Wickramasuriya, S. and Wang, M.E. (2018) Package ‘hts’.
[32] 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. · Zbl 07077309
[33] Kahn, K.B. (1998) Revisiting top-down versus bottom-up forecasting. The Journal of Business Forecasting, 17(2), 14.
[34] 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.
[35] 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.
[36] 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.
[37] 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.
[38] 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.
[39] 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.
[40] Lane, M. and Beckwith, R. (2017) Annual review and commentary for the four quarters, Q2 2016 to Q1 2017. Tech. Rep. Lane Finacial LLC.
[41] Lee, R. and Miller, T. (2001) Evaluating the performance of the lee-carter method for forecasting mortality.Demography, 38(4), 537-549.
[42] 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. · Zbl 1314.91143
[43] Lin, Y. and Cox, S.H. (2008) Securitization of catastrophe mortality risks. Insurance: Mathematics and Economics, 42(2), 628-637. · Zbl 1152.91593
[44] Lin, Y., Liu, S. and Yu, J. (2013) Pricing mortality securities with correlated mortality indexes. Journal of Risk and Insurance, 80(4), 921-948.
[45] MacMinn, R., Brockett, P. and Blake, D. (2006) Longevity risk and capital markets. Journal of Risk and Insurance, 73(4), 551-557.
[46] 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.
[47] 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.
[48] 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. · Zbl 1394.62146
[49] 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.
[50] Shlifer, E. and Wolff, R. (1979) Aggregation and proration in forecasting. Management Science, 25(6), 594-603. · Zbl 0428.62062
[51] Standard and Poor’s (2010) Presale information: Kortis capital ltd. Tech. Rep. Standard and Poors.
[52] 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.
[53] 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. · Zbl 1390.62222
[54] 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. · Zbl 1346.90181
[55] 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.
[56] 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.
[57] 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.
[58] 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. · Zbl 1420.62402
[59] Zellner, A. and Tobias, J. (2000) A note on aggregation, disaggregation and forecasting performance. Journal of Forecasting, 19(5), 457-465.
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