Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T23:58:35.498Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LOCALLY LINEAR CAIRNS–BLAKE–DOWD MODEL: A NOTE ON DELTA–NUGA HEDGING OF LONGEVITY RISK

Published online by Cambridge University Press:  09 November 2016

Yanxin Liu  and
Johnny Siu-Hang Li
Yanxin Liu
Affiliation:
Department of Finance, University of Nebraska-Lincoln, Lincoln, NE, USA E-Mail: [email protected]
Johnny Siu-Hang Li*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Although longevity risk arises from both the variation surrounding the trend in future mortality and the uncertainty about the trend itself, the latter is often left unmodeled. In this paper, we address this problem by introducing the locally linear CBD model, in which the drifts that govern the expected mortality trend are allowed to follow a stochastic process. This specification results in median forecasts that are more consistent with the recent trends and more robust relative to changes in the data sample period. It also yields wider prediction intervals that may better reflect the possibilities of future trend changes. The treatment of the drifts as a stochastic process naturally calls for nuga hedging, a method proposed by Cairns (2013) to hedge the risk associated with changes in drifts. To improve the existing nuga-hedging method, we propose a new hedging method which demands less stringent assumptions. The proposed method allows hedgers to extract more hedge effectiveness out of a hedging instrument, and is therefore useful when there are only a few traded longevity securities in the market.

Keywords

Nuga hedgingq-forwardsstate-space modelsthe CBD modeltrend changes
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 1 , January 2017 , pp. 79 - 151
Copyright
Copyright © Astin Bulletin 2016 

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.)

References

Ahmadi, S. and Li, J.S.-H. (2014) Coherent mortality forecasting with generalized linear models: A modified time-transformation approach. Insurance: Mathematics and Economics, 59, 194–221.Google Scholar
Booth, H., Maindonald, J. and Smith, L. (2002) Applying Lee-Carter under conditions of variable mortality decline. Population Studies, 56, 325–336.Google Scholar
Cairns, A.J.G. (2011) Modelling and management of longevity risk: Approximations to survival functions and dynamic hedging. Insurance: Mathematics and Economics, 49, 438–453.Google Scholar
Cairns, A.J.G. (2013) Robust hedging of longevity risk. Journal of Risk and Insurance, 80, 621–648.Google Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73, 687–718.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K. and Coughlan, G.D. (2014) Longevity hedge effectiveness: A decomposition. Quantitative Finance, 14, 217–235.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011a) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48, 355–367.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A. and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England and wales and the United States. North American Actuarial Journal, 13, 1–35.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D. and Khalaf-Allah, M. (2011b) Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin, 41, 29–59.Google Scholar
Canadian Institute of Actuaries (2014) Final report on Canadian pensioners' mortality. Available at http://www.cia-ica.ca/docs/default-source/2014/214013e.pdf.Google Scholar
Carter, L.R. (1996) Forecasting U.S. Mortality: A comparison of Box-Jenkins ARIMA and structural time series models. The Sociological Quarterly, 37, 127–144.Google Scholar
Cavanaugh, J. and Shumway, R. (1997) A bootstrap variant of AIC for state-space model selection. Statistica Sinica, 7, 473–496.Google Scholar
Coelho, E. and Nunes, L. (2011) Forecasting mortality in the event of a structural change. Journal of the Royal Statistical Society Series A, 174, 713–736.Google Scholar
Coughlan, G. (2009) Longevity risk transfer: Indices and capital market solutions. In The Handbook of Insurance Linked Securities (eds. Barrieu, P.M. and Albertini, L.), pp. 261–281. London: Wiley.Google Scholar
Coughlan, G., Blake, D., MacMinn, R., Cairns, A.J.G. and Dowd, K. (2013) Longevity risk and hedging solutions. In Handbook of Insurance (eds. Dionne, G.), pp. 997–1035. New York: Springer.Google Scholar
Coughlan, G.D., Khalaf-Allah, M., Ye, Y., Kumar, S., Cairns, A.J.G., Blake, D. and Dowd, K. (2011) Longevity hedging 101: A framework for longevity basis risk analysis and hedge effectiveness. North American Actuarial Journal, 15, 150–176.CrossRefGoogle Scholar
de Jong, P. and Tickle, L. (2006) Extending Lee-Carter mortality forecasting. Mathematical Population Studies, 13, 1–18.Google Scholar
Denuit, M. and Goderniaux, A. (2005) Closing and projecting lifetables using log-linear models. Bulletin of the Swiss Association of Actuaries, 1, 29–49.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010a) Evaluating the goodness of fit of stochastic mortality models. Insurance: Mathematics and Economics, 47, 255–265.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010b) Backtesting stochastic mortality models: An ex-post evaluation of multi-period-ahead density forecasts. North American Actuarial Journal, 14, 281–298.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) A gravity model of mortality rates for two related populations. North American Actuarial Journal, 15, 331–356.Google Scholar
Gallop, A. (2006) Mortality improvements and evolution of life expectancies. Paper presented at the Seminar on Demographic, Economic and Investment Perspectives for Canada, Office of the Superintendent of Financial Institutions Canada.Google Scholar
Girosi, F. and King, G. (2005) A reassessment of Lee-Carter mortality forecasting method. International Journal of Forecasting, 21, 249–260.Google Scholar
Hári, N., de Waegenaere, A., Melenberg, B.} and Nijman, T.E. (2008) Estimating the term structure of mortality. Insurance: Mathematics and Economics, 42, 492–504.Google Scholar
Harvey, A.C. (1990) Forecasting, Structural Time Series Models and the Kalman Filter. New York: Cambridge University Press.Google Scholar
Human Mortality Database (2014) University of California, Berkeley (USA), and Max Planck Institute of Demographic Research (Germany) Available at www.mortality.org or www.humanmortality.de (data downloaded on 1 April 2014).Google Scholar
Holmes, E.E. (2013) Derivation of an EM algorithm for constrained and unconstrained multivariate autoregressive state-space (MARSS) models. preprint (arXiv:1302.3919).Google Scholar
Imhof, J.P. (1961) Computing the distribution of quadratic forms in normal variables. Biometrika, 48, 419–426.Google Scholar
Kalman, R.E. (1960) A new approach to linear filtering and prediction problems. Journal of Fluids Engineering, 82, 35–45.Google Scholar
Kalman, R.E. and Bucy, R.S. (1961) New results in linear filtering and prediction theory. Journal of Fluids Engineering, 83, 95–108.Google Scholar
Kannisto, V., Lauristen, J., Thatcher, A.R. and Vaupel, J.W. (1994) Reduction in mortality at advanced ages: Several decades of evidence from 27 countries. Population Development Review, 20, 793–810.Google Scholar
LaMotte, L.R. and McWhorter, A. (1978) An exact test for the presence of random walk Coefficients in a linear regression model. Journal of the American Statistical Association, 364, 816–820.Google Scholar
Lee, R. and Carter, L. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–671.Google Scholar
Lee, R. and Miller, T. (2001) Evaluating the performance of the Lee-Carter method for forecasting mortality. Demography, 38, 537–549.CrossRefGoogle ScholarPubMed
Li, H., De Waegenaere, A. and Melenberg, B. (2015a) The choice of sample size for mortality forecasting: A Bayesian learning approach. Insurance: Mathematics and Economics, 63 4, 153–168.Google Scholar
Li, J.S.-H., Chan, W.S. and Cheung, S.H. (2011) Structural changes in the Lee-Carter mortality indexes: Detection and implications. North American Actuarial Journal, 15, 13–31.Google Scholar
Li, J.S.-H. and Hardy, M.R. (2011) Measuring basis risk in longevity hedges. North American Actuarial Journal, 15, 177–200.Google Scholar
Li, J.S.-H. and Luo, A. (2012) Key q-duration: A framework for hedging longevity risk. ASTIN Bulletin, 42, 413–452.Google Scholar
Li, J.S.-H., Zhou, R. and Hardy, M.R. (2015b) A step-by-step guide to building multi-population stochastic mortality models. Insurance: Mathematics and Economics, 63, 121–134.Google Scholar
Li, N. and Lee, R. (2005) Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography, 42, 575–594.Google Scholar
Li, S.H. and Chan, W.S. (2005) Outlier analysis and mortality forecasting: The United Kingdom and Scandinavian countries. Scandinavian Actuarial Journal, 3, 187–211.Google Scholar
Luciano, E., Regis, L. and Vigna, E. (2012) Delta-gamma hedging of mortality and interest rate risk. Insurance: Mathematics and Economics, 50, 402–412.Google Scholar
Mavros, G., Cairns, A.J.G., Kleinow, T. and Streftaris, G. (2015) A parsimonious approach to stochastic mortality modelling with dependent residuals, Working Paper, Heriot-Watt University.Google Scholar
Milidonis, A., Lin, Y. and Cox, S.H. (2011) Mortality regimes and pricing. North American Actuarial Journal, 15, 266–289.Google Scholar
Nyblom, J. and Mäkeläinen, T. (1983) Comparisons of tests for the presence of random walk coefficients in a simple linear model. Journal of the American Statistical Association, 384, 856–864.Google Scholar
Oeppen, J. and Vaupel, J.W. (2002) Broken limits to life expectancy. Science, 296, 1029–1031.Google Scholar
O'Hare, C. and Li, Y. (2015) Identifying structural breaks in stochastic mortality models. Journal of Risk and Uncertainty in Engineering Part B, doi: 10.1115/1.4029740.Google Scholar
Pedroza, C. (2006) A Bayesian forecasting model: Predicting U.S. male mortality. Biostatistics, 7, 530–550.CrossRefGoogle ScholarPubMed
Renshaw, A. and Haberman, S. (2003) Lee-Carter mortality forecasting: A parallel generalized linear modelling approach for England and wales mortality projections. Journal of the Royal Statistical Society. Series C (Applied Statistics), 52, 119–137.Google Scholar
Shumway, R.H. and Stoffer, D.S. (2006) Time Series Analysis and Its Applications: With R Examples. New York: Springer-Verlag.Google Scholar
Society of Actuaries. (2014) Mortality Improvement Scale MP-2014 Report. Available at https://www.soa.org/Files/Research/Exp-Study/research-2014-mp-report.pdf.Google Scholar
Stoffer, D.S. and Wall, K.D. (1991) Bootstrapping state-space models: Gaussian maximum likelihood estimation and the Kalman filter. Journal of the American Statistical Association, 86, 1024–1033.Google Scholar
Sweeting, P.J. 2011. A trend-change extension of the Cairns-Blake-Dowd model. Annals of Actuarial Science, 5, 143–162.Google Scholar
Tan, C.I., Li, J., Li, J.S.-H. and Balasooriya, U. (2014) Parametric mortality indexes: From index construction to hedging strategies. Insurance: Mathematics and Economics, 59, 285–299.Google Scholar
van Berkum, F., Antonio, K. and Vellekoop, M. (2014) The impact of multiple structural changes on mortality predictions. Scandinavian Actuarial Journal, doi: 10.1080/03461238.2014.987807.Google Scholar
Vaupel, J.W. (1997) The remarkable improvements in survival at older ages. Philosophical transactions of the Royal Society of London, B, 352, 1799–1804.Google Scholar
Wilmoth, J.R. Andreev, E., Jdanov, D.A. and Glei, D.A. (2005) Methods protocol for the human mortality database. Available at: www.mortality.org.Google Scholar
Zhou, K.Q. and Li, J.S.-H. (2014) Dynamic longevity hedging in the presence of population basis risk: A feasibility analysis from technical and economic perspectives. Paper presented at the 10th International Longevity Risk and Capital Markets Solutions Symposium, Santiago, Chile.Google Scholar
Zhou, R. and Li, J.S.H. (2013) A cautionary note on pricing longevity index swaps. Scandinavian Actuarial Journal, 2013, 1–23.Google Scholar