Dynamic hedging of longevity risk: the effect of trading frequency. (English) Zbl 1390.91194

Summary: This paper investigates dynamic hedging strategies for pension and annuity liabilities that are exposed to longevity risk. In particular, we consider a hedger who wishes to minimize the variance of her hedging error using index-based longevity-linked derivatives. To cope with the fact that liquidity of longevity-linked derivatives is still limited, we consider a liquidity constrained case where the hedger can only trade longevity-linked derivatives at a frequency lower than other assets. Time-consistent, closed-form solutions of optimal hedging strategies are obtained under a forward mortality framework. In the numerical illustration, we show that lowering the trading of the longevity-linked derivatives to a 2-year frequency only leads to a slight loss of the hedging performance. Moreover, even when the longevity-linked derivatives are traded at a very low (5-year) frequency, dynamic hedging strategies still significantly outperform the static one.


91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91D20 Mathematical geography and demography
Full Text: DOI


[1] Ang, A.; Papanikolaou, D.; Westerfield, M. M., Portfolio choice with illiquid assets, Management Science, 60, 2737-2761, (2014)
[2] Basak, S.; Chabakauri, G., Dynamic mean-variance asset allocation, Review of Financial Studies, 23, 2970-3016, (2010)
[3] Basak, S.; Chabakauri, G., Dynamic hedging in incomplete markets: A simple solution, Review of financial studies, 25, 1845-1896, (2012)
[4] Bauer, D.; Benth, F. E.; Kiesel, R., Modeling the forward surface of mortality, SIAM Journal on Financial Mathematics, 3, 639-666, (2012) · Zbl 1255.91443
[5] Bauer, D.; Börger, M.; Ruß, J., On the pricing of longevity-linked securities, Insurance: Mathematics and Economics, 46, 139-149, (2010) · Zbl 1231.91142
[6] Bauer, D.; Börger, M.; Ruß, J.; Zwiesler, H.-J., The volatility of mortality, Asia-Pacific Journal of Risk and Insurance, 3, 172-199, (2008)
[7] Bauer, D.; Ruß, J., (2006)
[8] Bayraktar, E.; Milevsky, M. A.; David Promislow, S.; Young, V. R., Valuation of mortality risk via the instantaneous sharpe ratio: Applications to life annuities, Journal of Economic Dynamics and Control, 33, 676-691, (2009) · Zbl 1170.91406
[9] Biffis, E., Affine processes for dynamic mortality and actuarial valuations, Insurance: Mathematics and Economics, 37, 443-468, (2005) · Zbl 1129.91024
[10] Biffis, E.; Denuit, M.; Devolder, P., Stochastic mortality under measure changes, Scandinavian Actuarial Journal, 2010, 284-311, (2010) · Zbl 1226.91022
[11] Biffs, B.; Blake, D., Keeping some skin in the game: How to start a capital market in longevity risk transfers, North American Actuarial Journal, 18, 14-21, (2014)
[12] Blackburn, C.; Sherris, M., Consistent dynamic affine mortality models for longevity risk applications, Insurance: Mathematics and Economics, 53, 64-73, (2013) · Zbl 1284.91208
[13] Blackburn, C.; Sherris, M., (2014)
[14] Blake, D.; Cairns, A. J.G.; Dowd, K., Living with mortality: Longevity bonds and other mortality-linked securities, British Actuarial Journal, 12, 153-197, (2006)
[15] Cairns, A. J.G., Modelling and management of longevity risk: Approximations to survivor functions and dynamic hedging, Insurance: Mathematics and Economics, 49, 438-453, (2011) · Zbl 1230.91068
[16] Cairns, A. J.G., Robust hedging of longevity risk, Journal of Risk and Insurance, 80, 621-648, (2013)
[17] Cairns, A. J.G.; Blake, D.; Dawson, P.; Dowd, K., Pricing the risk on longevity bonds, Life and Pensions, 1, 41-44, (2005)
[18] Cairns, A. J.G.; Dowd, K.; Blake, D.; Coughlan, G. D., Longevity hedge effectiveness: A decomposition, Quantitative Finance, 14, 217-235, (2014) · Zbl 1294.91072
[19] Caplin, A.; Leahy, J., The recursive approach to time inconsistency, Journal of Economic Theory, 131, 134-156, (2006) · Zbl 1142.90457
[20] (2010)
[21] Dahl, M.; Glar, S.; Møller, T., Mixed dynamic and static risk-minimization with an application to survivor swaps, European Actuarial Journal, 1, 233-260, (2011)
[22] Dahl, M.; Melchior, M.; Møller, T., On systematic mortality risk and risk-minimization with survivor swaps, Scandinavian Actuarial Journal, 2008, 114-146, (2008) · Zbl 1224.91054
[23] Dawson, P.; Dowd, K.; Cairns, A. J.G.; Blake, D., Survivor derivatives: A consistent pricing framework, Journal of Risk and Insurance, 77, 579-596, (2010)
[24] De Jong, F.; Santa-Clara, P., The dynamics of the forward interest rate curve: A formulation with state variables, Journal of Financial and Quantitative Analysis, 34, 131-157, (1999)
[25] De Rosa, C.; Luciano, E.; Regis, L., Basis risk in static versus dynamic longevity-risk hedging, Scandinavian Actuarial Journal, 2017, 343-365, (2017) · Zbl 1401.91129
[26] Driessen, J.; Klaassen, P.; Melenberg, B., The performance of multi-factor term structure models for pricing and hedging caps and swaptions, Journal of Financial and Quantitative Analysis, 38, 635-672, (2003)
[27] Heath, D.; Jarrow, R.; Morton, A., Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation., Econometrica: Journal of the Econometric Society, 60, 77-105, (1992) · Zbl 0751.90009
[28] Inui, K.; Kijima, M., A markovian framework in multi-factor heath-jarrow-morton models, Journal of Financial and Quantitative Analysis, 33, 423-440, (1998)
[29] Jeanblanc, M.; Yor, M.; Chesney, M., Mathematical Methods for Financial Markets, (2009), New York: Springer, New York · Zbl 1205.91003
[30] Kallsen, J.; Geman, H.; Madan, D.; Pliska, S. R.; Vorst, T., Mathematical Finance Bachelier Congress 2000, Utility-based derivative pricing in incomplete markets, 313-338, (2002), New York: Springer, New York
[31] Karatzas, I., Brownian Motion and Stochastic Calculus, (1991), New York: Springer, New York · Zbl 0734.60060
[32] Lee, R. D.; Carter, L. R., Modeling and forecasting us mortality, Journal of the American Statistical Association, 87, 659-671, (1992) · Zbl 1351.62186
[33] Li, H.; Waegenaere, A.; Melenberg, B., Robust mean-variance hedging of longevity risk, Journal of Risk and Insurance, 84, 459-475, (2017)
[34] Li, J. S.-H.; Hardy, M. R., Measuring basis risk in longevity hedges, North American Actuarial Journal, 15, 177-200, (2011) · Zbl 1228.91042
[35] Li, J. S.-H.; Luo, A., Key q-duration: A framework for hedging longevity risk, Astin Bulletin, 42, 413-452, (2012) · Zbl 1277.91089
[36] Lin, Y.; Cox, S. H., Securitization of mortality risks in life annuities, Journal of Risk and Insurance, 72, 227-252, (2005)
[37] Lin, Y.; Cox, S. H., Securitization of catastrophe mortality risks, Insurance: Mathematics and Economics, 42, 628-637, (2008) · Zbl 1152.91593
[38] Menoncin, F., The role of longevity bonds in optimal portfolios, Insurance: Mathematics and Economics, 42, 343-358, (2008) · Zbl 1141.91537
[39] (2013)
[40] Strotz, R. H., Myopia and inconsistency in dynamic utility maximization., The Review of Economic Studies, 23, 165-180, (1956)
[41] Wang, S., A universal framework for pricing financial and insurance risks, Astin Bulletin, 32, 213-234, (2002) · Zbl 1090.91555
[42] Wong, T. W.; Chiu, M. C.; Wong, H. Y., Time-consistent mean-variance hedging of longevity risk: Effect of cointegration, Insurance: Mathematics and Economics, 56, 56-67, (2014) · Zbl 1304.91136
[43] Wong, T. W.; Chiu, M. C.; Wong, H. Y., Managing mortality risk with longevity bonds when mortality rates are cointegrated, Journal of Risk and Insurance, (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.