The role of longevity bonds in optimal portfolios. (English) Zbl 1141.91537

Summary: We study the optimal consumption and portfolio for an agent maximizing the expected utility of his intertemporal consumption in a financial market with: (i) a riskless asset, (ii) a stock, (iii) a bond as a derivative on the stochastic interest rate, and (iv) a longevity bond whose coupons are proportional to the population (stochastic) survival rate. With a force of mortality instantaneously uncorrelated with the interest rate (but not necessarily independent), we demonstrate that the wealth invested in the longevity bond must be taken from the ordinary bond and the riskless asset proportionally to the duration of the two bonds. This result is valid for both a complete and an incomplete financial market.


91B30 Risk theory, insurance (MSC2010)
91B28 Finance etc. (MSC2000)
90C39 Dynamic programming
Full Text: DOI


[1] Azzoppardi, M., 2005. The longevity bond. In: First International Conference on Longevity Risk and Capital Markets Solutions
[2] Babbs, S.H.; Nowman, K.B., Econometric analysis of a continuous time multi-factor generalized vasicek term structure model: international evidence, Asian-Pacific financial market, 5, 159-183, (1998) · Zbl 1153.91698
[3] Babbs, S.H.; Nowman, K.B., Kalman filtering of generalized vasicek term structure models, Journal of financial and quantitative analysis, 34, 115-130, (1999)
[4] Boulier, J.-F.; Huang, S.J.; Taillard, G., Optimal management under stochastic interest, Insurance: mathematics and economics, 28, 173-189, (2001) · Zbl 0976.91034
[5] Chacko, G.; Viceira, L.M., Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, The review of financial studies, 18, 1369-1402, (2005)
[6] Cox, J.C.; Huang, C.F., Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of economic theory, 49, 33-83, (1989) · Zbl 0678.90011
[7] Cox, J.C.; Huang, C.F., A variational problem arising in financial economics, Journal of mathematical economics, 20, 465-487, (1991) · Zbl 0734.90009
[8] Cox, J.C.; Ingersoll, J.E.; Ross, S.A., A theory of the term structure of interest rates, Econometrica, 53, 385-407, (1985) · Zbl 1274.91447
[9] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: mathematics and economics, 35, 113-136, (2004) · Zbl 1075.62095
[10] Deelstra, G.; Grasselli, M.; Koehl, P.-F., Optimal investment strategies in a CIR framework, Journal of applied probability, 37, 1-12, (2000) · Zbl 0989.91040
[11] Hainaut, D., Devolder, P., 2006. A martingale approach applied to the management of life insurances. Institute of Actuarial Sciences, Université Catholique de Louvain, WP06-01
[12] Hainaut, D.; Devolder, P., Management of a pension fund under mortality and financial risk, Insurance: mathematics and economics, 41, 134-155, (2007) · Zbl 1119.91053
[13] Kim, T.S.; Omberg, E., Dynamic nonmyopic portfolio behavior, The review of financial studies, 9, 141-161, (1996)
[14] Lioui, A.; Poncet, P., On optimal portfolio choice under stochastic interest rates, Journal of economic dynamics and control, 25, 1841-1865, (2001) · Zbl 0979.91032
[15] Menoncin, F., Optimal portfolio and background risk: an exact and an approximated solution, Insurance: mathematics and economics, 31, 249-265, (2002) · Zbl 1055.91054
[16] Merton, R.C., Lifetime portfolio selection under uncertainty: the continuous-time case, Review of economics and statistics, 51, 247-257, (1969)
[17] Merton, R.C., 1970. A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm. Working Paper 497-70, A.P. Sloan School of Management, MIT
[18] Merton, R.C., Optimum consumption and portfolio rules in a continuous-time model, Journal of economic theory, 3, 373-413, (1971) · Zbl 1011.91502
[19] Milevsky, M., Optimal annuitization policies: analysis of the options, North American actuarial journal, 5, 57-69, (2001) · Zbl 1083.91522
[20] Richard, S., Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of financial economics, 2, 187-203, (1975)
[21] Rudolf, M.; Ziemba, W.T., Intertemporal surplus management, Journal of economic dynamics and control, 28, 975-990, (2004) · Zbl 1179.91236
[22] Vasiček, O., An equilibrium characterization of the term structure, Journal of financial economics, 5, 177-188, (1977) · Zbl 1372.91113
[23] Wachter, J.A., 1998. Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets. Working Paper, Harvard University
[24] Wachter, J.A., Risk aversion and allocation to long-term bonds, Journal of economic theory, 112, 325-333, (2003) · Zbl 1055.91035
[25] Zariphopoulou, T., A solution approach to valuation with unhedgeable risks, Finance and stochastics, 5, 61-82, (2001) · Zbl 0977.93081
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