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Heath-Jarrow-Morton modelling of longevity bonds and the risk minimization of life insurance portfolios. (English) Zbl 1140.91377

Summary: This paper has two parts. In the first, we apply the Heath-Jarrow-Morton (HJM) methodology to the modelling of longevity bond prices. The idea of using the HJM methodology is not new. We can cite A. J. Cairns et al. [Pricing death: framework for the valuation and the securitization of mortality risk. Astin Bull., 36 (1), 79–120 (2006)], K. R. Miltersen and S. A. Persson [Is mortality dead? Stochastic force of mortality determined by arbitrage? Working Paper, University of Bergen (2005)] and D. Bauer [An arbitrage-free family of longevity bonds. Working Paper, Ulm University (2996)]. Unfortunately, none of these papers properly defines the prices of the longevity bonds they are supposed to be studying. Accordingly, the main contribution of this section is to describe a coherent theoretical setting in which we can properly define these longevity bond prices. A second objective of this section is to describe a more realistic longevity bonds market model than in previous papers. In particular, we introduce an additional effect of the actual mortality on the longevity bond prices, that does not appear in the literature. We also study multiple term structures of longevity bonds instead of the usual single term structure. In this framework, we derive a no-arbitrage condition for the longevity bond financial market. We also discuss the links between such HJM based models and the intensity models for longevity bonds such as those of M. Dahl [Insur. Math. Econ. 35, No. 1, 113–136 (2004; Zbl 1075.62095)], E. Biffis [Insur. Math. Econ. 37, No. 3, 443–468 (2005; Zbl 1129.91024)], E. Luciano and E. Vigna [Non mean reverting affine processes for stochastic mortality. ICER working paper], D. F. Schrager [Insur. Math. Econ. 38, No. 1, 81–97 (2006; Zbl 1103.60063)] and D. Hainaut and P. Devolder [Mortality modelling with Lévy processes. Insur. Math. Econ. (2007), in press], and suggest the standard pricing formula of these intensity models could be extended to more general settings.
In the second part of this paper, we study the asset allocation problem of pure endowment and annuity portfolios. In order to solve this problem, we study the “risk-minimizing” strategies of such portfolios, when some but not all longevity bonds are available for trading. In this way, we introduce different basis risks.

MSC:

91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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