We unify two directions of generalization of Merton’s consumption-investment problem of an individual, important for the pension industry: Uncertain lifetime and recursive utility. This allows us to discuss both optimal insurance and consumption strategies in standard markets and optimal product design that makes the ‘buy-and-hold your pension contract’-strategy optimal. On one hand, lifetime risk in Merton’s problem is well-studied dating back to Richard (1975) and has since then generalized in various directions, see e.g. the generalization to multistate models in Kraft and Steffensen (2008). Typically, however, these generalizations assume time-additive utility, implicitly aligning the investor’s attitudes towards risk and inter-temporal aspects. On the other hand, there exists a vast literature on so-called recursive utility where the investor’s preferences with respect to risk and inter-temporal substitution, respectively, are disentangled. However, the concept is developed explicitly for diffusive markets, see Duffie and Epstein (1992) and little is known about how the concepts generalizes to e.g. the jump risk that formalizes the uncertain lifetime. Nevertheless, the long-term nature of pension products is characteristic and makes the special attention to inter-temporal aspects in connection with uncertain lifetimes particularly important for the pension industry. We 1) discuss the various notions at play, 2) solve the integrated problem with recursive utility and an uncertain lifetime and 3) illustrate our solution in terms of optimal pension advice and product design.

References:

• Duffie, D, and Epstein, L.G. (1992). Stochastic Differential Utility. Econometrica, 60(2):353-394.
• Kraft, H. and Steffensen, M. (2008). Optimal consumption and insurance: a continuous-time Markov chain approach, ASTIN Bulletin, 28:231-257.
• Richard, S. F. (1975). Optimal Consumption, Portfolio and Life Insurance Rules for an Uncertain lived Individual in a Continuous Time Model. Journal of Financial Economics, 2(2):187-203.