Asset Allocation, Longevity Risk, Annuitisation and Bequests

Consumption and portfolio optimisation during retirement has not received as much attention in financial research as optimisation prior to retirement. However, retirement planning is becoming more and more relevant for several reasons. The present paper is mainly based on Schiess (2008) and studies a pensioner deriving utility from a stream of consumption or an annuity and from bequeathing wealth to his heirs in a continuous-time framework. The task of finding the pensioner's optimal consumption, asset allocation and annuity decision rule leads to the interesting interplay of optimal control theory, optimal stopping theory and mortality issues or, technically speaking, to a combined optimal stopping and optimal control problem (COSOCP). Stabile (2006) solved this problem in an all-or-nothing framework assuming exponential mortality and power utility functions. In this paper we present the extensions of Schiess (2008) to the model of Stabile (2006):The essential inclusion of a bequest motive, the additional study of the economically interesting range of relative risk aversion levels greater than one and a new solution method for the COSOCP via duality arguments. For identical risk aversion levels Stabile (2006) finds that the pensioner either annuitises immediately or never which means that COSOCP reduces to a trivial or to a pure optimal control problem. In contrast to this, the annuitisation decision rule can become wealth-dependent in our more general model and consequently, a real COSOCP has to be dealt with. The main result is that longevity risk matters very much (quite attractive annuity market) even if we allow for a bequest motive.