Constrained portfolio choices in the de-cumulation phase of a pension plan

This paper deals with a constrained investment problem for a defined contribution pension fund where retirees are allowed to defer purchase of annuity at some future time after retirement. This flexibility, sometimes referred to as ``income drawdown option", reduces the annuity risk borne by members of the schemes, since it allows to postpone the conversion of the accumulated capital into pension within a given period of time beyond the retirement. During this period the capital is allocated dynamically while the pensioner withdraws periodic amounts of money to provide for daily life in accordance with restrictions imposed by the scheme's rules or by law. The aim of this work is to find the optimal portfolio choice to be adopted by the retiree until the purchase of the annuity becomes compulsory. The financial market is composed by a risky and a riskless asset and the allocation has constraints on the amount of money invested in the risky asset. The mathematical problem is naturally formulated as a stochastic control problem with constraints on the control variable, representing the investment on the risky asset, and is approached by the tool of dynamic programming. We explicitly compute the value function for the problem and give the optimal strategy in feedback form.