In this paper we propose and study a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon.In our model the dynamics of wealth takes directly into account the flows of contributions and benefits and the level of wealth is constrained to stay above a "solvency level". The fund manager can invest in a riskless asset and in a risky asset but borrowing and short selling are prohibited.We concentrate the analysis on the effect of the solvency constraint, analyzing in particular what happens when the fund wealth reaches the allowed minimum value represented by the solvency level.The model is naturally formulated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a regular solution of the associated Hamilton-Jacobi-Bellman equation. Then we apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties. We finally give a special example with explicit solution.
This paper deals with a constrained investment problem for a defined contribution (DC) pension fund where retirees are allowed to defer the purchase of the annuity at some future time after retirement.This problem has already been treated in the unconstrained case in a number of papers. The aim of this work is to deal with the more realistic case when constraints on the investment strategies and on the state variable are present. Due to the difficulty of the task, we consider, as a first step, the basic model of [Gerrard, Haberman & Vigna, 2004], where interim consumption and annuitization time are fixed. We extend their model by adding a no short-selling constraint on the control variable and a final capital requirement constraint on the state variable. This implies, in particular, no ruin.The mathematical problem is naturally formulated as a stochastic control problem with constraints on the control and the state variable, and is approached by the dynamic programming method. We write the non-linear Hamilton-Jacobi-Bellman equation for the problem and transform it into a dual one that is semi-linear, following a well-established duality procedure. In the special relevant case without running cost, we explicitly compute the value function for the problem and give the optimal strategy in feedback form. A numerical application ends the paper and shows the extent of applicability of the model to a DC pension fund in the decumulation phase.J.E.L. classification: C61, G11, G23. A.M.S. classification: 91B28, 93E20, 49L25.Keywords: pension fund, decumulation phase, constrained portfolio, stochastic optimal control, dynamic programming, Hamilton-Jacobi-Bellman equation.
Acknowledgements:We thank three anonymous reviewers for thorough reading and the editor for careful handling of the paper.
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