Constrained Portfolio Choices in the Decumulation Phase of a Pension Plan

53 Pages Posted: 8 May 2010

See all articles by Marina Di Giacinto

Marina Di Giacinto

University of Cassino

Salvatore Federico

Luiss Guido Carli University

Fausto Gozzi


Elena Vigna

University of Turin - Faculty of Economics; Collegio Carlo Alberto; CeRP

Date Written: April 27, 2010


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 the basic model of [Gerrard et al., 2004], where interim consumption and annuitization time are fixed. The main goal is to find the optimal portfolio choice to be adopted by the retiree from retirement to annuitization time in a Black and Scholes financial market. We define and study the problem at two different complexity levels. In the first level (problem P1), we only require no short-selling. In the second level (problem P2), we add a constraint on the state variable, by imposing that the final fund cannot be lower than a certain guaranteed safety level. 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 give a general result of existence and uniqueness of regular solutions for the Hamilton-Jacobi-Bellman equation and, in a special case, we explicitly compute the value function for the problem and give the optimal strategy in feedback form.

A numerical application of the special case - when explicit solutions are available - ends the paper and shows the extent of applicability of the model to a DC pension fund in the decumulation phase.

Keywords: pension fund, decumulation phase, constrained portfolio, stochastic optimal control, dynamic programming, Hamilton-Jacobi-Bellman equation

JEL Classification: C61, G11, G23

Suggested Citation

Di Giacinto, Marina and Federico, Salvatore and Gozzi, Fausto and Vigna, Elena, Constrained Portfolio Choices in the Decumulation Phase of a Pension Plan (April 27, 2010). Available at SSRN: or

Marina Di Giacinto (Contact Author)

University of Cassino ( email )

Via S. Angelo, Loc. Folcara
Cassino, Frosinone 03043

Salvatore Federico

Luiss Guido Carli University ( email )

Via O. Tommasini 1
Rome, Roma 00100

Fausto Gozzi

Luiss ( email )

Viale di Villa Massimo, 57
Rome, 00161


Elena Vigna

University of Turin - Faculty of Economics ( email )

Dipartimento di Statistica e Matematica Applicata
Corso Unione Sovietica 218 bis
Torino, 10134

Collegio Carlo Alberto ( email )

via Real Collegio 30
Moncalieri, Torino 10024

CeRP ( email )

Via Real Collegio, 30
Moncalieri, Turin 10024

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