Pension Funds with a Minimum Guarantee: A Stochastic Control Approach

44 Pages Posted: 31 Jan 2007 Last revised: 5 May 2010

Date Written: July 29, 2008


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. Usually, portfolio selection models for pension funds maximize the expected utility from final wealth over a finite horizon (the retirement time), whereas our target is to maximize the expected utility from current wealth over an infinite horizon since we adopt the point of view of the fund manager.

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 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.

Keywords: Defined contribution pension fund, minimum guarantee, stochastic optimal control, dynamic programming, Hamilton-Jacobi-Bellman equation, viscosity solutions

JEL Classification: C61, G11, G23

Suggested Citation

Di Giacinto, Marina and Gozzi, Fausto and Salvatore, Federico, Pension Funds with a Minimum Guarantee: A Stochastic Control Approach (July 29, 2008). Finance and Stochastics, Forthcoming, Available at SSRN: or

Marina Di Giacinto (Contact Author)

University of Cassino ( email )

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

Fausto Gozzi

Luiss ( email )

Viale di Villa Massimo, 57
Rome, 00161


Federico Salvatore

Luiss Guido Carli University ( email )

Via O. Tommasini 1
Rome, Roma 00100

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