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# Constrained portfolio choices in the decumulation phase of a pension plan

## Author

Listed:
• Marina Di Giacinto
• Salvatore Federico
• Fausto Gozzi
• Elena Vigna

## Abstract

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, Haberman & Vigna, 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.

## Suggested Citation

• Marina Di Giacinto & Salvatore Federico & Fausto Gozzi & Elena Vigna, 2010. "Constrained portfolio choices in the decumulation phase of a pension plan," Carlo Alberto Notebooks 155, Collegio Carlo Alberto.
• Handle: RePEc:cca:wpaper:155
as

File URL: http://www.carloalberto.org/assets/working-papers/no.155.pdf

## Citations

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Cited by:

1. Elena Vigna, 2009. "Mean-variance inefficiency of CRRA and CARA utility functions for portfolio selection in defined contribution pension schemes," CeRP Working Papers 89, Center for Research on Pensions and Welfare Policies, Turin (Italy).
2. Elena Vigna, 2010. "On efficiency of mean-variance based portfolio selection in DC pension schemes," Carlo Alberto Notebooks 154, Collegio Carlo Alberto, revised 2011.
3. Di Giacinto, Marina & Federico, Salvatore & Gozzi, Fausto & Vigna, Elena, 2014. "Income drawdown option with minimum guarantee," European Journal of Operational Research, Elsevier, vol. 234(3), pages 610-624.
4. Elena Vigna, 2009. "Mean-variance inefficiency of CRRA and CARA utility functions for portfolio selection in defined contribution pension schemes," Carlo Alberto Notebooks 108, Collegio Carlo Alberto, revised 2009.
5. Hassan Dadashi, 2018. "Optimal investment-consumption problem post-retirement with a minimum guarantee," Papers 1803.00611, arXiv.org.
6. Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2015. "Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation," Finance and Stochastics, Springer, vol. 19(2), pages 415-448, April.

### Keywords

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

### JEL classification:

• C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
• G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
• G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors

### NEP fields

This paper has been announced in the following NEP Reports:

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