Investing for retirement: Terminal wealth constraints or a desired wealth target?

• Published date: 01 November 2022
• Author: Catherine Donnelly,Gaurav Khemka,William Lim
• Date: 01 November 2022
• DOI: http://doi.org/10.1111/eufm.12351

DOI: 10.1111/eufm.12351

ORIGINAL ARTICLE

Investing for retirement: Terminal wealth

constraints or a desired wealth target?

Catherine Donnelly

1

|Gaurav Khemka

2,3

|William Lim

2

1

Department of Actuarial Mathematics

and Statistics, School of Mathematical

and Computer Sciences, Maxwell

Institute for Mathematical Sciences,

Heriot‐Watt University, Edinburgh, UK

2

Research School of Finance, Actuarial

Studies and Statistics, College of Business

and Economics, Australian National

University, Canberra, Australian

Capital Territory, Australia

3

Data61, Commonwealth Scientific and

Industrial Research Organisation,

Docklands, Victoria, Australia

Correspondence

William Lim, Research School of

Finance, Actuarial Studies and Statistics,

College of Business and Economics,

Australian National University,

Canberra, ACT 2601, Australia.

Email: william.lim@anu.edu.au

Abstract

We investigate how well different investment stra-

tegies can give pre‐retirees more certainty about

their income in retirement, whilst allowing them to

benefit from taking investment risk. Under an ex-

pected utility‐maximizing framework, we find that a

loss aversion utility function gives a high degree of

certainty about its desired wealth target and is ro-

bust to different market models. Imposing terminal

wealth constraints does not improve the certainty of

achieving the desired target enough to counter-

balance the increased chance of obtaining a lower

income. The power utility function is not robust to

different market models and becomes too risk‐

averse with wealth constraints.

Eur Financ Manag. 2022;28:1283–1307. wileyonlinelibrary.com/journal/eufm

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This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits

use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or

adaptations are made.

© 2022 The Authors. European Financial Management published by John Wiley & Sons Ltd.

The authors gratefully acknowledge the constructive comments of the Editor and the Associate Editor of European

Financial Management journal along with the thorough comments of an anonymous referee. We also have benefited

from comments made by Jenni Bettman, Alfred Chong, Marcos Escobar‐Anel, Mogens Steffensen and Rudi Zagst.

Furthermore, we also thank seminar participants at the 22nd International Congress on Insurance: Mathematics and

Economics School of Risk & Actuarial Studies, University of New South Wales, and the PhD Symposium (2018),

Research School of Finance, Actuarial Studies and Statistics, College of Business and Economics, Australia National

University for their comments on an earlier version of the paper. Catherine Donnelly gratefully acknowledges funding

from the Institute and Faculty of Actuaries' Actuarial Research Centre, as part of the research programme ‘Minimising

Longevity and Investment Risk while Optimising Future Pension Plans’. We further thank David Wilkie for providing

some of the UK financial market data.

KEYWORDS

asset allocation, constrained optimization, loss aversion utility,

numerical dynamic programming, retirement outcomes

JEL CLASSIFICATION

G11, G22, G51

1|INTRODUCTION

How to invest one's pension savings before retirement is a decision faced by defined‐

contribution (DC) pension plan members. A typical DC pension plan member both saves and

invests to build up a pension fund before retirement, from which they withdraw an income in

retirement. The member's DC fund value at retirement can be volatile as it depends on several

factors in the savings phase, such as their salary, the amount and frequency of contributions,

and, how the funds are invested. To cite (Merton, 2014, p. 1402), the primary concern of the

pre‐retiree is ‘Will I have sufficient income in retirement to live comfortably?’. This paper

studies whether the investor is better off with investment strategies that constrain their re-

tirement income, or equivalently fund value accumulated after the savings phase, so as to

protect them against extreme (negative) scenarios of income in retirement.

We compare several ways of formulating the investor's problem, by allowing for different

utility functions and terminal wealth boundary constraints and deriving optimal investment

strategies through the maximization of expected utility. Using these optimal investment stra-

tegies, we calculate, analyse and compare the distributions of income at retirement. Our

findings show that a loss aversion utility function gives a very attractive retirement income

distribution, with the distribution peaked at the investor's chosen income goal with some level

of robustness. In contrast, risk preferences expressed via a constant relative risk aversion

(CRRA) utility function give a much more spread out income distribution, providing the

investor less certainty on achieving a sufficient level of income in retirement. Imposing

terminal wealth boundary constraints, in both the utility function settings, result in strategies

that provide certainty of achieving the lower boundary but at the expense of significant re-

duction in the overall retirement outcome. We conclude that the investor can benefit from

following a loss aversion‐derived optimal investment strategy to target a sufficient level of

income at retirement.

The investment problem with terminal wealth constraints at retirement has been ex-

tensively studied in the utility literature, in particular, a lower boundary constraint that

guarantees the investor a minimum level of wealth at the cost of reducing the possible upside.

For example, (Korn, 2005) and Kraft and Steffensen (2013) consider a static lower constraint

while Teplá (2001) and Han and Hung (2012) consider minimum performance relative to a

benchmark strategy. On the other hand, Donnelly et al. (2015) demonstrate the benefit of an

upper constraint on terminal wealth in reducing the risk of poor retirement outcomes at the

expense of giving up the possibility of higher returns. The merits of an upside constraint, in the

form of capped investment earning rates, is also discussed by Mahayni and Schneider (2015).

Schütte (2017) and Donnelly et al. (2018) consider outcomes with both upper and lower con-

straints on the terminal wealth based on an exponential utility function and a power utility

function, respectively.

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