Retirement spending problem under Habit Formation Model S. Kirusheva1 H. Huang2 and T.S. Salisbury3

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Retirement spending problem under

Habit Formation Model

S. Kirusheva∗1, H. Huang2, and T.S. Salisbury†3

1,2,3Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

2Research Centre for Mathematics, Advanced Institute of Natural Sciences, Beijing Normal

University, Zhuhai, Guangdong, China

2BNU-HKBU United International College, Zhuhai, Guangdong, China

Abstract

In this paper we consider the problem of optimizing lifetime consumption

under a habit formation model. Our work diﬀers from previous results,

because we incorporate mortality and pension income. Lifetime utility of

consumption makes the problem time inhomogeneous, because of the eﬀect

of ageing. Considering habit formation means increasing the dimension of the

stochastic control problem, because one must track smoothed-consumption

using an additional variable, habit ¯c. Including exogenous pension income π

means that we cannot rely on a kind of scaling transformation to reduce the

dimension of the problem as in earlier work, therefore we solve it numerically,

using a ﬁnite diﬀerence scheme. We also explore how consumption changes

over time based on habit if the retiree follows the optimal strategy. Finally,

we answer the question of whether it is reasonable to annuitize wealth at the

time of retirement or not by varying parameters, such as asset allocation θ

and the smoothing factor η.

1. Introduction

1.1. Motivation

Nowadays, we observe a growing interest in investment plans that give a potential client

the conﬁdence of a stable income over the course of retirement. The main goal of

any such plan is to ﬁnd the strategy that minimizes the risk of ruin and, at the same

time, maximizes the level of consumption. Our current research deals with a retirement

∗skir@yorku.ca. Kirusheva’s research was supported in part by MITACS and CANNEX Inc.

†salt@yorku.ca. Salisbury’s and Huang’s research were suppoted in part by grants from NSERC.

arXiv:2210.06255v1 [q-fin.PM] 12 Oct 2022

spending problem (RSP) under dynamics that include the individual’s habit. In other

words, we take into consideration how much a retiree usually spends, i.e. we solve the

problem under a habit formation model (HFM). This postulate makes the model much

more diﬃcult to solve.

In this paper our goal is to explore how the presence of exogenous income in the model

that includes the client’s habit will aﬀect the optimal consumption and compare these

results with diﬀerent models, such as HFM without pension for two cases, with and

without asset allocation. Also we answer the question of how much the agent can spend

during retirement based on his initial wealth wand consumption habit ¯c.

There is one more interesting option for a retiree that we also discuss in this article.

The individual can convert some or all of his initial wealth into annuities. We brieﬂy

discuss this possibility, and if it is reasonble to do so at age of 65 depending on the client’s

habit. We assume that once he converts his wealth into annuities he can’t reverse the

transaction. We obtain numerical results for diﬀerent parameters, such as habit ¯c, asset

allocation θ, and smoothing factor η.

1.2. Literature review

Many articles have been written on this topic that consider various scenarios. Lately

more researchers are paying more attention to the HFM when they deal with ﬁnancial

questions. There are several articles that solve similar problems, somehow related to the

habit formation model, for example [1], [4], [5], [16], [17], [18]. All of them use diﬀerent

approaches and techniques. First, we should mention one of the earliest papers [6].

In that article the author tried to solve the equity premium puzzle (EPP) which was

ﬁrst formalized in a study by Rajnish Mehra and Edward C. Prescott [11]. Under

the assumption of rational expectations this problem was resolved. One of the issues

with that formulation is that the consumption should be always greater or equal to

the exponentially weighted average (EWA) of consumption which is hard to implement

in real situations. As a consequence, there are modiﬁcations of this work which are

discussed, for instance in the book [20] which introduces a novel form for the HFM

utility. Another attempt to resolve the EPP was made by the authors [22]. They

considered optimal portfolio and consumption selection problems with habit formation

in a jump diﬀusion incomplete market in continuous-time. One more pioneering work [18]

describes a model of consumer behaviour based on a speciﬁc class of utility functions,

the so-called “modiﬁed Bergson family”.

Another example of using HFM was covered in the following article [17] which explores

the implications of additive and endogenous habit formation preferences in the context of

a life-cycle model of consumption and portfolio choice for an investor who has stochastic

uninsurable labor income. In order to get a solution he derives analytically constraints

for habit and wealth and explains the relationship between the worst possible path of

future labor income and the habit strength parameter. He concludes that even a small

possibility of a very low income implies more conservative portfolios and higher savings

rates. The main implications of the model are robust to income smoothing through

borrowing or ﬂexible labor supply.

Finally, we would like to mention one more paper [7] where the authors proved

existence of optimal consumption-portfolio policies for utility functions for which the

marginal cost of consumption (MCC) interacted with the habit formation process and

satisﬁed a recursive integral equation with a forward functional Lipschitz integrand and

for utilities for which the MCC is independent of the standard of living and satisﬁed a

recursive integral equation with locally Lipschitz integrand.

There are a lot of ﬁnancial strategies which can help to plan how to spend money under

diﬀerent preferences but one of the most important targets is to arrange consumption

during retirement. There are many works devoted to retirement spending plans, such

as [1], [9], [10], [14]. For example, in the paper [1] the authors discuss consumption and

investment decisions in a life-cycle model under a habit formation model incorporating

stochastic wages and labor supply ﬂexibility. One of the results shown was that utilities

that exhibit habit formation and consumption-leisure complimentarities induce an im-

pact of past wages on the consumption of retirees. Hence the authors showed that it is

important to take into consideration habit and consumption-leisure complimentarities

when formulating life-cycle investment plans. In the next article [9] the authors consider

a model based on results from the article [14] where a similar problem was solved under

assumption of deterministic investment returns. In [9] the authors accept stochastic

returns and then compare optimal spending rates with the analytic approach from the

article [14]. When a potential client starts to think about a retirement spending plan

there is one more question that arises, namely under which conditions he can consider

investment into annuities for part or all of his wealth. To be precise, when we say “an-

nuities” we mean life annuities, insurance products that pay out a periodic amount for

as long as the annuitant is alive, in exchange for a premium (see [3]). This question has

been widely discussed in the literature, for example [13], [15] or [19].

In the recent article [9] RSP was solved for ﬁxed risky asset allocation θ= const. Here

we solve a similar problem following HFM, using the novel utility of [20].

1.3. Paper Agenda

The paper is organized as follows. In Section 2 we explain what the habit formation

model is and formulate our problem for two diﬀerent cases, without (see Section 2.2)

and with (see Section 2.3) pension. In Section 3 we discuss how the smoothing factor

ηaﬀects the numerical solution and provide a comparison between two diﬀerent cases,

without pension (Section 3.1) and when the client has constant pension income (Section

3.2.1 and 3.2.2).

Section 4 is devoted to discussing how the client can spend money during his retirement

based on a given initial amount of wealth wand a certain habit ¯c. In the Section 5 we

analyze the possibility of annuitizing wealth, entire or partially, at the age of 65.

As with most of these models, this one doesn’t have an analytical solution and has to

be approximated numerically. Many algorithms have been developed through the years.

Every algorithm has its own advantages and disadvantages, which diﬀer in accuracy

and eﬃciency. In this paper we chose a ﬁnite diﬀerence scheme for its simplicity and

accuracy. The detailed description of the approximation scheme, some error analysis as

well as some theoretical background are provided in the Appendices (see A, B, C). In

Appendix D we summarize numerical results obtained for diﬀerent sets of wealth wand

habit ¯c. Finally, in Appendix E, we describe some numerical results for diﬀerent sets

of asset allocations θand volatility σfor ﬁxed smoothing factor η= 1.0,as the most

interesting case for solving problem under HFM.

2. Model formulation

2.1. Introduction

When we think about the model we should think about a client, more precisely about a

retiree, who has a certain amount of wealth wand who wants to know what to do next

with his endowment. In order to decide how much he can spend we should understand

how wealth is changing over the time counting all possible income and expenses, as in

the following:

dwt= [θ(µ−r) + r]wtdt+θσwtdWt+πdt−ctdt

d¯ct=η(ct−¯ct)dt. (1)

We can provide the following explanation of the equations (1). There is a part of wealth

wwhich grows at the riskless rate r, there is a stochastic component represented by the

parameter θ, which is the fraction of wealth invested into risky assets (in our case, we

take it as a ﬁxed parameter θ), drift µ, volatility σand Wta Brownian Motion (BM).

Also assume that there is an exogenous ﬁxed income, pension π. We solve our problem

using a habit formation model. This means that the agent’s utility depends on the

consumption rate ctand on an EWA ¯ctof consumption rates over previous time periods.

We will consider the ﬁnite-horizon problem therefore the client’s objective function is

taken to be

V(t,w,¯c) = sup

EZT

e−ρs p

s xucs

¯csds|wt=w,¯ct= ¯c(2)

where ρis the personal time preference or subjective discount rate, p

s x is the probability

of survival from the retirement age xto x+s. We set up the probability of survival

based on the Gompertz Law of Mortality ( [12]), i.e.

s x =e−Rt

0λx+qdq.(3)

Here λis the biological hazard rate λx+q=1

be(x+q−m)/b where mis the modal value of

life (see p47, [12]), bis the dispersion coeﬃcient of the future lifetime random variable,

uis the CRRA utility function

u(c) = c1−γ

1−γ(4)

where γis the risk aversion parameter. Note that the formulation of utility in (2) is

due to [20] and diﬀers from much of the HFM literature. This changes the nature of the

solutions.

In order to solve this problem, we will use the value function approach that implies

we should derive an Hamilton-Jacobi-Bellman (HJB) equation for our model. Let us

consider two diﬀerent cases. The problem that we are going to solve ﬁrst, reﬂects an

agent’s expectations who doesn’t have any pension income which means that his wealth’s

growth results partly from investing in a bank account growing at the risk free rate and

partly from investing another portion of the wealth into risky assets. The second problem

involves the presence of pension income. In addition, the asset allocation will be ﬁxed.

2.2. Model without pension.

In this paragraph we consider wealth and habit dynamics (1) without pension and with

asset allocation as a parameter θ=const with the client’s objective function described

as follows (see Appendix A(29)). Since we use the value function approach, we need

to derive an HJB equation using a veriﬁcation theorem (A.1) but, ﬁrst, let’s change

variables. This will allow us to reduce the dimension of our problem by analogy with

one introduced in the book [20]

xt≡wt

¯ct

, qt=ct

¯ct

V(t,w,¯c) = Vt,w

¯c,1≡νt,w

¯c=ν(t,x)

(5)

where V(t,w,¯c) is the unknown value function deﬁned by equation (2). Then the dy-

namics of the new scaled wealth xwill be the following

dxt= d wt

¯ct=rxtdt+θ((µ−r)xtdt+σxtdWt)−(ηxt+ 1)qtdt+ηxtdt.

Omitting all calculations we provide only the ﬁnal HJB equation using the new variables

νt−(ρ+λ)ν+ανx+βνxx +u(q∗) = 0 (6)

where the optimal consumption qwill be

q∗= [(ηx + 1)νx]−1

γ.(7)

and the coeﬃcients α=α(x) and β=β(x) in the formula have the following form

α= (θ(µ−r) + r+η)x−(ηx + 1)q∗, β =1

2θ2σ2x2.

Boundary conditions for this problem are taken to be following: At the boundary

x= 0 assume that optimal consumption is q∗

t= 0 since there is not any other income.

Then at the other boundary where x=xmax assume that the value function gradually

approaches zero which implies that the value function derivative is zero, i.e. νx= 0.

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RetirementspendingproblemunderHabitFormationModelS.Kirusheva*1,H.Huang2,andT.S.Salisbury31,2,3DepartmentofMathematicsandStatistics,YorkUniversity,Toronto,Ontario,Canada2ResearchCentreforMathematics,AdvancedInstituteofNaturalSciences,BeijingNormalUniversity,Zhuhai,Guangdong,China2BNU-HKBUUnitedInter...

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Retirement spending problem under Habit Formation Model S. Kirusheva1 H. Huang2 and T.S. Salisbury3

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