optimal life insurance purchase, consumption and investment on a financial market with...
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Optimal life insurance purchase,consumption and investment ona financial market with multi-dimensional diffusive termsI. Duarte a , D. Pinheiro b , A.A. Pinto c & S.R. Pliska da Centro de Matemática da Universidade do Minho , Braga ,Portugalb CEMAPRE, ISEG , Universidade Técnica de Lisboa , Lisbon ,Portugalc LIAAD-INESC Porto LA, Department of Mathematics, Faculty ofScience , University of Porto , Portugald Department of Finance , University of Illinois at Chicago ,Chicago , IL 60607 , USAPublished online: 20 Mar 2012.
To cite this article: I. Duarte , D. Pinheiro , A.A. Pinto & S.R. Pliska (2012): Optimal life insurancepurchase, consumption and investment on a financial market with multi-dimensional diffusiveterms, Optimization: A Journal of Mathematical Programming and Operations Research, DOI:10.1080/02331934.2012.665054
To link to this article: http://dx.doi.org/10.1080/02331934.2012.665054
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Optimization2012, 1–24, iFirst
Optimal life insurance purchase, consumption and investment on a
financial market with multi-dimensional diffusive terms
I. Duartea, D. Pinheirob*, A.A. Pintoc and S.R. Pliskad
aCentro de Matematica da Universidade do Minho, Braga, Portugal; bCEMAPRE, ISEG,Universidade Tecnica de Lisboa, Lisbon, Portugal; cLIAAD-INESC Porto LA,Department of Mathematics, Faculty of Science, University of Porto, Portugal;
dDepartment of Finance, University of Illinois at Chicago, Chicago, IL 60607, USA
(Received 13 September 2011; final version received 26 January 2012)
We introduce an extension to Merton’s famous continuous time model ofoptimal consumption and investment, in the spirit of previous works byPliska and Ye, to allow for a wage earner to have a random lifetime and touse a portion of the income to purchase life insurance in order to providefor his estate, while investing his savings in a financial market comprised ofone risk-free security and an arbitrary number of risky securities driven bymulti-dimensional Brownian motion. We then provide a detailed analysisof the optimal consumption, investment and insurance purchase strategiesfor the wage earner whose goal is to maximize the expected utility obtainedfrom his family consumption, from the size of the estate in the event ofpremature death, and from the size of the estate at the time of retirement.We use dynamic programming methods to obtain explicit solutions for thecase of discounted constant relative risk aversion utility functions anddescribe new analytical results which are presented together with thecorresponding economic interpretations.
Keywords: stochastic optimal control; consumption-investment problems;life insurance
AMS Subject Classifications: 93E20; 90C39; 91G10; 91G80
1. Introduction
We consider the problem faced by a wage earner having to make decisionscontinuously about three strategies: consumption, investment and life insurancepurchase during a given interval of time [0,min{T, �}], where T is a fixed point in thefuture that we will consider to be the retirement time of the wage earner and � is arandom variable representing the wage earner’s time of death. We assume that thewage earner receives his income at a continuous rate i(t) and that this income isterminated when the wage earner dies or retires, whichever happens first. One of ourkey assumptions is that the wage earner’s lifetime � is a random variable and,therefore, the wage earner needs to buy life insurance to protect his family for the
*Corresponding author. Email: [email protected]
ISSN 0233–1934 print/ISSN 1029–4945 online
� 2012 Taylor & Francis
http://dx.doi.org/10.1080/02331934.2012.665054
http://www.tandfonline.com
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eventuality of premature death. The life insurance depends on an insurance premiumpayment rate p(t) such that if the insured pays p(t) � �t and dies during the ensuingshort-time interval of length �t then the insurance company will pay p(t)/�(t) dollarsto the insured’s estate, where �(t) is an amount set in advance by the insurancecompany. Hence this is like term insurance with an infinitesimal term. We alsoassume that the wage earner wants to maximize the satisfaction obtained from aconsumption process with rate c(t). In addition to consumption and purchase of alife insurance policy, we assume that the wage earner invests the full amount of hissavings in a financial market consisting of one risk-free security and a fixed numberN� 1 of risky securities with diffusive terms driven by M-dimensional Brownianmotion.
The wage earner is then faced with the problem of finding strategies thatmaximize the utility of (i) his family consumption for all t�min{T, �}; (ii) his wealthat retirement date T if he lives that long and (iii) the value of his estate in the event ofpremature death. Various quantitative models have been proposed to model andanalyse this kind of problem, problems having at least one of these three objectives.This literature is perhaps highlighted by Yarri [9] who considered the problem ofoptimal financial planning decisions for an individual with an uncertain lifetime aswell as by Merton [5,6] who emphasized optimal consumption and investmentdecisions but did not consider life insurance. These two approaches were combinedby Richard, whose impressive paper [8] uses sophisticated methods at an early datefor the analysis of a life-cycle life insurance and consumption–investment problem ina continuous time model. Later, Pliska and Ye [7] introduced a continuous-timemodel that combined the more realistic features of all those in the existing literatureand extended the model proposed previously by Richard, the main difference beingthe choice of the boundary condition, leading to somewhat different economicinterpretations of the underlying problem. More precisely, while Richard assumedthat the lifetime of the wage earner is limited by some fixed number, the modelintroduced by Pliska and Ye had the feature that the duration of life is a randomvariable which takes values in the interval ]0,1[ and is independent of the stochasticprocess defining the underlying financial market. Moreover, Pliska and Ye made thefollowing refinements to the theory: (i) the planning horizon T is now seen as themoment when the wage earner retires, contrary to Richard’s interpretation as a finiteupper bound on the lifetime; and (ii) the utility of the wage earner’s wealth at theplanning horizon T is taken into account as well as the utility of lifetimeconsumption and the utility of the bequest in the event of premature death. Paperby Blanchet-Scalliet et al. [1] deals with optimal portfolio selection with anuncertainty exit time for a suitable extension of the familiar optimal consumptioninvestment problem of Merton, without considering any kind of life insurancepurchase.
Whereas Pliska and Ye’s financial market involved only one security that wasrisky, in this article we study the extension where there is an arbitrary (but finite)number of risky securities. This provides a more general setup for the problem underconsideration where the extra risky securities give greater freedom for the wageearner to manage the interaction between his life insurance policies and the portfoliocontaining his savings invested in the financial market. More importantly, thisextended setup includes the general case of incomplete markets, where investors areunable to hedge some given contingent claims. Indeed, this is the real markets case,
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where market frictions may restrict individuals from transferring the desired level ofwealth among certain states of nature.
Following Pliska and Ye, we use the model of uncertain life found in reliabilitytheory, commonly used for industrial life-testing and actuarial science, to model theuncertain time of death for the wage earner. This enables us to replace Richard’sassumption that lifetimes are bounded with the assumption that lifetimes take valuesin the interval ]0,1[. We then set up the wage earner’s objective functionaldepending on a random horizon min{T, �} and transform it to an equivalent problemhaving a fixed planning horizon, that is, the wage earner who faces unpredictabledeath acts as if he will live until some time T, but with a subjective rate of timepreferences equal to his ‘force of mortality’ for his consumption and terminal wealth.This transformation to a fixed planning horizon enables us to state the dynamicprogramming principle and derive an associated Hamilton–Jacobi–Bellman (HJB)equation. We use the HJB equation to derive the optimal feedback control, that is,the optimal insurance, portfolio and consumption strategies. Furthermore, we obtainexplicit solutions for the family of discounted constant relative risk aversion (CRRA)utilities and examine the economic implications of such solutions.
In the case of discounted CRRA utilities, our results generalize those obtainedpreviously by Pliska and Ye. We provide a detailed analytic analysis for the optimalinsurance purchase and investment strategies under CRRA utilities and describetheir interactions. For instance, we obtain: (i) an economically reasonable descrip-tion for the optimal expenditure for insurance as a decreasing function of the wageearner’s overall wealth and a unimodal function of age, reaching a maximum at anintermediate age; (ii) a more controversial conclusion that possibly an optimalsolution calls for the wage earner to sell a life insurance policy on his own life late inhis career. Nonetheless, the extra risky securities in our model introduce novelfeatures to the wage earner’s portfolio and insurance management interaction suchas the following: a young wage earner with small wealth has an optimal portfoliowith larger values of volatility and higher expected returns, with the possibility ofhaving short positions in lower yielding securities.
Another novelty in this article is the comparison of the optimal consumption andinvestment strategies between wage earner who can buy life insurance policies andwage earner without such opportunity in the case of CRRA utilities. We prove that awage earner who can buy life insurance policies will choose a more conservativeportfolio than a wage earner who is without the opportunity to buy life insurance,the distinction being clearer for young wage earners with low wealth.
This article is organized as follows. In Section 2 we describe the problem wepropose to address. Namely, we introduce the underlying financial and insurancemarkets, as well as the problem formulation from the point of view of optimalcontrol. In Section 3 we see how to use the dynamic programming principle to reducethe optimal control of Section 2 to one with a fixed planning horizon and then derivean associated HJB equation. We devote Section 4 to the case of discounted CRRAutilities. We conclude in Section 5.
2. Problem formulation
Throughout this section, we define the setting in which the wage earner has to makehis decisions regarding consumption, investment and life insurance purchase.
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Namely, we introduce the specifications regarding the financial and insurance
markets available to the wage earner. We start by the financial market description,
followed by the insurance market and conclude with the definition of a wealth
process for the wage earner.
2.1. The financial market model
We consider a complete probability space (�,F ,P) on which a standard
M-dimensional Brownian motion W(t)¼ (W1(t), . . . ,WM(t))T is given and assume
that W(0)¼ 0 almost surely. We define F¼ {F t, t2 [0,T]} to be the P-augmentation
of the filtration generated by the Brownian motion W(t), �{W(s), s� t} for t� 0.
Each sub-�-algebra F t represents the information available to any given agent
observing the financial market until time t.The financial market under consideration is composed of one risk-free asset
and several risky-assets. Their respective prices (S0(t))0�t�T and (Sn(t))0�t�T for
n¼ 1, . . . ,N evolve according to the equations:
dS0ðtÞ ¼ rðtÞS0ðtÞdt, S0ð0Þ ¼ s0,
dSnðtÞ ¼ �nðtÞSnðtÞdtþ SnðtÞXMm¼1
�nmðtÞdWmðtÞ, Snð0Þ ¼ sn 4 0,
where r(t) is the riskless interest rate, �(t)¼ (�1(t), . . . ,�N(t))2RN is the vector of the
risky-assets appreciation rates and �(t)¼ (�nm(t))1�n�N,1�m�M is the matrix of risky-
assets volatilities.We assume that the coefficients r(t), �(t) and �(t) are deterministic continuous
functions on the interval [0,T ]. We also assume that the interest rate r(t) is positive
for all t2 [0,T ] and the matrix �(t) is such that ��T is nonsingular for Lebesgue
almost all t2 [0,T ] and satisfies the following integrability condition:
XNn¼1
XMm¼1
Z T
0
�2nmðtÞdt51:
Furthermore, we suppose that there exists an (F t)0�t�T -progressively measurable
process �(t)2RM (see [3]), called the market price of risk, such that for Lebesgue-
almost-every t2 [0,T ] the risk premium
�ðtÞ ¼ ð�1ðtÞ � rðtÞ, . . . ,�NðtÞ � rðtÞÞ 2RN ð1Þ
is related to �(t) by the equation
�ðtÞ ¼ �ðtÞ�ðtÞ a.s.
and is such that the following two conditions hold:Z T
0
�ðtÞ�� ��2 51 a.s.
E exp �
Z T
0
�ðsÞdWðsÞ �1
2
Z T
0
�ðsÞ�� ��2ds� �� �
¼ 1:
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The existence of such a process �(t) ensures the absence of arbitrage opportunities inthe financial market defined above. Note also that the conditions on the matrix �above do not imply market completeness. See [4] for further details on marketviability and completeness.
2.2. The life insurance market model
We assume that the wage earner is alive at time t¼ 0 and that his lifetime is a non-negative random variable � defined on the probability space (�,F ,P). Furthermore,we assume that the random variable � is independent of the filtration F and has adistribution function F: [0,1)! [0, 1] with density f : [0,1)!R
þ so that
FðtÞ ¼
Z t
0
f ðsÞds:
We define the survivor function F : ½0,1Þ ! ½0, 1� as the probability for the wageearner to survive at least until time t, i.e.
FðtÞ ¼ Pð� � tÞ ¼ 1� FðtÞ:
We shall make use of the hazard function, the conditional, instantaneous death ratefor the wage earner surviving to time t , that is
ðtÞ ¼ lim�t!0
Pðt � �5 tþ �tj� � tÞ
�t¼
f ðtÞ
FðtÞ: ð2Þ
Throughout this article, we will suppose that the hazard function : [0,1)!Rþ is a
continuous and deterministic function such thatZ 10
ðtÞdt ¼ 1:
These two concepts introduced above are standard in the context of reliability theoryand actuarial science. In our case, such concepts enable us to consider an optimalcontrol problem with a stochastic planning horizon and restate it as one with a fixedhorizon.
Due the uncertainty concerning his lifetime, the wage earner buys life insuranceto protect his family for the eventuality of premature death. The life insurance isavailable continuously and the wage earner buys it by paying an insurance premiumpayment rate p(t) to the insurance company. The insurance contract is like terminsurance, with an infinitesimally small term. If the wage earner dies at time �5Twhile buying insurance at the rate p(t), the insurance company pays an amountp(�)/�(�) to his estate, where �: [0,T ]!R
þ is a continuous and deterministicfunction which we call the insurance premium-payout ratio and is regarded as fixed bythe insurance company. The contract ends when the wage earner dies or achievesretirement age, whichever happens first. Therefore, the wage earner’s total legacy tohis estate in the event of a premature death at time �5T is given by
Zð�Þ ¼ Xð�Þ þpð�Þ
�ð�Þ,
where X(t) denotes the wage earner’s savings at time t.
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2.3. The wealth process
We assume that the wage earner receives an income i(t) at a continuous rate
during the period [0,min{T, �}], i.e. the income will be terminated either by his
death or his retirement, whichever happens first. Furthermore, we assume that i:
[0,T ]!Rþ is a deterministic Borel-measurable function satisfying the integrabil-
ity condition
Z T
0
iðtÞdt51:
The consumption process (c(t))0�t�T is a (F t)0�t�T-progressively measurable
nonnegative process satisfying the following integrability condition for the invest-
ment horizon T4 0
Z T
0
cðtÞdt51 a.s.
We also assume that the insurance premium payment rate (p(t))0�t�T is a (F t)0�t�T -
predictable process [3]. In an intuitive manner, a predictable process can be described
as such that its values are ‘known’ just in advance of time.For each n¼ 0, 1, . . . ,N and t2 [0,T ], let n(t) denote the fraction of the wage
earner’s wealth allocated to the asset Sn at time t. The portfolio process is then given
by �(t)¼ (0(t), 1(t), . . . , N(t))2RNþ1, where
XNn¼0
nðtÞ ¼ 1, 0 � t � T: ð3Þ
We assume that the portfolio process is (F t)0� t�T -progressively measurable and
that, for the fixed investment horizon T4 0, we have that
Z T
0
�ðtÞ�� ��2 dt51 a.s.
where k�k denotes the Euclidean norm in RNþ1.
The wealth process X(t), t2 [0,min{T, �}], is then defined by
XðtÞ ¼ xþ
Z t
0
iðsÞ � cðsÞ � pðsÞ½ �dsþXNn¼0
Z t
0
nðsÞXðsÞ
SnðsÞdSnðsÞ, ð4Þ
where x is the wage earner’s initial wealth. This last equation can be rewritten in the
differential form
dXðtÞ ¼ iðtÞ � cðtÞ � pðtÞ þ
�0ðtÞrðtÞ þ
XNn¼1
nðtÞ�nðtÞ
�XðtÞ
!dt
þXNn¼1
nðtÞXðtÞXMm¼1
�nmðtÞdWmðtÞ, ð5Þ
where 0� t�min{�,T}.
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Using relation (3) we can always write 0(t) in terms of 1(t), . . . , N(t), so from
now on we will define the portfolio process in terms of the reduced portfolio process
(t)2RN given by
ðtÞ ¼ 1ðtÞ, 2ðtÞ, . . . , NðtÞð Þ 2RN:
2.4. The optimal control problem
The wage earner is faced with the problem of finding strategies that maximize the
expected utility obtained from:
(a) his family consumption for all t�min{T, �};(b) his wealth at retirement date T if he lives that long;(c) the value of his estate in the event of premature death.
This problem can be formulated by means of optimal control theory: the wage
earner’s goal is to maximize some cost functional subject to (i) the (stochastic)
dynamics of the state variable, i.e. the dynamics of the wealth process X(t) given by
(4); (ii) constraints on the control variables, i.e. the consumption process c(t), the
premium insurance rate p(t) and the portfolio process (t) and (iii) boundary
conditions on the state variables.Let us denote byA(x) the set of all admissible decision strategies, i.e. all admissible
choices for the control variables �¼ (c, p, )2RNþ2 characterized above. The
dependence of A(x) on x denotes the restriction imposed on the wealth process by
the boundary condition X(0)¼x. In particular, A(x) must be such that for each
�2A(x) the corresponding wealth process satisfies X(t)� 0 for all t�min{T, �}. Note
that A(x) 6¼ ; since the strategy �0¼ (0, 0, {0})2A(x), corresponding to zero
consumption, zero life-insurance purchase and full investment in the risk free asset S0.The wage earner’s problem can then be restated as follows: find a strategy
�¼ (c, p, )2A(x) which maximizes the expected utility
VðxÞ ¼ sup�2AðxÞ
E0,x
Z T^�
0
UðcðsÞ, sÞdsþ BðZð�Þ, �ÞIf��Tg þWðXðT ÞÞIf�4Tg
� �, ð6Þ
where T6 �¼min{T, �}, IA denotes the indicator function of event A,U(c, �) is the
utility function describing the wage earner’s family preferences regarding consump-
tion in the time interval [0,min{T, �}], B(Z, �) is the utility function for the size of the
wage earner’s legacy in case ��T and W(X) is the utility function for the terminal
wealth at time t¼T in the case �4T.The notation E0,x is used to denote the expectation operator conditional on
X(0)¼ x, where X is the solution of the SDE (5) with control �2A(x) and initial
condition X(0)¼ x.We suppose that U and B are strictly concave on their first variable and that W is
strictly concave on its sole variable. In Section 4 we specialize our analysis to the case
where the wage earner’s preferences are described by discounted CRRA utility
functions.
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3. Stochastic optimal control
In this section we use the techniques in [7] to restate the stochastic optimal control
problem formulated in the preceding section as one with a fixed planning horizon
and to derive a dynamic programming principle and the corresponding HJB
equation.
3.1. Dynamic programming principle
Let us denote by A(t,x) the set of admissible decision strategies �¼ (c, p, ) for thedynamics of the wealth process with boundary condition X(t)¼ x, and by Et,x the
expectation operator conditional on X(t)¼ x, where X is the solution of the SDE (5)
with control �2A(t, x) and initial condition X(t)¼ x. For any �2A(t, x) we define
the functional
Jðt, x; �Þ ¼ Et,x
Z T^�
t
UðcðsÞ, sÞdsþ BðZð�Þ, �ÞIf��Tg þWðXðT ÞÞIf�4Tg
���� �4 t,F t
� �
and introduce the associated value function:
Vðt, xÞ ¼ sup�2Aðt, xÞ
Jðt, x; �Þ:
Given some initial condition (t, x)2 [0,T ]�R, we have that �*2A(t, x) is an
optimal control if V(t, x)¼ J(t, x; �*).The following lemma is the key tool to restating the above control problem as an
equivalent one with a fixed planning horizon (see [10] for a proof).
LEMMA 3.1 Suppose that the utility function U is either nonnegative or nonpositive.
If the random variable � is independent of the filtration F, then
Jðt,x; �Þ ¼ Et,x
Z T
t
Fðs, tÞUðcðsÞ, sÞ þ f ðs, tÞBðZðsÞ, sÞdsþ FðT, tÞWðXðT ÞÞ���F t
� �,
where Fðs, tÞ is the conditional probability distribution function for the wage earner to
be alive at time s conditional upon the wage earner being alive at time t� s and f(s, t) is
the conditional probability density function for the wage earner death to occur at time s
conditional upon the wage earner being alive at time t� s.
Using the previous lemma, one can state the following dynamic programming
principle, obtaining a recursive relationship for the maximum expected utility as a
function of the wage earner’s age and his wealth at that time. See [10] for a proof.
LEMMA 3.2 (Dynamic programming principle) For 0� t5 s5T, the maximum
expected utility V(t,x) satisfies the recursive relation
Vðt, xÞ ¼ sup�2Aðt, xÞ
Et,x
"exp �
Z s
t
ðuÞdu
� �Vðs,XðsÞÞ
þ
Z s
t
Fðu, tÞUðcðuÞ, uÞ þ f ðu, tÞBðZðuÞ, uÞdu���F t
#:
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The transformation to a fixed planning horizon can then be given the followinginterpretation: a wage earner facing unpredictable death acts as if he will live untiltime T, but with a subjective rate of time preferences equal to his ‘force of mortality’for the consumption of his family and his terminal wealth.
3.2. HJB equation
The dynamic programming principle enables us to state the HJB equation, a second-order partial differential equation whose ‘solution’ is the value function of the optimalcontrol problem under consideration here. The techniques used in the derivation ofthe HJB equation and the proof of the next theorem follow closely those in [2,10,11].
THEOREM 3.3 Suppose that the maximum expected utility V is of class C2. Then Vsatisfies the HJB equation
Vtðt, xÞ � ðtÞVðt, xÞ þ sup�2Aðt, xÞ
Hðt, x; �Þ ¼ 0
VðT, xÞ ¼WðxÞ,
(ð7Þ
where the Hamiltonian function H is given by
Hðt,x; �Þ ¼ iðtÞ � c� pþ rðtÞ þXNn¼1
nð�nðtÞ � rðtÞÞ
!x
!Vxðt, xÞ
þx2
2
XMm¼1
XNn¼1
n�nmðtÞ
!2
Vxxðt, xÞ þ ðtÞB xþp
�ðtÞ, t
� �þUðc, tÞ:
Moreover, an admissible strategy �*¼ (c*, p*, *) whose corresponding wealth is X* isoptimal if and only if for a.e. s2 [t,T ] and P-a.s. we have
Vtðs,X�ðsÞÞ � ðsÞVðs,X�ðsÞÞ þ Hðs,X�ðsÞ; ��Þ ¼ 0: ð8Þ
Proof We divide the proof into two parts: we start by establishing the HJBEquation (7) and then we will prove that the equality (8) holds.
Recall that the wealth process X(t), t2 [0,min{T, �], satisfies the stochasticdifferential equation (5). Using Ito’s lemma, we obtain
Vðtþ h,Xðtþ hÞÞ ¼Vðt,XðtÞÞþ
Z tþh
t
aðu,XðuÞÞduþXMm¼1
Z tþh
t
bmðu,XðuÞÞdWmðuÞ, ð9Þ
where the integrand functions a and bm, m¼ 1, . . . ,M, are given by
aðt,XðtÞÞ ¼ Vtðt,XðtÞÞ þ iðtÞ � c� pþ XðtÞ rðtÞ þXNn¼1
n �nðtÞ � rðtÞð Þ
! !Vxðt,XðtÞÞ
þ1
2X2ðtÞVxxðt,XðtÞÞ
XMm¼1
XNn¼1
n�nmðtÞ
!2
bmðt,XðtÞÞ ¼ XðtÞVxðt,XðtÞÞXNn¼1
n�nmðtÞ, m ¼ 1, . . . ,M: ð10Þ
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Using the dynamic programming principle of Lemma 3.2 and setting s¼ tþ h, we get
the identity
Vðt, xÞ ¼ sup�2Aðt, xÞ
Et,x
"exp �
Z tþh
t
ðuÞdu
� �Vðtþ h,Xðtþ hÞÞ
þ
Z tþh
t
Fðu, tÞUðcðuÞ, uÞ þ f ðu, tÞBðZðuÞ, uÞdu���F t
#: ð11Þ
We now note that for small enough values of h the following inequalities hold:
exp �
Z tþh
t
ðvÞdv
� �2 1� ðtÞhOðh2Þ, ð12Þ
i.e. there exist �4 0 and M4 0 such that for all jhj5 � we have that
exp �
Z tþh
t
ðvÞdv
� �� ð1� ðtÞhÞ
���������� �Mh2:
Combining equality (11) with inequalities (12), we obtain
02 sup�2Aðt, xÞ
Et,x
"ð1� ðtÞhOðh2ÞÞVðtþ h,Xðtþ hÞÞ � Vðt, xÞ
þ
Z tþh
t
Fðu, tÞUðcðuÞ, uÞ þ f ðu, tÞBðZðuÞ, uÞdu���F t
#:
Substituting (9) in the last equation, we obtain
02 sup�2Aðt, xÞ
Et,x
"1� ðtÞhOðh2Þ� �
Vðt,XðtÞÞ þ
Z tþh
t
aðu,XðuÞÞdu
�
þ 1� ðtÞhOðh2Þ� XM
m¼1
Z tþh
t
bmðu,XðuÞÞdWmðuÞ � Vðt,xÞ
þ
Z tþh
t
ð1� Fðu, tÞÞUðcðuÞ, uÞ þ ðtÞð1� Fðu, tÞÞBðZðuÞ, uÞdu���F t
#:
Dividing the previous equation by h and letting h go to zero, we obtain the equality
0 ¼ sup�2Aðt, xÞ
"Vtðt, xÞ � ðtÞVðt, xÞ
þ iðtÞ � c� pþ rðtÞ þXNn¼1
nð�nðtÞ � rðtÞÞ
!x
!Vxðt, xÞ
þx2
2
XMm¼1
XNn¼1
n�nmðtÞ
!2
Vxxðt, xÞ þ ðtÞB ZðtÞ, tð Þ þUðc, tÞ���F t
#:
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Letting ZðtÞ ¼ xþ p�ðtÞ, the dynamic programming equation becomes
0 ¼ sup�2Aðt,xÞ
"Vtðt,xÞ � ðtÞVðt, xÞ
þ iðtÞ � c� pþ rðtÞ þXNn¼1
nð�nðtÞ � rðtÞÞ
!x
!Vxðt,xÞ
þx2
2
XMm¼1
XNn¼1
n�nmðtÞ
!2
Vxxðt, xÞ þ ðtÞB xþp
�ðtÞ, t
� �þUðc, tÞ
���F t
#:
Finally, noting that Vt(t, x)� (t)V(t, x) does not depend on �, we obtain the HJB
equation of (7), thus concluding the proof of the first part of the theorem.Regarding the second part of the theorem, we start by letting �2A(t, x) and
apply Ito’s lemma to
exp �
Z s
t
ðvÞdv
� �Vðs,XðsÞÞ:
We obtain that
Vðt, xÞ ¼ exp �
Z T
t
ðvÞdv
� �WðXðT ÞÞ
�
Z T
t
exp �
Z u
t
ðvÞdv
� �aðu,XðuÞÞ � ðuÞVðu,XðuÞÞð Þdu
�XMm¼1
Z T
t
exp �
Z u
t
ðvÞdv
� �bmðu,XðuÞÞdWmðuÞ,
where a(t, x) and bm(t, x), m¼ 1, . . . ,M are as given in (10). Applying the expectation
operator Et,x[�] to both sides of the above equality and recalling that the expected
value of the Ito integral is equal to zero, we get
Vðt, xÞ ¼ Et,x
"exp �
Z T
t
ðvÞdv
� �WðXðT ÞÞ
�
Z T
t
exp �
Z u
t
ðvÞdv
� �aðu,XðuÞÞ � ðuÞVðu,XðuÞÞð Þ du
���F t
#: ð13Þ
From the definition of the hazard function in (2), we obtain that the conditional
probability Fðs, tÞ for the wage earner to be alive at time s conditional upon the wage
earner being alive at time t� s is given by
Fðs, tÞ ¼ exp �
Z s
t
ðvÞdv
� �: ð14Þ
Similarly, we obtain that the conditional probability density function f(s, t) for
the death to occur at time s conditional upon the wage earner being alive at time
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t� s is given by
f ðs, tÞ ¼ ðsÞexp �
Z s
t
ðvÞdv
� �: ð15Þ
Using (14) and (15), we rewrite (13) as
Vðt,xÞ ¼ Et,x FðT, tÞWðXðT ÞÞ �
Z T
t
Fðu, tÞ aðu,XðuÞÞ � ðuÞVðu,XðuÞÞð Þdu���F t
� �: ð16Þ
Using Lemma 3.1, we rearrange (16) to obtain
Vðt,xÞ ¼ Jðt,x;�Þ
�Et,x
Z T
t
Fðu, tÞ Vtðu,XðuÞÞ � ðuÞVðu,XðuÞÞ þHðu,XðuÞ;�Þð Þdu���F t
� �: ð17Þ
Now consider an optimal admissible strategy �*¼ (c*, p*, *) whose correspond-ing wealth is X*. From Equation (17) we have
Vðt,xÞ¼ Jðt,x;��Þ
�Et,x
Z T
t
Fðu,tÞ Vtðu,X�ðuÞÞ�ðuÞVðu,X�ðuÞÞþHðu,X�ðuÞ;��Þð Þdu
���F t
� �ð18Þ
and from the HJB Equation (7), we have
Vtðu,X�ðuÞÞ � ðuÞVðu,X�ðuÞÞ þ Hðu,X�ðuÞ; ��Þ � 0: ð19Þ
Combining (18) and (19), we obtain that �* is optimal if and only if the value
function V satisfies (8), which concludes the proof. g
The second part of the above theorem provides a clear approach for the
computation of optimal insurance, portfolio and consumption strategies. In
particular, we obtain the existence of such optimal strategies under rather weak
conditions on the utility functions.
COROLLARY 3.4 Suppose that the maximum expected utility V is of class C2 and that
the utility functions U and B are strictly concave with respect to their first variable. Then
the Hamiltonian function H has a regular interior maximum �*¼ (c*, p*, *)2A(t, x).
Proof Using the second part of Theorem 3.3, an optimal admissible strategy
�*¼ (c*, p*, *) with wealth process X* must satisfy (8). Therefore, �* must be such
that H attains its maximum value. We start by remarking that the condition to
obtain the maximum for H decouples into three independent conditions, as seen in
the following:
sup�Hðt,x;�Þ ¼ ðrðtÞxþ iðtÞÞVxðt,xÞ þ sup
c
Uðc, tÞ � cVxðt,xÞ
�
þ supp
ðtÞB
�xþ
p
�ðtÞ, t
�� pVxðt,xÞ
�
þ sup
x2
2
XMm¼1
XNn¼1
n�nmðtÞ
!2
Vxxðt,xÞ þXNn¼1
nð�nðtÞ � rðtÞÞxVxðt,xÞ
�:
ð20Þ
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Therefore, it is enough to study the variation of H with respect to each one of the
variables c, p and independently. Thus, computing the first-order conditions for a
regular interior maximum of H with respect to c, p and , we obtain, respectively, thefollowing three conditions:
�Vxðt,xÞ þUcðc�, tÞ ¼ 0
�Vxðt, xÞ þðtÞ
�ðtÞBZ
�xþ
p�
�ðtÞ, t
�¼ 0
xVxðt, xÞ�þ x2Vxxðt, xÞ��T� ¼ 0
RN ,
ð21Þ
where the subscripts in U and B denote differentiation with respect to each function’s
first variable, � denotes the risk premium (1) and 0RN denotes the origin of RN.
Computing the second derivative with respect to each variable (or the Hessian matrix
in the case of ), we obtain
Hccðt, x; ��Þ ¼ Uccðc�, tÞ
Hppðt, x; ��Þ ¼ðtÞ
�2ðtÞBZZ
�xþ
p�
�ðtÞ, t
�Hðt, x; ��Þ ¼ x2Vxxðt, xÞ��
T:
ð22Þ
Note that Hcc(t, x;�*) is negative since U is strictly concave on its first variable
and that Hpp(t, x; �*) is negative since (t) is positive for every 0� t�T and B is
strictly concave on its first variable. To see thatH(t,x; �*) is negative definite, recallthat ��T is assumed to be non-singular and, thus, positive definite. Moreover, note
that Vxx(t, x) must be negative: if Vxx(t, x) is positive, then H would not be bounded
above and as a consequence of the HJB equation either Vt(t, x) or V(t, x) would have
to be infinity, contradicting the smoothness assumption on V. Therefore, H is
negative definite and H has a regular interior maximum. g
4. The family of discounted CRRA utilities
In this section we describe the special case where the wage earner has the same
discounted CRRA utility functions for the consumption of his family, the size of his
legacy and the size of his terminal wealth, i.e. from now on we assume that the utility
functions are given by
Uðc, tÞ ¼ e��tc
, BðZ, tÞ ¼ e��t
Z
, WðXÞ ¼ e��T
X
, ð23Þ
where the risk aversion parameter is such that 5 1, 6¼ 0, and the discount rate �is positive.
4.1. The optimal strategies
Using the optimality criteria provided in Theorem 3.3, we obtain the following
optimal strategies for discounted CRRA utility functions.
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PROPOSITION 4.1 Let � denote the non-singular square matrix given by (��T)�1. Theoptimal strategies in the case of discounted constant relative risk aversion utility
functions are given by
c�ðt, xÞ ¼1
eðtÞðxþ bðtÞÞ
p�ðt, xÞ ¼ �ðtÞ DðtÞ � 1ð ÞxþDðtÞbðtÞð Þ
�ðt, xÞ ¼1
xð1� Þðxþ bðtÞÞ��ðtÞ,
where
bðtÞ ¼
Z T
t
iðsÞ exp �
Z s
t
rðvÞ þ �ðvÞdv
� �ds
DðtÞ ¼1
eðtÞ
ðtÞ
�ðtÞ
� �1=ð1� Þ
eðtÞ ¼ exp �
Z T
t
HðvÞdv
� �þ
Z T
t
exp �
Z s
t
HðvÞdv
� �KðsÞds
HðtÞ ¼ðtÞ þ �
1� �
�ðtÞ
ð1� Þ2�
1� ðrðtÞ þ �ðtÞÞ
KðtÞ ¼ððtÞÞ1=ð1� Þ
ð�ðtÞÞ =ð1� Þþ 1
�ðtÞ ¼ �TðtÞ��ðtÞ �1
2k�T��ðtÞk2:
Proof Assume that the utility functions U,B and W are as given in (23). Using the
first-order conditions in (21), we obtain that the optimal strategies depending on the
value function V are given by
c�ðt, xÞ ¼ e�tVxðt, xÞ� �1=ð1� Þ
p�ðt, xÞ ¼ �ðtÞ�ðtÞe�tVxðt, xÞ
ðtÞ
� ��1=ð1� Þ�x
!
�ðtÞ ¼ �Vxðt, xÞ
xVxxðt, xÞ��ðtÞ:
ð24Þ
We are now going to find an explicit solution for the HJB equation (7). We
substitute c, p and in the HJB equation by the optimal strategies in (24) and
combine similar terms to arrive at the following partial differential equation:
Vtðt, xÞ � ðtÞVðt, xÞ þ ðrðtÞ þ �ðtÞÞxþ iðtÞð ÞVxðt,xÞ
��ðtÞðVxðt, xÞÞ
2
Vxxðt, xÞþ1�
e��t=ð1� ÞKðtÞðVxðt, xÞÞ
� =ð1� Þ¼ 0, ð25Þ
where �(t) and K(t) are as given in the statement of this proposition and the terminal
condition is given by
VðT, xÞ ¼WðxÞ: ð26Þ
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We consider an ansatz of the form
Vðt, xÞ ¼aðtÞ
ðxþ bðtÞÞ , ð27Þ
and substitute it in (25) so that a(t) and b(t) are determined by the differential
equation
1
daðtÞ
dtþ
aðtÞ
xþ bðtÞ
dbðtÞ
dt� ðtÞ
aðtÞ
þ½ðrðtÞ þ �ðtÞÞxþ iðtÞ�aðtÞ
xþ bðtÞ
þ�ðtÞaðtÞ
1� þ1�
e��t=ð1� ÞKðtÞðaðtÞÞ� =ð1� Þ ¼ 0:
Now note that the previous differential equation and the terminal condition (26)
decouples into two independent boundary value problems for a(t) and b(t) which are
given, respectively, by
1
daðtÞ
dtþ rðtÞ þ �ðtÞ �
ðtÞ
þ
�ðtÞ
1�
� �aðtÞ
þ1�
e��t=ð1� ÞKðtÞðaðtÞÞ� =ð1� Þ ¼ 0
aðT Þ ¼ e��T,
ð28Þ
and
dbðtÞ
dt� ðrðtÞ þ �ðtÞÞbðtÞ þ iðtÞ ¼ 0:
bðT Þ ¼ 0:ð29Þ
To find a solution for the boundary value problem (28), we write a(t) in the form
aðtÞ ¼ e��tðeðtÞÞ1� ,
obtaining a new boundary value problem for the function e(t) of the form
deðtÞ
dt�HðtÞeðtÞ þ KðtÞ ¼ 0
eðT Þ ¼ 1,
ð30Þ
where K(t) and H(t) are as given in the statement of this proposition. Since
Equation (30) is a linear, non-autonomous, first-order ordinary differential equation,
it clearly has an explicit solution of the form
eðtÞ ¼ exp �
Z T
t
HðvÞdv
� �þ
Z T
t
exp �
Z s
t
HðvÞdv
� �KðsÞds:
Therefore, we obtain that the solution of (28) is given by
aðtÞ ¼ e��t exp �
Z T
t
HðvÞdv
� �þ
Z T
t
exp �
Z s
t
HðvÞdv
� �KðsÞds
� �1�
: ð31Þ
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To find a solution for the boundary value problem (29), we just note that this is
again a linear, non-autonomous, first-order differential equation and its solution is
given by
bðtÞ ¼
Z T
t
iðsÞexp �
Z s
t
rðvÞ þ �ðvÞdv
� �ds ð32Þ
as required.Combining (24) with (27), (31) and (32), we obtain that the optimal strategies in
the case of CRRA utilities are then given by
c�ðt, xÞ ¼1
eðtÞðxþ bðtÞÞ
p�ðt, xÞ ¼ �ðtÞððDðtÞ � 1ÞxþDðtÞbðtÞÞ
�ðt, xÞ ¼xþ bðtÞ
xð1� Þ��ðtÞ,
where D(t) is as given in the statement of this proposition, which concludes the
proof. g
Note that the quantities b(t) and xþ b(t) are of essential relevance for the
definition of the optimal strategies in Proposition 4.1. The quantity b(t), that we will
refer to as human capital following the nomenclature introduced in [8], should be seen
as representing the fair value at time t of the wage earner’s future income from time t
to time T, while the quantity xþ b(t) should be thought of as the full wealth (present
wealth plus future income) of the wage earner at time t. It is then natural that these
two quantities play a central role in the choice of optimal strategies, since they
determine the present and future wealth available for the wage earner and his family.
Remark 1 Noting that r(t), �(t) and i(t) are positive functions and considering the
boundary value problem (32), if r(t)þ �(t) is small enough, we can deduce that
human capital function b(t) has the following properties:
(a) it is a positive function for all 0� t5T;(b) it is concave.
Moreover, we have that b(t) is either:
(i) a decreasing function for all t2 [0,T ]; or(ii) a unimodal map of t, i.e. there exists some t*2 (0,T ) such that b(t) is
increasing for all 05 t5 t*, decreasing for all t*5 t5T. Furthermore,
we have that the graph of b(t) intersects the graph of the function
i(t)/(r(t)þ �(t)) at t¼ t*.
From the explicit knowledge of the optimal strategies, several economically
relevant conclusions can be obtained. See Figure 1 for a graphical representation of
the optimal life-insurance purchase as a function of age and ‘full wealth’ xþ b(t) of
the wage earner. We start by proving an auxiliary lemma before moving on to the
statement and proof of some of the optimal strategies properties.
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LEMMA 4.2 Suppose that for all t2 [0,min{T, �}] the following two conditions are
satisfied:
(a) (t)� �(t);(b) H(t)� 1.
Then, the inequality D(t)5 1 holds for every t2 [0,min{T, �}].
Proof Recall that D(t) is given by
DðtÞ ¼1
eðtÞ
ðtÞ
�ðtÞ
� �1=ð1� Þ
:
Using condition (b) and noting that K(t) is positive for all t2 [0,min{T, �}], we
have that
eðtÞ ¼ exp �
Z T
t
HðvÞdv
� �þ
Z T
t
exp �
Z s
t
HðvÞdv
� �KðsÞds
� exp �
Z T
t
1 dv
� �þ
Z T
t
exp �
Z s
t
1 dv
� �KðsÞds4 1:
Putting together the previous inequality and condition (a), we obtain the required
inequality. g
The next result provides a qualitative characterization of the optimal life
insurance purchase strategy.
COROLLARY 4.3 Assume that the conditions of Lemma 4.2 are satisfied. Then, the
optimal insurance purchase strategy p*(t, x) has the following properties:
(a) it is a decreasing function of the wealth x;(b) it is an increasing function of the wage earner’s human capital b(t);
3040
5060
t
0
1000
2000
3000
x+b
–10
–5
0
5
10p
Figure 1. The optimal life-insurance purchase for a wage earner that starts working at age 25and retires 40 years later. The parameters of the model were taken as N¼M¼ 2, i(t)¼50000exp(0.03t), r¼ 0.04, �¼ 0.03, ¼�3, (t)¼ 0.001þ exp(�9.5þ 0.1t), �(t)¼ 1.05(t),�1¼ 0.07, �2¼ 0.11, �11¼ 0.19, �12¼ 0.15, �21¼ 0.17 and �22¼ 0.21.
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(c) it is negative for suitable pairs of wealth x and ‘age’ t;(d) if the wage earner’s wealth x is small enough and �(t) is non-decreasing, the
function t � p*(t, x� b(t)) has the same monotonicity as the human capital
function b(t).
Proof Recall from Proposition 4.1 that the optimal insurance purchase strategy
p*(t, x) is given by
p�ðt, xÞ ¼ �ðtÞ DðtÞ � 1ð ÞxþDðtÞbðtÞð Þ ð33Þ
for all t2 [0,min{T, �}].Items (a) and (b) follow from Lemma 4.2, since D(t) is a positive function such
that D(t)5 1 for all t2 [0,min{T, �}].For the proof of item (c), note that p*(t,x) is negative for all (t, x)2 [0,T ]�R
þ
such that
x4DðtÞ
1�DðtÞbðtÞ
¼ðtÞ1=ð1� Þ
eðtÞ�ðtÞ1=ð1� Þ � ðtÞ1=ð1� ÞbðtÞ4 0,
and positive otherwise.Item (d) follows from (33) and the fact that �(t) is a non-decreasing. g
Some comments regarding the assumptions in Lemma 4.2 (and Corollary 4.3)
seem necessary. Starting with condition (a), the life insurance company must
establish the premium-insurance �(t) in such a way that (t)� �(t) in order to make a
profit (the insurance policy being fair whenever (t)� �(t)). Regarding condition (b),
we note that the quantities r, �, � and are usually very small in the real world and,
moreover, the relative risk aversion of the wage earner is negative in general. This is
consistent with the assumption that H(t) is bounded above by some positive
constant.Apart from studying how optimal life insurance purchase varies with age and
wealth, it is also relevant to understand how the remaining parameters which define
the financial and insurance markets influence life insurance purchase.
COROLLARY 4.4 With all other parameters, including t and x constant, the optimal
life insurance purchase rate p*(t, x) is an increasing function of the discount rate �.
Proof To study the influence of the discount rate � on the optimal optimal life
insurance purchase, we consider two different values of � and compare the
corresponding values of the optimal life insurance purchase. We distinguish the
functions associated with each of the two parameter values by their subscript.Assume that �1 and �2 are such that �15 �2. Recall the definitions of p*(t, x),
H(t), e(t) and D(t) given in Proposition 4.1. Then, it is clear that the inequalities
H1ðtÞ5H2ðtÞ
e1ðtÞ4 e2ðtÞ
D1ðtÞ5D2ðtÞ
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hold for all t2 [0,min{T, �}], where the subscripts correspond to �1 and �2 in an
obvious manner. Rewriting p*(t, x) as
p�ðt, xÞ ¼ �ðtÞðDðtÞðxþ bðtÞÞ � xÞ,
it follows from the preceding inequalities that
p�1ðt, xÞ5 p�2ðt, xÞ
for all t2 [0,min{T, �}], concluding the proof of the statement. g
Remark 2 The variation of the optimal life insurance purchase rate p*(t, x) with
respect to the interest rate r(t), the risk aversion parameter , the hazard rate (t) andthe insurance premium–payout ratio �(t) is non-trivial. However, by studying the
function p*(t, x) given in Proposition 4.1, we can make the following observations:
(i) p*(t, x) is a decreasing function of the interest rate r(t), except for large
values of x and t close enough to T;(ii) p*(t, x) is a decreasing function of the risk aversion parameter , except for
values of t close enough to T;(iii) p*(t, x) is an increasing function of the hazard rate (t) and the insurance
premium–payout ratio �(t) for small enough values of wealth x and a
decreasing function for large values of x.
The extra risky securities in our model introduce novel features to the wage
earner’s portfolio management, as is exemplified in the following result.
COROLLARY 4.5 Let � denote the non-singular square matrix given by (��T)�1 and let
(��(t))n denote the n-th component of the vector ��(t). The optimal portfolio process
�ðt, xÞ ¼ �1, . . . , , �N�
is such that for every n2 {1, . . . ,N}:
(a) �n has the same sign as (��(t))n;(b) �n is a decreasing function of the total wealth x if (��(t))n4 0 for all t2 [0,
min{T, �}] and an increasing function of x if (��(t))n5 0 for all
t2 [0,min{T, �}];(c) �n is an increasing function of the wage earner’s human capital b(t) if
(��(t))n4 0 for all t2 [0,min{T, �}] and a decreasing function of b(t) if
(��(t))n5 0 for all t2 [0,min{T, �}].
Furthermore, for every n, m2 {1, . . . ,N} the following equalities hold:
limx!0þ
�nðt, xÞ ¼ þ1 limx!0þ
�nðt, xÞ
�mðt,xÞ¼ð��ðtÞÞnð��ðtÞÞm
limx!1
�nðt, xÞ ¼ð��ðtÞÞn1�
limt!T
�nðt,xÞ ¼ð��ðT ÞÞn1�
:
Proof Items (a), (b) and (c) on the first part of the corollary follow from the form of
�n , n2 {1, . . . ,N}, given in the statement of Proposition 4.1 and positivity of b(t).
The limiting behaviours on the second part of the corollary also follow from the
form of �n , n2 {1, . . . ,N}. g
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Remark 3 Corollary 4.5 is a mutual fund result: the relative proportions among the
risky securities are independent of all parameters except for the interest rate and the
risky assets appreciation rates and volatilities since, for any n,m2 {1, . . . ,N} we have
�nðt, xÞ
�mðt,xÞ¼ð��ðtÞÞnð��ðtÞÞm
:
The quantities (��(t))n, n2 {1, . . . ,N} can be thought as ‘weighted risk premiums’
for the risky assets S1, . . . ,SN, where the weights are provided by (quadratic)
functions on the coefficients of the matrix of risky-assets volatilities �.
Under the assumption that the ‘weighted risk premiums’ (��(t))n are positive forall n2 {1, . . . ,N} and t2 [0,min{T, �}], we obtain the following interesting conse-
quence of the previous corollary.
COROLLARY 4.6 Let � denote the non-singular square matrix given by (��T)�1 and let
(��(t))n denote the n-th component of the vector ��(t). Assume that for every
n2 {1, . . . ,N} we have (��(t))n4 0 for all t2 [0,min{T, �}]. Then the optimal strategy
for wage earners with small enough wealth x is to shorten the risk-free security and hold
higher amounts of risky assets.
Proof The corollary follows from Corollary 4.5 and the fact that the wage earner
has no budget limitations, thus allowing him to get into short positions on the risk
free asset S0 of arbitrary size. g
We conclude this session with a result concerning some qualitative properties of
the optimal consumption strategy. Its proof follows trivially from the form of the
optimal consumption c*(t, x) given in Proposition 4.1.
COROLLARY 4.7 The optimal consumption rate c*(t) is an increasing function of both
the wealth x and the human capital b(t).
4.2. The interaction between life insurance purchase and portfolio management
In this section we compare the optimal life-insurance strategies for a wage earner
who faces the following two situations:
(a) in the first case, we assume that the wage earner has access to an insurance
market as described above and that his goal is to maximize the combined
utility of his family consumption for all t�min{T, �}, his wealth at
retirement date T if he lives that long, and the value of his estate in the
event of premature death. The optimal strategies for the wage earner in this
setting are given in Proposition 4.1.(b) in the second case, we assume that the wage earner is without the opportunity
of buying life insurance. His goal is to maximize the combined utility of his
family consumption for all t�min{T, �} and his wealth at retirement date T
if he lives that long. Similar to what we have done previously, we translate
this situation to the language of stochastic optimal control and derive explicit
solutions in the case of discounted CRRA utilities.
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We concentrate on the case (b) described above for the moment. Similar to what
was done in case (a), this problem can be formulated by means of optimal control
theory. The wage earner’s goal is then to maximize a new cost functional subject to:
. the dynamics of the state variable, i.e. the dynamics of a wealth process X0(t)
given by
X0ðtÞ ¼ xþ
Z t
0
iðsÞ � c0ðsÞdsþXNn¼0
Z t
0
0nðsÞX0ðsÞ
SnðsÞdSnðsÞ,
where t2 [0,min{T, �}] and x is the wage earner’s initial wealth.. constraints on the remaining control variables, i.e. the consumption process
c0(t) and the reduced portfolio process 0ðtÞ ¼ 01ðtÞ, . . . , 0NðtÞ�
2RN; and
. boundary conditions on the state variables.
Let us denote by A0(x) the set of all admissible decision strategies, i.e. all
admissible choices for the control variables �0¼ (c0, 0)2RNþ1. The dependence of
A0(x) on x denotes the restriction imposed on the wealth process by the boundary
condition X0(0)¼x.The wage earner’s problem can then be stated as follows: find a strategy
�0¼ (c0, 0)2A0(x) which maximizes the expected utility
V0ðxÞ ¼ sup�0 2A0
ðxÞ
E0,x
Z T^�
0
Uðc0ðsÞ, sÞdsþWðX0ðT ÞÞIf�4Tg
� �, ð34Þ
where U(c0, �) is again the utility function describing the wage earner’s family
preferences regarding consumption in the time interval [0,min{T, �}] and B(X0, t) is
the utility function for the terminal wealth at time t¼T6�. As before, we restrict
ourselves to the special case where the wage earner has the same discounted CRRA
utility functions for the consumption of his family and his terminal wealth given in
(23). The optimal strategies are given in the next result
PROPOSITION 4.8 Let � denote the non-singular square matrix given by (��T)�1. Theoptimal strategies for problem (34) in the case where U(c0, �) and B(X0, �) are the
discounted constant relative risk aversion utility functions in (23) are given by
c0�ðt, xÞ ¼
1
e0ðtÞðxþ b0ðtÞÞ
0�ðt, xÞ ¼
1
xð1� Þðxþ b0ðtÞÞ��ðtÞ,
where
b0ðtÞ ¼
Z T
t
iðsÞ exp �
Z s
t
rðvÞdv
� �ds
e0ðtÞ ¼ exp �
Z T
t
H0ðvÞdv
� �þ
Z T
t
exp �
Z s
t
H0ðvÞ dv
� �ds
H0ðtÞ ¼ðtÞ þ �
1� �
�ðtÞ
ð1� Þ2�
1� rðtÞ
and �(t) is as given in the statement of Proposition 4.1.
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We skip the proof of the previous proposition, since it is of the same nature as the
proofs of Theorem 3.3 and Proposition 4.1.We should point that it would be preferable from an economic standpoint to
maximize also the utility of wealth at the time of premature death. In fact, it is easy
to check that this corresponds to the addition of a constraint of the form p(t, x)¼ 0 to
our original problem. However, the corresponding HJB equation would contain an
extra term of the form (t)x / . The presence of this additional term in the HJB
equation makes it impossible for us to obtain closed-form solutions. On the other
hand, a strategy optimizing the final wealth at retirement should be close to an
optimal strategy which also maximizes the wealth for the case of an eventual death of
the wage earner before retirement time. We plan to address such a comparison in
future research.If we make the wage earner income i(t) equal to zero, then we get a solution
which is close to Merton’s classical solution, but it still depends on the hazard
function (t), i.e. even in the absence of a life insurance policy the uncertainty
regarding wage earner’s lifetime still plays a role on the determination of the optimal
consumption and investment strategies.Propositions 4.1 and 4.8 provide us with optimal portfolio processes for the two
settings (a) and (b) described above. In the next theorem we show how these optimal
portfolio processes compare.
THEOREM 4.9 Let � denote the non-singular square matrix given by (��T)�1 and
(��(t))n the n-th component of the vector ��(t). For each n2 {1, . . . ,N}, we have that
0n�ðt, xÞ4 �nðt, xÞ if and only if (��(t))n4 0.
Proof Recall the definitions of 0�
and * from Propositions 4.8 and 4.1,
respectively. Note that
0n�ðt, xÞ � �nðt, xÞ ¼
1
xð1� Þðxþ b0ðtÞÞð��ðtÞÞn �
1
xð1� Þðxþ bðtÞÞð��ðtÞÞn
¼ð��ðtÞÞnxð1� Þ
ðb0ðtÞ � bðtÞÞ: ð35Þ
Using the definitions of b0(t) and b(t) given in the statements of Propositions 4.8
and 4.1, respectively, we obtain that
b0ðtÞ � bðtÞ ¼
Z T
t
iðsÞexp �
Z s
t
rðvÞdv
� �1� exp �
Z s
t
�ðvÞdv
� �� �ds: ð36Þ
Since �(t) is a positive function, we get thatZ s
t
�ðvÞdv4 0
and therefore, the inequality
1� exp �
Z s
t
�ðvÞdv
� �4 0 ð37Þ
holds for all 0� t� s�T. Therefore, combining (35), (36) and (37) and recalling that
5 1, we obtain that the sign of 0n�ðt, xÞ � �nðt, xÞ is the same as the sign of (��(t))n
for all t2 [0,min{T, �}]. g
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The economic implications of the theorem above are made clear in the followingresult.
COROLLARY 4.10 Let � denote the non-singular square matrix given by (��T)�1 and(��(t))n the n-th component of the vector ��(t). Assume that for every n2 {1, . . . ,N} wehave (��(t))n4 0. Then, the optimal portfolio of a wage earner with the possibility ofbuying a life insurance policy is more conservative than the optimal portfolio of thesame wage earner if he does not have the opportunity to buy life insurance.
5. Conclusions
We have introduced a model for optimal insurance purchase, consumption andinvestment for a wage earner with an uncertain lifetime with an underlying financialmarket consisting of one risk-free security and a fixed number of risky securities withdiffusive terms driven by multidimensional Brownian motion.
When we restrict ourselves to the case where the wage earner has the samediscounted CRRA utility functions for the consumption of his family, the size of hislegacy and his terminal wealth, we obtain explicit optimal strategies and describe newproperties of these optimal strategies. Namely, we obtain economically relevantconclusions such as: (i) a young wage earner with smaller wealth has an optimalportfolio with larger values of volatility and higher expected returns and (ii) a wageearner who can buy life insurance policies will choose a more conservative portfoliothan a similar wage earner who is without the opportunity to buy life insurance.
It is worth noting that the model described in this article can be improved byadding further ingredients such as, for instance, some form of stochasticity on theincome function or the hazard rate, some constraints on the possibility of trading ofstocks on margin, the existence of additional life insurance products on the market oreven adding some correlation between the wage earner’s mortality rate and theunderlying financial market. We plan to address at least some of these issues in afuture publication.
Acknowledgements
We thank the Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI and POSI by FCTand Ministerio da Ciencia, Tecnologia e Ensino Superior, CEMAPRE, LIAAD-INESC PortoLA, Centro de Matematica da Universidade do Minho and Centro de Matematica daUniversidade do Porto for their financial support. D. Pinheiro’s research was supported byFCT – Fundacao para a Ciencia e Tecnologia program ‘Ciencia 2007’ and project‘Randomness in Deterministic Dynamical Systems and Applications’ (PTDC/MAT/105448/2008). I. Duarte’s research was supported by FCT – Fundacao para a Ciencia e Tecnologiagrant with reference SFRH/BD/33502/2008.
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