optimal investment and consumption decision of a family with life insurance

Insurance: Mathematics and Economics 48 (2011) 176–188

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Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Optimal investment and consumption decision of a family with life insuranceMinsuk Kwak a,∗, Yong Hyun Shin b, U Jin Choi aa Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 305701, Republic of Koreab Department of Mathematics, Hannam University, Daejeon, 306791, Republic of Korea

a r t i c l e i n f o

Article history:Received September 2009Received in revised formSeptember 2010Accepted 25 October 2010

IME Subject and Insurance Branch Category:IE13IB11

JEL classification:D91E21G11G22

Keywords:Life insuranceOptimal investment/consumptionLabor incomeUtility maximizationMartingale method

a b s t r a c t

We study an optimal portfolio and consumption choice problem of a family that combines life insurancefor parents who receive deterministic labor income until the fixed time T . We consider utility functionsof parents and children separately and assume that parents have an uncertain lifetime. If parents diebefore time T , children have no labor income and they choose the optimal consumption and portfoliowith remaining wealth and life insurance benefit. The object of the family is to maximize the weightedaverage of utility of parents and that of children. We obtain analytic solutions for the value function andthe optimal policies, and then analyze how the changes of the weight of the parents’ utility function andother factors affect the optimal policies.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

We investigate an optimal investment and consumptiondecision problem of the family with life insurance for parents. Weassume that parents have an uncertain lifetime and while theyare alive, they receive deterministic labor income until the fixedtime T > 0. If parents die before time T , children have no laborincome until time T and they choose their optimal policies withremaining wealth and life insurance benefit. We consider utilityfunctions of parents and children separately. The object of thefamily is to maximize the weighted average of utility of parentsand that of children. It is assumed that the utility functions ofboth parents and children belong to HARA utility class. UsingHARAutility we impose the condition that instantaneous consumptionrate should be above a given lower bound. Using the martingalemethod, analytic solutions for the value function and the optimalpolicies are derived. We analyze how the changes of the weightof the parents’ utility function and other factors, such as the

∗ Corresponding author. Tel.: +82 42 350 2766.E-mail addresses: [email protected] (M. Kwak), [email protected]

(Y.H. Shin), [email protected] (U.J. Choi).

0167-6687/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2010.10.012

family’s currentwealth level and the fair discounted value of futurelabor income, affect the optimal policies and also illustrate somenumerical examples.

Since the 1960s, there have beenmuch research on the analysisof the demand for life insurance. Yaari (1965) considered aconsumption and life insurance premium choice problem andshowed that if an individual has an uncertain lifetime andno bequest motive, it is optimal to annuitize all of savings.Hakansson (1969) examined life-cycle pattern of consumptionand saving in a discrete time environment and found conditionsunder which zero insurance premium is optimal. Fischer (1973)also examined the optimal life insurance purchase by using thedynamic programming methods and obtained the formula ofpresent value of future incomewhich is different from the formulaunder a certain lifetime. Richard (1975) combined life insurancewith the well-known continuous-time model of Merton (1969,1971) for the optimal consumption and investment. Richard (1975)assumed that the individual’s lifetime is bounded and imposeda condition that the individual cannot purchase life insurance atthe end of his lifetime. Also Campbell (1980), Lewis (1989), Hurd(1989) and Babbel and Ohtsuka (1989) examined the life insuranceproblems from different perspectives and gave us numerousinsights into the demand for life insurance.

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M. Kwak et al. / Insurance: Mathematics and Economics 48 (2011) 176–188 177

Recently, Moore and Young (2006) studied the optimal con-sumption, investment, and insurance strategies for an individual.Pliska and Ye (2007) studied the optimal life insurance and con-sumption rules for a wage earner whose lifetime is random andNielsen and Steffensen (2008) investigated optimal investment,and life insurance strategies under minimum and maximum con-straints. Ye (2006, 2007) considered an optimal life insurance, con-sumption and portfolio choice problem with an uncertain lifetimeand solved the problem by using the martingale method whichwe use to solve our problem. Huang et al. (2008), and Huangand Milevsky (2008) investigated an optimal life insurance, con-sumption and portfolio choice problem with a stochastic incomeprocess. They focused on the effect of correlation between thedynamics of financial capital and human capital on the optimalpolicies. They used a similarity reduction technique to find thenumerical solution efficiently and found that the life insurancehedges human capital and is insensitive to the risk aversion.

In addition to Merton (1969, 1971), many studies aboutcontinuous-time optimal investment and consumption choiceproblems have been carried out by Karatzas et al. (1986), Pliska(1986), Karatzas et al. (1987), and Cox and Huang (1989) and so on.Recently, portfolio selection problems under a preference changeare investigated by Choi and Koo (2005), Stabile (2006), and Kwaket al. (2009).

The rest of this paper proceeds as follows. In Section 2,we introduce our financial market and insurance market setupand derive the budget constraint of the family. We present theoptimization problem of the family in Section 3. We derive theanalytic solution for the value function of the family’s optimizationproblem and give the optimal policies in explicit forms. In Section 4we analyze the effects of varying parameters on the optimalsolutions with numerical examples, and Section 5 concludes. Alldetailed proofs in this paper are given in the appendices.

2. The economy

We consider an optimization problem of one family until thefixed parents’ retirement time T > 0 in the financial market andthe insurance market. In the financial market, it is assumed thatthere are one risk-free asset and one risky asset which evolveaccording to the ordinary differential equation (ODE) and to thestochastic differential equation (SDE)

dS0t = rS0t dt and dS1t = µS1t dt + σ S1t dWt ,

respectively, where r, µ and σ are constants, Wt is a standardBrownian motion on a complete probability space (Ω, F , P), andFt

Tt=0 is the P-augmentation of the natural filtration generated

byWt .1Let cp(t) be the consumption rate of parents at time t, cc(t)

be the consumption rate of children at time t and π(t) be theamount invested in the risky asset S1t at time t . It is assumed thatthe consumption rate processes cp(t) and cc(t) are nonnegative,Ft-progressively measurable, satisfying, respectively,∫ T

0cp(t)dt < ∞, almost surely (a.s.) and∫ T

0cc(t)dt < ∞, a.s.,

and the portfolio process π(t) is Ft-adapted, satisfying∫ T

0π(t)2dt < ∞, a.s.

1 For more details, refer Section 1.1 of Karatzas and Shreve (1998).

Let τ be the death time of parents who have the deterministiclabor income wt until time T .2 It is assumed that the lifetime τ isindependent of the filtration Ft.

In the insurance market, let λy+t be an instantaneous force ofmortality curve, where y is the age of the breadwinner at an initialtime of the model. Then the conditional probability of survival,from age y to age y+ t , under the law of mortality defined by λy+t ,can be computed by

tpy , e− t0 λy+sds. (2.1)

It is assumed that if the family’s insurance premium rate at time t isI(t), then they receive lump sumpayment I(τ )/λy+τ at the parents’death time τ .3Let Xt be the family’s wealth at time t until a randomtime τm , min(τ , T ), then the total legacyM(t), when the parentsdie at time t with a wealth Xt , is

M(t) , Xt +I(t)λy+t

,

where the family’s wealth dynamics Xt satisfies the following SDE:dXt =

rXt + (µ − r)π(t) − cp(t) − cc(t) − I(t) + wt

dt

+ σπ(t)dWt , for 0 ≤ t < τm. (2.2)Now let us define the market-price-of-risk, the discount

process, the exponentialmartingale process, and the pricing kernelprocess (or state-price-density process), respectively, by

θ ,µ − r

σ, ζt , e−

t0 (λy+s+r)ds,

Zt , e−θWt−12 θ2t and Ht , ζtZt .

The equivalent martingale measure is defined byP(A) , E[ZT1A],for all A ∈ FT and the Girsanov theorem implies that Wt ,Wt + θ t, 0 ≤ t ≤ T , is also a standard Brownianmotion under thenew measureP. So the wealth dynamics (2.2) until time τm can berewritten asdXt =

rXt − cp(t) − cc(t) − I(t) + wt

dt + σπ(t)dWt

=

(r + λy+t)Xt − cp(t) − cc(t) − λy+tM(t) + wt

dt

+ σπ(t)dWt , for 0 ≤ t < τm.

For 0 ≤ t < u, applying Itô’s formula to ζtXt , we obtain

ζuXu +

∫ u

tζscp(s) + cc(s) + λy+sM(s)

ds

= ζtXt +

∫ u

tζswsds +

∫ u

tζsσπ(s)dWs. (2.3)

If we define

bu ,

∫ T

uws

ζs

ζuds, 4

and add ζubu on both sides of Eq. (2.3), then we have

ζu(Xu + bu) +

∫ u

tζscp(s) + cc(s) + λy+sM(s)

ds

= ζt(Xt + bt) +

∫ u

tζsσπ(s)dWs. (2.4)

Taking the conditional expectationEt [·] = E[·|Ft ] on both sidesof Eq. (2.4) and using Bayes rule, we derive the following budget

2 In this paper, it is assumed that parents represent children’s father or motherwith labor income. It is also assumed that children have no labor income.3 We focus on the case with zero premium loading. The premium loading can be

incorporated into our model by introducing a continuous function η : [0, T ] → R+

which satisfies η(t) ≥ λy+t and replacing the insurance benefit I(τ )

λy+τwith I(τ )

η(τ ).

4 bt is the fair discounted value of the parents’ future labor income from t to τm .

178 M. Kwak et al. / Insurance: Mathematics and Economics 48 (2011) 176–188

constraint:

Et

∫ T

tHscp(s)ds +

∫ T

tHscc(s)ds +

∫ T

tλy+sHsM(s)ds + HTXT

≤ Ht(Xt + bt), for 0 ≤ t < τm. (2.5)

(See Ye (2006, 2007).)

3. The optimization problem

Now we describe our main problem. The family’s expectedutility function U(t, Xt; cp, cc, π, I) with an initial endowment Xtat time t, t < τm, is given by

U(t, Xt; cp, cc, π, I)

= Et

α1

∫ τm

te−δ(s−t)up(cp(s))ds + α2

∫ T

te−δ(s−t)uc(cc(s))ds

= Et

∫ τm

te−δ(s−t) α1up(cp(s)) + α2uc(cc(s))

ds

+ α21τ<T

∫ T

τ

e−δ(s−t)uc(cc(s))ds

, (3.1)

where δ > 0 is a constant subjective discount rate, up(c) anduc(c) are utility functions of parents and children, respectively.Weassume that up(c) and uc(c) are given by

up(c) ,

(c − Rp)

1−γp

1 − γp, c > Rp,

limz↓Rp

(z − Rp)1−γp

1 − γp, c = Rp,

−∞, c < Rp,

and

uc(c) ,

(c − Rc)

1−γc

1 − γc, c > Rc,

limz↓Rc

(z − Rc)1−γc

1 − γc, c = Rc,

−∞, c < Rc,

where γp > 0 (γp = 1) and γc > 0 (γc = 1) are the parents’and children’s coefficients of relative risk aversion, respectively,and Rp ≥ 0 and Rc ≥ 0 represent constant consumption floors ofparents and children, respectively. α1 ≥ 0 and α2 ≥ 0 are constantweights of utility functions of parents and children, respectively,which satisfy

α1 + α2 = 1.

Assumption 1. We define the Merton constant Ki, i = p, c , andassume that it is always positive, that is,

Ki , r +δ − r

γi+

γi − 12γ 2

iθ2 > 0, i = p, c.

If parents die before T , that is, τ < T , then wt = 0 for τm ≤

t ≤ T . Since I(t) is life insurance premium rate on parents,I(t) = 0 for τm ≤ t ≤ T . Therefore, after parents’ death time τ ,children have only two control variables: their consumption cc(t),and investment π(t). Then the family’s expected utility functionUc(t, Xt; cc, π)with an initial endowmentXt at time t, τm ≤ t ≤ T ,is given by

Uc(t, Xt; cc, π) = Et

[∫ T


]. (3.2)

For τm ≤ t ≤ T , let Ac(t, Xt) be the admissible class of the pair(cc, π) at time t for which the family’s expected utility function

(3.2) is well-defined, that is,

Et

[∫ T

te−δ(s−t)uc(cc(s))−ds

]< ∞,

where x− , max(−x, 0).Then we have the following lemma.

Lemma 1. For τm ≤ t ≤ T , the value function Vc(t, Xt) is given by

Vc(t, Xt) , sup(cc ,π)∈Ac (t,Xt )

Uc(t, Xt; cc, π) = eδtΦ(t, Xt),

where

Φ(t, Xt) , e−δt g(t)γc

1 − γc

Xt −

Rc

r

1 − e−r(T−t)1−γc

and

g(t) ,1 − e−Kc (T−t)

Kc.

And the optimal policies are given by

c∗

c (t) =1

g(t)

Xt −

Rc

r

1 − e−r(T−t)

+ Rc

and

π∗(t) =θ

σγc

Xt −

Rc

r

1 − e−r(T−t) .

Proof. See Appendix A.

So the value function of the family at time t < τm, V (t, Xt), isdefined as follows:

V (t, Xt) , sup(cp,cc ,π,I)∈A(t,Xt )

Et

∫ τm

te−δ(s−t)

α1up(cp(s))

+ α2uc(cc(s))ds + α21τ<T eδtΦ(τ ,M(τ ))

subject to the budget constraint (2.5), where A(t, Xt) is the admis-sible class of the quadruple (cp, cc, π, I) at time t < τm for whichthe family’s expected utility function (3.1) is well-defined, that is,

Et

∫ τm

te−δ(s−t) α1up(cp(s)) + α2uc(cc(s))

− ds

+ α21τ<T

∫ T

τ

e−δ(s−t)uc(cc(s))−ds

< ∞.

Let us defineJ(t, Xt) , e−δtV (t, Xt),

then

J(t, Xt) = sup(cp,cc ,π,I)∈A(t,Xt )

Et

α1

∫ τm

te−δsup(cp(s))ds

+ α2

∫ τm

te−δsuc(cc(s))ds + α21τ<T Φ(τ ,M(τ ))

. (3.3)

Note that, from the definition of tpy in (2.1), the conditionaldistribution of τ is Pt(τ > s) = e−

st λy+udu, for t ≤ s. Thus the

first term except α1 of right-hand side of Eq. (3.3) is given by

Et

[∫ τm

te−δsup(cp(s))ds

]= Et

[∫ τ

te−δsup(cp(s))ds1τ≤T +

∫ T

te−δsup(cp(s))ds1τ>T

]= Et

[∫ T

te−(δs+

st λy+udu)up(cp(s))ds

]


and similarly, the second term except α2 of right-hand side of Eq.(3.3) is given by

Et

[∫ τm

te−δsuc(cc(s))ds

]= Et

[∫ T

te−(δs+

st λy+udu)uc(cc(s))ds

].

So we have

J(t, Xt) = sup(cp,cc ,π,I)∈A(t,Xt )

Et

α1

∫ T

te−(δs+


+ α2

∫ T

te−(δs+


+ α2

∫ T

tλy+se−

st λy+uduΦ(s,M(s))ds

.

For a Lagrange multiplier ν > 0, let us define a dual valuefunctionJ(ν, t)

, sup(cp,cc ,π,I)∈A(t,Xt )

Et

α1

∫ T

te−(δs+


+ α2

∫ T

te−(δs+


+ α2

∫ T

tλy+se−

st λy+uduΦ(s,M(s))ds

− νe t0 λy+uduEt

∫ T

tHscp(s)ds

+

∫ T

tHscc(s)ds +

∫ T

tλy+sHsM(s)ds + HTXT

= e t0 λy+uduEt

∫ T

tDsu1(Y ν

s )ds

+

∫ T

tDsu2(Y ν

s )ds +

∫ T

tλy+sDsu3(s, Y ν

s )ds

, (3.4)

where

Dt , exp−

∫ t

0(λy+u + δ)du

,

Y νt , νHt/Dt = ν exp

δ − r −

12θ2t − θWt

,

u1(y) = supcp>0

α1

(cp − Rp)1−γp

1 − γp− ycp

= α

1γp1

γp

1 − γpy

γp−1γp − Rpy,

u2(y) = supcc>0

α2

(cc − Rc)1−γc

1 − γc− ycc

= α

1γc2

γc

1 − γcy

γc−1γc − Rcy,

and

u3(t, y) = supM>0

α2

g(t)γc

1 − γc

M −

Rc

r

1 − e−r(T−t)1−γc

− My

= α1γc2 g(t)

γc

1 − γcy

γc−1γc −

Rc

r

1 − e−r(T−t) y.

Consequently we derive the optimal policies5

c∗

p (t) = α1γp1

Y νt

− 1γp + Rp, (3.5)

c∗

c (t) = α1γc2

Y νt

− 1γc + Rc, (3.6)

M∗(t) = α1γc2 g(t)

Y νt

− 1γc +

Rc

r

1 − e−r(T−t) , (3.7)

and

X∗

T = 0. (3.8)

Now we can derive the value function V (t, Xt) fromJ(ν, t) byLegendre transform inverse formula

V (t, Xt) = eδt infYνt >0

J(ν, t) + νe t0 λy+uduHt(Xt + bt)

= inf

Yνt >0

eδtJ(ν, t) + Y ν

t (Xt + bt). (3.9)

The following theorem gives the value function V (t, Xt) and theoptimal policies.

Theorem 2. The value function V (t, Xt) is given by

V (t, Xt) = α1γp1

γp

1 − γp

Y ν∗

t

γp−1γp∫ T

te−

st (λy+u+Kp)duds

+ α1γc2

γc

1 − γc

Y ν∗

t

γc−1γc g(t)

+ Y ν∗

t

Xt + bt − Rp

∫ T

te−

st (λy+u+r)duds

−Rc

r

1 − e−r(T−t),

where Y ν∗

t satisfies the following equation:

Xt = α1γp1

Y ν∗

t

−1γp∫ T

te−

st (λy+u+Kp)duds

+ α1γc2

Y ν∗

t

−1γc g(t) + Rp

∫ T

te−

st (λy+u+r)duds

+Rc

r

1 − e−r(T−t)

− bt . (3.10)

And, for 0 ≤ t < τm, the optimal policies are given by

c∗

p (t) = α1γp1

Y ν∗

t

−1γp

+ Rp, c∗

c (t) = α1γc2

Y ν∗

t

−1γc

+ Rc,

π∗(t) =θ

σγc

Xt + bt − Rp

∫ T

te−

st (λy+u+r)duds

−Rc

r

1 − e−r(T−t)

+θ

σ

γc − γp

γpγcα

1γp1

Y ν∗

t

−1γp

×

∫ T

te−

st (λy+u+Kp)duds, (3.11)

5 We can always obtain the optimal strategies (c∗p (t), c

∗c (t), π

∗(t), I∗(t)) ∈

A(t, Xt ), which satisfy the above Eqs. (3.5)–(3.8). See Chapter 3 of Karatzas andShreve (1998).


andI∗(t)λy+t

= M∗(t) − Xt

= bt − Rp

∫ T

te−

st (λy+u+r)duds − α

1γp1

Y ν∗

t

−1γp

×

∫ T

te−

st (λy+u+Kp)duds. (3.12)

Proof. See Appendix B.

4. Properties of optimal solutions and numerical examples

In this section we examine the properties of the optimalsolutions, and analyze how the changes of parameters affect theoptimal policies. Especially, the following propositions summarizethe effects of the weight of parents’ utility function α1 on theoptimal life insurance premium rate and on the optimal portfolio.Every numerical example is computed under the followingassumption.

Assumption 2. Throughout the numerical examples in this sec-tion, we assume that

wt = Cwekw t ,

λy+t = AL + BLt,

for some positive constants Cw, kw, AL and BL.6

As we have mentioned in footnote 3, we focus on the case withzero premium loading. The following Proposition 1 holds if thepremium loading is zero. If the premium loading is positive, theeffect of current wealth level Xt on the optimal life insurancepremium rate can be positive or negative depending on the otherparameters. However, other propositions except Proposition 1(Propositions 2–5) still hold even though the premium loading ispositive.7

Proposition 1. If α1 ∈ (0, 1], the optimal life insurance premiumrate I∗(t) decreases as the wealth level Xt increases. If α1 = 0, I∗(t)is not affected by Xt .Proof. See Appendix C.

Proposition 2. If α1 ∈ [0, 1), the optimal life insurance premiumrate I∗(t) increases as the fair discounted value of future labor incomebt increases. If α1 = 1, I∗(t) is not affected by bt .Proof. See Appendix C.

It is easily verified that both Xt and bt have positive effects on theoptimal policies except the optimal life insurance premium rateI∗(t). In other words, the optimal consumption rate of parents, theoptimal consumption rate of children, and the optimal investmentincrease as Xt + bt increases.8Unless α1 = 0, the optimalconsumption rate of parents c∗

p (t) increases as Xt or bt increases.When α1 = 0, we only consider children’s utility, and thereforec∗p (t) is equal to the minimum level Rp regardless of Xt and bt . Theoptimal consumption rate of children c∗

c (t) decreases as Xt or btincreases, unless α1 = 1. If α1 = 1, we do not care about children’sutility, and therefore c∗

c (t) is equal to the minimum level Rc . Theoptimal investment π∗(t) increases as bt or Xt increases for anyα1 ∈ [0, 1].

6 Pliska and Ye (2007) used the same types of labor income and hazard rate fortheir numerical examples.7 For details, see Appendix D.8 Xt + bt is the sum of the current wealth level Xt and the fair discounted value

of future labor income bt .

On the other hand, the effect of Xt on the optimal life insurancepremium rate are different from that of bt . As we have mentionedin Proposition 1, the optimal life insurance premium rate I∗(t)decreases as Xt increases if α1 ∈ (0, 1]. If α1 = 0,

I∗(t)λy+t

α1=0

= bt − Rp

∫ T

te−

st (λy+u+r)duds

≥ bt − Rp

∫ T

te−

st (λy+u+r)duds − α

1γp1

Y ν∗

t

−1γp

×

∫ T

te−

st (λy+u+Kp)duds,

and therefore I∗(t)|α1=0 is not affected by Xt and greater than I∗(t)of other cases with α1 ∈ (0, 1].

As we have stated in Proposition 2, I∗(t) increases as btincreases if α1 ∈ [0, 1). For the case of α1 = 1,

I∗(t)λy+t

α1=1

= bt − Rp

∫ T

te−

st (λy+u+r)duds

−

Y ν∗

t

−1γp∫ T

te−

st (λy+u+Kp)duds

≤ bt − Rp

∫ T

te−

st (λy+u+r)duds

− α1γp1

Y ν∗

t

−1γp∫ T

te−

st (λy+u+Kp)duds,

and therefore I∗(t)|α1=1 is not affected by bt and less than I∗(t) ofother cases with α1 ∈ [0, 1). Note that

M∗(t)α1=1 = Xt +

I∗(t)λy+t

α1=1

=Rc

r

1 − e−r(T−t)

=

∫ T

te−r(s−t)Rcds. (4.1)

Eq. (4.1) implies that the total legacy M∗(t) with the optimal lifeinsurance premium rate I∗(t) enables children to maintain theirminimum consumption rate Rc from t to T .

In Fig. 1, solid lines represent the relations between the optimalpolicies and Xt , whereas dotted lines represent the relationsbetween the optimal policies and bt . As we have mentioned, wecan observe that optimal policies except the optimal life insurancepremium rate I∗(t) are determined by Xt +bt . Unlike other optimalpolicies, I∗(t) is not determinedbyXt+bt . FromFig. 1, it can be seenthat the currentwealth level Xt has a negative effect on the optimallife insurance premium rate. On the other hand, bt has a positiveeffect on the optimal life insurance premium rate. Therefore, theoptimal life insurance premium rate is determined not by Xt + bt ,but by both Xt and bt .

Proposition 3. The optimal life insurance premium rate I∗(t)decreases as the weight of the utility function of parents α1 increasesfrom 0 to 1.

Proof. See Appendix C.

Proposition 4. The optimal investment π∗(t) decreases as α1increases from 0 to 1 if γp > γc , and increases as α1 increasesfrom 0 to 1 if γp < γc . If γp = γc, π

∗(t) is not affected by α1.

Proof. See Appendix C.

Figs. 2–5 illustrate the relations between the weight of parents’utility function α1 and the optimal policies for different risk


Fig. 1. Relations between the optimal policies and Xt + bt . (t = 0, Rp = 2, Rc = 1, δ = 0.04, r = 0.04, µ = 0.06, σ = 0.3, AL = 0.005, BL = 0.001125, T = 30).

Fig. 2. Relations between α1 and the optimal policies when γp = 2, γc = 2. (t = 0, X0 = 10, Rp = 2, Rc = 1, δ = 0.04, r = 0.04, µ = 0.06, σ = 0.3, Cw = 5.0,kw = 0.03, AL = 0.005, BL = 0.001125, T = 30).


Fig. 3. Relations between α1 and the optimal policies when γp = 2, γc = 3. (t = 0, X0 = 10, Rp = 2, Rc = 1, δ = 0.04, r = 0.04, µ = 0.06, σ = 0.3, Cw = 5.0, kw =

0.03, AL = 0.005, BL = 0.001125, T = 30).

Fig. 4. Relations between α1 and the optimal policies when γp = 3, γc = 2. (t = 0, X0 = 10, Rp = 2, Rc = 1, δ = 0.04, r = 0.04, µ = 0.06, σ = 0.3, Cw = 5.0, kw =

0.03, AL = 0.005, BL = 0.001125, T = 30).


Fig. 5. Relations between α1 and the optimal policies when γp = 1/2, γc = 1/2. (t = 0, X0 = 10, Rp = 2, Rc = 1, δ = 0.04, r = 0.04, µ = 0.06, σ = 0.3, Cw = 5.0, kw =

0.03, AL = 0.005, BL = 0.001125, T = 30).

aversion coefficients of parents and children. It is obvious thatthe optimal consumption of parents c∗

p (t) increases and theoptimal consumption of children c∗

c (t) decreases as α1 increases.As we have mentioned in Proposition 3, the optimal life insurancepremium rate I∗(t) decreases as α1 increases. Bequest motive isweaker as α1 is larger. Unlike other optimal policies, the effectof α1 on the optimal investment π∗(t) depends on γp and γc ,risk aversion coefficients of parents and children, respectively. Ifγp > γc, π

∗(t) decreases as α1 increases, and π∗(t) increases asα1 increases when γp > γc . In other words, if the risk aversioncoefficients of parents and children are different, the optimalinvestment increases when the relative importance of less riskaverse family member’s utility increases. If γp = γc, π

∗(t) is notaffected by α1.

Proposition 5. If α1 ∈ [0, 1), the optimal life insurance premiumrate I∗(t) decreases as the consumption floor of parents Rp increases. Ifα1 = 1, I∗(t) is not affected by Rp. On the other hand, I∗(t) increasesas the consumption floor of children Rc increases if α1 ∈ (0, 1]. Ifα1 = 0, I∗(t) is not affected by Rc . The optimal investment π∗(t)decreases as Rp or Rc increases, regardless of α1.Proof. See Appendix C.

Huang et al. (2008) highlighted that the role of life insuranceis a hedge against the loss of human capital, future labor incomeof parents in our model. Hence, determining the optimal lifeinsurance premium rate I∗(t) is equivalent to determining theoptimal total legacy M∗(t) or determining the optimal level ofinsurance benefit I∗(t)/λy+t . From (3.10), it is easy to verify that∂Y ν∗

t /∂Rp > 0 and this shows that, if α1 ∈ [0, 1), ∂M∗(t)/∂Rp < 0from (3.7). Therefore, the optimal life insurance premium rate I∗(t)decreases as Rp increases if α1 ∈ [0, 1). However, if α1 = 1,

M∗(t)α1=1 =

Rc

r

1 − e−r(T−t)

=

∫ T

te−r(s−t)Rcds

which is the wealth needed at time t to maintain children’sminimum consumption Rc from t to T . Therefore, if α1 = 1,∂M∗(t)/∂Rp = 0 and this implies that I∗(t) is not affected by Rp.From (3.10), it is straightforward to show that ∂Y ν∗

t /∂Rc > 0. Thisimplies that, if α1 ∈ (0, 1], ∂(I∗(t)/λy+t)/∂Rc > 0 from (3.12).Therefore, I∗(t) increases as Rc increases if α1 ∈ (0, 1]. If α1 = 0,

I∗(t)λy+t

α1=0

= bt − Rp

∫ T

te−

st (λy+u+r)duds

= bt − E[∫ τm

te−r(s−t)Rpds

]which is the fair discounted value of the parents’ future laborincome less the expected wealth needed to maintain parents’minimum consumption Rp from t to τm. Consequently, if α1 =

0, ∂(I∗(t)/λy+t)/∂Rc = 0 and this means that I∗(t) is not affectedby Rc .

5. Concluding remarks

We investigate an optimal portfolio, consumption and lifeinsurance premium choice problem of the family. Analyticsolutions for the value function and the optimal policies arederived by the martingale method. We analyze the properties ofthe optimal policies, where the emphasis is placed on the role ofα1 which is the weight of the parents’ utility function.

Acknowledgements

An earlier version of this paper was presented at the 3rdKorean Mathematical Society Probability Workshop, KangwonNational University, Chuncheon, Korea, June 12–13, 2009, theSociety for Computational Economics 15th International Con-ference on Computing in Economics and Finance, University of


Technology, Sydney, Australia, July 15–17, 2009, the 41st ISCIEInternational Symposium on Stochastic Systems Theory and ItsApplications, Konan University, Kobe, Japan, November 13–14,2009, the Workshop and Spring School on Stochastic Calculus andApplications, Institute of Mathematics, Academia Sinica, Taipei,Taiwan, April 9–17, 2010, the 6th World Congress of the Bache-lier Finance Society, Toronto, Canada, June 22–26, 2010, and theInternational Workshop on Recent Trends in Learning, Computa-tion, and Finance, POSCO International Center, POSTECH, Pohang,Korea, August 30–31, 2010. We are grateful to an anonymous ref-eree and editor for valuable comments and advice, which improveour paper greatly. We also thank Hyeng Keun Koo, Jaeyoung Sung,Stanley R. Pliska, Ji Hee Yoon, and Jacek Krawczyk for useful com-ments and discussion.

The first author’s work was partially supported by the KoreaStudent Aid Foundation (KOSAF) Grant funded by the KoreanGovernment (MEST) (Grant No. S2-2009-000-01695-1) and byBK21 project of Department of Mathematical Sciences, KAIST.

The second author’s work was supported by NRF of KoreaGrant funded by the Korean Government (MEST) (Grant No.2009-0072021).

The third author’s work was supported by BK21 project ofDepartment of Mathematical Sciences, KAIST.

Appendix A. Proof of Lemma 1

For t ≥ τm, the value function of the family at time t is

Vc(t, Xt) , sup(cc ,π)∈Ac (t,Xt )

Et

[∫ T


],

subject to the wealth process (cf. the wealth process (2.2))dXt = [rXt + (µ − r)π(t) − cc(t)] dt + σπ(t)dWt . (A.1)Let us define the discount process, and the pricing kernel process,respectively, byηt , e−rt , Gt , ηtZt .From (A.1), we can derive the budget constraint

Et

[GTXT +

∫ T

tGscc(s)ds

]≤ GtXt .

For a Lagrange multiplier ν > 0, let us define a dual value function

Vc(ν, t) , supcc ,π

Et

∫ T

te−δsuc(cc(s))ds

− νEt

GTXT +

∫ T

tGscc(s)ds

= supcc ,π

Et

∫ T

te−δs

(cc(s) − Rc)

1−γc

1 − γc

− Y νs cc(s)

ds − νGTXT

= e−δt

γc

1 − γc

Y νt

γc−1γc g(t) − RcY ν

t1 − e−r(T−t)

r

,

whereY νt , νeδtGt

and

g(t) =1 − e−Kc (T−t)

Kc.

Consequently we derive the optimal policies

c∗

c (s) = (Y νs )

−1γc + Rc and X∗

T = 0.

It is easily verified that the value function is given by

Vc(t, Xt) = eδt infYνt >0

Vc(ν, t) + νGtXt

= infYνt >0

γc

1 − γc

Y νt

γc−1γc g(t) − RcY ν

t1 − e−r(T−t)

r+ Y ν

t Xt

=g(t)γc

1 − γc

Xt −

Rc

r

1 − e−r(T−t)1−γc

,

where

Y ν∗

t =

Xt −

Rc

r

1 − e−r(T−t)−γc

g(t)γc . (A.2)

So the optimal consumption c∗c (t) is given by

c∗

c (t) =

Y ν∗

t

−1γc

+ Rc =1

g(t)

Xt −

Rc

r

1 − e−r(T−t)

+ Rc .

Note that Eq. (A.2) is equivalent to

Xt = g(t)Y ν∗

t

−1γc

+Rc

r

1 − e−r(T−t) .

Itô’s formula to the above optimal wealth Xt implies

dXt =

g ′(t)

Y ν∗

t

−1γc

− Rce−r(T−t)dt

−1γc

g(t)Y ν∗

t

−1γc −1

dY ν∗

t

+1 + γc

2γ 2c

g(t)Y ν∗

t

−1γc −2

dY ν∗

t

2=

Y ν∗

t

−1γcrg(t) − 1 +

θ2

γcg(t)

− Rce−r(T−t)

dt

+ σθ

σγcg(t)

Y ν∗

t

−1γc dWt

=

rXt − c∗

c (t) + (µ − r)θ

σγcg(t)

Y ν∗

t

−1γcdt

+ σθ

σγcg(t)

Y ν∗

t

−1γc dWt .

Therefore, a comparison of the drift term and the volatility termof the optimal wealth process dXt with Eq. (A.1) gives the optimalportfolio π∗(t) as follows:

π∗(t) =θ

σγcg(t)

Y ν∗

t

−1γc

=θ

σγc

Xt −

Rc

r

1 − e−r(T−t) .

Appendix B. Proof of Theorem 2

From (3.4), we derive the following equation,

eδtJ(ν, t) = α1γp1

γp

1 − γp

Y νt

γp−1γp

∫ T

te−

st (λy+u+Kp)duds

+ α1γc2

γc

1 − γc

Y νt

γc−1γc g(t)

− Y νt

Rp

∫ T

te−

st (λy+u+r)duds +

Rc

r

1 − e−r(T−t) .

Using the first order condition for Eq. (3.9) with respect to Y νt ,

then we obtain the optimal wealth Xt in (3.10), where ν∗ indicatesthe optimal Lagrange multiplier. Thus we can derive the value


function V (t, Xt) using proper substitution. Using Itô’s formulato the optimal wealth Xt in (3.10) and comparing the drift termand the volatility term of the optimal wealth process dXt withEq. (2.2) aswehavedone inAppendixA, thenweobtain the optimalportfolio π∗(t) as follows:

π∗(t) =θ

σγc

Xt + bt − Rp

∫ T

te−

st (λy+u+r)duds

−Rc

r

1 − e−r(T−t)

+θ

σ

γc − γp

γpγcα

1γp1

Y ν∗

t

−1γp∫ T

te−

st (λy+u+Kp)duds.

Appendix C. Proofs of Propositions 1–5

Proof of Proposition 1. By differentiating both sides of (3.10)with respect to Xt , we have the following inequality

∂Y ν∗

t

∂Xt

−1

= −1γp

α1γp1

Y ν∗

t

−1γp −1

∫ T

te−

st (λy+u+Kp)duds

−1γc

α1γc2

Y ν∗

t

−1γc −1

g(t)

< 0.

Also by differentiating both sides of (3.12) with respect to Xt , wehave∂(I∗(t)/λy+t)

∂Xt

=

1γp

α1γp1

Y ν∗

t

−1γp −1

∫ T

te−

st (λy+u+Kp)duds

∂Y ν∗

t

∂Xt.

Therefore, we have∂(I∗(t)/λy+t)

∂Xt= 0, if α1 = 0,

∂(I∗(t)/λy+t)

∂Xt< 0, if α1 ∈ (0, 1].

Proof of Proposition 2. By differentiating both sides of (3.10)with respect to bt , we have the following inequality

∂Y ν∗

t

∂bt

−1

= −1γp

α1γp1

Y ν∗

t

−1γp −1

∫ T

te−

st (λy+u+Kp)duds

−1γc

α1γc2

Y ν∗

t

−1γc −1

g(t) < 0.

Also by differentiating both sides of (3.12) with respect to bt , wehave∂(I∗(t)/λy+t)

∂bt=

∂M∗(t)∂bt

= −1γc

∂(I∗(t)/λy+t)

∂btg(t)(Y ν∗

t )−

1γc −1 ∂Y ν∗

t

∂bt.

Therefore, we have∂(I∗(t)/λy+t)

∂bt= 0, if α1 = 1,

∂(I∗(t)/λy+t)

∂bt> 0, if α1 ∈ [0, 1).

Proof of Proposition 3. From (3.7) and (3.12), we know that

I∗(t)/λy+t = M∗(t) − Xt

= α1γc2 g(t)

Y ν∗

t

−1γc

+Rc

r

1 − e−r(T−t)

− Xt , (C.1)

and

∂ I∗(t)/λy+t

∂α1= −

1γc

α1γc −12 g(t)

Y ν∗

t

−1γc −1

α2

∂Y ν∗

t

∂α1+ Y ν∗

t

.

By differentiating both sides of Eq. (3.10) with respect to α1, weobtain

∂Y ν∗

t

∂α1=

1γp

α1γp −1

1

Y ν∗

t

−1γp∫ T

te−

ut (λy+u+Kp)duds

−1γc

(1 − α1)1γc −1

Y ν∗

t

−1γc g(t)

1γp

α1γp1

Y ν∗

t

−1γp −1

∫ T

te−

ut (λy+u+Kp)duds

+1γc

(1 − α1)1γc

Y ν∗

t

−1γc −1

g(t)

. (C.2)

By Eq. (C.2), we have the following inequality

α2∂Y ν∗

t

∂α1+ Y ν∗

t

=

1γp

α1γp −1

1

Y ν∗

t

−1γp

×

∫ T

te−

ut (λy+u+Kp)duds

1γp

α1γp1

Y ν∗

t

−1γp −1

×

∫ T

te−

ut (λy+u+Kp)duds +

1γc

α1γc2

Y ν∗

t

−1γc −1

g(t)

> 0,

if α1 ∈ (0, 1).

Therefore, we have

For α1 ∈ (0, 1),∂ I∗(t)/λy+t

∂α1< 0.

Proof of Proposition 4. From the optimal investment (3.11) wehave

∂π∗(t)∂α1

=θ

σ

γc − γp

γpγc

1γp

α1γp −1

1

Y ν∗

t

−1γp

−1γp

α1γp1

Y ν∗

t

−1γp −1 ∂Y ν∗

t

∂α1

×

∫ T

te−

st (λy+u+Kp)duds

=θ

σ

γc − γp

γ 2p γc

α1γp −1

1

Y ν∗

t

1γp −1

Y ν∗

t − α1∂Y ν∗

t

∂α1

×

∫ T

te−

st (λy+u+Kp)duds.


.4)

.5)

∂Y ν∗

t

∂Rp=

Tt e−

st (λy+u+r)duds

1γp

α1γp1

Y ν∗

t− 1

γp −1 Tt e−

st (λy+u+Kp)duds +

1γc

α1γc2

Y ν∗

t− 1

γc −1 g(t)> 0, (C

∂Y ν∗

t

∂Rc=

1r

1 − e−r(T−t)

1γp

α1γp1

Y ν∗

t− 1

γp −1 Tt e−

st (λy+u+Kp)duds +

1γc

α1γc2

Y ν∗

t− 1

γc −1 g(t)> 0. (C

Box I.

∂π∗(t)∂Rp

= −θ

σγc

∫ T

te−

st (λy+u+r)duds −

θ

σγp

γc − γp

γpγcα

1γp1

Y ν∗

t

−1γp −1

∫ T

te−

st (λy+u+Kp)duds

∂Y ν∗

t

∂Rp

=

− Tt e−

st (λy+u+r)duds

θ

σγ 2cα

1γc2

Y ν∗

t

− 1γc −1

g(t) +θ

σγ 2pα

1γp1

Y ν∗

t

− 1γp −1 T

t e− st (λy+u+Kp)duds

1γp

α1γp1

Y ν∗

t− 1

γp −1 Tt e−

st (λy+u+Kp)duds +

1γc

α1γc2

Y ν∗

t− 1

γc −1 g(t)

< 0.

Box II.

∂π∗(t)∂Rc

= −θ

σγc

∫ T

te−

st rduds −

θ

σγp

γc − γp

γpγcα

1γp1

Y ν∗

t

−1γp −1

∫ T

te−

st (λy+u+Kp)duds

∂Y ν∗

t

∂Rc

=

− Tt e−

st rduds

θ

σγ 2cα

1γc2

Y ν∗

t

− 1γc −1

g(t) +θ

σγ 2pα

1γp1

Y ν∗

t

− 1γp −1 T

t e− st (λy+u+Kp)duds

1γp

α1γp1

Y ν∗

t− 1

γp −1 Tt e−

st (λy+u+Kp)duds +

1γc

α1γc2

Y ν∗

t− 1

γc −1 g(t)

< 0.

Box III.

From Eq. (C.2), we have

Y ν∗

t − α1∂Y ν∗

t

∂α1

=

1γc

α1γc −12

Y ν∗

t

−1γc g(t)

1γp

α1γp1

Y ν∗

t

−1γp −1

×

∫ T

te−

ut (λy+u+Kp)duds +

1γc

α1γc2

Y ν∗

t

−1γc −1

g(t)

> 0, if α1 ∈ (0, 1). (C.3)

Since inequality (C.3) holds, we conclude that

For α1 ∈ (0, 1),∂π∗(t)∂α1

< 0, if γp > γc,

∂π∗(t)∂α1

> 0, if γc > γp,

∂π∗(t)∂α1

= 0, if γp = γc .

Proof of Proposition 5. By differentiating both sides of (3.10)with respect to Rp and Rc , respectively, we obtain the inequalities(C.4) and (C.5) given in Box I. By differentiating both sides of (C.1)with respect to Rp, we have

∂ I∗(t)/λy+t

∂Rp=

∂M∗(t)∂Rp

= −1γc

α1γc2

Y ν∗

t

−1γc −1

g(t)∂Y ν∗

t

∂Rp. (C.6)

By differentiating both sides of (3.12) with respect to Rc , we have

∂ I∗(t)/λy+t

∂Rc=

1γp

α1γp1

Y ν∗

t

−1γp −1

×

∫ T

te−

st (λy+u+Kp)duds

∂Y ν∗

t

∂Rc. (C.7)

From (C.4), (C.5), (C.6), and (C.7), we conclude that

∂ I∗(t)/λy+t

∂Rp< 0,

∂ I∗(t)/λy+t

∂Rc= 0, if α1 = 0,

∂ I∗(t)/λy+t

∂Rp< 0,

∂ I∗(t)/λy+t

∂Rc> 0, if α1 ∈ (0, 1),

∂ I∗(t)/λy+t

∂Rp= 0,

∂ I∗(t)/λy+t

∂Rc> 0, if α1 = 1.

By differentiating both sides of (3.11) with respect to Rp andusing (C.4), we have the expression given in Box II. Similarly, bydifferentiating both sides of (3.11) with respect to Rc and using(C.5), we obtain the expression in Box III.

Appendix D. The case with positive premium loading

The premium loading can be incorporated into our model byintroducing a continuous function η : [0, T ] → R+ which satisfiesη(t) ≥ λy+t and replacing the insurance benefit I(τ )

λy+τwith I(τ )

η(τ ).

Thenwehave to redefine the discount process ζt , the pricing kernelHt , discounted value of parents’ future labor income bt , and Y ν

t , as


follows:

ζt = e− t0 (ηy+s+r)ds, Ht = ζtZt , bt =

∫ T

tws

ζs

ζuds,

Y νt = νHt/Dt .

The following theorem can be obtained using almost the samemethod as used in Section 3.

Theorem 3. The value function V (t, Xt) is given by

V (t, Xt) = α1γp1

γp

1 − γp

Y ν∗

t

γp−1γp∫ T

te−

st

Kp+

λy+u−η(u)γp +η(u)

duds

+ α1γc2

γc

1 − γc

Y ν∗

t

γc−1γc∫ T

t

Kc + λy+s1 − e−Kc (T−s)

Kc

× e− st

Kc+


duds

+ Y ν∗

t

Xt + bt − Rp

∫ T

te−

st (η(u)+r)duds

− Rc

∫ T

t

r + λy+s1 − e−r(T−s)

r

e− st (η(u)+r)duds

,

where Y ν∗

t satisfies the following equation:

Xt = α1γp1

Y ν∗

t

−1γp∫ T

te−

st

Kp+


duds

+ α1γc2

Y ν∗

t

−1γc∫ T

t


Kc

× e− st

Kc+


duds + Rp

∫ T

te−

st (η(u)+r)duds

+ Rc

∫ T

t


r

e− st (η(u)+r)duds − bt . (D.1)

And, for 0 ≤ t < τm, the optimal policies are given by

c∗

p (t) = α1γp1

Y ν∗

t

−1γp

+ Rp, c∗

c (t) = α1γc2

Y ν∗

t

−1γc

+ Rc,

π∗(t) =θ

σ

γc − γp

γpγcα

1γp1

Y ν∗

t

−1γp∫ T

te−

st

Kp+


duds

+θ

σγc

Xt + bt − Rp

∫ T

te−

st (η(u)+r)duds

− Rc

∫ T

t


r

e− st (η(u)+r)duds

,

andI∗(t)η(t)

= M∗(t) − Xt

= α1γc2 g(t)

Y ν∗

t

−1γc

+ Rc

1 − e−r(T−t)

r

− Xt . (D.2)

There are many studies on the relation between wealth and lifeinsurance. Fortune (1973) concludes that the amount of nonhuman

wealth has a negative relation with the optimal amount of net lifeinsurance. However, several studies including Hau (2000) founda positive relation between the net worth and the life insurancedemand. In Proposition 1 we have shown that there is a negativerelation between the current wealth and the optimal life insurancepremium rate if the premium loading is zero. But if we considerthe case with positive premium loading, the relation between thecurrent wealth level and the optimal life insurance premium ratecan be positive or negative depending on the other parameters. Thefollowing proposition illustrates certain conditions for the positiverelation between the current wealth level and the optimal lifeinsurance premium rate.

Proposition 6. If α1 = 0, γc > 1, and η(u) > λy+u, the optimal lifeinsurance premium rate I∗(t) increases as the current wealth level Xtincreases.

Proof. By differentiating both sides of (D.1) with respect to Xt , wehave the following inequality

∂ Y ν∗

t

∂Xt

−1

= −1γp

α1γp1

Y ν∗

t

−1γp −1

∫ T

te−

st

Kp+


duds

−1γc

α1γc2

Y ν∗

t

−1γc −1

∫ T

t


Kc

× e− st

Kc+


duds

< 0. (D.3)

Also by differentiating both sides of (D.2) with respect to Xt , wehave

∂(I∗(t)/η(t))∂Xt

=1γc

α1γc2

Y ν∗

t

−1γc −1 ∂ Y ν∗

t

∂Xt

×

∫ T

t


Kc

× e− st

Kc+


duds − g(t)

,

if α1 = 0. (D.4)

Note thatKc +

λy+u − η(u)γp

+ η(u)

−Kc + λy+u

= (η(u) − λy+u)

1 −

1γc

> 0, if γc > 1 and η(u) > λy+u.

Therefore, we have∫ T

t


Kc

e− st

Kc+


duds

<

∫ T

t


Kc

e− st (Kc+λy+u)duds

= g(t). (D.5)

From (D.3)–(D.5), we conclude that

∂(I∗(t)/η(t))∂Xt

> 0, if α1 = 0, γc > 1, and η(u) > λy+u.


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