Optimal investment and consumption decision of a family with life insurance

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  • biReceived 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

    of 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:minsuk.kwak@gmail.com (M. Kwak), yhshin@hnu.kr

    (Y.H. Shin), ujinchoi@kaist.ac.kr (U.J. Choi).

    familys 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 individuals 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.Insurance: Mathematics and E

    Contents lists availa

    Insurance: Mathema

    journal homepage: www

    Optimal investment and consumption deMinsuk Kwak a,, Yong Hyun Shin b, U Jin Choi aa Department of Mathematical Sciences, Korea Advanced Institute of Science and Technolob 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 2009

    a b s t r a c t

    We study an optimal portfolfor parents who receive dete0167-6687/$ see front matter 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2010.10.012conomics 48 (2011) 176188

    le at ScienceDirect

    tics and Economics

    .elsevier.com/locate/ime

    cision of a family with life insurance

    gy (KAIST), Daejeon, 305701, Republic of Korea

    o and consumption choice problem of a family that combines life insurancerministic labor income until the fixed time T . We consider utility functions

  • iM. Kwak et al. / Insurance: Mathemat

    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 familys 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 = S1t dt + S1t dWt ,respectively, where r, and are constants, Wt is a standardBrownian motion on a complete probability space (,F , P), and{Ft}Tt=0 is the P-augmentation of the natural filtration generatedbyWt .1

    Let cp(t) be the consumption rate of parents at time t, cc(t)be the consumption rate of children at time t and (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, T0

    cp(t)dt

  • i178 M. Kwak et al. / Insurance: Mathemat

    constraint:

    Et

    Tt

    Hscp(s)ds+ Tt

    Hscc(s)ds+ Tty+sHsM(s)ds+ HTXT

    Ht(Xt + bt), for 0 t < m. (2.5)

    (See Ye (2006, 2007).)

    3. The optimization problem

    Now we describe our main problem. The familys expectedutility function U(t, Xt; cp, cc, , I) with an initial endowment Xtat time t, t < m, is given by

    U(t, Xt; cp, cc, , I)

    = Et1

    mt

    e(st)up(cp(s))ds+ 2 Tt

    e(st)uc(cc(s))ds

    = Et m

    te(st)

    1up(cp(s))+ 2uc(cc(s))

    ds

    +21{ 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)1p

    1 p , c > Rp,

    limzRp

    (z Rp)1p1 p , c = Rp,, c < Rp,

    and

    uc(c) ,

    (c Rc)1c

    1 c , c > Rc,

    limzRc

    (z Rc)1c1 c , c = Rc,, c < Rc,

    where p > 0 (p = 1) and c > 0 (c = 1) are the parentsand childrens coefficients of relative risk aversion, respectively,and Rp 0 and Rc 0 represent constant consumption floors ofparents and children, respectively. 1 0 and 2 0 are constantweights of utility functions of parents and children, respectively,which satisfy

    1 + 2 = 1.Assumption 1. We define the Merton constant Ki, i = p, c , andassume that it is always positive, that is,

    Ki , r + ri

    + i 12 2i

    2 > 0, i = p, c.

    If parents die before T , that is, < T , then wt = 0 for m t T . Since I(t) is life insurance premium rate on parents,I(t) = 0 for m t T . Therefore, after parents death time ,children have only two control variables: their consumption cc(t),and investment (t). Then the familys expected utility functionUc(t, Xt; cc, )with an initial endowmentXt at time t, m t T ,is given by

    Uc(t, Xt; cc, ) = Et[ T

    te(st)uc(cc(s))ds

    ]. (3.2)For m t T , let Ac(t, Xt) be the admissible class of the pair(cc, ) at time t for which the familys expected utility functioncs and Economics 48 (2011) 176188

    (3.2) is well-defined, that is,

    Et

    [ Tt

    e(st)uc(cc(s))ds]

  • iM. Kwak et al. / Insurance: Mathemat

    and similarly, the second term except 2 of right-hand side of Eq.(3.3) is given by

    Et

    [ mt

    esuc(cc(s))ds]= Et

    [ Tt

    e(s+ st y+udu)uc(cc(s))ds

    ].

    So we have

    J(t, Xt) = sup(cp,cc ,,I)A(t,Xt )

    Et

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