Optimal investment and consumption decision of a family with life insurance

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biReceived in revised formSeptember 2010Accepted 25 October 2010IME Subject and Insurance Branch Category:IE13IB11JEL classification:D91E21G11G22Keywords:Life insuranceOptimal investment/consumptionLabor incomeUtility maximizationMartingale methodof 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. IntroductionWe 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 EContents lists availaInsurance: Mathemajournal homepage: wwwOptimal 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 Koreaa r t i c l e i n f oArticle history:Received September 2009a b s t r a c tWe 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) 176188le at ScienceDirecttics and Economics.elsevier.com/locate/imecision of a family with life insurancegy (KAIST), Daejeon, 305701, Republic of Koreao and consumption choice problem of a family that combines life insurancerministic labor income until the fixed time T . We consider utility functionsiM. Kwak et al. / Insurance: MathematRecently, 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 economyWe 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 .1Let 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, T0cp(t)dt i178 M. Kwak et al. / Insurance: Mathematconstraint:Et TtHscp(s)ds+ TtHscc(s)ds+ Tty+sHsM(s)ds+ HTXT Ht(Xt + bt), for 0 t < m. (2.5)(See Ye (2006, 2007).)3. The optimization problemNow 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 byU(t, Xt; cp, cc, , I)= Et1 mte(st)up(cp(s))ds+ 2 Tte(st)uc(cc(s))ds= Et mte(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 byup(c) ,(c Rp)1p1 p , c > Rp,limzRp(z Rp)1p1 p , c = Rp,, c < Rp,anduc(c) ,(c Rc)1c1 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 satisfy1 + 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 2i2 > 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 byUc(t, Xt; cc, ) = Et[ Tte(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[ Tte(st)uc(cc(s))ds]iM. Kwak et al. / Insurance: Mathematand similarly, the second term except 2 of right-hand side of Eq.(3.3) is given byEt[ mtesuc(cc(s))ds]= Et[ Tte(s+ st y+udu)uc(cc(s))ds].So we haveJ(t, Xt) = sup(cp,cc ,,I)A(t,Xt )Et1 Tte(s+ st y+udu)up(cp(s))ds+2 Tte(s+ st y+udu)uc(cc(s))ds+2 Tty+se st y+udu(s,M(s))ds.For a Lagrange multiplier > 0, let us define a dual valuefunctionJ(, t), sup(cp,cc ,,I)A(t,Xt )Et1 Tte(s+ st y+udu)up(cp(s))ds+2 Tte(s+ st y+udu)uc(cc(s))ds+2 Tty+se st y+udu(s,M(s))ds e t0 y+uduEt TtHscp(s)ds+ TtHscc(s)ds+ Tty+sHsM(s)ds+ HTXT= e t0 y+uduEt TtDsu1(Y s )ds+ TtDsu2(Y s )ds+ Tty+sDsu3(s, Y s )ds, (3.4)whereDt , exp t0(y+u + )du,Y t , Ht/Dt = exp r 122t Wt,u1(y) = supcp>01(cp Rp)1p1 p ycp= 1p1p1 p yp1p Rpy,u2(y) = supcc>02(cc Rc)1c1 c ycc= 1c2c1 c yc1c Rcy,andu3(t, y) = supM>02g(t)c1 cM Rcr1 er(Tt)1c My= 1c2 g(t)c1 c yc1c Rcr1 er(Tt) y.cs and Economics 48 (2011) 176188 179Consequently we derive the optimal policies5cp (t) = 1p1Y t 1p + Rp, (3.5)cc (t) = 1c2Y t 1c + Rc, (3.6)M(t) = 1c2 g(t)Y t 1c + Rcr1 er(Tt) , (3.7)andXT = 0. (3.8)Now we can derive the value function V (t, Xt) fromJ(, t) byLegendre transform inverse formulaV (t, Xt) = et infYt >0J(, t)+ e t0 y+uduHt(Xt + bt)= infYt >0etJ(, t)+ Y 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 byV (t, Xt) = 1p1p1 pY t p1p Tte st (y+u+Kp)duds+1c2c1 cY t c1c g(t)+ Y tXt + bt Rp Tte st (y+u+r)duds Rcr1 er(Tt),where Y t satisfies the following equation:Xt = 1p1Y t 1p Tte st (y+u+Kp)duds+1c2Y t 1c g(t)+ Rp Tte st (y+u+r)duds+ Rcr1 er(Tt) bt . (3.10)And, for 0 t < m, the optimal policies are given bycp (t) = 1p1Y t 1p + Rp, cc (t) = 1c2 Y t 1c + Rc,(t) = cXt + bt Rp Tte st (y+u+r)duds Rcr1 er(Tt)+ c ppc1p1Y t 1p Tte st (y+u+Kp)duds, (3.11)5 We can always obtain the optimal strategies (cp (t), cc (t), (t), I(t)) A(t, Xt ), which satisfy the above Eqs. (3.5)(3.8). See Chapter 3 of Karatzas andShreve (1998).i180 M. Kwak et al. / Insurance: MathematandI(t)y+t= M(t) Xt= bt Rp Tte st (y+u+r)duds 1p1Y t 1p Tte st (y+u+Kp)duds. (3.12)Proof. See Appendix B. 4. Properties of optimal solutions and numerical examplesIn 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 1 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 thatwt = Cwekw t ,y+t = AL + BLt,for some positive constants Cw, kw, AL and BL.6As 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 25) still hold even though the premium loading ispositive.7Proposition 1. If 1 (0, 1], the optimal life insurance premiumrate I(t) decreases as the wealth level Xt increases. If 1 = 0, I(t)is not affected by Xt .Proof. See Appendix C. Proposition 2. If 1 [0, 1), the optimal life insurance premiumrate I(t) increases as the fair discounted value of future labor incomebt increases. If 1 = 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 1 = 0, the optimalconsumption rate of parents cp (t) increases as Xt or bt increases.When 1 = 0, we only consider childrens utility, and thereforecp (t) is equal to the minimum level Rp regardless of Xt and bt . Theoptimal consumption rate of children cc (t) decreases as Xt or btincreases, unless 1 = 1. If 1 = 1, we do not care about childrensutility, and therefore cc (t) is equal to the minimum level Rc . Theoptimal investment (t) increases as bt or Xt increases for any1 [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 valueof future labor income bt .cs and Economics 48 (2011) 176188On 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 1 (0, 1]. If 1 = 0,I(t)y+t1=0= bt Rp Tte st (y+u+r)duds bt Rp Tte st (y+u+r)duds 1p1Y t 1p Tte st (y+u+Kp)duds,and therefore I(t)|1=0 is not affected by Xt and greater than I(t)of other cases with 1 (0, 1].As we have stated in Proposition 2, I(t) increases as btincreases if 1 [0, 1). For the case of 1 = 1,I(t)y+t1=1= bt Rp Tte st (y+u+r)dudsY t 1p Tte st (y+u+Kp)duds bt Rp Tte st (y+u+r)duds1p1Y t 1p Tte st (y+u+Kp)duds,and therefore I(t)|1=1 is not affected by bt and less than I(t) ofother cases with 1 [0, 1). Note thatM(t)1=1 = Xt +I(t)y+t1=1= Rcr1 er(Tt) = Tter(st)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 1 increasesfrom 0 to 1.Proof. See Appendix C. Proposition 4. The optimal investment (t) decreases as 1increases from 0 to 1 if p > c , and increases as 1 increasesfrom 0 to 1 if p < c . If p = c, (t) is not affected by 1.Proof. See Appendix C. Figs. 25 illustrate the relations between the weight of parentsutility function 1 and the optimal policies for different riskFig. 1. Relations between the optimal policies and Xt + bt . (t = 0, Rp = 2, Rc = 1, = 0.04, r = 0.04, = 0.06, = 0.3, AL = 0.005, BL = 0.001125, T = 30).M. Kwak et al. / Insurance: Mathematics and Economics 48 (2011) 176188 181Fig. 2. Relations between 1 and the optimal policies when p = 2, c = 2. (t = 0, X0 = 10, Rp = 2, Rc = 1, = 0.04, r = 0.04, = 0.06, = 0.3, Cw = 5.0,kw = 0.03, AL = 0.005, BL = 0.001125, T = 30).Fig. 3. Relations between 1 and the optimal policies when p = 2, c = 3. (t = 0, X0 = 10, Rp = 2, Rc = 1, = 0.04, r = 0.04, = 0.06, = 0.3, Cw = 5.0, kw =0.03, AL = 0.005, BL = 0.001125, T = 30).182 M. Kwak et al. / Insurance: Mathematics and Economics 48 (2011) 176188Fig. 4. Relations between 1 and the optimal policies when p = 3, c = 2. (t = 0, X0 = 10, Rp = 2, Rc = 1, = 0.04, r = 0.04, = 0.06, = 0.3, Cw = 5.0, kw =0.03, AL = 0.005, BL = 0.001125, T = 30).iaversion coefficients of parents and children. It is obvious thatthe optimal consumption of parents cp (t) increases and theoptimal consumption of children cc (t) decreases as 1 increases.As we have mentioned in Proposition 3, the optimal life insurancepremium rate I(t) decreases as 1 increases. Bequest motive isweaker as 1 is larger. Unlike other optimal policies, the effectof 1 on the optimal investment (t) depends on p and c ,risk aversion coefficients of parents and children, respectively. Ifp > c, (t) decreases as 1 increases, and (t) increases as1 increases when p > c . In other words, if the risk aversioncoefficients of parents and children are different, the optimalinvestment increases when the relative importance of less riskaverse family members utility increases. If p = c, (t) is notaffected by 1.Proposition 5. If 1 [0, 1), the optimal life insurance premiumrate I(t) decreases as the consumption floor of parents Rp increases. If1 = 1, I(t) is not affected by Rp. On the other hand, I(t) increasesas the consumption floor of children Rc increases if 1 (0, 1]. If1 = 0, I(t) is not affected by Rc . The optimal investment (t)decreases as Rp or Rc increases, regardless of 1.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)/y+t . From (3.10), it is easy to verify thatY t /Rp > 0 and this shows that, if 1 [0, 1), M(t)/Rp < 0from (3.7). Therefore, the optimal life insurance premium rate I(t)decreases as Rp increases if 1 [0, 1). However, if 1 = 1,which is the wealth needed at time t to maintain childrensminimum consumption Rc from t to T . Therefore, if 1 = 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 t /Rc > 0. Thisimplies that, if 1 (0, 1], (I(t)/y+t)/Rc > 0 from (3.12).Therefore, I(t) increases as Rc increases if 1 (0, 1]. If 1 = 0,I(t)y+t1=0= bt Rp Tte st (y+u+r)duds= bt E[ mter(st)Rpds]which is the fair discounted value of the parents future laborincome less the expected wealth needed to maintain parentsminimum consumption Rp from t to m. Consequently, if 1 =0, (I(t)/y+t)/Rc = 0 and this means that I(t) is not affectedby Rc .5. Concluding remarksWe 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 of1 which is the weight of the parents utility function.AcknowledgementsAn earlier version of this paper was presented at the 3rdKorean Mathematical Society Probability Workshop, KangwonNational University, Chuncheon, Korea, June 1213, 2009, theM. Kwak et al. / Insurance: MathematFig. 5. Relations between 1 and the optimal policies when p = 1/2, c = 1/2. (t =0.03, AL = 0.005, BL = 0.001125, T = 30).M(t)1=1 =Rcr1 er(Tt) = Tter(st)Rcdscs and Economics 48 (2011) 176188 1830, X0 = 10, Rp = 2, Rc = 1, = 0.04, r = 0.04, = 0.06, = 0.3, Cw = 5.0, kw =Society for Computational Economics 15th International Con-ference on Computing in Economics and Finance, University ofi184 M. Kwak et al. / Insurance: MathematTechnology, Sydney, Australia, July 1517, 2009, the 41st ISCIEInternational Symposium on Stochastic Systems Theory and ItsApplications, Konan University, Kobe, Japan, November 1314,2009, the Workshop and Spring School on Stochastic Calculus andApplications, Institute of Mathematics, Academia Sinica, Taipei,Taiwan, April 917, 2010, the 6th World Congress of the Bache-lier Finance Society, Toronto, Canada, June 2226, 2010, and theInternational Workshop on Recent Trends in Learning, Computa-tion, and Finance, POSCO International Center, POSTECH, Pohang,Korea, August 3031, 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 authors 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 authors work was supported by NRF of KoreaGrant funded by the Korean Government (MEST) (Grant No.2009-0072021).The third authors work was supported by BK21 project ofDepartment of Mathematical Sciences, KAIST.Appendix A. Proof of Lemma 1For t m, the value function of the family at time t isVc(t, Xt) , sup(cc ,)Ac (t,Xt )Et[ Tte(st)uc(cc(s))ds],subject to the wealth process (cf. the wealth process (2.2))dXt = [rXt + ( r)(t) cc(t)] dt + (t)dWt . (A.1)Let us define the discount process, and the pricing kernel process,respectively, byt , ert , Gt , tZt .From (A.1), we can derive the budget constraintEt[GTXT + TtGscc(s)ds] GtXt .For a Lagrange multiplier > 0, let us define a dual value functionVc(, t) , sup{cc ,}Et Ttesuc(cc(s))ds EtGTXT + TtGscc(s)ds= sup{cc ,}Et Ttes(cc(s) Rc)1c1 c Y s cc(s)ds GTXT= etc1 cY t c1c g(t) RcY t1 er(Tt)r,whereY t , etGtandg(t) = 1 eKc (Tt)Kc.Consequently we derive the optimal policiescc (s) = (Y s )1c + Rc and XT = 0.cs and Economics 48 (2011) 176188It is easily verified that the value function is given byVc(t, Xt) = et infYt >0Vc(, t)+ GtXt= infYt >0c1 cY t c1c g(t) RcY t1 er(Tt)r+ Y t Xt= g(t)c1 cXt Rcr1 er(Tt)1c ,whereY t =Xt Rcr1 er(Tt)c g(t)c . (A.2)So the optimal consumption cc (t) is given bycc (t) =Y t 1c + Rc = 1g(t)Xt Rcr1 er(Tt)+ Rc .Note that Eq. (A.2) is equivalent toXt = g(t)Y t 1c + Rcr1 er(Tt) .Its formula to the above optimal wealth Xt impliesdXt =g (t)Y t 1c Rcer(Tt) dt 1cg(t)Y t 1c 1 dY t+ 1+ c2 2cg(t)Y t 1c 2 dY t 2=Y t 1c rg(t) 1+ 2cg(t) Rcer(Tt)dt+ cg(t)Y t 1c dWt=rXt cc (t)+ ( r)cg(t)Y t 1c dt+ cg(t)Y t 1c dWt .Therefore, a comparison of the drift term and the volatility termof the optimal wealth process dXt with Eq. (A.1) gives the optimalportfolio (t) as follows:(t) = cg(t)Y t 1c = cXt Rcr1 er(Tt) .Appendix B. Proof of Theorem 2From (3.4), we derive the following equation,etJ(, t) = 1p1 p1 p Y t p1p Tte st (y+u+Kp)duds+1c2c1 cY t c1c g(t) Y tRp Tte st (y+u+r)duds+ Rcr1 er(Tt) .Using the first order condition for Eq. (3.9) with respect to Y t ,then we obtain the optimal wealth Xt in (3.10), where indicatesthe optimal Lagrange multiplier. Thus we can derive the valueiM. Kwak et al. / Insurance: Mathematfunction V (t, Xt) using proper substitution. Using Its 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 (t) as follows:(t) = cXt + bt Rp Tte st (y+u+r)duds Rcr1 er(Tt)+ c ppc1p1Y t 1p Tte st (y+u+Kp)duds.Appendix C. Proofs of Propositions 15Proof of Proposition 1. By differentiating both sides of (3.10)with respect to Xt , we have the following inequalityY tXt1= 1p1p1Y t 1p 1 Tte st (y+u+Kp)duds 1c1c2Y t 1c 1 g(t)< 0.Also by differentiating both sides of (3.12) with respect to Xt , wehave(I(t)/y+t)Xt=1p1p1Y t 1p 1 Tte st (y+u+Kp)dudsY tXt.Therefore, we have(I(t)/y+t)Xt= 0, if 1 = 0,(I(t)/y+t)Xt< 0, if 1 (0, 1]. Proof of Proposition 2. By differentiating both sides of (3.10)with respect to bt , we have the following inequalityY tbt1= 1p1p1Y t 1p 1 Tte st (y+u+Kp)duds 1c1c2Y t 1c 1 g(t) < 0.Also by differentiating both sides of (3.12) with respect to bt , wehave(I(t)/y+t)bt= M(t)bt= 1c(I(t)/y+t)btg(t)(Y t ) 1c 1 Ytbt.Therefore, we have(I(t)/y+t)bt= 0, if 1 = 1,(I (t)/y+t)bt> 0, if 1 [0, 1). cs and Economics 48 (2011) 176188 185Proof of Proposition 3. From (3.7) and (3.12), we know thatI(t)/y+t = M(t) Xt= 1c2 g(t)Y t 1c + Rcr1 er(Tt) Xt , (C.1)and I(t)/y+t1= 1c1c 12 g(t)Y t 1c 1 2Y t1+ Y t.By differentiating both sides of Eq. (3.10) with respect to 1, weobtainY t1=1p1p 11Y t 1p Tte ut (y+u+Kp)duds 1c(1 1) 1c 1Y t 1c g(t)1p1p1Y t 1p 1 Tte ut (y+u+Kp)duds+ 1c(1 1) 1cY t 1c 1 g(t). (C.2)By Eq. (C.2), we have the following inequality2Y t1+ Y t=1p1p 11Y t 1p Tte ut (y+u+Kp)duds1p1p1Y t 1p 1 Tte ut (y+u+Kp)duds+ 1c1c2Y t 1c 1 g(t) > 0,if 1 (0, 1).Therefore, we haveFor 1 (0, 1), I(t)/y+t1< 0. Proof of Proposition 4. From the optimal investment (3.11) wehave(t)1= c ppc1p1p 11Y t 1p 1p1p1Y t 1p 1 Y t1 Tte st (y+u+Kp)duds= c p 2p c1p 11Y t 1p 1Y t 1Y t1Tte st (y+u+Kp)duds.iue.7)ndbyngbyies) ).(t)1= 0, if p = c . 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)/y+tRp= M(t)Rp 1 By differentiating both sides of (3.11) with respect to Rp ausing (C.4), we have the expression given in Box II. Similarly,differentiating both sides of (3.11) with respect to Rc and usi(C.5), we obtain the expression in Box III. Appendix D. The case with positive premium loadingThe premium loading can be incorporated into our modelintroducing a continuous function : [0, T ] R+ which satisf(t) y+t and replacing the insurance benefit I( )y+ with I(( Tte ut (y+u+Kp)duds+ 1c1c2Y t 1c 1 g(t)> 0, if 1 (0, 1). (C.3)Since inequality (C.3) holds, we conclude thatFor 1 (0, 1), (t)1< 0, if p > c,(t)1> 0, if c > p,From (C.4), (C.5), (C.6), and (C.7), we conclude that I(t)/y+tRp< 0, I(t)/y+tRc= 0, if 1 = 0, I(t)/y+tRp< 0, I(t)/y+tRc> 0, if 1 (0, 1), I(t)/y+tRp= 0, I(t)/y+tRc> 0, if 1 = 1.< 0.Box III.From Eq. (C.2), we haveY t 1Y t1=1c1c 12Y t 1c g(t) 1p1p1Y t 1p 1By differentiating both sides of (3.12) with respect to Rc , we hav I(t)/y+tRc= 1p1p1Y t 1p 1 Tte st (y+u+Kp)dudsY tRc. (C186 M. Kwak et al. / Insurance: MathematY tRp= Tt e st (y+u+r)duds1p1p1Y t 1p 1 Tt e st (y+u+Kp)duds+ 1c1c2Y tY tRc=1r1 er(Tt)1p1p1Y t 1p 1 Tt e st (y+u+Kp)duds+ 1c1c2Y tBox I.(t)Rp= c Tte st (y+u+r)duds pc ppc1p1Y t= Tt e st (y+u+r)duds 2c 1c2 Y t 1c 1 g(t)+1p1p1Y t 1p 1 Tt e st (y+u+Kp)d< 0.Box II.(t)Rc= c Tte st rduds pc ppc1p1Y t 1p= Tt e st rduds 2c 1c2 Y t 1c 1 g(t)+ 2p 11p1p1Y t 1p 1 Tt e st (y+u+Kp)duds= 1c1c2 Yt c 1 g(t)YtRp. (C.6)cs and Economics 48 (2011) 176188 1c 1 g(t) > 0, (C.4) 1c 1 g(t) > 0. (C.5) 1p 1 Tte st (y+u+Kp)dudsY tRp 2p1p1Y t 1p 1 Tt e st (y+u+Kp)dudsds+ 1c1c2Y t 1c 1 g(t)1 Tte st (y+u+Kp)dudsY tRc1p Y t 1p 1 Tt e st (y+u+Kp)duds+ 1c1c2Y t 1c 1 g(t)Thenwehave to redefine the discount process t , the pricing kernelHt , discounted value of parents future labor income bt , and Y t , asiM. Kwak et al. / Insurance: Mathematfollows:t = e t0 (y+s+r)ds, Ht = tZt , bt = Ttwssuds,Y t = Ht/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 byV (t, Xt) = 1p1p1 pY t p1p Tte stKp+ y+u(u)p +(u)duds+1c2c1 cY t c1c TtKc + y+s1 eKc (Ts)Kc e stKc+ y+u(u)p +(u)duds+ Y tXt + bt Rp Tte st ((u)+r)duds Rc Ttr + y+s1 er(Ts)re st ((u)+r)duds,where Y t satisfies the following equation:Xt = 1p1Y t 1p Tte stKp+ y+u(u)p +(u)duds+1c2Y t 1c TtKc + y+s1 eKc (Ts)Kc e stKc+ y+u(u)p +(u)duds+ Rp Tte st ((u)+r)duds+ Rc Ttr + y+s1 er(Ts)re st ((u)+r)duds bt . (D.1)And, for 0 t < m, the optimal policies are given bycp (t) = 1p1Y t 1p + Rp, cc (t) = 1c2 Y t 1c + Rc,(t) = c ppc1p1Y t 1p Tte stKp+ y+u(u)p +(u)duds+ cXt + bt Rp Tte st ((u)+r)duds Rc Ttr + y+s1 er(Ts)re st ((u)+r)duds,andI(t)(t)= M(t) Xt= 1c2 g(t)Y t 1c + Rc 1 er(Tt)r Xt . (D.2)There are many studies on the relation between wealth and lifeinsurance. Fortune (1973) concludes that the amount of nonhumancs and Economics 48 (2011) 176188 187wealth 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 1 = 0, c > 1, and (u) > y+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 tXt1= 1p1p1Y t 1p 1 Tte stKp+ y+u(u)p +(u)duds 1c1c2Y t 1c 1 TtKc + y+s1 eKc (Ts)Kc e stKc+ y+u(u)p +(u)duds< 0. (D.3)Also by differentiating both sides of (D.2) with respect to Xt , wehave(I(t)/(t))Xt= 1c1c2Y t 1c 1 Y tXt TtKc + y+s1 eKc (Ts)Kc e stKc+ y+u(u)p +(u)duds g(t),if 1 = 0. 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Optimal life insurance purchase, consumption, and portfoliounder an uncertain life. Ph.D. Dissertation. University of Illinois at Chicago,Chicago.Ye, J., 2007. Optimal life insurance purchase, consumption and portfolio underuncertainty: martingale methods. In: Proceedings of 2007 American ControlConference. pp. 11031109.Optimal investment and consumption decision of a family with life insuranceIntroductionThe economyThe optimization problemProperties of optimal solutions and numerical examplesConcluding remarksAcknowledgementsProof of Lemma 1Proof of Theorem 2Proofs of Propositions 1--5The case with positive premium loadingReferences

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