Optimal Life Insurance Purchase, Consumption and Investment

Transcription

1 Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Absrac Meron s famous coninuous-ime model of opimal consumpion and invesmen is exended o allow for a wage earner o have a random lifeime and o use a porion of he income o purchase life insurance in order o provide for his or her esae. Wih consan relaive risk aversion uiliy funcions, he wage earner s problem is o find he opimal consumpion, invesmen, and insurance purchase decisions in order o maximize expeced uiliy of consumpion, of he size of he esae in he even of premaure deah, and of he size of he esae a he ime of reiremen. Dynamic programming mehods are used o obain explici soluions. Numerical resuls are presened, from which are drawn economic implicaions and undersanding. Key words: Life Insurance, Invesmen/Consumpion Model, HJB Equaion, CRRA Uiliies 1 Inroducion In his paper we consider he opimal life insurance purchase, consumpion and porfolio managemen sraegies for a wage earner subjec o moraliy risk in a coninuous ime economy. Decisions are made coninuously abou hese hree sraegies for all ime [, min(τ, T )], where he fixed planning We are graeful for heir helpful commens o David Wilkie and oher paricipans a he Phelim Boyle Feskolloquium a he Universiy of Waerloo. Corresponding auhor. Tel.: ; fax: addresses: jinch ye(*)yahoo.com (Jinchun Ye), srpliska(*)uic.edu (Sanley R. Pliska). Preprin submied o Insurance:Mahemaics and Economics

2 horizon T can be inerpreed as he planned reiremen ime of he wage earner and where τ is he random ime when he or she dies. The wage earner receives his/her income a rae i() coninuously, bu his is erminaed by he wage earner s deah or reiremen, whichever happens firs. The wage earner needs o buy life insurance o proec his/her family due o his/her premaure deah. Needless o say, he bigger he insurance premium paid by he wage earner, he bigger he claim paid o his/her family upon premaure deah. As will be seen, he insurance policy is like erm insurance, bu wih an arbirarily small erm because he wage earner buys insurance a a coninuous rae. Aside from consumpion and life insurance purchase, he wage earner also has he opporuniy o inves in he capial marke which consiss of a riskless securiy and a risky securiy. Alhough we only consider one risky securiy in his paper, i is easy and no echnically difficul o include muliple risky securiies in our model. The problem is o find he sraegies ha are bes in erms of boh he family s consumpion for all min(τ, T ) as well as he size of he esae a ime min(τ, T ). Roughly speaking, mos of he research leading up o our model ended o ignore one or more imporan aspecs of he problem we consider. As we review he exising lieraure we will ry o poin ou shorcomings of exising models and he conribuions of ours. In one line of research he emphasis was on he demand for life insurance for an individual wih an uncerain lifeime, a he expense of opimal invesmen decisions. Yarri (1965) migh have been he firs o consider opimal financial planning decisions for an individual wih an uncerain lifeime. His objecive, subjec o a nonnegaive wealh consrain, was o maximize τ E U(c())d, where τ, he individual s lifeime, is a bounded random variable ha was assumed o ake values in [, T ] for some fixed, posiive number T, where c() is he consumpion rae a ime, and where U is a uiliy funcion. Yarri considered four cases defined by he availbiliy of a life insurance marke and by he choice of he uiliy funcion. However, for all four cases, Yarri was unable o obain any closed-form soluion. Considerable lieraure (Hakansson(1969), Fischer (1973) and Richard (1975), ec) has been buil on Yaari s pioneering work. In paricular, Leung (1994) 2

3 poined ou ha Yaari s model canno have an inerior soluion which lass unil he maximum lifeime for he opimal consumpion. In a discree-ime environmen, Hakansson (1969) examined life-cycle paerns of consumpion and saving when he individual s lifeime is a discree random variable aking values in [, T ] and having a known probabiliy disribuion. The individual s objecive is o maximize he expeced uiliy from consumpion as long as he lives and from he beques lef upon his deah. Hakansson also briefly considered he opimal insurance purchase. Wihin a similar, discree-ime framework, Fischer (1973) examined life-cycle paerns of opimal insurance purchase in deail using he dynamic programming echnique and obained he formula for he presen value of he fuure income, a formula ha is differen from he one under a cerain lifeime. Noe he maximal possible lifeime for he individual is T periods, so if he individual is alive a he beginning of he period T he will definiely be dead a he end of his period, and hen he individual is inclined o buy infinie life insurance for he las period. In order o address his difficuly, Fischer imposed he arificial condiion ha he individual canno buy insurance in he las period of his life. Bu dynamic programming for he discree model works backward, so his arificial condiion changes he whole srucure of soluions. In a second line of relevan research he emphasis was on opimal consumpion and invesmen decisions while ending o ignore he possibiliy of he individual s premaure deah as well as he opporuniy o purchase life insurance. Noeworhy in his caegory is he pioneering work by Meron (1969, 1971). In his model mos relevan o ours he individual sars wih a specified endowmen and has a specified income, he planning horizon T is fixed, and he individual seeks he consumpion and invesmen sraegies ha maximize he expecaion of he uiliy of consumpion plus he uiliy of erminal wealh. He used dynamic programming mehods and obained, in he case of consan relaive risk aversion uiliy funcions, explici soluions. These wo lines of research converged when Richard (1975) combined Yaari s seing for an uncerain lifeime and Meron s dynamic programming approach o consider a life-cycle life insurance and consumpion/invesmen problem wih a budge consrain and objecive τ max E U(c(), )d + B(Z(τ), τ), where τ is a bounded random lifeime aking values on [, T ] and having a known probabiliy disribuion, U is he uiliy for consumpion, and B is he uiliy for he beques Z(τ). He also inroduced he concep of a coninuousime life insurance marke where he wage earner coninuously buys erm life insurance, he erm being infiniesimally shor. Richard employed he dynamic 3

4 programming echnique o aack his problem. To do ha, Richard rewroe he objecive funcional in he dynamic form: T x J(W, ) max E π(x, ) U(c(s), s)ds + B(Z(x), x) dx F, (1) where π(x, ) is he condiional probabiliy densiy for deah a ime x condiional upon he invesor being alive a ime and where W is he individual s ime- wealh. Noe ha Equaion (1) is well-defined for < T, bu (1) is undefined for = T since he lengh of he inegral inerval is zero bu π(x, T ) can be infinie. We know dynamic programming works backward (we can clearly see his poin in he discree-ime environmen), so he erminal condiion is very imporan in dynamic programming. The assumpion abou he individual s behavior a = T and hus he dynamic program s boundary condiion criically affecs he resuling soluions. To obain explici soluions for Consan Relaive Risk Aversion (CRRA) uiliy funcions, Richard imposed an arificial condiion: he individual canno purchase life insurance a ime T if he individual is alive a ha ime. Bu his soluion shows ha he individual is inclined o purchase infinie life insurance jus before his maximal possible lifeime T. Tha violaes his imposed condiion and is unrealisic, hereby compromising his resuls. Subsequenly, Campbell (198), Lewis (1989) and Iwaki e al. (24) examined he demand for life insurance from differen perspecives. Campbell (198) considered he insurance problem in a very shor ime inerval (viz., [, + ]), used a local analysis (Taylor expansion) o grealy simplify he problem, and hen derived he insurance policy in erms of he presen value of he fuure income, he curren wealh, and oher parameers. Since he local analysis ignored informaion abou he presen value of he fuure income even if one knew he income sream, Campbell had o assume ha he presen value of he fuure income is given exogenously. Lewis (1989) examined he demand for life insurance from he perspecive of beneficiaries. Iwaki e al. (24) used maringale mehods o consider he opimal insurance problem from he perspecive of a household. Bu households are only allowed o buy life insurance a ime in heir model, and hey subsequenly canno change he amoun of life insurance regardless of wha happens afer ime, and so he exisence of a dynamic marke for life insurance is ignored. Blanche-Scallie, El Karouri, Jeanblanc, and Marellini (23) considered a consumpion/invesmen model wih a sochasic ime-horizon, bu wihou life insurance and exogenous income. For CRRA uiliies, hey obained explici soluions similar o he classical Meron s resuls. Blanche-Scallie, El Karouri and Marellini (25) addressed he problem of pricing and hedging a random cash-flow received a a random dae using arbirage argumen wihou considering insurance. 4

5 In our paper we develop a coninuous-ime model ha combines he bes and mos realisic feaures of all hose in he lieraure. In paricular, our model should be viewed as a naural exension of Richard (1975) since we borrowed his seing of a coninuous-ime life insurance marke. To model he wage earner s lifeime we use he concep of uncerain life found in reliabiliy heory; his approach is commonly used for indusrial life-esing and acuarial science. In his way we overcome a criical shorcoming of Richard s (1975) work, replacing his assumpion ha lifeimes are bounded wih he assumpion ha lifeimes simply ake values in all of (, ). Moreover, we inerpre he planning horizon T as he wage earner s reiremen ime, insead of he maximal lengh of his lifeime, and we inroduce a uiliy for his wealh a his ime. So our model urns ou o be an ineremporal model insead of a life-cycle model. Bu in doing his we also assume ha uncerainy abou he wage earner s lifeime is independen of uncerainy abou he financial marke. In secion 2 we describe our model in deail, showing ha i is an ineremporal model insead of a life-cycle model. There we also inroduce some conceps from he survival analysis and reliabiliy heory lieraures, necessary for our modeling of he wage earner s lifeime. In secion 3 we use he dynamic programming echnique o obain explici soluions for he wage earner s opimal sraegies. There i will be seen ha we do no impose any arificial condiion o obain he dynamic program s erminal boundary condiion; i comes naurally from he invesor s objecive funcional. There i will also be seen ha our soluion differs from Richard s (1975) even hough his model is a special case or ours. In secion 4 we provide some numerical examples of opimal sraegies, hereby providing some economic insighs abou basic relaionships, such as how insurance paymens migh depend upon age and wealh. Many such findings are in accordance wih a priori inuiion, bu some are no obvious and hus are hough provoking. Finally, we conclude in secion 5 wih a brief summary of our main resuls, a comparison of our resuls wih Richard s (1975) and some houghs abou fuure research. We should add ha much of his paper is based upon work by Ye (26). Moreover, Pliska and Ye (27), which is also based upon work by Ye (26), examined economic implicaions using a simplified version of his paper s model, namely, where here is no risky asse and so he only uncerainy is he wage earner s lifeime. Ye (27) summarized a series of fundamenal resuls which were obained using maringale approaches in Ye (26). 5

6 2 The Model Le W () be a sandard 1-dimensional Brownian moion defined on a given probabiliy space (Ω, F, P ). Le T < be a fixed planning horizon, convenienly inerpreed as he wage earner s reiremen ime. Le F = {F, [, T ]} be he P-augmenaion of he filraion σ{w (s), s }, [, T ], so F represens he informaion known o he wage earner a ime. The coninuous-ime economy consiss of a financial marke and an insurance marke. We are going o describe heir deails separaely, beginning wih he financial marke. We assume here is a risk-free securiy in he financial marke whose ime- price is denoed by S (). I evolves according o ds () S () = r()d, (2) where S () = s is a given posiive consan and where r() : [, T ] R + is a given coninuous funcion describing he riskless ineres rae as a funcion of ime. Also here is a risky securiy in he financial marke 1. I evolves according o he linear sochasic differenial equaion ds 1 () S 1 () = µ()d + σ()dw (), (3) where S 1 () = s 1 is a given posiive consan, µ : [, T ] R is coninuous process, and σ : [, T ] R is a coninuous funcion saisfying σ 2 () k, [, T ], for some posiive consan k. Suppose he wage earner is alive a ime = and has a lifeime denoed τ, a non-negaive random variable defined on he probabiliy space (Ω, F, P ). We assume he random variable τ is independen of he filraion F and has a probabiliy disribuion wih underlying probabiliy densiy funcion f() and disribuion funcion given by F () P (τ < ) = f(u)du. (4) 1 Our model can be easily exended o N risky securiies 6

7 Turning o he life insurance marke, we assume ha life insurance is available coninuously and he wage earner can ener a life insurance conrac by paying premiums a he rae p a each poin in ime. In compensaion, if he wage earner dies a ime when he premium paymen rae is p, hen he insurance company pays an insurance amoun p/η(). This is like erm insurance where he erm is infiniesimally small. Here η : [, T ] R + is a coninuous, deerminisic, specified funcion ha is called he insurance premium-payou raio. For example, if he wage earner dies while paying a he rae p = 1 dollars per year, and if η() =.1, hen he insurance company pays he esae $1,. Now suppose he wage earner is endowed wih he iniial wealh x and will receive an income a rae i() during he period [, min{t, τ}], so he income will be erminaed by he wage earner s deah or reiremen a T, whichever happens firs. Here i : [, T ] R + is a given Borel measurable funcion ha saisfies T i(u)du <. The wage earner s decisions can be fully described by hree processes: c() Consumpion rae (e.g., dollars per year) a ime, an {F }- progressively measurable, nonnegaive process p() Premium rae (e.g., dollars per year) a ime, an {F }-predicable process θ() Dollar amoun in he risky securiy a ime, an {F }-progressively measurable process We le A(x) denoe he se of all admissible 3-uples (c, p, θ), ha is, he se of all admissible decision sraegies. Given porfolio process θ, consumpion process c, premium rae process p, and income process i, he wealh process X() on [, min{t, τ}] is defined by X() = x + c(s)ds p(s)ds + X(s) θ(s) ds (s) + S (s) i(s)ds θ(s) S 1 (s) ds 1(s) on min{τ, T }. (5) Using (2) and (3), we wrie (5) as he sochasic differenial equaion 7

8 dx() = r()x()d c()d p()d + i()d +θ()[(µ() r())d + σ()dw ()] on min{τ, T }. (6) If he wage earner dies a ime, < T, he esae will ge he insurance amoun p()/η(). Then he wage earner s oal legacy when he/she dies a ime wih wealh X(), ha is, he oal value of he esae, is Z() = X() + p() η() on {τ = }. (7) The wage earner s problem is o choose life insurance purchase, consumpion, and porfolio invesmen sraegies so as o maximize he expecaion of uiliy from hree sources: from consumpion during [, min{τ, T }], from he legacy if he/she dies before ime T, and from he erminal wealh if he/she is alive a ime T. We assume he wage earner s preferences are based upon discouned Consan Relaive Risk Aversion (CRRA) uiliy funcions. Thus he maximum expeced uiliy is expressed as V (x) sup (c,p,θ) A(x) E [ T τ e ρs γ (c(s))γ ds + e ρτ γ (Z(τ))γ 1 {τ T } + e ρt γ (X(T ))γ 1 {τ>t } ], (8) where T τ min{t, τ}. We assume ha γ < 1 and ρ >. I is well acceped ha he uiliy funcions are logarihmic when γ =. We show how o solve his problem in he following secion. To do so we shall model he wage earner s lifeime by making imporan use of some conceps from he reliabiliy heory and survival analysis lieraures (see, e.g., Colle, 23), so i is appropriae o conclude his secion by inroducing some relevan noaion. The funcion F (), which is called he survivor funcion, is defined o be he probabiliy ha he lifeime is greaer han or equal o, i.e., F () P (τ ) = 1 F (). (9) The hazard funcion represens he insananeous deah rae for he wage earner who has survived o ime, and i is defined by λ() lim δ P ( τ < + δ τ ) δ = f() F (), (1) 8

9 where he las equaliy follows from definiions and basic relaionships. From his i follows ha λ() = d d (ln F ()), in which case he survivor funcion is given by F () = exp λ(u)du (11) and he probabiliy densiy funcion is relaed o he hazard rae by f() = λ() exp λ(u)du. (12) From hese equaions we see here is a correspondence beween hazard funcions and densiy funcions. Hence hroughou he res of his paper we suppose ha he hazard funcion λ() is given and λ() : [, ] R + is a coninuous, deerminisic funcion which saisfies λ()d =. Then he probabiliy densiy of τ is given by (12) and he survivor funcion is given by (11). We now inroduce some addiional noaion associaed wih he random variable τ. Denoe by f(s, ) he condiional probabiliy densiy for deah a ime s condiional upon he wage earner being alive a ime s, so ha f(s, ) f(s) F () s = λ(s) exp λ(u)du. (13) And denoe by F (s, ) he condiional probabiliy for he wage earner being alive a ime s condiional upon being alive a ime s, so ha F (s, ) F (s) F () = exp s λ(u)du. (14) 3 Sochasic Dynamic Programming In his secion we shall use sochasic dynamic programming mehods o derive explici formulas for he wage earner s opimal decision sraegies. In paricular, we shall use he opimaliy principle o se up he Hamilon-Jacobi- Bellman (HJB) equaion, and hen we shall solve his equaion for he opimal sraegies. 9

10 To derive he HJB equaion we need o resae (8) in a dynamic programming form. To begin, for any (c, p, θ) define he corresponding expeced uiliy, saring wih wealh x a ime : J(, x; c, p, θ) E [ T τ e ρs γ (c(s))γ ds + e ρτ γ (Z(τ))γ 1 {τ T } ] + e ρt γ (X(T ))γ 1 {τ>t }. (15) The nex sep is o ransform our problem having a random planning horizon o an equivalen problem having a fixed planning horizon, hereby permiing us o proceed wih he HJB equaion. Since τ is independen of he filraion F, we can achieve he necessary ransformaion (refer o Ye (26)) by using an equivalen form of J(, x; c, p, θ): [ T J(, x; c, p, θ) = E [f(u, ) e ρu γ + F (T, ) e ρt γ (Z(u))γ + F (u, ) e ρu γ (c(u))γ ]du (X(T ))γ F ], (16) where f(u, ) is given by (13) and F (u, ) is given by (14). From his resul we see ha he wage earner who faces unpredicable deah acs as if he or she will live unil ime T, bu wih a subjecive rae of ime preferences equal o his or her force of moraliy for his/her consumpion and erminal wealh. From he mahemaical poin of view, his resul enables us o sae he dynamic programming principle. This is a recursive relaionship for he maximum expeced uiliy as a funcion of he wage earner s age and his/her wealh a ha ime: [ s V (, x) = sup E exp λ(v)dv V (s, X(s)) (c,p,θ) A(,x) s + f(u, ) e ρu γ (Z(u))γ + F (u, ) e ρu γ (c(u))γ du F ]. Here he definiion of he se of admissible sraegies A(, x) is similar o he definiion A(x), excep ha he saring ime is ime and he wealh a ime is x. 1

11 We are now in a posiion o presen he dynamic programming equaion, ha is, he HJB equaion: V (, x) λ()v (, x) + sup (c,p,θ) Ψ(, x; c, p, θ) = V (T, x) = e ρt γ x γ, (17) where Ψ(, x; c, p, θ) (r()x + θ(µ() r()) + i() c p)v x (, x) σ2 ()θ 2 V xx (, x) + λ() e ρ γ (x + p/η())γ + e ρ γ cγ. We refer o Ye (26) for deriving his HJB equaion. We should poin ou ha he PDE par of his HJB equaion is he same as Richard (1975). The boundary condiion is naurally obained by seing o T in (16), while Richard arificially se he boundary condiion o since he canno obain he boundary condiion from his model. This HJB equaion enables us o derive he opimal insurance, porfolio and consumpion sraegies. According o Ye (26), if V is a soluion of he HJB equaion (17) and if an admissible 3-uple (c, p, θ ) saisfies = V (, x) λ()v (, x) + Ψ(, x; c, p, θ ) = V (, x) λ()v (, x) + sup Ψ(, x; c, p, θ), (c,p,θ) (18) hen V is he maximum expeced uiliy funcion and (c, p, θ ) are he opimal sraegies. To exploi his, we firs use he firs-order condiions for a regular inerior maximum o (18) in order o derive he following expressions for he opimal sraegies in erms of he soluion V : [ ] c 1 1/(1 γ) () =, (19) V x e ρ ( ) 1/(1 γ) x + p () 1 η() = λ(), (2) V x e ρ η() and θ = (µ() r())v x σ 2 ()V xx. (21) 11

12 We now plug (19)-(21) in (17) and combine he similar erms o ge λ()v + V 1 ( µ() r() 2 σ() + 1 γ [ ] (λ()) 1/(1 γ) γ e ρ/(1 γ) (η()) + 1 γ/(1 γ) ) 2 Vx 2 + [(r() + η())x + i()]v x V xx V γ/(1 γ) x =. (22) I hus remains o deermine he soluion V of differenial equaion (22) subjec o he boundary condiion in (17). The ensuing analysis is raher lenghy and echnical (see Ye (26) for he deails), so we proceed righ o he soluion: V (, x) = a() γ (x + b())γ, (23) where a() = e ρ (e()) 1 γ, (24) b() = T s i(s) exp [r(v) + η(v)]dv ds, (25) T e() = exp T H(v)dv + s exp H(v)dv K(s)ds, (26) H() λ() + ρ 1 γ 1 ( ) 2 µ() r() 2 γ γ (r() + η()), (27) (1 γ)σ() 1 γ and K() (λ())1/(1 γ) + 1. (28) (η()) γ/(1 γ) Consequenly, from (19)-(21), (23) and (24) he opimal consumpion, beques, and porfolio sraegies can be explicily wrien as c () = 1 (x + b()), (29) e() Z () = x + p () η() = ( ) 1/(1 γ) λ() 1 (x + b()), (3) η() e() 12

13 and θ () = µ() r() (x + b()). (31) (1 γ)σ2 From (3), he opimal sraegy for insurance premium paymens is given by ( ) 1/(1 γ) p () = η() λ() 1 η() e() 1 x + ( λ() η() ) 1/(1 γ) b() e(). (32) Noice ha all hese opimal sraegies have been presened in feedback form, depending no only on he curren ime bu also on he ime- wealh x. I is worh remarking ha he funcion b() represens he fair value a ime of he wage earner s fuure income from ime o ime T. In oher words, his is he maximum amoun a bank will lend he individual in a riskless loan, hedging he risk of premaure deah by purchasing a suiable amoun of life insurance a he borrower s expense. Because of is inerpreaion and imporance, we call b() he value of human capial. The formula for b() were derived wih arbirage argumens by Ye (26). The sum of he curren wealh and he value of human capial, viz., x + b(), is a quaniy ha sands ou in all hree of he formulas (29)-(31) for he opimal sraegies. Because of is apparen imporance we shall refer o his sum as he wage earner s oal overall wealh. Noe from (29) ha he opimal consumpion as a fracion of he oal overall wealh is simply equal o he deerminisic funcion of ime e() 1. And he same can be said abou he opimal beques Z if he payou raio η is a simple muliple of he hazard rae λ. Finally, noe ha if he risky asse s appreciaion rae µ and he riskless ineres rae r are consans, hen he opimal fracion of he overall wealh ha should be invesed in he risky asse is a consan, independen of age. This las resul is enirely consisen wih well known resuls for classical porfolio managemen problems (see Meron (1969, 1971)) in he case of CRRA uiliy funcions. 4 Numerical Resuls and Economic Implicaions In his secion we use explici soluions of a numerical example o examine economic implicaions of our model. Throughou we consider a wage earner who sars work a age 25, is planning o reire a age 65, and whose iniial wage a age 25 is $5,, growing a he rae 3% per year. Oher parameers are given in Table 1. 13

14 Table 1 The Parameers Para. T r µ σ ρ λ() η() γ Val e λ() 3 Since i plays such an imporan role in our resuls, we show in Figure 1 a graph of b(), he arbirage value of human capial for his example. In he remaining figures we varied one and only one parameer each ime in order o examine each parameer s effec on opimal insurance purchase decisions. For each parameer we produce he new opimal life insurance paymen funcion and hen subrac he baseline opimal life insurance paymen funcion o produce Figures 2 8. For Figure 2 we doubled he riskless ineres rae from 4% o 8%. The wage earner hen buys less life insurance, probably because he riskless invesmen has become more aracive. For Figure 3 we doubled he discoun rae ρ from 3% o 6%. This makes lile difference when he wealh is small, bu wealhy persons nearing reiremen should purchase more life insurance, ha is, hey should no sell as much. For Figure 4 we changed he risk aversion parameer γ from -3 o -1, hereby making he wage earner less risk averse. This has lile effec unil he wage earner nears reiremen, a which ime he or she should spend less on insurance, i.e., should sell more insurance. For Figure 5 we doubled he hazard rae λ, so he wage earner s life expecancy is considerably reduced. For he mos par here is lile difference in he amoun of insurance purchased, bu he less healhy wage earner nearing reiremen will wan o buy more life insurance han he healhy one. For Figure 6 we doubled he insurance premium-payou raio η from 1.5 o 1.1 imes λ, hereby making he insurance policy more expensive for he wage earner. We see his change has an undeermined effec on he wage earner s life insurance purchase. For small levels of wealh he wage earner faced wih a more expensive policy will bie he bulle and spend more on insurance, bu for larger levels of wealh his wage earner will acually spend less. For Figure 7 we doubled he risky securiy s expeced rae of reurn µ from 9% o 18%. We see he wage earner wih he beer invesmen opporuniy will buy more life insurance, especially when nearing reiremen wih a large wealh. Perhaps his is because his wage earner finds i unnecessary o save as much in order o achieve similar invesmen objecives. Finally, for Figure 8 we doubled he volailiy σ of he risky securiy from 18% 14

15 o 36%. We see he wage earner wih he more volaile invesmen opporuniy will buy less life insurance, especially for wage earners in heir middle years. Perhaps his is because his wage earner will find i necessary o save more in order o achieve similar invesmen objecives. 5 Discussion In his paper we have developed a comprehensive, coninuous-ime model of lifeime opimal consumpion, opimal invesmen, and opimal life insurance purchase for a wage earner. Our model can be viewed as a naural exension of Meron s (1969), (1971) model of opimal consumpion and invesmen for a wage earner having a specified income, he key exensions being he random lifeime of he wage earner, he opporuniy o purchase life insurance, and he uiliy of he beques upon he wage earner s deah before reiremen. By assuming he wage earner has CRRA uiliy funcions we were able o derive explici formulas for he opimal sraegies. We hen used hese formulas in conjuncion wih a numerical example o sudy he economic implicaions of our resuls. Our model clearly resembles ha of Richard (1975), o he exen ha he sochasic differenial equaions for his and our wealh processes are he same and even he PDE for his and our dynamic programs are he same. However, here are some imporan differences. His wage earner has a bounded life ime, and he planning horizon is he leas upper bound on his lifeime. In our case he wage earner s lifeime is no necessarily bounded, and he planning horizon is explained as he wage earner s planned ime of reiremen, wih a uiliy of his wealh a ha ime. Looking a his and our explici soluions, one migh be emped o conclude ha hey coincide, since hey have exacly he same form (see (23)). However, his is no he case, because his funcion a() is differen from ours. In fac, he has a(t ) = for T equal o his planning horizon, where our funcion saisfies a() > for all. For he mos par he implicaions of our analysis are in accordance wih economic inuiion. For example, he opimal expendiure for insurance is a decreasing funcion of he wage earner s overall wealh and is a unimodal funcion of age, reaching a maximum a an inermediae age. On he oher hand, our model has revealed some relaionships whose economic explanaions are unclear, such as why an increase in he discoun rae ρ would cause he wage earner o purchase more life insurance. A perhaps conroversial aspec of our model is he fac ha opimal soluions can call for he wage earner o sell a life insurance policy on his or her own life. This is imporan and informaive, because i cerainly suggess pracical 15

16 circumsances when a person should no purchase any life insurance, bu i is hardly realisic. To deal wih his i is necessary o impose a consrain requiring he insurance purchase rae p() o be nonnegaive. Bu doing so seriously damages he hopes of obaining explici soluions, making i necessary o use numerical mehods o solve he corresponding HJB equaion. Some numerical work in his direcion has been carried ou by Ye (26). References [1] Campbell, R. A., 198. The Demand for Life Insurance: An Applicaion of he Economics of Uncerainy, Journal of Finance, 35, [2] Chrisophee Blanche-Scallie, Nicole El Karouri, 21. Monique Jeanblanc and Lionel Marellini, Opimal Invesmen and Consumpion Decisions when Time-Horizon is Uncerain, Working Paper, Marshall School of Business, USC. [3] Chrisophee Blanche-Scallie, Nicole El Karouri, and Lionel Marellini, 25. Dynamic Asse Pricing Theory wih Uncerain Time-Horizon, Journal of Economic Dynamics and Conrol, 29, [4] Colle, D., 23. Modelling Survival Daa in Medical Research, Second Ediion, Chapman&Hall, London. [5] Fischer, S., A Life Cycle Model of Life Insurance Purchase, Inernaional Economics Review, 14, [6] Hakansson, N.H.., Opimal Invesmen and Consumpion Sraegies under Risk, an Uncerain Lifeime, and Insurance, Inernaional Economics Review 1, 3, [7] Iwaki, H. and Komoribayashi, K., 24. Opimal Life Insurance for a Household, Inernaional Finance Workshop a Nanzan Universiy (preprin). [8] Leung, S.F., Uncerain Lifeime, he Theory of he Consumer, and he Life Cycle Hypohesis, Economerica 62, [9] Lewis, F.D., Dependens and he Demand for Life Insurance, American Economic Review, 79, [1] Meron, R.C., Lifeime Porfolio Selecion under Uncerainy: The Coninuous Time Case, Review of Economics and Saisics, 51, [11] Meron, R.C., Opimum Consumpion and Porfolio Rules in a Coninuous-Time Model, Journal of Economic Theory, 3, [12] Pliska, S.R., and Ye, J., 27. Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime, Journal of Banking and Finance, 31, [13] Richard, S.F., Opimal Consumpion, Porfolio and Life Insurance Rules for an Uncerain Lived Individual in a Coninuous Time Model, Journal of Financial Economics, 2, [14] Yaari, M.E., Uncerain Lifeime, Life Insurance, and he Theory of 16

17 he Consumer, Review of Economic Sudies, 32, [15] Ye, J., 26. Opimal Life Insurance Purchase, Consumpion and Porfolio under an Uncerain Life, Ph.D. Thesis, Universiy of Illinois a Chicago, Chicago. [16] Ye, J., 27. Opimal Life Insurance, Consumpion and Porfolio under Uncerainy: Maringale Mehods, he 26h American Conrol Conference. 17

18 18 Arbirage Value of Human Capial 16 Human Capial b() (in Thousand Dollars) Age (in Years) Fig. 1. The fair value of he human capial Fig. 2. The difference of life insurance raes beween he case of wice he base ineres rae and he case of he base ineres rae 18

19 Fig. 3. The difference of life insurance raes beween he case of wice he base uiliy discouned rae and he case of he base uiliy discouned rae Fig. 4. The difference of life insurance raes beween he risk aversion parameer γ = 1 and he case of he base risk aversion parameer 19

20 Fig. 5. The difference of life insurance raes beween he case of wice he base hazard rae and he case of he base hazard rae Fig. 6. The difference of life insurance raes beween he case of wice he base insurance premium - payou raio and he case of he base insurance premium - payou raio 2

21 Fig. 7. The difference of life insurance raes beween he case of wice he base expeced reurn of he risky securiy and he case of he base expeced reurn of he risky securiy Fig. 8. The difference of life insurance raes beween he case of wice he base volailiy of he risky securiy and he case of he base volailiy of he risky securiy 21