Optimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach

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1 28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 11-13, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A coninuous-ime model of opimal life insurance, consumpion and porfolio is examined by dynamic programming echnique. The Hamilon-Jacobi- Bellman (HJB in shor) equaion wih he absorbing boundary condiion is derived. Then explici soluions for Consan Relaive Risk Aversion (CRRA in shor) uiliies wih subsisence levels are obained. Asympoic analysis is used o analyze he model. Key words: life insurance, consumpion/invesmen, HJB equaion, absorbing boundary condiion, CRRA uiliies wih subsisence levels, asympoic analysis. I. INTRODUCTION This paper considers 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 [,T], where he fixed planning horizon T can be inerpreed as he reiremen ime of he wage earner. The wage earner receives his income a rae i() coninuously, bu his is erminaed by he wage earner s deah or reiremen, whichever happens firs. We use a random variable o model he wage earner s lifeime. The wage earner needs o buy life insurance o proec his family due o his premaure deah. 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. Yarri [8] provided he key idea for research on life insurance, consumpion and/or porfolio decisions under an uncerain lifeime, ha is, he problem under he random horizon can be convered o one under he fixed horizon. he uncerainy of lifeime is he sole source of uncerainy in his model, and he used a nonnegaive bounded random variable o model uncerain lifeime. Leung [3] poined ou ha Yaari s model canno have an inerior soluion which lass unil he maximum lifeime for he opimal consumpion. Ye [9] showed ha using a nonnegaive bounded random variavle o model uncerain lifeime will produce abnormal behavior of he model from he perspecive of dynamic programming. Meron [5] briefly considered consumpion/porfolio under an uncerain life deparmen of mahemaics, saisics and compuer science, universiy of illinois a chicago. I hanks wo referees for beneficial commens. All errors are mine. jinch using he dynamic programming approach, bu he ignored life insurance. Richard [7] used Yaari s seing for an uncerain lifeime and dynamic programming o consider a life-cycle life insurance and consumpion/invesmen problem. Richard inroduced he concep of a coninuous-ime life insurance marke where he wage earner coninuously buys erm life insurance, he erm being infiniesimally shor. Bu he erminal condiion for his model is in quesion. Recenly, Pliska and Ye [6] used a comparaive echnique o sudy he demands of life insurance. Ye [1] summarized he resuls from he maringale approach in Ye [9]. Zhu [11] sudied individual consumpion, life insurance, and porfolio decisions in one-period environmen. This paper is organized as follows. The nex secion describes he inemporal model proposed in Ye [9]. Secion 3 derives he HJB equaion wih he absorbing boundary condiion which is imporan in numerical research, and hen derives he opimal feedback conrol. In Secion 4 we obain explici soluions for he family of CRRA uiliies wih subsisence requiremens, and asympoic analysis is used o analyze he model. We conclude wih some remarks in Secion 5. II. 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, here inerpreed as he wage earner s reiremen ime. Le F = F, [,T] be he P-augmenaion of he filraion σw(s), s, [,T], so F represens he informaion a ime. The coninuous-ime economy consiss of a fricionless financial marke and an insurance marke. We are going o describe heir deails separaely in he following. We assume ha 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, (1) where r( ) is a funcion of ime saisfying r() d <. This condiion ensures he above equaion is welldefined. There is a risky securiy in he financial marke. I evolves according o he linear sochasic differenial /8/$ AACC. 356

2 equaion ds 1 () S 1 () = µ()d σ()dw(), (2) where µ( ) and σ( ) are funcions of ime saisfying (µ() r())/σ() <. This condiion ensures he financial marke is complee (see Chaper 1, Karazas and Shreve [2]). We suppose he wage earner is alive a ime = and his lifeime is denoed by a random variable τ. The hazard rae funcion of τ is denoed by λ(), [,T]. According o Colle [1], he condiional probabiliy densiy funcion, f(s,), for he deah a ime s condiional upon he wage earner being alive a ime s is given by f(s,) λ(s)exp s λ(u)du. (3) Suppose life insurance is offered coninuously and he wage earner eners 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(). Here η() is called he insurance premium-payou raio. Suppose he wage earner is endowed wih he iniial wealh x and will receive he income a rae i() during he period [, mint, τ], ha is, during a period which will be erminaed by he wage earner s deah or reiremen a T, whichever happens firs. We now define some processes describing he wage earner s decisions: c() Consumpion rae a ime, which is an F -progressively measurable, nonnegaive process saisfying c()d < almos surely. p() Premium rae (e.g., dollars per annum) a ime, which is an F -predicable process saisfying p() d < almos surely. θ() Dollar amoun in he risky securiy a ime, which is an F -progressively measurable process saisfying σ2 ()θ 2 ()d < almos surely. For a wage earner s decision, (c,p, θ), he wealh process X() on [, mint, 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). (4) Using (1) and (2), we wrie (4) as he sochasic differenial equaion dx() = r()x()d c()d p()d i()d θ()[(µ() r())d σ()dw()]. If he wage earner dies a ime, < T, he esae will ge he insurance amoun p()/η(). Then he wage earner s oal beques when he dies a ime wih wealh X() is Z() = X() p() η() (5) on τ =. (6) We denoe by A(x) he conrol se of all 3-uples (c,p,θ) such ha X() b() and Z(), [,T], where b() is defined as s b() = i(s)exp [r(v) η(v)]dv ds. (7) A (c,p, θ) A(x) is called as an admissible decision. Remark 2.1: Here we give an economic meaning for b(). Suppose he wage earner wans o borrow money from a financial insiuion using his fuure income as a morgage. The quesion is how much he wage earner can borrow from he financial insiuion. We analyze his problem as follows. The financial insiuion issues a loan o he wage earner a ime and he wage earner ransfers all of his fuure income i(s), s T, o he financial insiuion. However, he wage earner s life is uncerain, so he financial insiuion buys life insurance for he wage earner in order o preven a loss due o he wage earner s premaure deah before he pays off he loan. Hence he financial insiuion uses he wage earner s fuure income o pay he loan and pay he life insurance premiums. If he wage earner dies a ime s, where < s < T, hen he insured amoun mus be enough o pay off he loan, and if he is alive a he ime T, his wage accumulaing from ime o ime T mus be enough o pay off he insurance premiums accumulaing from ime o ime T and pay off he loan which is coninuously compounded by he ineres rae r( ) (in general, he loan rae is higher han he risk-free rae in realiy. The mehodology in his iem of his remark can sill be applied if you use he acual loan rae.). Le l(s), < s T, be he ime s principal balance for he loan process described above, so l(s) = s r(u)l(u)du s (i(u) p(u))du on τ s, l(s) p(s) η(s) on τ = s, l(t) on τ > T, where p(s) is he insurance premiums paid a ime s by he financial insiuion o hedge he 357

3 wage earner s moraliy risk. Rewriing he above equaion, l(s) s (r(u) η(u))l(u)du s i(u)du on τ s, l(t) on τ > T. Inroduce he variable subsiuion w = T s, use Grönwall s inequaliy, and do some algebra, hen we have u l() i(u)exp [r(v) η(v)]dv du = b(). Hence he wage earner can borrow b() a mos from he financial insiuion using he fuure income as he morgage. Now i is clear ha he funcion b() represens he value a ime of he wage earner s fuure income from ime o ime T. We inerpre X()b() as he oal wealh a ime, so i is reasonable ha we require X()b() o be nonnegaive. In fac, we can show once he wage earner s wealh X() reaches b(), hen no furher consumpion for he wage earner can ake place. Moreover, if he dies beween ime and ime T wih X() = b() a ime, his beques is almos surely, and his erminal wealh X(T) is almos surely (see Ye [9]). In oher words, X() = b() is an absorbing sae for he wealh process X( ) when (c,p,θ) A(x). Hence A(x) is an empy se when x < b(). Suppose ha he wage earner s preference srucure is given by (U 1,U 2, U 3 ). U 1 (,) is a uiliy funcion for he consumpion wih he subsisence consumpion c() assumed o be a nonnegaive funcion of ime, U 2 (, ) is a uiliy funcion for he beques wih he subsisence beques Z() assumed o be a nonnegaive funcion of ime, and U 3 ( ) is a uiliy funcion for he erminal wealher wih he subsisence erminal wealh X assumed o be a nonnegaive number. Remark 2.2: One refers o Ye [9], [1] for he mahemaical definiion of a preference srucure. The subsisence levels c() and Z() a ime mean he wage earner does no wan his consumpion c() and he beques Z() lower han c() and Z() a ime, respecively, and he subsisence level X means he does no wan his erminal wealh X(T) lower han X. From he perspecive of mahemaics, he subsisence levels impose implici consrains on he wage earner s decisions, viz., c() c(), [,T], Z() Z(), [,T], and X(T) X. As shown in Ye [9], hese implici consrains can be saisfied only if X() b() b(), [,T], (8) where b() is defined as b() ) s = ( c(s) η(s) Z(s) exp [r(v) η(v)]dv ds X exp [r(s) η(s)] ds. (9) Furhermore, once he wage earner s wealh X() reaches b() b(), hen c(s) = c(s) and Z(s) = Z(s) for any s saisfying s T, and X(T) = X. Hence we inerpre X() b() b() as he oal available wealh a ime while x() b() is he oal wealh a ime. Noe ha c( ), Z( ), and X are nonnegaive, hen he subsisence levels impose a more resricive consrain on he wealh process han admissible decisions do. The wage earner s problem is o choose life insurance purchase and consumpion/porfolio invesmen sraegies so as o maximize he expeced uiliy [ τ V (x) sup E (c,p,θ) A 1 (x) U 2 (Z(τ), τ)1 τ T ] U 3 (X(T))1 τ>t U 1 (c(s),s)ds where T τ mint,τ, and where A 1 (x) (c, p,θ) A(x); [ τ E U1 (s,c(s))ds U 2 (τ, Z(τ))1 τ T ] U3 (X(T))1 τ>t > (1) where U i min, U i, for i = 1,2, 3. The funcion of A 1 (x) is o pick ou every admissible conrol which saisfies he subsisence requiremens (see Ye [9], [1]). III. STOCHASTIC DYNAMIC PROGRAMMING In his secion we use he sochasic dynamic programming echnique o derive he HJB equaion, and hen derive he opimal feedback conrol from he HJB equaion. We resae (1) in a dynamic programming form. For any (c,p,θ) A 1 (, x), where he definiion of A 1 (,x) is similar o he definiion A 1 (x) excep ha he saring ime is ime and he wealh a ime is x, define J(, x; c, p, θ) [ τ E U 1(c(s), s)ds U 2(Z(τ), τ)1 τ T ] τ U 3(X(T))1 τ>t >, F (11) 358

4 and V (,x) sup J(,x; c,p,θ). (12) c,p,θ A 1 (,x) The above value funcion is nonradiional due o he random horizon. Using Fubini-Tonelli heorem (see Ye [9]), we can J(,x; c,p, θ) as follows: Lemma 3.1: If he deah ime τ is independen of he filraion F. For each (c,p, θ) A 1 (x,i), J(,x;c, p,θ) [ = E [ F(u,)U 1 (c(u), u) f(u,)u 2 (Z(u),u)]du F(T, )U 3 (X(T)) F ], (13) where F(u,) exp s λ(u)du and f(u,) is given by (3). From Lemma 3.1, we know ha he wage earner who faces unpredicable deah acs as if he will live a leas unil ime T, bu wih a subjecive rae of ime preferences equal o his force of moraliy for his consumpion and erminal wealh. From he mahemaical poin of view, his lemma enables us o conver he opimizaion problem wih a random erminal ime o a problem wih a fixed erminal ime. According o Ye [9], ha is, se up he opimaliy principal and use Iô s lemma, we derive so-called HJB equaion V (,x) λ()v (,x) sup (c c(),p η()( Z() x),θ) Ψ(,x; c,p,θ) =, V (T, x) = U 3 (x), (14) on he domain D (,x) [,T] (, ),x > b() b(). Where Ψ(,x; c,p,θ) (r()x θ(µ() r()) i() c p)v x (, x) 1 2 σ2 ()θ 2 V xx (,x) U 1 (c,) λ()u 2 (x p/η(),). (15) Moreover, V saisfies he absorbing boundary condiion (see (8)) V (, b() b()) = [ F(u, )U 1 ( c(u),u) f(u,)u 2 ( Z(u),u)]du F(T,)U 3 ( X). (16) The boundary condiion (16) for V follows from Remark 2.2 and Lemma 3.1. The firs-order condiions for a regular inerior maximum o (15) are = Ψ c (,x; c,p,θ ) = V x (,x) U 1,c (c,), (17) and = Ψ p (,x; c,p,θ ) = V x (,x) λ() η() U 2,Z (x p /η(),), (18) = Ψ θ (,x; c,p,θ ) = (µ() r())v x (,x) σ 2 ()θ V xx (,x). (19) A se of sufficien condiions for a regular inerior maximum is Ψ cc = U 1,cc (c,) <, Ψ pp = λ() η 2 () U 2,ZZ(Z,) <, Ψ θθ = σ 2 ()V xx (,x) <. Noe ha he firs wo condiions are auomaically saisfied according o he definiion of uiliy funcions. Thus a sufficien condiion for a maximum is V xx (,x) <. IV. THE CASE OF CONSTANT RELATIVE RISK AVERSION In his secion we derive explici soluions for he case where he wage earner has he same consan relaive risk aversion for he consumpion, he beques and he erminal wealh. Assume for γ < 1, ρ >, a i () >, [,T], i = 1,2, and a 3 > ha and U 1 (c,) = e ρ γ a 1()(c c()) γ, U 2 (Z, ) = e ρ γ a 2()(Z Z()) γ, U 3 (x) = e ρt γ a 3(x X) γ, where c c() and Z Z() for all [,T], and x X. Here γ = represens he logarihmic uiliy funcions. From (17), (18), and (19), we have ha ( ) 1/(1 γ) c a1 () () = c() V x e ρ, (2) ( ) 1/(1 γ) x p () η() = Z() a2 () λ() V x e ρ, (21) η() θ () = (µ() r())v x σ 2 ()V xx. (22) We now plug (2)-(22) in (14) and ake as a rial soluion V (,x) = a() γ (x b() b()) γ, x b() b(), (23) where a( ) is a funcion o be deermined. Then a() mus saisfy he following ordinary differenial equaion: da() d = ( λ() γ 2(1 γ) ( µ() r() σ() (1 γ)e ρ/(1 γ) K()(a()) γ/(1 γ) ) 2 γ(r() η())) a() 359

5 wih a(t) = a 3 e ρt, where K() (a 1 ()) 1/(1 γ) (a 2 ()) 1/(1 γ) (λ())1/(1 γ). (24) (η()) γ/(1 γ) To solve for a(), suppose for some new funcion m( ) ha a() has he form: and define H() λ() ρ 1 γ 1 2 γ Then dm() d a() = e ρ (m()) 1 γ, (25) H()m() K() =, ( ) 2 µ() r() γ (1 γ)σ() 1 γ (r()η()). (26) m(t) = a 1/(1 γ) 3. (27) Solving (27) by an inegraing facor, we ge ha m() = a 1/(1 γ) 3 exp exp s Hence [ a() = e ρ a 1/(1 γ) 3 exp exp s H(v)dv H(v)dv K(s)ds. (28) H(v)dv 1 γ H(v)dv K(s)ds]. (29) From (2)-(22), (23) and (25) he opimal consumpion, life insurance and porfolio rules can be explicily wrien in feedback form as and c () = c() (a1())1/(1 γ) (x b() m() b()), (3) Z () = x p () η() ( ) 1/(1 γ) = Z() λ() (a 2 ()) 1/(1 γ) (x b() η() m() b()), (31) θ () = µ() r() (1 γ)σ 2 () (x b() b()). (32) The above formulas for c and θ are consisen wih he classical resuls in Meron [4]. In paricular, he opimal porfolio fracion µ() r() (1 γ)σ 2 () is he same as Meron s. This means, under he assumpion of independence beween he moraliy risk and sock reurn risk, he moraliy risk will no affec he risky invesmen. From (3) and (31), he wage earner s opimal decisions c and Z consis of a compulsory par and free choice par. The free choice par depends on his spare money x b() b(). The wage earner s opimal porfolio decisions is made based on his spare money x b() b(). We sudy he following cases using asympoic analysis: Z () Z() as λ() for any [,T]. This means ha a long-lived wage earner will jus mainain he subsisence level of he beques. c () c() as a 1 () for any [,T]. This means ha he consumpion becomes unimporan o he wage earner as he weigh a 1 approaches o zero, hen he wage earner will jus mainain he subsisence level of consumpion. Z () Z() as a 2 () for any [,T]. This means ha he beques becomes unimporan o he wage earner as he weigh a 2 approaches o zero, hen he wage earner will jus mainain he subsisence level of he beques. X (T) X as a 3. This means ha saving more for he afer-reiremen life becomes unimporan, hen he wage earner will jus mainain he subsisence level of he erminal wealh. In his case, seing c() = and Z() =, for all [,T], and X =, he soluions are he same as Richard s [7] if some errors in his soluions are correced, alhough he model is an inemporal model while Richard s is a life-cycle model. From (31), he policy of insurance premiums is given by [ (λ() ) ] 1/(1 γ) p (a 2()) 1/(1 γ) () = η() 1 x η() m() ( ) 1/(1 γ) λ() (a 2 ()) 1/(1 γ) b() b() η() m() Z(). From he above formula, he fuure income has a posiive effec on life insurance purchase as we expec. The subsisence levels of consumpion and erminal wealh have a negaive effec on life insurance purchase, his means he wage earner ends o buy less life insurance as hese wo subsisence levels increase. As we carefully examine he effec of he subsisence level of he beques on life insurance purchase, we find ha he curren subsisence level of he beques has a posiive effec, while he fuure subsisence level of he beques has a negaive effec. Now le s consider a wage earner who sars o work a age 25 (he iniial ime), his expeced reiremen ime is age 65 (T=65-25=4), and his iniial wage a age 25 is $5,, growing a he rae 3% every year. His risk aversion parameer γ = 3, he uiliy discouned rae ρ =.3, all uiliy weighs are 1, and all subsisence levels are zero. The hazard rae for him is 1/2 9/8. The marke parameers are given by Table I. Figure 1 was compued via (3) and shows he opimal consumpion proporion, viz., c ()/(xb()), as a funcion of age and he oal overall wealh xb(). We (33) 36

6 TABLE I THE PARAMETERS Parameers r µ σ η() 1 Value Curren Wealh (in Thousand Dollars) Differen Regions for Life Insurance Purchase p* > p* < Age (in Years) Fig. 3. The criical level curve Fig. 1. Opimal consumpion proporion wihou consrains using exac soluion see his proporion is consan wih respec o overall wealh, i is relaively small in he early years, and i rises wih respec o age. We used formula (33) o produce Figure 2, which shows he opimal life insurance purchase amoun p () in erms of age and he oal overall wealh. We see ha for small values of overall wealh he opimal insurance paymen is increasing wih respec o age up o a cerain poin, and hen he paymen declines as he wage earner approaches reiremen. Moreover, wih age fixed he opimal insurance paymen is a decreasing funcion of he overall wealh. Finally, for large values of overall wealh we see ha he opimal insurance paymen is acually negaive. In paricular, when he oal overall wealh exceeds a criical level which varies wih he wage earner s age, i becomes opimal for he wage earner o sell a life insurance policy on his own life. We plo his criical level in erms of he overall wealh and age in Figure 3. When he overall wealh is below he criical level he wage earner buys life insurance, bu when he overall wealh is above he criical level he wage earner sells life insurance. In paricular, he wage earner will buy life insurance in he early years since he criical level of he wealh is very high a ha ime. Bu he wage earner migh find i opimal o sell life insurance close o reiremen ime. V. DISCUSSION We derived HJB equaion wih he absorbing boundary condiion for he model. Explici soluions were found for a rich family of CRRA uiliies wih subsisence levels. Several economic implicaions were undersood via inerpreing he model seing and he soluions. We also used asympoic analysis o inerpre he model. One poin worh menioning is ha he asympoic resuls in Secion 4 provides an unified perspecive for invesigaing varians of he objecive funcional (1). For example, if we le a 2 () for each [,T], Z() = for each [, T], a 3, and X =, hen he asympoic resuls correspond o maximizing he expeced uiliy from consumpion. We considered he financial marke which consiss of a riskless and a risky securiies and he insurance marke which is allowed o sell life insurance. This combinaion of he financial marke and he insurance marke is complee in ha he wage earner s any reasonable financial plan (c,z( ),X(T)), X(T) can be viewed as he pension plan, can be replicaed in hese wo markes (see Ye [9]). I is no echnically difficul o include muliple risky securiies in he model assuming he financial marke is complee. However, eiher incompleeness of he financial marke or prohibiion of selling life insurance makes he combinaion incomplee. The incomplee financial marke has been exensively sudied in he lieraure. The numerical mehod has been carried ou in Ye [9] o deal wih he consrain on life insurance. REFERENCES Fig. 2. Opimal life insurance rule wihou consrains using exac soluion [1] Colle, D., Modelling Survival Daa in Medical Research, Second Ediion, Chapman&Hall,

7 [2] Karazas, I. and Shreve, S.E., Mehods of Mahemaical Finance, Springer-Verlag, New York, [3] Leung, S.F., Uncerain Lifeime, he Theory of he Consumer, and he Life Cycle Hypohesis, Economerica, 62(1994), [4] Meron, R.C., Lifeime Porfolio Selecion under Uncerainy: The Coninuous Time Case, Review of Economics and Saisics, 51(1969), [5] Meron, R.C., Opimum Consumpion and Porfolio Rules in a Coninuous-Time Model, Journal of Economic Theory, 3(1971), [6] Pliska, S.R. and Ye, J. Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime, Journal of Banking and Finance, 5(27), [7] 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(1975), [8] Yaari, M.E., Uncerain Lifeime, Life Insurance, and he Theory of he Consumer, Review of Economic Sudies, 32(1965), pp [9] Ye, J., Opimal Life Insurance Purchase, Consumpion and Porfolio under an Uncerain Life, PhD Thesis, Universiy of Illinois a Chicago, Chicago, 26. [1] Ye, J., Opimal Life Insurance Purchase, Consumpion and Porfolio under Uncerainy: Maringale Mehods, Proceedings of 27 American Conrol Conference, [11] Zhu,Y., One-Period Model of Individual Consumpion, Life Insurance, and Invesmen Decisions, The Journal of Risk & Insurance, 74(27),

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