OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTIDIMENSIONAL DIFFUSIVE TERMS


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1 OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTIDIMENSIONAL DIFFUSIVE TERMS I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Absrac. We inroduce an exension o Meron s famous coninuous ime model of opimal consumpion and invesmen, in he spiri of previous works by Pliska and Ye, 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 esae, while invesing his savings in a financial marke comprised of one riskfree securiy and an arbirary number of risky securiies driven by mulidimensional Brownian moion. We hen provide a deailed analysis of he opimal consumpion, invesmen, and insurance purchase sraegies for he wage earner whose goal is o maximize he expeced uiliy obained from his family consumpion, from he size of he esae in he even of premaure deah, and from he size of he esae a he ime of reiremen. We use dynamic programming mehods o obain explici soluions for he case of discouned consan relaive risk aversion uiliy funcions and describe new analyical resuls which are presened ogeher wih he corresponding economic inerpreaions. 1. Inroducion We consider he problem faced by a wage earner having o make decisions coninuously abou hree sraegies: consumpion, invesmen and life insurance purchase during a given inerval of ime [,min{t,τ}], where T is a fixed poin in he fuure ha we will consider o be he reiremen ime of he wage earner and τ is a random variable represening he wage earner s ime of deah. We assume ha he wage earner receives his income a a coninuous rae i and ha his income is erminaed when he wage earner dies or reires, whichever happens firs. One of our key assumpions is ha he wage earner s lifeime τ is a random variable and, herefore, he wage earner needs o buy life insurance o proec his family for he evenualiy of premaure deah. The life insurance depends on a insurance premium paymen rae p such ha if he insured pays p δ and dies during he ensuing shor ime inerval of lengh δ hen he insurance company will pay p/η dollars o he insured s esae, where η is an amoun se in advance by he insurance company. Hence his is like erm insurance wih an infiniesimal erm. We also assume ha he wage earner wans o maximize he saisfacion obained from a consumpion process wih rae c. In addiion o consumpion and purchase of a life insurance policy, we assume ha he wage earner invess he full amoun of his savings in a financial marke consising of one riskfree securiy and a fixed number N 1 of risky securiies wih diffusive erms driven by Mdimensional Brownian moion. The wage earner is hen faced wih he problem of finding sraegies ha maximize he uiliy of i his family consumpion for all min{t,τ}; ii his wealh a reiremen dae T if he lives ha long; and iii he value of his esae in he even of premaure deah. Various quaniaive models have been proposed o model and analyze his kind of problem, a leas problems having a leas one of hese hree objecives. This lieraure is perhaps highlighed by Yarri [9] who considered he problem of opimal financial planning decisions for an individual wih an uncerain lifeime as well as by Meron [4, 5] who emphasized opimal consumpion and 2 Mahemaics Subjec Classificaion. 91G1 91G8 93E2 9C39. Key words and phrases. sochasic opimal conrol; consumpioninvesmen problems; lifeinsurance. 1
2 2 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA invesmen decisions bu did no consider life insurance. These wo approaches were combined by Richard, whose impressive paper [8] uses sophisicaed mehods a an early dae for he analysis of a lifecycle life insurance and consumpioninvesmen problem in a coninuous ime model. Laer, Pliska and Ye [6, 7] inroduced a coninuousime model ha combined he more realisic feaures of all hose in he exising lieraure and exended he model proposed previously by Richard, he main difference being he choice of he boundary condiion, leading o somewha differen economic inerpreaions of he underlying problem. More precisely, while Richard assumed ha he lifeime of he wage earner is limied by some fixed number, he model inroduced by Pliska and Ye had he feaure ha he duraion of life is a random variable which akes values in he inerval ], [ and is independen of he sochasic process defining he underlying financial marke. Moreover, Pliska and Ye made he following refinemens o he heory: i he planning horizon T is now seen as he momen when he wage earner reires, conrary o Richard s inerpreaion as a finie upper bound on he lifeime; and ii he uiliy of he wage earner s wealh a he planning horizon T is aken ino accoun as well as he uiliy of lifeime consumpion and he uiliy of he beques in he even of premaure deah. Blanche Scallie e al. paper [1] deals wih opimal porfolio selecion wih an uncerainy exi ime for a suiable exension of he familiar opimal consumpion invesmen problem of Meron, wihou considering any kind of life insurance purchase. Whereas Pliska and Ye s financial marke involved only one securiy ha was risky, in he presen paper we sudy he exension where here is an arbirary bu finie number of risky securiies. The exisence of hese exra risky securiies gives greaer freedom for he wage earner o manage he ineracion beween his life insurance policies and he porfolio conaining his savings invesed in he financial marke. Some examples of hese ineracions are described below. Following Pliska and Ye, we use he model of uncerain life found in reliabiliy heory, commonly used for indusrial lifeesing and acuarial science, o model he uncerain ime of deah for he wage earner. This enables us o replace Richard s assumpion ha lifeimes are bounded wih he assumpion ha lifeimes ake values in he inerval ], [. We hen se up he wage earner s objecive funcional depending on a random horizon min{t, τ} and ransform i o an equivalen problem having a fixed planning horizon, ha is, he wage earner who faces unpredicable deah acs as if he will live unil some ime T, bu wih a subjecive rae of ime preferences equal o his force of moraliy for his consumpion and erminal wealh. This ransformaion o a fixed planning horizon enables us o sae he dynamic programming principle and derive an associaed HamilonJacobiBellman HJB equaion. We use he HJB equaion o derive he opimal feedback conrol, ha is, he opimal insurance, porfolio and consumpion sraegies. Furhermore, we obain explici soluions for he family of discouned Consan Relaive Risk Aversion CRRA uiliies and examine he economic implicaions of such soluions. In he case of discouned CRRA uiliies our resuls generalize hose obained previously by Pliska and Ye. For insance, we obain: i an economically reasonable descripion for he opimal expendiure for insurance as a decreasing funcion of he wage earner s overall wealh and a unimodal funcion of age, reaching a maximum a an inermediae age; ii a more conroversial conclusion ha possibly an opimal soluion calls for he wage earner o sell a life insurance policy on his own life lae in his career. Noneheless, he exra risky securiies in our model inroduce novel feaures o he wage earner s porfolio and insurance managemen ineracion such as: i a young wage earner wih small wealh has an opimal porfolio wih larger values of volailiy and higher expeced reurns, wih he possibiliy of having shor posiions in lower yielding securiies; and ii a wage earner who can buy life insurance policies will choose a more conservaive porfolio han a wage earner who is wihou he opporuniy o buy life insurance, he disincion being clearer for young wage earners wih low wealh.
3 OPTIMAL LIFE INSURANCE PURCHASE 3 Thispaperisorganizedasfollows. Insecion2wedescribeheproblemweproposeoaddress. Namely, we inroduce he underlying financial and insurance markes, as well as he problem formulaion from he poin of view of opimal conrol. In secion 3 we see how o use he dynamic programming principle o reduce he opimal conrol of secion 2 o one wih a fixed planning horizon and hen derive an associaed HJB equaion. We devoe secion 4 o he case of discouned CRRA uiliies. We conclude in secion Problem formulaion Throughou his secion, we define he seing in which he wage earner has o make his decisions regarding consumpion, invesmen and life insurance purchase. Namely, we inroduce he specificaions regarding he financial and insurance markes available o he wage earner. We sar by he financial marke descripion, followed by he insurance marke and conclude wih he definiion of a wealh process for he wage earner The financial marke model. We consider a financial marke consising of one riskfree asse and several riskyasses. Their respecive prices S T and S n T for n = 1,...,N evolve according o he equaions: ds = rs d, S = s, M ds n = µ n S n d+s n σ nm dw m, S n = s n >, m=1 where W = W 1,...,W M T is a sandard Mdimensional Brownian moion on a probabiliy space Ω,F,P, r is he riskless ineres rae, µ = µ 1,...,µ N R N is he vecor of he riskyasses appreciaion raes and σ = σ nm 1 n N,1 m M is he marix of riskyasses volailiies. We assume ha he coefficiens r, µ and σ are deerminisic coninuous funcions on he inerval [,T]. We also assume ha he ineres rae r is posiive for all [,T] and he marix σ is such ha σσ T is nonsingular for Lebesgue almos all [,T] and saisfies he following inegrabiliy condiion N M σnmd 2 <. m=1 Furhermore, we suppose ha here exiss an F T progressively measurable process π R M, called he marke price of risk, such ha for Lebesguealmosevery [,T] he risk premium α = µ 1 r,...,µ N r R N 1 is relaed o π by he equaion α = σπ and is such ha he following wo condiions hold E π 2 < a.s. a.s. [ exp πsdws 1 ] πs 2 ds = 1. 2 The exisence of such a process π ensures he absence of arbirage opporuniies in he financial marke defined above. Noe also ha he condiions on he marix σ above do no imply marke compleeness. See [3] for furher deails on marke viabiliy and compleeness.
4 4 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Moreover, hroughou he paper we will assume ha Ω,F,P is a filered probabiliy space and ha is filraion F = {F, [,T]} is he Paugmenaion of he filraion generaed by he Brownian moion W, σ{ws,s } for. Each subσalgebra F represens he informaion available o any given agen observing he financial marke unil ime The life insurance marke model. We assume ha he wage earner is alive a ime = and ha his lifeime is a nonnegaive random variable τ defined on he probabiliy space Ω, F, P. Furhermore, we assume ha he random variable τ is independen of he filraion F and has a disribuion funcion F : [, [,1] wih densiy f : [, R + so ha F = fs ds. We define he survivor funcion F : [, [,1] as he probabiliy for he wage earner o survive a leas unil ime, i.e. F = Pτ = 1 F. We shall make use of he hazard funcion, he condiional, insananeous deah rae for he wage earner surviving o ime, ha is P τ < +δ τ λ = lim = f δ δ F. 2 Throughou he paper, we will suppose ha he hazard funcion λ : [, R + is a coninuous and deerminisic funcion such ha λ d =. These wo conceps inroduced above are sandard in he conex of reliabiliy heory and acuarial science. In our case, such conceps enable us o consider an opimal conrol problem wih a sochasic planning horizon and resae i as one wih a fixed horizon. Due he uncerainy concerning his lifeime, he wage earner buys life insurance o proec his family for he evenualiy of premaure deah. The life insurance is available coninuously and he wage earner buys i by paying a insurance premium paymen rae p o he insurance company. The insurance conrac is like erm insurance, wih an infiniesimally small erm. If he wage earner dies a ime τ < T while buying insurance a he rae p, he insurance company pays an amoun pτ/ητ o his esae, where η : [,T] R + is a coninuous and deerminisic funcion which we call he insurance premiumpayou raio and is regarded as fixed by he insurance company. The conrac ends when he wage earner dies or achieves reiremen age, whichever happens firs. Therefore, he wage earner s oal legacy o his esae in he even of a premaure deah a ime τ < T is given by Zτ = Xτ+ pτ ητ, where X denoes he wage earner s savings a ime The wealh process. We assume ha he wage earner receives an income i a a coninuous rae during he period [,min{t,τ}], i.e., he income will be erminaed eiher by his deah or his reiremen, whichever happens firs. Furhermore, we assume ha i : [,T] R + is a deerminisic Borelmeasurable funcion saisfying he inegrabiliy condiion i d <.
5 OPTIMAL LIFE INSURANCE PURCHASE 5 The consumpion process c T is a F T progressively measurable nonnegaive process saisfying he following inegrabiliy condiion for he invesmen horizon T > c d < a.s.. We assume also ha he insurance premium paymen rae p T is a F T predicable process, i.e., p is measurable wih respec o he smalles σalgebra on R + Ω such ha all lefconinuous and adaped processes are measurable. In a inuiive manner, a predicable process can be described as such ha is values are known jus in advance of ime. For each n =,1,...,N and [,T], le θ n denoe he fracion of he wage earner s wealh allocaed o he asse S n a ime. The porfolio process is hen given by Θ = θ,θ 1,,θ N R N+1, where N θ n = 1, T. 3 n= We assume ha he porfolio process is F T progressively measurable and ha, for he fixed invesmen horizon T >, we have ha Θ 2 d < a.s., where denoes he Euclidean norm in R N+1. The wealh process X, [,min{t,τ}], is hen defined by X = x+ [is cs ps] ds+ N n= θ n sxs S n s ds n s, 4 where x is he wage earner s iniial wealh. This las equaion can be rewrien in he differenial form N dx = i c p+ θ r+ θ n µ n X d + N θ n X M σ nm dw m, 5 m=1 where min{τ,t}. Using relaion 3 we can always wrie θ in erms of θ 1,...,θ N, so from now on we will define he porfolio process in erms of he reduced porfolio process θ R N given by θ = θ 1,θ 2,,θ N R N The opimal conrol problem. The wage earner is faced wih he problem of finding sraegies ha maximize he expeced uiliy obained from: a his family consumpion for all min{t,τ}; b his wealh a reiremen dae T if he lives ha long; c he value of his esae in he even of premaure deah. This problem can be formulaed by means of opimal conrol heory: he wage earner s goal is o maximize some cos funcional subjec o i he sochasic dynamics of he sae variable, i.e., he dynamics of he wealh process X given by4; ii consrains on he conrol variables, i.e., he consumpion process c, he premium insurance rae p and he porfolio process θ; and iii boundary condiions on he sae variables.
6 6 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Le us denoe by Ax he se of all admissible decision sraegies, i.e., all admissible choices for he conrol variables ν = c,p,θ R N+2. The dependence of Ax on x denoes he resricion imposed on he wealh process by he boundary condiion X = x. In paricular, Ax mus be such ha for each ν Ax he corresponding wealh process saisfies X for all min{t,τ}. The wage earner s problem can hen be resaed as follows: find a sraegy ν = c,p,θ Ax which maximizes he expeced uiliy [ τ ] Vx = sup E,x Ucs,s ds+bzτ,τi {τ T} +WXTI {τ>t}, 6 ν Ax where T τ = min{t,τ}, I A denoes he indicaor funcion of even A, Uc, is he uiliy funcion describing he wage earner s family preferences regarding consumpion in he ime inerval [,min{t,τ}], BZ, is he uiliy funcion for he size of he wage earners s legacy in case τ T, and WX is he uiliy funcion for he erminal wealh a ime = T in he case τ > T. We suppose ha U and B are sricly concave on heir firs variable and ha W is sricly concave on is sole variable. In secion 4 we specialize our analysis o he case where he wage earner s preferences are described by discouned CRRA uiliy funcions. 3. Sochasic opimal conrol In his secion we use he echniques in [6, 7] o resae he sochasic opimal conrol problem formulaed in he preceding secion as one wih a fixed planning horizon and o derive a dynamic programming principle and he corresponding HJB equaion Dynamic programming principle. Le us denoe by A, x he se of admissible decision sraegies ν = c,p,θ for he dynamics of he wealh process wih boundary condiion X = x. For any ν A,x we define he funcional [ τ ] J,x;ν = E,x Ucs,s ds+bzτ,τi {τ T} +WXTI {τ>t} τ >,F and inroduce he associaed value funcion: V,x = sup J,x;ν. ν A,x Given some iniial condiion,x [,T] R, we have ha ν A,x is an opimal conrol if V,x = J,x;ν. The following lemma is he key ool o resaing he conrol problem above as an equivalen one wih a fixed planning horizon. See [1] for a proof. Lemma 3.1. Suppose ha he uiliy funcion U is eiher nonnegaive or nonposiive. If he random variable τ is independen of he filraion F, hen [ ] J,x;ν = E,x Fs,Ucs,s+fs,BZs,s ds+ft,wxt F, where Fs, is he condiional probabiliy for he wage earner s deah o occur a ime s condiional upon he wage earner being alive a ime s and fs, is he corresponding condiional probabiliy densiy funcion. Using he previous lemma, one can sae he following dynamic programming principle, obaining a recursive relaionship for he maximum expeced uiliy as a funcion of he wage earner s age and his wealh a ha ime. See [1] for a proof.
7 OPTIMAL LIFE INSURANCE PURCHASE 7 Lemma 3.2 Dynamic programming principle. For < s < T, he maximum expeced uiliy V, x saisfies he recursive relaion V,x = sup E ν A,x [ exp + λu du Vs,Xs Fu,Ucu,u+fu,BZu,u du ] F. The ransformaion o a fixed planning horizon can hen be given he following inerpreaion: a wage earner facing unpredicable deah acs as if he will live unil ime T, bu wih a subjecive rae of ime preferences equal o his force of moraliy for he consumpion of his family and his erminal wealh HamilonJacobiBellman equaion. The dynamic programming principle enables us o sae he HJB equaion, a secondorder parial differenial equaion whose soluion is he value funcion of he opimal conrol problem under consideraion here. The echniques used in he derivaion of he HJB equaion and he proof of he nex heorem follow closely hose in [2, 1, 11]. Theorem 3.3. Suppose ha he maximum expeced uiliy V is of class C 2. Then V saisfies he HamilonJacobiBellman equaion V,x λv,x+ sup VT,x = Wx where he Hamilonian funcion H is given by H,x;ν = i c p+ + x2 2 r+ ν A,x H,x;ν = N θ n µ n r x V x,x M N 2 θ n σ nm V xx,x+λb m=1, 7 x+ p η, +Uc,. Moreover, an admissible sraegy ν = c,p,θ whose corresponding wealh is X is opimal if and only if for a.e. s [,T] and Pa.s. we have V s,x s λsvs,x s+hs,x s;ν =. 8 Proof. We divide he proof in wo pars: we sar by esablishing he HJB equaion 7 and hen we will prove ha he equaliy 8 holds. Recall ha he wealh process X, [, min{t, τ}], saisfies he sochasic differenial equaion 5. Using Iô s lemma, we obain ha V+h,X+h = V,X+ +h au,xu du+ M m=1 +h b m u,xu dw m u, 9
8 8 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA where he inegrand funcions a and b m, m = 1,...,M, are given by N a,x = V,X+ i c p+x r+ θ n µ n r V x,x X2 V xx,x b m,x = XV x,x V,x = sup E ν A,x M N 2 θ n σ nm 1 m=1 N θ n σ nm, m = 1,...,M. Using he dynamic programming principle of Lemma 3.2 and seing s = +h we ge he ideniy [ +h exp λu du V+h,X+h + +h Fu,Ucu,u+fu,BZu,u du Noing ha for small enough values of h he following inequaliies hold exp +h ] F. 11 λv dv 1 λh±oh 2 12 and combining he equaliy 11 wih he inequaliies 12 above, we obain [ sup E ν A,x + +h Subsiuing 9 in he las equaion we obain 1 λh±oh 2 V+h,X+h V,x Fu,Ucu,u+fu,BZu,u du 1 λh±oh sup E[ 2 V,X+ ν A,x + 1 λh±oh 2 M V,x+ +h m=1 +h +h b m u,xu dw m u ] F. au, Xu du 1 Fu,Ucu,u+λ1 Fu,BZu,u du ] F. Dividing he previous equaion by h and leing h go o zero we obain he equaliy [ N = sup V,x λv,x+ i c p+ r+ θ n µ n r x V x,x ν A,x M N 2 ] + x2 θ n σ nm V xx,x+λbz,+uc, F. 2 m=1
9 OPTIMAL LIFE INSURANCE PURCHASE 9 Leing Z = x+ p η, he dynamic programming equaion becomes [ N = sup V,x λv,x+ i c p+ r+ θ n µ n r x V x,x ν A,x M N 2 + x2 θ n σ nm V xx,x+λb x+ p ] 2 η, +Uc, F. m=1 Finally, noing ha V,x λv,x does no depend on ν, we obain he HJB equaion of 7, hus concluding he proof of he firs par of he heorem. Regarding he second par of he heorem, we sar by leing ν A,x and apply Iô s lemma o exp λv dv Vs,Xs. We obain ha V,x = exp exp M m=1 λv dv WXT exp u λv dv au,xu λuvu,xu du u λv dv b m u,xu dw m u, where a,x and b m,x, m = 1,...,M are as given in 1. From he previous equaliy, we ge V,x = E,x [exp λv dv WXT exp u λv dv au,xu λuvu,xu du ] F. 13 From he definiion of he hazard funcion in 2, we obain ha he condiional probabiliy Fs, for he wage earner s deah o occur a ime s condiional upon he wage earner being alive a ime s is given by Fs, = exp λv dv. 14 Similarly, we obain ha he condiional probabiliy densiy funcion fs, for he deah o occur a ime s condiional upon he wage earner being alive a ime s is given by fs, = λsexp λv dv. 15 Using 14 and 15, we rewrie 13 as V,x = E,x [FT,WXT Using Lemma 3.1, we rearrange 16 o obain Fu,au,Xu λuvu,xu du ] F. 16 V,x = J,x;ν [ ] E,x Fu,V u,xu λuvu,xu+hu,xu;ν du F. 17
10 1 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Consider now an opimal admissible sraegy ν = c,p,θ whose corresponding wealh is X. From equaion 17 we have ha V,x = J,x;ν [ ] E,x Fu,V u,x u λuvu,x u+hu,x u;ν du F and from he HJB equaion 7, we have 18 V u,x u λuvu,x u+hu,x u;ν. 19 Combining 18 and 19, we obain ha ν is opimal if and only if he value funcion V saisfies 8, which concludes he proof. The second par of he heorem above provides a clear approach for he compuaion of opimal insurance, porfolio and consumpion sraegies. In paricular, we obain he exisence of such opimal sraegies under raher weak condiions on he uiliy funcions. Corollary 3.4. Suppose ha he maximum expeced uiliy V is of class C 2 and ha he uiliy funcions U and B are sricly concave wih respec o heir firs variable. Then he Hamilonian funcion H has a regular inerior maximum ν = c,p,θ A,x. Proof. Using he second par of heorem 3.3, an opimal admissible sraegy ν = c,p,θ wihwealhprocessx mussaisfy 8. Therefore, ν musbesuchhahaainsismaximum value. We sar by remarking ha he condiion o obain he maximum for H decouples ino hree independen condiions, as seen in he following: { } sup ν H,x;ν = rx+iv x,x+sup Uc, cv x,x c { +sup λb x+ p } p η, pv x,x { x 2 M N 2 N +sup θ n σ nm V xx,x+ θ 2 m=1 2 } θ n µ n rxv x,x. Therefore, i is enough o sudy he variaion of H wih respec o each one of he variables c, p and θ independenly. Thus, compuing he firsorder condiions for a regular inerior maximum of H wih respec o c, p and θ we obain, respecively, he following hree condiions V x,x+u c c, = V x,x+ λ η B Z x+ p η, = 21 xv x,xα+x 2 V xx,xσσ T θ = R N, where he subscrips in U and B denoe differeniaion wih respec o each funcion s firs variable, α denoes he risk premium 1, and R N denoes he origin of R N. Compuing he second derivaive wih respec o each variable or he Hessian marix in he case of θ, we obain H cc,x;ν = U cc c, H pp,x;ν = λ η 2 B ZZ x+ p η, H θθ,x;ν = x 2 V xx,xσσ T. Noe ha H cc,x;ν is negaive since U is sricly concave on is firs variable and ha H pp,x;ν is negaive since λ is posiive for every T and B is sricly concave on is 22
11 OPTIMAL LIFE INSURANCE PURCHASE 11 firs variable. To see ha H θθ,x;ν is negaive definie, recall ha σσ T is assumed o be nonsingular and, hus, posiive definie. Moreover, noe ha V xx,x mus be negaive: if V xx,x is posiive, hen H would no be bounded above and as a consequence of he HJB equaion eiher V,x or V,x would have o be infiniy, conradicing he smoohness assumpion on V. Therefore, H θθ is negaive definie and H has a regular inerior maximum. 4. The family of discouned CRRA uiliies In his secion we describe he special case where he wage earner has he same discouned CRRA uiliy funcions for he consumpion of his family, he size of his legacy, and he size of his erminal wealh, i.e., from now on we assume ha he uiliy funcions are given by Uc, = e ρcγ γ, BZ, = e ρzγ γ, Xγ WX = e ρt γ, 23 where he risk aversion parameer γ is such ha γ < 1, γ, and he discoun rae ρ is posiive The opimal sraegies. Using he opimaliy crieria provided in heorem 3.3, we obain he following opimal sraegies for discouned CRRA uiliy funcions. Proposiion 4.1. Le ξ denoe he nonsingular square marix given by σσ T 1. The opimal sraegies in he case of discouned consan relaive risk aversion uiliy funcions are given by c 1,x = e x+b where b = D = e = exp p,x = ηd 1x+Db θ 1,x = x1 γ x+bξα, is exp 1 λ e η H = λ+ρ 1 γ 1/1 γ K = λ1/1 γ +1 η γ/1 γ rv+ηv dv ds Hv dv + exp γ Σ 1 γ 2 γ 1 γ r+η Σ = α T ξα 1 2 σt ξα 2. Hv dv Ks ds Proof. Assume ha he uiliy funcions U, B and W are as given in 23. Using he firs order condiions in 21, we obain ha he opimal sraegies depending on he value funcion V are given by c,x = e ρ V x,x 1/1 γ ηe p ρ 1/1 γ V x,x,x = η x λ θ = V x,x xv xx,x ξα. 24
12 12 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA We are now going o find an explici soluion for he HJB equaion 7. We subsiue c, p and θ in he HJB equaion by he opimal sraegies in 24 and combine similar erms o arrive a he following parial differenial equaion V,x λv,x+r+ηx+iv x,x Σ V x,x 2 V xx,x + 1 γ γ e ρ/1 γ KV x,x γ/1 γ =, 25 where Σ and K are as given in he saemen of his proposiion and he erminal condiion is given by VT,x = Wx. 26 We consider an ansaz of he form V,x = a γ x+bγ, 27 and subsiue i in 25 so ha a and b are deermined by he differenial equaion 1 da + a db λ a γ d x+b d γ + [r+ηx+i]a x+b +Σ a 1 γ + 1 γ γ e ρ/1 γ Ka γ/1 γ =. Noe now ha he previous differenial equaion and he erminal condiion 26 decouples ino wo independen boundary value problems for a and b which are given, respecively, by 1 da + r+η λ γ d γ + Σ a+ 1 γ 1 γ γ e ρ/1 γ Ka γ/1 γ = and at = e ρt, 28 db r+ηb+i = d bt =. 29 To find a soluion for he boundary value problem 28, we wrie a in he form a = e ρ e 1 γ, obaining a new boundary value problem for he funcion e of he form de He+K = d et = 1, 3 where K and H are as given in he saemen of his proposiion. Since equaion 3 is a linear, nonauonomous, firs order ordinary differenial equaion, i clearly has an explici soluion of he form e = exp Hv dv + exp Therefore, we obain ha he soluion of 28 is given by a = e exp ρ Hv dv + exp Hv dv Ks ds. Hv dv Ks ds 1 γ. 31
13 OPTIMAL LIFE INSURANCE PURCHASE 13 To find a soluion for he boundary value problem 29, we jus noe ha his is again a linear, nonauonomous, firs order differenial equaion and is soluion is given by b = is exp rv+ηv dv ds 32 as required. Combining 24 wih 27, 31 and 32, we obain ha he opimal sraegies in he case of CRRA uiliies are hen given by c 1,x = e x+b p,x = ηd 1x+Db θ,x = x+b x1 γ ξα, where D is as given in he saemen of his proposiion, which concludes he proof. Noe ha he quaniies b and x + b are of essenial relevance for he definiion of he opimal sraegies in proposiion 4.1. The quaniy b, ha we will refer o as human capial following he nomenclaure inroduced in [6], should be seen as represening he fair value a ime of he wage earner s fuure income from ime o ime T, while he quaniy x + b should be hough of as he full wealh presen wealh plus fuure income of he wage earner a ime. I is hen naural ha hese wo quaniies play a cenral role in he choice of opimal sraegies, since hey deermine he presen and fuure wealh available for he wage earner and his family. Remark 4.2. Noing ha r, η and i are posiive funcions and considering he boundary value problem 32, if r + η is small enough, we can deduce ha human capial funcion b has he following properies: a i is a posiive funcion for all < T; b i is concave. Moreover, we have ha b is eiher: i a decreasing funcion for all [,T]; or ii a unimodal map of, i.e. here exiss some,t such ha b is increasing for all < <, decreasing for all < < T; Furhermore, we have ha he graph of b inersecs he graph of he funcion i/r+η a =. From he explici knowledge of he opimal sraegies, several economically relevan conclusions can be obained. See Fig. 1 for a graphical represenaion of he opimal lifeinsurance purchase as a funcion of age and full wealh x+b of he wage earner. We sar by proving an auxiliary lemma before moving on o he saemen and proof of some of he opimal sraegies properies. Lemma 4.3. Suppose ha for all [,min{t,τ}] he following wo condiions are saisfied: a λ η; b H 1. Then, he inequaliy D < 1 holds for every [,min{t,τ}]. Proof. Recall ha D is given by D = 1 e λ 1/1 γ. η
14 14 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA 1 5 p,x x+b Figure 1. The opimal lifeinsurance purchase for a wage earner ha sars working a age 25 and reires 4 years laer. The parameers of he model were aken as N = M = 2, i = 5exp.3, r =.4, ρ =.3, γ = 3, λ =.1+exp , η = 1.5λ, µ 1 =.7, µ 2 =.11, σ 11 =.19, σ 12 =.15, σ 21 =.17 and σ 22 =.21. Using condiion b and noing ha K is posiive for all [,min{t,τ}], we have ha e = exp Hv dv + exp Hv dv Ks ds exp 1 dv + exp 1 dv Ks ds > 1. Puing ogeher he previous inequaliy and condiion a, we obain he required inequaliy. The nex resul provides a qualiaive characerizaion of he opimal life insurance purchase sraegy. Corollary 4.4. Assume ha he condiions of lemma 4.3 are saisfied. Then, he opimal insurance purchase sraegy p,x has he following properies: a i is a decreasing funcion of he wealh x; b i is an increasing funcion of he wage earner s human capial b; c i is negaive for suiable pairs of wealh x and age ; d if he wage earner s wealh x is small enough and η is nondecreasing, he funcion p,x b has he same monooniciy as he human capial funcion b. Proof. Recall from proposiion 4.1 ha he opimal insurance purchase sraegy p,x is given by for all [,min{t,τ}]. p,x = ηd 1x+Db 33
15 OPTIMAL LIFE INSURANCE PURCHASE 15 Iems a and b follow from lemma 4.3, since D is a posiive funcion such ha D < 1 for all [,min{t,τ}]. For he proof of iem c, noe ha p,x is negaive for all,x [,T] R + such ha x > = D 1 D b λ 1/1 γ eη 1/1 γ λ 1/1 γb >, and posiive oherwise. Iem d follows from 33 and he fac ha η is a nondecreasing. Some commens regarding he assumpions in lemma 4.3 and corollary 4.4 seem necessary. Saring wih condiion a, he life insurance company mus esablish he premiuminsurance η in such a way ha λ η in order o make a profi he insurance policy being fair whenever λ η. Regarding condiion b, we noe ha he quaniies r, ρ, η and λ are usually very small in he real world and, moreover, he relaive risk aversion of he wage earner is negaive in general. This is consisen wih he assumpion ha H is bounded above by some posiive consan. Apar from sudying how opimal life insurance purchase varies wih age and wealh, i is also relevan o undersand how he remaining parameers which define he financial and insurance markes influence life insurance purchase. Corollary 4.5. Wih all oher parameers, including and x consan, he opimal life insurance purchase rae p,x is an increasing funcion of he discoun rae ρ. Proof. To sudy he influence of he discoun rae ρ on he opimal opimal life insurance purchase, we consider wo differen values of ρ and compare he corresponding values of he opimal life insurance purchase. We disinguish he funcions associaed wih each of he wo parameer values by heir subscrip. Assume ha ρ 1 and ρ 2 are such ha ρ 1 < ρ 2. Recall he definiions of p,x, H, e and D given in proposiion 4.1. Then, i is clear ha he inequaliies H 1 < H 2 e 1 > e 2 D 1 < D 2 hold for all [,min{t,τ}], where he subscrips correspond o ρ 1 and ρ 2 in an obvious manner. Rewriing p,x as i follows from he preceding inequaliies ha p,x = ηdx+b x, p 1,x < p 2,x for all [,min{t,τ}], concluding he proof of he saemen. Remark 4.6. The variaion of he opimal life insurance purchase rae p,x wih respec o he ineres rae r, he risk aversion parameer γ, he hazard rae λ and he insurance premiumpayou raio η is nonrivial. However, by sudying he funcion p,x given in proposiion 4.1 we can make he following observaions: i p,x is a decreasing funcion of he ineres rae r, excep for large values of x and close enough o T; ii p,x is a decreasing funcion of he risk aversion parameer γ, excep for values of close enough o T;
16 16 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA iii p,x is an increasing funcion of he hazard rae λ and he insurance premiumpayou raio η for small enough values of wealh x and a decreasing funcion for large values of x. The exra risky securiies in our model inroduce novel feaures o he wage earner s porfolio managemen, as is exemplified in he following resul. Corollary 4.7. Le ξ denoe he nonsingular square marix given by σσ T 1 and le ξα n denoe he nh componen of he vecor ξα. The opimal porfolio process θ,x = θ 1,...,θ N is such ha for every n {1,...,N}: a θ n has he same sign as ξα n ; b θ n is a decreasing funcion of he oal wealh x if ξα n > for all [,min{t,τ}] and an increasing funcion of x if ξα n < for all [,min{t,τ}]; c θ n is an increasing funcion of he wage earner s human capial b if ξα n > for all [,min{t,τ}] and a decreasing funcion of b if ξα n < for all [,min{t,τ}]. Furhermore, for every n,m {1,...,N} he following equaliies hold lim x +θ n,x = + lim x θ n,x = ξα n 1 γ θ lim n,x x + θm,x = ξα n ξα m lim T θ n,x = ξαt n 1 γ Proof. Iems a, b and c on he firs par of he corollary follow from he form of θn, n {1,...,N}, given in he saemen of proposiion 4.1 and posiiviy of b. The limiing behaviours on he second par of he corollary also follow from he form of θn, n {1,...,N}. Remark 4.8. Corollary 4.7 is a muual fund resul: he relaive proporions among he risky securiies are independen of all parameers excep for he ineres rae and he risky asses appreciaion raes and volailiies since, for any n,m {1,...,N} we have θ n,x θ m,x = ξα n ξα m. The quaniies ξα n, n {1,...,N} can be hough as weighed risk premiums for he risky asses S 1,...,S N, where he weighs are provided by quadraic funcions on he coefficiens of he marix of riskyasses volailiies σ. Under he assumpion ha he weighed risk premiums ξα n are posiive for all n {1,...,N} and [,min{t,τ}], we obain he following ineresing consequence of he previous corollary. Corollary 4.9. Le ξ denoe he nonsingular square marix given by σσ T 1 and le ξα n denoe he nh componen of he vecor ξα. Assume ha for every n {1,...,N} we have ξα n > for all [,min{t,τ}]. Then he opimal sraegy for wage earners wih small enough wealh x is o shor he riskfree securiy and hold higher amouns of risky asses. Proof. The corollary follows from corollary 4.7 and he fac ha he wage earner has no budge limiaions, hus allowing him o ge ino shor posiions on he risk free asse S of arbirary size. We conclude his session wih a resul concerning some qualiaive properies of he opimal consumpion sraegy. Is proof follows rivially from he form of he opimal consumpion c,x given in proposiion 4.1. Corollary 4.1. The opimal consumpion rae c is an increasing funcion of boh he wealh x and he human capial b..
17 OPTIMAL LIFE INSURANCE PURCHASE The ineracion beween life insurance purchase and porfolio managemen. In his secion we compare he opimal lifeinsurance sraegies for a wage earner who faces he following wo siuaions: a in he firs case, we assume ha he wage earner has access o an insurance marke as described above and ha his goal is o maximize he combined uiliy of his family consumpion for all min{t,τ}, his wealh a reiremen dae T if he lives ha long, and he value of his esae in he even of premaure deah. The opimal sraegies for he wage earner in his seing are given in Proposiion 4.1. b in he second case, we assume ha he wage earner is wihou he opporuniy of buying life insurance. His goal is o maximize he combined uiliy of his family consumpion for all min{t,τ} and his wealh a reiremen dae T if he lives ha long. Similarly o wha we have done previously, we ranslae his siuaion o he language of sochasic opimal conrol and derive explici soluions in he case of discouned CRRA uiliies. We concenrae on he case b described above for he momen. Similarly o wha was done in case a, his problem can be formulaed by means of opimal conrol heory. The wage earner s goal is hen o maximize a new cos funcional subjec o: he dynamics of he sae variable, i.e., he dynamics of a wealh process X given by X = x+ is c s ds+ N n= θnsx s ds n s, S n s where [,min{t,τ}] and x is he wage earner s iniial wealh. consrains on he remaining conrol variables, i.e., he consumpion process c and he reduced porfolio process θ = θ 1,,θ N R N ; and boundary condiions on he sae variables. Le us denoe by A x he se of all admissible decision sraegies, i.e. all admissible choices for he conrol variables ν = c,θ R N+1. The dependence of A x on x denoes he resricion imposed on he wealh process by he boundary condiion X = x. The wage earner s problem can hen be saed as follows: find a sraegy ν = c,θ A x which maximizes he expeced uiliy [ ] T τ V x = sup E,x Uc s,s ds+wx TI {τ>t}, 34 ν A x where Uc, is again he uiliy funcion describing he wage earner s family preferences regarding consumpion in he ime inerval [,min{t,τ}] and BX, is he uiliy funcion for he erminal wealh a ime = T τ. As before, we resric ourselves o he special case where he wage earner has he same discouned CRRA uiliy funcions for he consumpion of his family and his erminal wealh given in 23. The opimal sraegies are given in he nex resul Proposiion Le ξ denoe he nonsingular square marix given by σσ T 1. The opimal sraegies for problem 34 in he case where Uc, and BX, are he discouned consan relaive risk aversion uiliy funcions in 23 are given by c,x = 1 e x+b θ,x = 1 x1 γ x+b ξα,
18 18 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA where b = e = exp H = λ+ρ 1 γ is exp H v dv rv dv ds + exp γ Σ 1 γ 2 γ 1 γ r H v dv ds and Σ is as given in he saemen of proposiion 4.1. We skip he proof of he previous proposiion, since i is of he same naure as he proofs of heorem 3.3 and proposiion 4.1. We should poin ha i would be preferable from an economic sandpoin o maximize also he uiliy of wealh a he ime of premaure deah. In fac, i is easy o check ha his corresponds o he addiion of a consrain of he form p,x = o our original problem. However, he corresponding HJB equaion would conain an exra erm of he form λ x γ /γ. The presence of his addiional erm in he HJB equaion makes i impossible for us o obain closed form soluions. On he oher hand, a sraegy opimizing he final wealh a reiremen should be close o an opimal sraegy which maximizes also he wealh for he case of an evenual deah of he wage earner before reiremen ime. We plan o address such a comparison in fuure research. If we make he wage earner income i equal o zero, hen we ge a soluion which is close o Meron s classical soluion, bu i sill depends on he hazard funcion λ, i.e., even in he absence of a life insurance policy he uncerainy regarding wage earner s lifeime sill plays a role on he deerminaion of he opimal consumpion and invesmen sraegies. Proposiions 4.1 and 4.11 provide us wih opimal porfolio processes for he wo seings a and b described above. In he nex heorem we show how hese opimal porfolio processes compare. Theorem Le ξ denoe he nonsingular square marix given by σσ T 1 and ξα n he nh componen of he vecor ξα. For each n {1,...,N}, we have ha θ n,x > θ n,x if and only if ξα n >. Proof. Recall he definiions of θ and θ from proposiions 4.11 and 4.1, respecively. Noe ha θn,x θ 1 1 n,x = x1 γ x+b ξα n x1 γ x+bξα n = ξα n x1 γ b b. 35 Using he definiions of b and b given in he saemens of proposiions 4.11 and 4.1, respecively, we obain ha b b = isexp rv dv 1 exp ηv dv ds. 36 Since η is a posiive funcion, we ge ha and herefore, he inequaliy 1 exp ηv dv > ηv dv > 37
19 OPTIMAL LIFE INSURANCE PURCHASE 19 holds for all s T. Therefore, combining 35, 36 and 37 and recalling ha γ < 1, we obain ha he sign of θn,x θ n,x is he same as he sign of ξα n for all [,min{t,τ}]. The economic implicaions of he heorem above are made clear in he following resul. Corollary Le ξ denoe he nonsingular square marix given by σσ T 1 and ξα n he nh componen of he vecor ξα. Assume ha for every n {1,...,N} we have ha ξα n >. Then, he opimal porfolio of a wage earner wih he possibiliy of buying a life insurance policy is more conservaive han he opimal porfolio of he same wage earner if he does no have he opporuniy o buy life insurance. 5. Conclusions We have inroduced a model for opimal insurance purchase, consumpion and invesmen for a wage earner wih an uncerain lifeime wih an underlying financial marke consising of one riskfree securiy and a fixed number of risky securiies wih diffusive erms driven by mulidimensional Brownian moion. When we resric ourselves o he case where he wage earner has he same discouned CRRA uiliy funcions for he consumpion of his family, he size of his legacy and his erminal wealh, we obain explici opimal sraegies and describe new properies of hese opimal sraegies. Namely, we obain economically relevan conclusions such as: i a young wage earner wih smaller wealh has an opimal porfolio wih larger values of volailiy and higher expeced reurns; andii a wage earner who can buy life insurance policies will choose a more conservaive porfolio han a similar wage earner who is wihou he opporuniy o buy life insurance. I is worh noing ha he model described in his paper can be improved by adding furher ingrediens such as, for insance, some form of sochasiciy on he income funcion or he hazard rae, some consrains on he possibiliy of rading of socks on margin, he exisence of addiional life insurance producs on he marke or even adding some correlaion beween he wage earner s moraliy rae and he underlying financial marke. We plan o address a leas some of hese issues in a fuure publicaion. Acknowledgmens We hank he Calouse Gulbenkian Foundaion, PRODYNESF, POCTI, and POSI by FCT and Minisério da Ciência, Tecnologia e Ensino Superior, CEMAPRE, LIAADINESC Poro LA, Cenro de Maemáica da Universidade do Minho and Cenro de Maemáica da Universidade do Poro for heir financial suppor. D. Pinheiro s research was suppored by FCT  Fundação para a Ciência e Tecnologia program Ciência 27. I. Duare s research was suppored by FCT  Fundação para a Ciência e Tecnologia gran wih reference SFRH / BD / 3352 / 28. References [1] C. BlancheScallie, N. El Karoui, M. Jeanblanc, and L. Marellini. Opimal invesmen decisions when imehorizon is uncerain. Journal of Mahemaical Economics, 4411: , 28. [2] W. H. Fleming and H. M. Soner. Conrolled Markov Processes and Viscosiy Soluions. SpringerVerlag, New York, 2nd ediion, 26. [3] I. Karazas and S. Shreve. Mehods of Mahemaical Finance. Springer, [4] R. C. Meron. Lifeime porfolio selecion under uncerainy: he coninuous ime case. Review of Economics and Saisics, 51: , [5] R. C. Meron. Opimum consumpion and porfolio rules in a coninuousime model. Journal of Economic Theory, 3: , [6] S. R. Pliska and J. Ye. Opimal life insurance purchase and consumpion/invesmen under uncerain lifeime. Journal of Banking and Finance, 31: , 27. [7] S. R. Pliska and J. Ye. Opimal life insurance purchase, consumpion and invesmen. Preprin, 29.
20 2 I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA [8] S. F. Richard. Opimal consumpion, porfolio and life insurance rules for an uncerain lived individual in a coninuousime model. Journal of Financial Economics, 2:187 23, [9] M. E. Yaari. Uncerain lifeime, life insurance and he heory of he consumer. Review of Economic Sudies, 32:137 15, [1] J. Ye. Opimal Life Insurance Purchase, Consumpion and Porfolio Under an Uncerain Life. PhD hesis, Universiy of Illinois a Chicago, 26. [11] J. Yong and X. Y. Zhou. Sochasic Conrols: Hamilonian Sysems and HJB Equaions. SpringerVerlag, New York, I. Duare Cenro de Maemáica da Universidade do Minho, Braga, Porugal address: D. Pinheiro CEMAPRE, ISEG, Technical Universiy of Lisbon, Lisbon, Porugal address: A. A. Pino LIAADINESC Poro LA and Dep of Mahemaics, Faculy of Science, Universiy of Poro, Porugal address: S. R. Pliska Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA address:
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