Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime Sanley R. Pliska a,, a Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Jinchun Ye b b Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA Absrac In his paper we consider opimal insurance and consumpion rules for a wage earner whose lifeime is random. The wage earner is endowed wih an iniial wealh, and he also receives an income coninuously, bu his may be erminaed by he wage earner s premaure deah. We use dynamic programming o analyze his problem and derive he opimal insurance and consumpion rules. Explici soluions are found for he family of CRRA uiliies, and he demand for life insurance is sudied by examining our soluions and doing numerical experimens. JEL Classificaion: C51, C61, D91, G11, G22 Key words: Life Insurance, Invesmen/Consumpion Model, HJB Equaion, CRRA Uiliies 1 Inroducion In his paper we consider he opimal life insurance purchase and consumpion sraegies for a wage earner subjec o moraliy risk in a coninuous ime economy. Decisions are made coninuously abou hese wo sraegies for all ime Corresponding auhor. Tel.: +1-312-996-717; fax: +1-312-996-717. Email addresses: srpliska@uic.edu (Sanley R. Pliska), jinch ye@yahoo.com (Jinchun Ye). Preprin submied o Journal of Banking and Finance
[, T ], where he fixed planning horizon T can be inerpreed as he reiremen ime of he wage earner. The wage earner also receives his/her income a rae y() 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 life insurance offered has an insananeous erm: he bigger he insurance premium rae paid by he wage earner, he bigger he claim paid o his/her family upon premaure deah. Income no consumed or used o buy insurance is invesed a a riskless ineres rae. The problem is o find he sraegies ha are bes in erms of boh he family s consumpion for all T as well as he erminal ime-t wealh. Beginning in he 196 s, many researchers consruced quaniaive models o analyze he demand for life insurance and he rae of invesmen for an individual under uncerainy. We are going o review some papers which conribued o his effor. Yarri (1965) is a saring poin for modern research on he demand for life insurance. Yarri considered he problem of life insurance under an uncerain lifeime for an individual; his is he sole source of he uncerainy. The individual s objecive was o maximize T E[ U(c())d], where T, he individual s lifeime, is he random variable which akes values on [, T ], T is some given posiive number which represens for he maximum possible lifeime for he consumer, and U is a uiliy funcion. Noe ha he horizon is random in he above funcional, bu he above funcional can be rewrien ino he following equivalen form: T F ()U(c())d where F () is he probabiliy ha he individual will be alive a ime. Noe ha now he horizon is a fixed ime. This simple idea provides a useful mehod o analyze he opimizaion problem wih a random life ime. Since hen, numerous lieraure has been buil on Yaari s pioneering work. However, Leung (1994) poined ou ha Yaari s model canno have an inerior soluion which lass unil he maximum lifeime for he opimal consumpion. Wha s more, we can no employ dynamic programming o analyze his kind of model since we can no appropriaely define he erminal condiion for he HJB equaion wihin he frame of Yaari s model (e.g., Ye, 26). Campbell (198), Lewis (1989), and Iwaki and Komoribayashi (24) examined he demand for life insurance from differen perspecives. Campbell (198) considered he insurance problem in a very shor ime, [, + ], used a local 2
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 has no informaion abou he presen value of he fuure income even if one knows 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 he beneficiaries. Iwaki and Komoribayashi (24) considered he opimal insurance from he perspecive of households using a maringale mehod. Households are only allowed o buy life insurance a ime in heir model, and so he households canno change he amoun of life insurance regardless of wha happens afer ime, alhough he marke for life insurance exiss. Then how do we incorporae he uncerain life ino he financial model such ha i is heoreically consisen and has pracical use. In our paper we use he model of uncerain life found in reliabiliy heory; his kind of model is commonly used for indusrial life-esing and acuarial science. We se up he wage earner s objecive (see (13) below) and conver he random horizon o fixed horizon (see Lemma 1 below). Then we employ he echnique of dynamic programming o analyze our model. This paper is organized as follows. In he nex secion we se up our model. In secion 3, we sae our HJB equaion and hen derive he opimal feedback conrol. In secion 4 we obain he explici soluions for he family of CRRA uiliies, and examine he economic implicaions of our soluions. We conclude wih some final remarks in secion 5. 2 The Model The coninuous-ime economy consiss of a financial marke and an insurance marke. 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 S () = s is a given posiive consan and where r() : [, T ] R + is a coninuous deerminisic funcion. We 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 ). Now suppose ha he random variable τ has a probabiliy disribuion wih underlying probabiliy densiy funcion f() and disribuion funcion given 3
by F () P (τ < ) = f(u)du. (2) The funcion F (), which is called he survivor funcion in he survival analysis lieraure (e.g., Colle, 23), is defined o be he probabiliy ha he survival ime is greaer han or equal o, i.e., F () P (τ ) = 1 F (). (3) The survivor funcion can herefore be used o represen he probabiliy ha he wage earner survives from he ime origin o some ime beyond. The hazard funcion herefore represens he insananeous deah rae for he wage earner surviving o ime, and i is defined by P ( τ < + δ τ ) λ() lim. (4) δ δ From his definiion we can obain some useful relaionships beween f() and λ(). Noe ha he condiional probabiliy in he definiion of he hazard funcion in equaion (4) is P ( τ<+δ) which is equal o. Then P (τ ) I hen follows ha λ() = lim δ F ( + δ) F () δ λ() = d d (ln F ()), in which case he survivor funcion is given by F (+δ) F () F () 1 F () = f() F (). (5) F () = exp{ λ(u)du} (6) and he probabiliy densiy funcion is relaed o he hazard rae by f() = λ() exp{ λ(u)du}. (7) From (5) and (7), we see here is a correspondence beween hazard funcion and densiy funcion. Hence hroughou our paper, we suppose ha he hazard funcion λ() is given, and λ() : [, ] R + is a coninuous, deerminisic funcion, which saisfies λ()d = (his condiion ensures he funcion f() given by (7) is a probabiliy densiy funcion). Then he probabiliy densiy of τ is given by (7) and he survivor funcion is given by (6). We now inroduce some noaion associaed wih he random variable τ. (i) Denoe by f(s, ) he condiional probabiliy densiy for he deah a ime 4
s condiional upon he wage earner being alive a ime s, so ha f(s, ) f(s) s F () = λ(s) exp{ λ(u)du}. (8) (ii) Denoe by F (s, ) he condiional probabiliy for he wage earner be alive a ime s condiional upon being alive a ime s, so ha F (s, ) F (s) s F () = exp{ λ(u)du}. (9) Now ha we have modeled he ime of he wage earner s deah, we are going o describe he life insurance marke. We assume ha he 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 p η() company pays an insurance amoun. Hence his is erm insurance where he lengh of he erm is infiniesimally small. Here η : [, T ] R + is a coninuous, deerminisic, specified funcion ha is called as he premiuminsurance raio. Suppose he wage earner is endowed wih he iniial wealh x and will receive he income a rae y() during he period [, min{t, τ}]. Here he specified funcion y : [, T ] R + is Borel measurable and saisfies T y(u)du <. Define c() Consumpion rae a ime. p() Premium rae a ime. A(x) he se of all such pairs of funcions (c(.), p(.)). Given he consumpion process c, he premium rae process p, and income process y, he wealh process X() on [, min{t, τ}] is defined by X() = x c(s)ds p(s)ds + y(s)ds + X(s) S (s) ds (s). (1) Wriing his equaion as a differenial equaion using (1), dx() = r()x()d c()d p()d + y()d on min{τ, T }. (11) If he wage earner dies a ime, < T, hen his/her family will ge he insurance amoun p(). Thus he wage earner s oal legacy when he/she dies η() 5
a ime wih wealh X() is Z() = X() + p() η() on {τ = }. (12) The wage earner s problem is o choose consumpion and life insurance purchase sraegies so as o maximize expeced uiliy from consumpion, from he legacy if he/she die before ime T, and from he erminal wealh if he/she is alive a ime T. Thus he maximum expeced uiliy is expressed as V (x) sup (c,p) A(x) T τ E,x [ U(c(s), s)ds + B(Z(τ), τ)1 {τ T } + L(X(T ))1 {τ>t } ], (13) where T τ min{t, τ}, 1 A denoes he indicaor funcion of even A meaning ha 1 A = 1 if A is rue and 1 A = oherwise, U(c,.) is he uiliy funcion for consumpion and is assumed o be sricly concave in c, B(Z,.) is he uiliy funcion for he legacy and is assumed o be sricly concave in Z, and L(X) is he uiliy funcion for he erminal wealh and is assumed o be sricly concave in X. 3 Dynamic Programming In his secion we will use he echnique of dynamic programming o derive he HJB equaion and hen derive he opimal feedback conrol from his equaion. To use he dynamic programming principle, we resae (13) in a dynamic programming form. For any (c, p), define T τ J(, x; c, p) E,x [ U(c(s), s)ds + B(Z(τ), τ)1 {τ T } +L(X(T ))1 {τ>t } τ > ] (14) and V (, x) sup J(, x; c, p). (15) {c,p} A(,x) Here he definiion of A(, x) is similar o he definiion A(x), excep ha he saring ime is ime and he wealh a ime is x. Lemma 1 Suppose ha U(, ) is nonnegaive or nonposiive. Then 6
T J(, x; c, p) = [f(u, )B(Z(u), u) + F (u, )U(c(u), u)]du + F (T, )L(X(T )) (16) where f(u, ) is given by (8) and F (u, ) is given by (9). We refer o Ye (26) for he proof of Lemma 1. From Lemma 1, we know ha he wage earner who faces 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 his/her consumpion and erminal wealh. From he mahemaical poin of view, his lemma enables us o conver he problem of opimizaion wih a random erminal ime o a problem wih a fixed erminal ime. Define Ψ(, x; c, p) (r()x + y() c p)v x (, x) + λ()b(x + p, ) + U(c, ). η() Using he echnique of dynamic programming, i becomes clear ha V (, x) mus saisfy he following HJB equaion: V (, x) λ()v (, x) + sup (c,p) Ψ(, x; c, p) = V (T, x) = L(x). (17) Furhermore, opimaliy is guaraneed by he following heorem: Theorem 2 Suppose he funcion V is smooh. Then an admissible pair (c, p ), whose corresponding wealh is X, is opimal if and only if V (, X ()) λ()v (, X ()) + sup Ψ(, X (); c, p) =. (c,p) A(,X ()) (18) This is a kind of deerminisic conrol problem. We refer o Fleming and Soner (1993) or Yong and Zhou (1999) for deriving he HJB equaion and he proof of Theorem 2. We are now in a posiion o derive he opimal insurance and consumpion rules. According o Theorem 2, if an admissible pair (c, p ) is opimal, hen 7
= V (, x) λ()v (, x) + Ψ(, x; c, p ) = V (, x) λ()v (, x) + sup Ψ(, x; c, p) (19) (c,p) The firs-order condiions for a regular inerior maximum o (19) are and Ψ c (, x; c, p ) = = V x (, x) + U c (c, ), (2) Ψ p (, x; c, p ) = = V x (, x) + λ() η() B Z(x + p, ). (21) η() A se of sufficien condiions for a regular inerior maximum is Ψ cc = U cc (c, ) <, Ψ pp = λ() η 2 () B ZZ(Z, ) <. Noe ha since our uiliy funcions are assumed o be sricly concave, he above wo condiions are auomaically saisfied. 4 The Case of Consan Relaive Risk Aversion In his secion we derive he explici soluion for he case where he wage earner has he same consan relaive risk aversion for he consumpion, he beques, and he erminal wealh. Assume ha γ < 1 and ρ >. Le U(c, ) = e ρ γ cγ, B(Z, ) = e ρ γ Zγ, and L(x) = e ρt γ xγ. From (2) we have ha c 1 () = [ V x e ] 1 ρ. (22) Similarly, from (21) we have ha x + p () η() = ( 1 V x e ρ λ() η() ) 1. (23) We now plug (22)-(23) in (17) and combine he similar erms, hereby obaining 8
Define V λ()v + [(r() + η())x + y()]v x + 1 γ γ e ρ λ 1 () [ + 1]V η γ () γ x =. K() λ 1 () η γ () + 1. (25) Then noe he erminal condiion and rewrie (24) as follows (24) V λ()v + [(r() + η())x + y()]v x + e ρ γ γ K()V x = V (T, x) = e ρt γ x γ. (26) To solve his equaion, ake as a rial soluion V (, x) = e ρ a() (x + b()) γ (27) γ where a(.) and b(.) are deerminisic funcions o be deermined. By subsiuion of he rial soluion ino (26) we see ha we mus have and b() = T s y(s) exp{ [r(v) + η(v)]dv}ds (28) T T s a() = [exp{ H(v)dv} + exp{ H(v)dv}K(s)ds] (29) where H() is defined as H() λ() + ρ 1 γ γ (r() + η()). (3) 1 γ From (22)-(23) he opimal consumpion and life insurance rules can now be explicily wrien as c 1 () = (x + b()) (31) [a()] 1 and Z () = x + p () η() = (λ() η() ) 1 1 (x + b()). (32) [a()] 1 The funcion b() represens he fair value a ime of he wage earner s fuure income from ime o ime T ; his formulae is differen from he one wihou 9
moraliy risk. From (32) we obain he life insurance rule: p () = η(){[( λ() η() ) 1 1 [a()] 1 1]x + ( λ() η() ) 1 1 [a()] 1 b()}. (33) To explore he economic implicaions of (33), we inroduce he following lemma: Lemma 3 If λ() η(), and H() 1, hen D ( λ() η() ) 1 1 [a()] 1 < 1 Proof. From (29) we ge [a()] 1 Now D < 1 is obvious. T exp{ 1dv} + T s exp{ 1dv}K(s)ds > 1. We should poin ou he condiions in Lemma 3 are reasonable in he real world, because he parameers r, ρ, η and λ are very small in he real world and, moreover, he relaive risk aversion of he wage earner is negaive in general. Hence he assumpion H() 1 for all [, T ] is realisic. Furhermore, he life insurance company mus esablish he premium-insurance raio η() > λ() in order o make a profi. In fac, he insurance is fair when η() = λ(); ha is because he expeced profi rae of he life insurance is hen p() η() λ() η() =. From (33), we obain he following insurance principles: Under assumpions of Lemma 3, he curren wealh of he wage earner has a negaive effec on his life insurance purchase. The more wealhy he is, he less incenive he has o buy life insurance. The fuure income of he wage earner has a posiive effec on his life insurance purchase. The bigger he fuure income, he more he incenive o buy life insurance. In he following, we are going o do some numerical experimens o examine he economic implicaions of our explici soluions. Le T = 4 years, and he wage earner s iniial wealh is. The baseline parameers are given in he Table 1. We vary one and only one parameer each ime and hen produce Fig. 1 Fig. 6. The baseline case is always shown wih a solid line; a perurbed case is always shown wih a doed line. From Fig. 1, we can see ha he wage 1
earner buys less life insurance overall as he ineres rae r increases. We know ha he wage earner ends o save more money for he fuure as he ineres rae increases. In oher words, he wage earner spends less o consume and o buy life insurance as he ineres rae increases. Hence he ineres rae has a negaive effec on he life insurance purchase for he wage earner. From Fig. 2, he uiliy discoun rae ρ has a posiive effec on he life insurance purchase for he wage earner. We know ha he wage earner ends o consume more and save less as he uiliy discoun rae increases. So he wage earner needs o spend more o buy life insurance in order o proec his/her family. From Fig. 3, we see ha he risk-aversion parameer γ has a negaive effec on he life insurance purchase for he wage earner. Noe ha he wage earner s relaive risk aversion is, so he wage earner wih more risk aversion ends o buy more life insurance. From Fig. 4, he hazard rae λ has a posiive effec on he life insurance purchase. Noe he moraliy risk becomes larger as he hazard rae λ increases, so he wage earner ends o spend more o buy life insurance when he/she faces a larger moraliy risk. How does he premium-insurance raio η() affec he life insurance purchase? We firs inroduced he loading facor which is defined as L() η(). From he λ() perspecive of life insurance companies, he larger he expeced reurn on he life insurance business, he larger he loading facor. Noe he baseline loading facor is always 1, and he perurbed loading facors are 3 and 4 in Fig. 5 and Fig. 6, respecively. From Fig. 5, we can see ha he wage earner will spend more o buy life insurance excep for a shor ime around he erminal ime as he loading facor increases o 3. Bu from Fig. 6, as he loading facor increases o 4, he enhusiasm for wage earners o buy life insurance is diminished. Hence, as he loading facors is increased moderaely from one, he wage earner will spend more on life insurance, alhough he payou upon premaure deah will decline. Bu as he loading facor increase beyond a cerain poin, he wage earner s expendiure on insurance will acually decline. So from he perspecive of insurance companies, here is an opimal loading facor. For our Table 1 The Baseline Parameers Parameers Value Income y() 4, e.3 Ineres Rae r.4 Uiliy Discoun Rae ρ.3 Hazard Rae λ().5 +.1125 Premium-Insurance Raio η().5 +.1125 Risk-Aversion Parameer γ 3 11
Table 2 Effecs of Parameers on Life Insurance Purchase Parameers Curren Wealh x Income y Ineres Rae r Uiliy Discoun Rae ρ Hazard Rae λ Premium-Insurance Raio η Risk-Aversion Parameer γ Effecs on Life Insurance Purchase Negaive Posiive Negaive Posiive Posiive Undeermined Negaive case, insurance company should choose he loading facor around 3. o make maximal profi. Now we summarize our observaions ino he able 2. 5 Conclusions We model he opimal insurance purchase and consumpion under an uncerain lifeime for a wage earner in a simple economic environmen, obaining successfully he explici soluions in he case of CRRA uiliies, and explaining how facors affec he demand of life insurance purchase via numerical experimens. Our numerical experimens show in paricular ha insurance companies should increase he loading facor moderaely from one in order o acquire maximal possible profi from he life insurance business. I should be noed from Fig. 5 and Fig. 6 ha he premium paymen rae p can be negaive, especially when he wage earner is nearing reiremen. This means he wage earner is collecing money a he rae p, bu his or her esae pays ou a lump sum upon his or her deah. Bu his negaive paymen rae is limied since he beques is nonnegaive from (32). This requiremen rules ou he possibiliy ha he wage earner migh be indebed o insurance companies if premaure deah occurs. For example, if he wage earner s curren wealh is $1,, and he premium-insurance raio is.1, hen his or her paymen rae mus saisfy p $1. This amoun relaive o he curren wealh is ypically very small, bu he penaly is very large if premaure deah occurs. So allowing he wage earner o sell life insurance changes his or her behavior lile. I is reasonable ha he wage earner would find i opimal o sell life insurance close o reiremen ime. Fuure research will invesigae his problem wih he addiion of a nonnegaive consrain on he premium 12
paymen rae. Nowadays, he merger of he capial and insurance markes is acceleraing due o financial services deregulaion and globalizaion of financial services. In a forhcoming paper, we will examine opimal life insurance purchase, consumpion and porfolio invesmen rules for a wage earner wih an uncerain lifeime in a more complex economic environmen where invesmen opporuniies are sochasic. References [1] Campbell, R. A., 198. The Demand for Life Insurance: An Applicaion of he Economics of Uncerainy, Journal of Finance 35, 1155-1172. [2] Colle, D., 23. Modelling Survival Daa in Medical Research, Second Ediion, Chapman&Hall. [3] Fischer, S., 1973. A Life Cycle Model of Life Insurance Purchase, Inernaional Economics Review 14, 132-152. [4] Fleming, W.H., Soner, H.M., 1993. Conrolled Markov Processes and Viscosiy Soluions, Springer-Verlag, New York. [5] Folland,G.B., 1999. Real Analysis: Modern Techniques and Their Applicaions, Second Ediion, Wiley-Inerscience. [6] Iwaki, H., Komoribayashi, K., 24. Opimal Life Insurance for a Household, Inernaional Finance Workshop a Nanzan Universiy(Preprin). [7] Lewis, Frank D., 1989. Dependens and he Demand for Life Insurance, American Economic Review 79, 452-466. [8] Leung, S.F., 1994. Uncerain Lifeime, he Theory of he Consumer, and he Life Cycle Hypohesis, Economerica 62, 1233-1239. [9] Meron, R.C., 1969. Lifeime Porfolio Selecion under Uncerainy: The Coninuous Time Case, Review of Economics and Saisics 51, 247-257. [1] Meron, R.C., 1971. Opimum Consumpion and Porfolio Rules in a Coninuous-Time Model, Journal of Economic Theory 3, 372-413. [11] Richard, S.F., 1975. Opimal Consumpion, Porfolio and Life Insurance Rules for an Uncerain Lived Individual in a Coninuous Time Model, Journal of Financial Economics 2, 187-23. [12] Yaari, M.E., 1965. Uncerain Lifeime, Life Insurance, and he Theory of he Consumer, Review of Economic Sudies 32, 137-15. [13] Ye, J., 26. Opimal Life Insurance Purchase, Consumpion, and Porfolio under an Uncerain Life, Ph.D. Disseraion, Universiy of Illinois a Chicago, Chicago. [14] Yong,J., Zhou, X., 1999. Sochasic conrols: Hamilonian sysems and HJB equaions, Springer-Verlag, New York. 13
Opimal Life Insurance(in Thousand Dollars) 14 12 1 8 6 4 2 Comparison of Opimal Life Insurance Rules r=.4 r=.8 5 1 15 2 25 3 35 4, Time(in Years) Fig. 1. Differen Ineres Raes r Opimal Life Insurance(in Thousand Dollars) 18 16 14 12 1 8 6 4 2 Comparison of Opimal Life Insurance Rules rho=.3 rho=.6 5 1 15 2 25 3 35 4, Time(in Years) Fig. 2. Differen Uiliy Discoun Raes ρ 14
Opimal Life Insurance(in Thousand Dollars) 14 12 1 8 6 4 2 Comparison of Opimal Life Insurance Rules gamma= 3 gamma= 1 5 1 15 2 25 3 35 4, Time(in Years) Fig. 3. Differen Risk-Aversion Parameers γ Opimal Life Insurance(in Thousand Dollars) 18 16 14 12 1 8 6 4 2 Comparison of Opimal Life Insurance Rules lambda=.1125+.5 lambda=.225+.1 5 1 15 2 25 3 35 4, Time(in Years) Fig. 4. Differen Hazard Raes λ 15
Opimal Life Insurance(in Thousand Dollars) 2 15 1 5 Comparison of Opimal Life Insurance Rules ea=.1125+.5 ea=.3375+.15 5 5 1 15 2 25 3 35 4, Time(in Years) Fig. 5. Differen Premium-Insurance Raios η (1) Opimal Life Insurance(in Thousand Dollars) 15 1 5 5 Comparison of Opimal Life Insurance Rules ea=.1125+.5 ea=.45+.2 1 5 1 15 2 25 3 35 4, Time(in Years) Fig. 6. Differen Premium-Insurance Raios η (2) 16