Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109

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1 Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio, Canada, M3J 1P3 Viginia R. Young Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, Vesion: 20 Febuay 2014 Abstact: We detemine how an individual can use life insuance to meet a bequest goal. We assume that the individual s consumption is met by an income, such as a pension, life annuity, o Social Secuity. Then, we conside the wealth that the individual wants to devote towads heis sepaate fom any wealth elated to the afoe-mentioned income) and find the optimal stategy fo buying life insuance to maximize the pobability of eaching a given bequest goal. We conside life insuance puchased by a single pemium, with and without cash value available. We also conside ievesible and evesible life insuance puchased by a continuously paid pemium; one can view the latte as instantaneous) tem life insuance. Keywods: Tem life insuance, whole life insuance, bequest motive, deteministic contol. 1. Intoduction Life insuance helps in estate planning, specifically, in poviding bequests fo childen, gandchilden, o chaitable oganizations. With this pupose in mind, we detemine how an individual can use life insuance to meet a bequest goal. We assume that the individual s consumption is met by an income, such as a pension, life annuity, o Social Secuity. Then, we conside the wealth that the individual wants to devote towads heis sepaate fom any wealth elated to the afoe-mentioned income) and find the optimal stategy fo buying life insuance to maximize the pobability of eaching a given bequest goal. In this pape, we join two hitheto unconnected steams of liteatue. The fist steam is that of optimal puchasing of life insuance, and most of the aticles in this aea maximize utility of consumption, bequest, o both. The seminal aticle in this aea is Richad 1975); please see Bayakta and Young 2013) fo some ecent efeences elevant to the poblem of maximizing utility of household consumption by using life insuance. The second steam is that of maximizing the pobability of eaching a paticula taget. This poblem has been studied in pobability poblems elated to gambling, as in the text Dubins and Savage 1965, 1976). Fo an impotant extension of the wok of Dubins and Savage see Pestien and Suddeth 1985), in which they contol a diffusion pocess to each a taget befoe uining. Fo elated papes

2 see Suddeth and Weeasinghe 1989), Kulldoff 1993), and Bowne 1997, 1999a, 1999b). Instead of contolling a diffusion, we maximize the pobability of eaching a paticula goal and allow the individual to puchase life insuance to help each that goal, while adding a andom deadline namely, death). The est of the pape is oganized as follows: In Section 2.1, we conside the case fo which the individual buys whole life insuance via a single pemium with no cash value available, while in Section 2.2, she can suende any o all of he whole life insuance fo a cash value. In both cases, we compute he expected wealth at death because he goal is to each a given bequest, so expected wealth at death is elevant. Section 3 paallels Section 2 fo the case in which insuance is puchased via a continuouslypaid pemium; howeve, we evese the ode of the topics as compaed with the ode in Section 2. In Section 3.1, the individual is allowed to change the amount of he insuance at any time; in ou time-homogeneous setting, this amounts to instantaneous tem life insuance. By contast, in Section 3.2, we do not allow the individual to teminate life insuance, so fo the emainde of he life, she has to pay fo any life insuance she buys. The solution of the poblem in Section 3.1 is simple than and infoms the solution to the poblem in Section 3.2, so we pesent the simple poblem fist. Section 4 concludes the pape. 2. Single-Pemium Life Insuance We begin this section by stating the optimization poblem that the individual faces. In Section 2.1, we conside the case fo which the individual buys whole life insuance via a single pemium with no cash value available, so she neve suendes he life insuance policy; she may only buy moe. In Section 2.2, we incopoate a non-zeo cash value and find the optimal insuance puchasing and suendeing policies in that case. At the end of each of Sections 2.1 and 2.2, we compute he expected wealth at death No cash value available We assume that the individual has an investment account that she uses to each a given bequest goal b. This account is sepaate fom the money that she uses to cove he living expenses. The individual may invest in a iskless asset inteest eaning at the continuous ate > 0, which actuaies call the foce of inteest, o she may puchase whole life insuance. Denote the futue lifetime andom vaiable of the individual by τ d. We assume that τ d follows an exponential distibution with mean 1 λ. In othe wods, the individual is subject to a constant foce of motality, o hazad ate, λ.) The individual buys life insuance that pays at time τ d. This insuance acts as a means fo achieving the bequest motive. In this time-homogeneous model, we assume that a dolla death benefit payable at time τ d costs H at any time. Wite the single pemium as follows: λ H = 1 + θ)āx = 1 + θ) + λ, 2.1) in which θ 0 is the popotional isk loading. Assume that θ is small enough so that H < 1; othewise, if H 1, then the buye would not pay a dolla o moe fo one dolla of death benefit. In this section and in Section 2.2, we suppose that the pemium is payable at the moment of the contact; as stated above, H is the single pemium pe dolla of death benefit. In Section 3, we conside the case fo which the insuance pemium is payable continuously. 2

3 3 Let W t denote the wealth in this sepaate investment account at time t 0. Let D t denote the amount of death benefit payable at time τ d puchased at o befoe time t 0. Thus, with single-pemium life insuance, wealth follows the dynamics { dwt = W t dt H dd t, 0 t < τ d, W τd = W τd + D τd. 2.2) An insuance puchasing stategy D = {D t } t 0 is admissible if i) D is a non-negative, nondeceasing pocess, and ii) if wealth unde this pocess is non-negative fo all t 0. We include the latte condition to pevent the individual fom boowing against he life insuance. Remak 2.1. By equiing that D be non-deceasing ove time, we effectively assume the individual cannot suende any life insuance once she has bought it. In the eal wold, whole life insuance has a suende value that the individual can withdaw, and in Section 2.2, we include that featue of whole life insuance. We assume that the individual seeks to maximize the pobability that W τd b, by optimizing ove admissible contols D. The coesponding value function is given by φw, D) = sup P w,d W τd b), 2.3) D in which P w,d denotes conditional pobability given W 0 = w 0 and D 0 = D 0. We call φ the maximum pobability of eaching the bequest goal. If D b, then the individual has aleady eached he bequest goal of b; thus, hencefoth, in this section, we assume that D < b. If wealth equals Hb D), the so-called safe level, then it is optimal fo the individual to spend all of he wealth to puchase life insuance of b D so that he total death benefit becomes b = b D) + D. It follows that φw, D) = 1 fo w Hb D) and 0 D < b. Thus, it emains only to detemine the maximum pobability of eaching the bequest on R = {w, D) : 0 w Hb D), 0 D < b}. We next pove a veification lemma that states that a nice solution to a vaiational inequality associated with the maximization poblem in 2.3) is the value function φ. Theefoe, we can educe ou poblem to one of solving a vaiational inequality. We state the veification lemma without poof because its poof is simila to othes in the liteatue; see, fo example, Wang and Young 2012a, 2012b) fo elated poofs in a financial maket that includes a isky asset. Lemma 2.1. Let Φ = Φw, D) be a function that is non-deceasing and diffeentiable with espect to both w and D on R = {w, D) : 0 w Hb D), 0 D < b}, except that Φ might have infinite deivative with espect to w at w = 0. Suppose Φ satisfies the following vaiational inequality on R, except possibly when w = 0: maxw Φ w λ Φ w, Φ D H Φ w ) = ) Additionally, suppose ΦHb D), D) = 1. Then, on R, φ = Φ.

4 4 The egion R 1 = {w, D) R : φ D w, D) H φ w w, D) < 0} is called the continuation egion because when the wealth and life insuance benefit lie in the inteio of R 1, the individual does not puchase additional life insuance; athe, she continues with he cuent benefit and invests he wealth in the iskless asset. Indeed, φ D < H φ w means that the maginal benefit of buying moe life insuance φ D ) is less than the maginal cost of doing so H φ w ). On the closue of that egion in R, witten clr 1 ), the following equation holds: wφ w λφ = 0. To help us solve the vaiational inequality 2.4), we ecall that in simila poblems fo example, puchasing life annuities to minimize the pobability of lifetime uin, as descibed in Milevsky et al. 2006)), the optimal stategy is to act only at the safe level. In ou case, that tanslates into buying life insuance only when wealth eaches Hb D) so that φ solves the following bounday-value poblem fo 0 w Hb D) and 0 D < b: { wφw λφ = 0, φhb D), D) = ) Buying life insuance only when wealth eaches Hb D) is indeed optimal, as we pove in the following poposition. Poposition 2.2. The maximum pobability of eaching the bequest goal on R = {w, D) : 0 w Hb D), 0 D < b} is given by φw, D) = w Hb D). 2.6) The associated optimal life insuance puchasing stategy is not to puchase additional life insuance until wealth eaches the safe level Hb D), at which time, it is optimal to buy additional life insuance of b D. Poof. We use Lemma 2.1 to pove this poposition. Fist, note that φ in 2.6) is inceasing and diffeentiable with espect to both w and D on R. Because φ solves the bounday-value poblem 2.5), we have wφ w λφ = 0 on R. Next, we show that φ D H φ w 0 on R: φ D w, D) H φ w w, D) = λ H w Hb D) 1 [ w b D) 2 H b D ] w Hb D) 0. We have, thus, shown that the expession fo φ in 2.6) satisfies the vaiational inequality 2.4). The continuation egion equals R 1 = {w, D) : 0 w < Hb D), D < b}; theefoe, the optimal insuance puchasing stategy is to buy additional insuance of b D when wealth eaches the safe level Hb D). Remak 2.2. We fully anticipate that the esults of this section will hold when one consides othe models, such as moe geneal financial and motality models, including those that ae not time homogeneous. Specifically, we expect that when insuance is puchased by a single pemium with no cash value available, then it will be optimal to wait until wealth eaches the safe level to buy additional life insuance.

5 Remak 2.3. Optimally contolled wealth is invested in the iskless asset until it eaches Hb D); thus, wealth at time t, befoe eaching the safe level, equals W t) = we t, fo a given initial wealth w < Hb D). The time that wealth eaches the safe level, denoted by τ Hb D), is given by τ Hb D) = 1 ) Hb D) ln. w The individual eaches he bequest motive if she dies afte time τ Hb D) ; this occus with pobability e λτ Hb D), which equals the expession given in 2.6), as expected. Because we ae maximizing the pobability that wealth at death equals b, it is of inteest to detemine the expected wealth at death. Coollay 2.3. Expected wealth at death, Ew, D) = E w,d W τd ), fo an individual who follows the optimal life insuance puchasing stategy of Poposition 2.2 is given by Ew, D) = [ w [ b D) 1 λh λ ] 1 H + ln Hb D) w w Hb D) + λw λ + D, if λ, )] + D, if λ = ) Poof. Expected wealth at death equals Ew, D) = E w,d W τd + D τd ); thus, fom the discussion in Remak 2.3, we have Ew, D) = and the expessions in 2.7) follow. τhb D) 0 we t + De λt dt + b φw, D), Remak 2.4. An expectation such as E w,d W τd ) satisfies a diffeential equation with bounday conditions. Indeed, via a standad veification lemma, one can show that E w,d W τd ) uniquely solves the following bounday-value poblem BVP) fo 0 w Hb D) and 0 D < b: { λe w + D)) = w Ew, EHb D), D) = b. Because the expession in equation 2.7) solves this BVP, we confim that it is the coect expession fo E w,d W τd ) Cash value available Standad nonfofeitue laws ensue that an individual who owns a whole life insuance policy can exchange the policy fo its cash value. In this section, we incopoate that featue of whole life insuance into the model in Section 2.1. Theefoe, we allow the pocess D to decease, although it is still equied to be non-negative. We assume that when the individual suendes he death benefit, she eceives a popotion of the puchase pice. If the cash value is detemined accoding to some othe method, such as a popotion

6 6 of the eseve, then one can still expess it as a popotion of the puchase pice.) Let ρ [0, 1] be the popotional suende chage, so that the individual eceives 1 ρ)h fo each dolla of death benefit that she suendes. The case in which ρ = 1 is equivalent to the case fo which no cash value is available, as in Section 2.1. Wite φ s fo the maximum pobability of wealth at death eaching the bequest b when whole life insuance can be suendeed. We use a supescipt s to denote that insuance can be suendeed.) The coesponding veification lemma is as follows. Lemma 2.4. Let Φ s = Φ s w, D) be a function that is non-deceasing, continuous, and piecewise diffeentiable with espect to both w and D on R = {w, D) : 0 w Hb D), 0 D < b}. Suppose Φ s satisfies the following vaiational inequality on R: maxw Φ s w λ Φ s, Φ s D H Φ s w, 1 ρ)h Φ s w Φ s D) = 0, 2.8) in which we use one-sided deivatives, if needed. Additionally, suppose Φ s Hb D), D) = 1. Then, on R, φ s = Φ s. The egion R 1 = {w, D) R : φ s D w, D) H φs ww, D) < 0 and 1 ρ)h φ s ww, D) φ s D w, D) < 0} is the continuation egion because when the wealth and life insuance benefit lie in the inteio of R 1, the individual does not puchase no suende life insuance; she continues with he cuent benefit. Afte Lemma 2.1, we discussed the inequality φ s D < H φs w; to eview, it means that the maginal benefit of buying moe life insuance φ s D ) is less than the maginal cost of doing so H φs w). Similaly, inequality 1 ρ)h φ s w < φ s D means that the maginal benefit of suendeing life insuance 1 ρ)h φs w) is less than the maginal cost of doing so φ s D ). On the closue of that egion in R, witten clr 1), the following equation holds: wφ s w λφ s = 0. To find φ s, we hypothesize that the optimal puchasing stategy is identical to the one in Section 2.1. Specifically, the individual does not buy additional insuance until wealth eaches the safe level Hb D). Futhemoe, we hypothesize that it is optimal to suende life insuance fo wealth small enough, so that the individual liquidates he assets in ode to take advantage of the iskless etun. It tuns out that this hypothesis is coect, and we pove this assetion in the following poposition. Poposition 2.5. The maximum pobability of eaching the bequest goal on R = {w, D) : 0 w Hb D), 0 D < b} is given by φ s w, D) = w+1 ρ)hd Hb, if 0 w < 1 ρ)hb D), w Hb D), if 1 ρ)hb D) w Hb D). The associated optimal life insuance suendeing and puchasing stategies ae as follows: a) If wealth is less than 1 ρ)hb D), then suende all life insuance. Theeafte, invest all wealth in the iskless asset until wealth eaches the safe level Hb, at which time, it is optimal to buy life insuance of b. 2.9)

7 b) If wealth is geate than o equal to 1 ρ)hb D), then invest all wealth in the iskless asset until Poof. wealth eaches the safe level Hb D), at which time, it is optimal to buy additional life insuance of b D. We use Lemma 2.4 to pove this poposition. Fist, note that φ s in 2.9) is inceasing, continuous, and piecewise diffeentiable with espect to both w and D on R. When 0 w < 1 ρ)hb D), w φ s w λφ s λ1 ρ)d = b w + 1 ρ)hd Hb 1 0. In fact, this inequality holds stictly, except when D = 0, in which case the individual has no death benefit to suende. Fo 1 ρ)hb D) w Hb D), φ s solves the diffeential equation in 2.5); thus, w φ s w λ φ s 0 on R. Next, obseve that 1 ρ)h φ s w φ s D 0 on R. Indeed, when 0 w < 1 ρ)hb D), the inequality holds with equality, while w φ s w λφ s < 0 fo D 0; thus, it is optimal to suende all one s life insuance when wealth is less than 1 ρ)hb D). When 1 ρ)hb D) < w Hb D), the inequality holds stictly; thus, we deduce that it is not optimal to suende any life insuance when wealth is geate than 1 ρ)hb D). When w = 1 ρ)hb D), the individual is indiffeent between suendeing all he life insuance and suendeing none of it, as fa as maximizing the pobability that she will die with wealth equal to b. We assume that she suendes none of he life insuance when w = 1 ρ)hb D) because, fo that wealth, expected wealth at death is geate when she does not suende he life insuance; see Coollay 2.6 below. Finally, obseve that φ s D w, D) H φs ww, D) 0 on R. Indeed, when 0 w < Hb D), the inequality holds stictly; thus, we deduce that it is not optimal to buy additional life insuance until wealth eaches the safe level. vaiational inequality 2.8). We have, thus, shown that the expession fo φ s in 2.9) satisfies the Remak 2.5. We anticipate that the esults of this section will hold when one consides othe models, such as moe geneal financial and motality models, including those that ae not time homogeneous. Specifically, we expect that when insuance is puchased by a single pemium with cash value available, then it will be optimal to wait until wealth eaches the safe level to buy additional life insuance, and it will be optimal to suende life insuance when wealth is low enough. As in Section 2.1, it is of inteest to detemine the expected wealth at death fo someone who is allowed to suende he life insuance in exchange fo its cash value. Coollay 2.6. Expected wealth at death, E s w, D) = E w,d W τd ), fo an individual who follows the optimal life insuance puchasing and suendeing stategies of Poposition 2.5 is given by the following if λ : E s w, D) = [ b 1 λh λ [ b D) 1 λh λ ] w+1 ρ)hd Hb + λw+1 ρ)hd) λ, if 0 w < 1 ρ)hb D), ] w Hb D) + λw λ + D, if 1 ρ)hb D) w Hb D) )

8 If λ =, then expected wealth at death is given by [ 1 w + 1 ρ)hd) E s H + ln w, D) = [ 1 w H + ln Hb D) w Poof. Hb w+1 ρ)hd )], if 0 w < 1 ρ)hb D), )] + D, if 1 ρ)hb D) w Hb D) ) If 0 w < 1 ρ)hb D), it is optimal fo the individual to suende all he life insuance and theeafte invest he money in the iskless asset until wealth eaches the safe level Hb. Thus, fo wealth in this ange, E s w, D) = Ew + 1 ρ)hd, 0). If 1 ρ)hb D) w Hb D), the individual neve suendes he life insuance and buys additional life insuance when he wealth eaches the safe level Hb D). Thus, fo wealth in this ange, E s w, D) = Ew, D). These obsevations lead to the expessions in equations 2.10) and 2.11). Remak 2.6. Note that E s in 2.10) and 2.11) is not continuous at w = 1 ρ)hb D), which is due to the diffeence between the optimal suendeing stategy of the individual fo wealth less than vesus geate than 1 ρ)hb D). One can show that E s 1 ρ)hb D), D) E s 1 ρ)hb D)+, D); thus, fom the standpoint of expected wealth at death, it is bette fo the individual not to suende he life insuance when w = 1 ρ)hb D), even though the pobability of eaching b is the same whethe she suendes all life insuance o suendes none at that level of wealth. 3. Insuance Puchased by a Continuously Paid Pemium Section 3 paallels Section 2 fo the case in which insuance is puchased via a continuously-paid pemium; howeve, we evese the ode of the subsections. In Section 3.1, the individual is allowed to change the amount of he insuance at any time; in ou time-homogeneous setting, this amounts to instantaneous tem life insuance. By contast, in Section 3.2, we do not allow the individual to teminate life insuance, so fo the emainde of he life, she has to pay fo any life insuance she buys. The solution of the poblem in Section 3.1 is simple than and infoms the solution to the poblem in Section 3.2, so we pesent the simple poblem fist. 3.1 Instantaneous tem life In this section, we assume that the individual buys life insuance via a pemium paid continuously at the ate of h = 1 + θ)λ pe dolla of insuance fo some θ 0. Futhemoe, we assume that the individual can change the amount of he insuance coveage at any time. The popotional loading coves expenses, pofit, and isk magin; theefoe, we assume that no eseve accumulates. Thus, the set up in this section is equivalent to the individual puchasing instantaneous tem life insuance. With continuously paid pemium fo instantaneous tem life insuance, wealth follows the dynamics { dwt = W t hd t ) dt, 0 t < τ d, W τd = W τd + D τd. 3.1) Fo this section, an admissible insuance stategy D = {D t } t 0 is any non-negative pocess. We do not insist admissible stategies be such that W t 0 fo all t 0 with pobability one because of the

9 constant dain on wealth by the negative dift tem hd t. Theefoe, we modify the definition of the maximized pobability of eaching the bequest by effectively ending the game if wealth eaches 0 befoe the individual dies. Define τ 0 = inf{t 0 : W t 0}, and define the value function by φ t w) = sup P w W τd τ 0 b), 3.2) D in which we maximize ove admissible stategies D. We use a ba to denote that the pemium is payable continuously, and we use a supescipt t to indicate that the insuance is tem life.) We efe to φ t as the maximum pobability of eaching the bequest goal befoe uining. To motivate the veification lemma fo this poblem, we pesent the following infomal discussion. Because D is an instantaneous contol, we anticipate that φ t solves the following contol equation: λ φ t = w φ t [ w + max λ1{w+d b} hd φ t ] w, 3.3) D in which the indicato function 1 {w+d b} equals 1 if w + D b and equals 0 othewise. We did not encounte this indicato function in the poblem in Section 2 because fo 0 w Hb D) and 0 D < b, we automatically have w + D < b. In 3.3), the indicato function equals 0 o 1, and coesponding to each of those values, we choose D to be a minimum because of the tem hd φ t w. Specifically, if the indicato equals 0, then the optimal insuance is D = 0; if it equals 1, then the optimal insuance is D = b w. Thus, we can eplace equation 3.3) with the equivalent expession: λ φ t = w φ t w + max [ λ hb w) φ t w, 0 ]. 3.4) Denote the safe level fo this poblem by w t. We obtain w t by aguing as follows: the income w t can fund a death benefit of wt h, and we equie that sum of this death benefit and the existing wealth w t equals the goal b; that is, wt h + wt = b, o equivalently, w t =. These obsevations lead to the following veification lemma. Lemma 3.1. Let Φ t = Φ t w) be a function that is non-deceasing, continuous, and piecewise diffeentiable on [0, w t ], in which w t = + h, 3.5) except that Φ t might not be diffeentiable at 0. Suppose Φ s satisfies the following vaiational inequality on 0, w t ]: λ Φ t = w Φ t w + max [ λ hb w) Φ t w, 0 ], 3.6) in which we use one-sided deivatives, if needed. Additionally, suppose Φ t w t ) = 1. Then, on [0, w t ], φ t = Φ t. 9 So that what follows does not seem like mathematical magic, we discuss how we obtained the solution to ou maximization poblem. Because we have a bounday condition at w = w t, we woked backwads fom that point. At any wealth level w, the individual chooses eithe to buy insuance of

10 b w o to buy no insuance. Fist, suppose that in a neigohood of w t, the individual buys full insuance of b w; denote the esulting solution of λ φ t 1) = + h)w) φ t w, with φ t w t ) = 1, by φ t f. Then, φ t f is given by + h)w φ t f w) = 1 k, fo w nea w t. Hee, k > 0 is some unknown) constant. Next, suppose that in a neigohood of w t, the individual buys no insuance; denote the esulting solution of λ φ t = w φ t w, with φ t w t ) = 1, by φ t 0. Then, φ t 0 is given by fo w nea w t. λ + h)w φ t 0w) =, To detemine which of φ t 0 and φ t f is lage fo w nea wt, compae thei deivatives at w t. Because < 1, lim w w t φ t f ) ww) =, while φ t 0) w w t ) is positive, but finite. Thus, fo wealth nea w t, φ t 0 φ t f. It might be that on some inteval of wealth [w, w ), we have φ t 0 φ t f. Howeve, the existence of such an inteval depends on whethe λ o λ >. If λ, then φ t 0 φ t f fo all 0 w wt ; and, in Poposition 3.2, we pove that φ t = φ t 0. If λ >, then thee is a wealth level w 0, w t ) such that φ t 0 φ t f on [0, w ) and φ t 0 φ t f on [w, w t ]; and, in Poposition 3.5, we pove that φ t = φ t f on [0, w ) and φ t = φ t 0 on [w, w t ], with k chosen to make φ t continuous at w. Poposition 3.2. If λ, then the maximum pobability of eaching the bequest goal befoe uining is given by + h)w φ t w) =, 3.7) fo initial wealth w [0, w t ]. The associated optimal life insuance puchasing stategy is not to puchase any life insuance until wealth eaches the safe level w t, at which time it is optimal to buy life insuance of b w t = Poof. b. We use Lemma 3.1 to pove this poposition. Fist, note that φ t in 3.7) is continuous and inceasing on [0, w t ], φ t is diffeentiable on 0, w t ], and φ t w t ) = 1. Next, note that λ φ t = w φ t w, 10 on 0, w t ]. The inequality λ hb w) φ t w 0 holds on 0, w t ] if and only if 1 + h x λ 1 + h x λ 0, 3.8) fo all 0 < x 1. Fo λ =, inequality 3.8) is clealy tue. To show inequality 3.8) fo λ <, define f on 0, 1] by fx) = 1 c + 1)x a 1 + cx a, in which c := h > 0 and a := λ 0, 1), so it is enough to show that fx) 0 on 0, 1]. To this end, obseve that lim x 0+ fx) =, and f1) = 0, so it is enough to show that f is inceasing on 0, 1]. f x) = c + 1)1 a)x a 2 + cax a 1

11 is positive on 0, 1] if and only if c + 1)1 a) 0, which is tue. Theefoe, we have shown that φ t in 3.7) satisfies the vaiational inequality 3.6). The optimal insuance stategy follows fom the fact that φ t solves the contol poblem 3.3) with D Remak 3.1. When the foce of motality is less than o equal to the foce of inteest, the individual feels as if she has time to each the safe level; theefoe, it is optimal fo the individual to invest in the iskless asset and wait until she eaches the safe level befoe she buys any life insuance. Fo initial wealth w, wealth at time t equals W t) = we t, and the time that wealth eaches the safe level equals τ w t = 1 ) ln. + h)w The pobability of eaching the safe level befoe dying equals e λτ w t, which equals φ t in 3.7), as expected. Coollay 3.3. If λ, then expected wealth at death, Ēt w) = E w W τd ), fo an individual who follows the optimal life insuance puchasing stategy of Poposition 3.2 is given by [ ] Ē t b 1 + h λ )w λ λw λ, if λ <, w) = [ )] w h + ln )w, if λ =. 3.9) Poof. Fom the discussion in Remak 3.1, expected wealth at death equals and the expessions in 3.9) follow. Ē t w) = τ w t 0 we t λe λt dt + b φ t w), Remak 3.2. Note that Ēt in 3.9) uniquely solves the following BVP on 0, w t ]: { λ Ē t w) = w Ēt w, Ē t w t ) = b. Next, we conside the slightly moe complicated case of λ > and pesent a helpful lemma. Lemma 3.4. Suppose c and a ae constants such that 0 < c < 1 < a. Then, the following thee statements hold: a) The function f 1 on [0, 1] defined by f 1 x) = x a + 1 x) c 1 has a unique zeo x in the inteio 0, 1). Futhemoe, f 1 x) 0 fo 0 x x, and f 1 x) 0 fo x x 1. b) The function f 2 on [0, 1) defined by f 2 x) = 1 c a 1 x)c 1 1 c ) 1 x) c a

12 12 is non-negative on [0, x ]. c) The function f 3 on [0, 1] defined by is non-positive on [x, 1]. f 3 x) = 1 a a ) c xa 1 + c 1 x a Poof. Poof of a). Obseve that f 1 0) = f 1 1) = 0. Also, f 1x) = a x a 1 c 1 x) c 1, and note that f 10) = c < 0 and lim x 1 f 1x) =. Thus, f 1 has an odd numbe of zeos in the inteio 0, 1), say, 2k 1, fo some k = 1, 2,.... This fact implies that f 1 has 2k zeos; thus, to show that k = 1, it is enough to show that f 1 has at most two zeos in 0, 1). The zeos of f 1 ae those points x that solve x a 1 1 x) 1 c = c a. So, if we define g by gx) = x a 1 1 x) 1 c c a, then to show that f 1 has at most two zeos in 0, 1), it is enough to show that g has one zeo in 0, 1) because g0) = g1) = c a < 0. This esult follows fom g x) = x a 2 1 x) c [a 1) a c)x], which has a unique zeo at x = a 1 a c 0, 1). Thus, we have poved that f 1 has a unique zeo in 0, 1). Poof of b). Obseve that f 2 0) = 0 and lim x 1 f 2 x) =. Also, [ thus, f 2 inceases on 0, a 1 a c f 2x) = c a 1 x)c 2 [a 1) a c)x] ; ) and deceases on is, theefoe, enough to show that f 2 x ) 0. a 1 a c, 1 ). To show that f 2 is non-negative on [0, x ), it To this end, ecall that 1 x) c 1 cx because the left side of this inequality is concave in x so lies below its tangents) and the ight side is the tangent of 1 x) c at x = 0. Fom pat a), we know that 1 x ) a = 1 x ) c ; thus, we conclude that cx x ) a. Inequality f 2 x ) 0 is equivalent to which will follow if we show the stonge inequality a1 x ) a1 x ) + cx 1 x ) c, 3.10) a1 x ) a1 x ) + x ) a 1 x ) a, 3.11) in which we use 1 x ) a = 1 x ) c and cx x ) a. Inequality 3.11) is equivalent to x ) a + a1 x ) 1 0,

13 which holds on [0, 1) because the left side deceases with espect to x and equals 0 if x = 1. Thus, we have poved that f 2 is non-negative on [0, x ). Poof of c). Obseve that f 3 0) = 1 and f 3 1) = 0. Also, [ thus, f 3 deceases on 0, a 1 a c f 3x) = a c xa 2 [ a 1) + a c)x] ; ) and inceases on 13 a 1 a c, 1 ). To show that f 3 is non-positive on [x, 1], it is, theefoe, enough to show that f 3 x ) 0. Inequality f 3 x ) 0 is equivalent to cx a1 x ) + cx x ) a, which is equivalent to inequality 3.10) because x ) a = 1 1 x ) c. Thus, we have poved that f 3 is non-positive on [x, 1]. Poposition 3.5. If λ >, then the maximum pobability of eaching the bequest goal befoe uining is given by 1 φ t w) = )w )w, if 0 w < w,, if w w w t =, 3.12) fo initial wealth w [0, w t ]. Hee, w is the unique zeo in 0, w t ) of the following expession: + h)w The associated optimal life insuance puchasing stategy is as follows: a) If wealth w is less than w, then puchase life insuance of b w. + h)w ) b) If wealth is geate than o equal to w, then do not puchase life insuance until wealth eaches the Poof. safe level w t, at which time it is optimal to buy life insuance of b w t = this end, let a = λ > 1, c = b. Fist, use Lemma 3.4a) to pove that the expession in 3.13) has a unique zeo in 0, w t ). To λ )w 0, 1), and x = ; then, the expession in 3.13) becomes f 1 in Lemma 3.4a). We know that f 1 has a unique zeo x in 0, 1); thus, w = x 3.13) in 0, w t ). is the unique zeo of Next, note that φ t is non-deceasing, continuous, and piecewise diffeentiable on [0, w t ], with φ t w t ) = 1. On [0, w ), λ 1) φ t = + h)w b) φ t w, 3.14) and the inequality λ hb w) φ t w ) holds if and only if 1 b w b In inequality 3.16), let a = λ > 1, c = λ 1 f 2 x) 0 on [0, x ), which we know is tue fom Lemma 3.4b). + h)w ) )w 0, 1), and x =, as befoe; then, 3.16) becomes Thus, we have poved inequality

14 3.15) on [0, w ). Fom equation 3.14) and inequality 3.15), it follows that φ t satisfies the vaiational inequality 3.6) in Lemma 3.1 on [0, w ). Because φ t satisfies 3.3) with Dw) = b w when 0 w < w, we deduce that fo wealth less than w, it is optimal to buy insuance in ode to each the bequest goal b. On [w, w t ], and the inequality holds if and only if 1 + h In inequality 3.19), let a = λ > 1, c = 14 λ φ t = w φ t w, 3.17) λ hb w) φ t w ) b w b λ + h)w f 3 x) 0 on [x, 1], which we know is tue fom Lemma 3.4c) ) )w 0, 1), and x =, as befoe; then, 3.19) becomes Thus, we have poved inequality 3.18) on [w, w t ]. Fom equation 3.17) and inequality 3.18), it follows that φ t satisfies the vaiational inequality 3.6) in Lemma 3.1 on [w, w t ]. Because φ t satisfies 3.3) with Dw) = 0 when w w 1, we deduce that fo wealth geate than o equal to w, it is optimal not to buy insuance. Instead, it is optimal to wait until wealth eaches the safe level w t = insuance of b., at which time the individual will buy Remak 3.3. Fo wealth equal to w, the pobability that wealth at death equals b is the same whethe the individual buys full insuance Dw) = b w until wealth eaches 0 o whethe she buys no insuance until wealth eaches the safe level. So, she is indiffeent between these two stategies, and we picked the buy-no-insuance stategy because he expected wealth at death is geate unde that stategy; see Coollay 3.6 below. Remak 3.4. When λ > and when initial wealth w [0, w ), optimally contolled wealth at time t equals W t) = ) + h + h w e )t, which continually deceases and might each zeo befoe the individual dies. The time that wealth hits zeo depends on w: τ 0 = 1 ) + h ln. + h)w The pobability that the individual dies with wealth at death including death benefit) equal to b equals the pobability that the individual dies befoe time τ 0, o 1 e λτ 0, which equals φ t, as expected. When initial wealth w [w, w t ], then the individual invests all he wealth in the iskless asset, so that wealth at time t equals we t, and she does not buy insuance until wealth eaches the safe level w t =. Coollay 3.6. If λ >, then expected wealth at death, Ēt w) = E w W τd ), fo an individual who follows the optimal life insuance puchasing stategy of Poposition 3.4 is given by [ ] b 1 )w Ē t, if 0 w < w, w) = [ ] b 1 h λ )w λ + λw λ, if w w w t. 3.20)

15 15 Moeove, Ēt w ) < Ēt w +). Poof. Fo 0 w < w, wealth at death is eithe b o 0, so expected wealth at death equals b φ t w). Fom the discussion at the end of Remak 3.4, fo initial wealth in [w, w t ], it follows that expected wealth at death equals the fist expession given in 3.9). One can show that Ēt w ) < Ēt w +) by using the fact that 1 x x = )w. Remak 3.5. On [0, w ), Ē t in 3.20) uniquely solves the following BVP: { λ Ē t b) = + h)w ) Ēt w, Ē t 0) = 0. On [w, w t ], Ē t in 3.20) uniquely solves the following BVP, as in Remak 3.2: { λ Ē t w) = w Ēt w, Ē t w t ) = b. = 1 x, in which Next, we pesent popeties that the dividing point w possesses. In the inteest of space, we omit the poof of this coollay but invite the inteested eade to povide it. Coollay 3.7. When λ >, the dividing point w between full insuance Dw) = b w fo w < w and no insuance Dw) = 0 fo w w satisfies the following popeties: a) w inceases popotionally with espect to b, the bequest goal. b) w inceases with espect to λ, the foce of motality. c) w deceases with espect to, the iskless ate of etun. Remak 3.6. a) It is clea that w changes popotionally with espect to b. Indeed, x = )w solves 1 x = 1 x. This equation is independent of b, so x does not change with b; thus, w b is constant. b) Thee ae competing effects of λ on w. Fo the pemium ate h fixed, w inceases with λ because the individual is moe likely to die befoe eaching the safe level w t =. Thus, fo h fixed, she is moe likely to want to buy full insuance now instead of waiting to each the safe level. Howeve, the pemium ate h inceases with λ, so we have to conside how w changes with h. The safe level inceases with h, which makes the individual less willing to wait to each the safe level. Howeve, the pemium becomes moe expensive; thus, the individual s desie to buy full insuance is dampened. The net of these effects is to incease w with λ; that is, the exta cost of the pemium is not enough to fully eliminate the individual s geate willingness to buy full insuance now. c) Thee ae two e-enfocing effects of on w. Fist, the safe level deceases with, so the individual does not have to wait as long to each the safe level. Second, if inceases, then the individual s money inceases at a faste ate namely, ) and eaches any level soone. Thus, w deceases with because the individual is moe willing to wait to each the safe level.

16 16 d) We found examples that demonstate that w might decease with θ o might incease with θ. Thee ae two competing effects of θ on w as discussed in pat b) above: inceasing the safe level vesus inceasing the pemium. If the effect of inceasing the pemium is lage than that of inceasing the safe level, then w deceases with θ, and vice vesa. Remak 3.7. The following povides a summay of what we have leaned in this section, as well as a claification of some of the esults. Suppose you decide to stat buying full insuance at a wealth level w that is less than the safe level. This is a winning move if you die befoe time τ 0, the time at which you wealth is depleted to zeo; on the othe hand, waiting to buy until afte eaching the safe level is the winning move if you live to time τ w t. Theefoe, by letting pt) = e λt denote the pobability of living to time t, the bette stategy is to buy full insuance if 1 pτ 0 ) pτ w t), that is, pτ 0 ) + pτ w t) 1, while the bette stategy is to wait if the inequality goes the othe way. We see, theefoe, that w is pecisely the wealth level that esults in pτ 0 ) + pτ w t) = 1. We can see fom this equation that any changes that cause both τ 0 and τ w t to incease will decease both pobabilities and theeby incease w, while changes that cause both times to decease such as an incease in ) will decease w. Fo changes that cause the two times to move in diffeent diections, the effect can be uncetain, as we noticed above fo an incease in h, which causes τ w t to incease but τ 0 to decease. 3.2 Ievesible whole life In this section, once the individual buys a given amount of insuance D, then she must pay pemium at the ate of hd fo the emainde of he life. She cannot evese this puchase. Wealth follows the pocess given in 3.1). Denote the maximum pobability of dying with wealth at least b befoe uining by φ; it is defined as in 3.2), except that the definition of admissible stategy diffes in this case. Indeed, an insuance puchasing stategy D = {D t } t 0 is admissible if D is a non-negative and non-deceasing pocess. In the case fo ievesible whole life insuance with pemium payable continuously, the safe level diffes depending on the existing amount of life insuance D. Indeed, fo a given level of wealth w, the individual can safely invest it in the iskless asset and ean investment income at the ate of w. Because the individual aleady has a death benefit of D, at the safe level, this income must be sufficient to cove the insuance pemium; that is, w hd, o equivalently, w hd. Moeove, if D is less than the safe level when life insuance is ievesible is given by b, then we have the safe level fom Section 3.1, namely. Thus, [ wd) = max + h, hd ] = { hd b, if D, b, if D >. 3.21) The veification lemma fo φ is as follows.

17 17 Lemma 3.8. Let Φ = Φw, D) be a function that is non-deceasing, continuous, and piecewise diffeentiable with espect to both w and D on R = {w, D) : 0 w wd), D 0}. Suppose Φ satisfies the following vaiational inequality on R: max w hd) Φ w λ Φ ) ) 1{w+D b}, ΦD = 0, 3.22) in which we use one-sided deivatives, if needed. Additionally, suppose Φ wd), D) = 1. Then, on R, φ = Φ. When D b, the optimal life insuance puchasing stategy is not to buy any additional life insuance because D aleady meets the tageted bequest. The goal fo the individual is not to uin while paying the pemium ate hd. Thus, in this case, φ solves the following BVP: λ ) φ 1 = w hd) φw, ) hd φ, D = ) We give the solution to this BVP in the next poposition and pove that it satisfies the conditions of Lemma 3.8. Poposition 3.9. On R 0 = {w, D) : 0 w hd bequest goal befoe uining is given by, D b}, the maximum pobability of eaching the hd w φw, D) = ) hd The associated optimal insuance puchasing stategy is not to buy any additional insuance. Poof. The function φ given in equation 3.24) satisfies the BVP 3.23), and it is non-deceasing and continuously diffeentiable with espect to w and D. Theefoe, to complete the poof of this poposition, we only need to show that φ D 0. To that end, note that φ D w, D) = λw hd 2 hd w hd 1 0. As a final obsevation, because φ D is stictly negative fo wealth less than hd additional life insuance., it is not optimal to buy Remak 3.8. When D b and when initial wealth w lies in [ 0, hd time t equals W t) = hd hd w ) e t, ], optimally contolled wealth at which deceases ove time. Thus, wealth will neve each the safe level, and the elevant hitting time is the hitting time of zeo wealth, τ 0, which equals τ 0 = 1 ) hd ln. hd w

18 The pobability that the individual dies with wealth at death equal to at least b is the pobability that she dies befoe time τ 0, o 1 e λτ 0, which equals 3.24), as expected. Thee is an inteesting analogy between the case fo which D b and the case, in Section 3.1, fo which λ > and initial wealth w [0, w ). Indeed, by examining the above expession fo W t) with the one given in Remak 3.4, we see that we can get the fome fom the latte by eplacing D and with b and + h, espectively. The hitting times of zeo similaly coespond, as do the pobabilities of dying befoe wealth eaches 0. In othe wods, we can get 3.24) fom the fist expession in 3.12) by eplacing b and + h with D and, espectively. Coollay If D b, then expected wealth at death, Ēw, D) = Ew,D W τd ), fo an individual who follows the optimal life insuance puchasing stategy of Poposition 3.9 is given by ) D 1 h λ 1 ) hd w hd + λw λ, if λ, Ēw, D) = hd w ln ) hd w hd + )w h, if λ =. Poof. Fom the discussion in Remak 3.8, expected wealth at death is given by τ0 ) ) hd hd Ēw, D) = w e t + D λe λt dt, fom which the expessions in 3.25) follow. 0 Remak 3.9. If D b, then Ē in 3.25) uniquely solves the following BVP fo 0 w hd : { λ Ē w + D)) = w hd) Ēw, Ē 0, D) = ) Hencefoth, we assume that D < b, and we will equie that φ be continuous acoss D = b. We popose the following ansatz fo optimally puchasing life insuance, in which w and D ae initial wealth and death benefit, espectively: a) Suppose b D w hd and b < D < b; then, hypothesize that the individual will buy no additional life insuance if w > b D. If wealth eaches the value b D, then, via instantaneous contol of the death benefit, wealth and death benefit will stay on the line w + D = b, moving towad the point w, D ) = 0, b). b) Suppose 0 D < b w and 0 w. i) Hypothesize that if w, D) is close enough to the line w + D = b, then the individual will buy additional life insuance of b w + D) and theeafte will keep wealth and death benefit on the line w + D = b. We expect points on the line w = hd to lie in this jump egion; othewise, fom the diffeential equation in 3.22), we have φ = 0 along w = hd, which is not tue. ii) Hypothesize that if w is close enough to the safe level, then the individual will buy no additional insuance until he wealth eaches the safe level. Inheent in this pat of the ansatz is that D < b, so that the safe level equals w =.

19 We will slightly) abuse notation below by efeing to φ as the solution of vaious bounday-value poblems esulting fom the above ansatz. Howeve, as we pogess, we will pove that the φ we thus obtain is indeed the maximum pobability of eaching the bequest goal befoe uining. Region R a = {w, D) : b D w hd, b < D < b}: Based on pat a) of the ansatz, the maximum pobability of eaching the bequest befoe uining solves the following BVP: λ φ 1) = w hd) φ w, φ D b D, D) = 0, ) ) 3.26) λ hd w φ, D = 1, lim φw, D) = 1. D b The condition that φ D = 0 at w = b D aises fom the ansatz that the individual puchases insuance continuously along that line. The last condition comes fom equiing continuity at D = b; altenatively, we could simply equie that lim w,d) 0+,b ) φw, D) = 0 and then check that continuity at D = b holds. The solution of 3.26) is given by + h)d b φw, D) = 1 hd w + h)d b ) In the next poposition, we show that φ in 3.27) equals the maximum pobability of eaching the bequest goal. Poposition On R a = {w, D) : b D w hd, b < D < b}, the maximum pobability of eaching the bequest goal befoe uining is given by φ in 3.27). The associated optimal insuance puchasing stategy is to buy additional insuance only when wealth eaches b D, afte which continually buy additional insuance to ensue that the sum of wealth and death benefit equals b. Poof. We use Lemma 3.8 to pove this poposition. Because φ in 3.27) satisfies the BVP 3.26), we only need to show that φ D 0 on R a. The inequality φ D w, D) 0 holds if and only if hd w + h)d b 1, which is equivalent to w + D b. Thus, φ in 3.27) is the maximum pobability of eaching the bequest goal befoe uining. Remak Fo initial wealth and death benefit lying in the inteio of Ra, optimally contolled wealth at time t equals W t) = hd hd w ) e t, which deceases ove time. Thus, wealth will neve each the safe level, and the fist elevant hitting time is the time that wealth eaches b D, τ b D, which equals τ b D = 1 ) + h)d b ln. hd w

20 Afte wealth eaches b D, the individual continually buys life insuance to keep wealth plus death benefit equal to b. It follows fom Remak 3.4, that at time t = τ b D + s fo s 0, optimally contolled wealth equals W t) = + h D b ) e )s, + h which deceases ove time. Thus, the second elevant hitting time is the hitting time of zeo, τ 0 = 1 ) + h ln, + h)d b which we measue fom time τ b D. The pobability that the individual dies with wealth at death of at least b equals the pobability that she dies befoe time τ b D plus the pobability that she dies befoe time τ 0 given that she dies afte time τ b D, o 1 e λτ b D) + e λτ b D 1 e λτ 0 ) = 1 e λτ 0 +τ b D ), which equals the expession in 3.27), as expected. Coollay Fo w, D) R a, expected wealth at death fo an individual who follows the optimal life insuance puchasing stategy of Poposition 3.11 is given by [ { hd w )D )D b λ b λ + )D b Ēw, D) = ) w + D hd w ln )D b hd w hd w h Poof. )D b 20 }] [ ] + D 1 h λ + λw λ, if λ, ) h, if λ =. Fom the discussion in Remak 3.10, it follows that τb D ) ) hd hd Ēw, D) = w e t + D λe λt dt + be λτ b D 1 e λτ 0 ), fom which the expessions in 3.28) follow. 0 Remak Fo w, D) R a, Ē in 3.28) uniquely solves the following BVP: { λ Ē w + D)) = w hd) Ēw, 3.28) Ē D b D, D) = 0, Ēw, b) = ēw), in which ēw) = ) b 1 h λ w ln w 1 w ) + λw λ, if λ, ) + )w h, if λ =. The bounday condition at D = b comes fom continuity of Ē acoss D = b; thus, ē is obtained via the expessions in 2.22a) with D = b. Altenatively, we could simply equie that lim w,d) 0+,b ) Ēw, D) = 0 and then check that continuity at D = b holds. Region R b = {w, D) : 0 D < b w, 0 w }: Based on pat b)i) of the ansatz, fo w, D) close enough to the line w + D = b, the individual immediately buys additional life insuance of b w + D). Thus, φ is given by φw, D) = φw, b w), in which the ight side is given by 3.27). Thus, + h)w φw, D) = )

21 21 D < Based on pat b)ii) of the ansatz, fo w close enough to the safe level b, φ solves the following BVP: λ φ = w hd) φ w, ) φ + h, D = 1., assuming that 3.30) The solution of 3.30) is given by φw, D) = w hd ) b h D λ. 3.31) To obtain 3.31), we assume that w hd > 0 when w < ; othewise, if the line w = hd is in the continuation egion, the diffeential equation in 3.30) implies that φ = 0 along w = hd, which is not tue. Also, in witing 3.31), we mean that φ = 1 if w = and D = safe egion. b because that point is in the Next, we find the bounday between the jump egion undelying the expession in 3.29) and the continuation egion undelying the expession in 3.31). It tuns out that we can expess this bounday as a function D = D j w); subscipt j fo jump. We equie φ to be continuous along that bounday; that is, we equie + h)w 1 Solving this equation fo D j fo 0 w < yields = w hd jw) ) b h D jw) D j w) = w f jw), 3.32) h 1 f j w) in which f j is given by [ + h)w f j w) = 1 ) ) Because f j = 1, we define D j by continuity; specifically, set D j [ ] comes late, it is impotant to undestand the gaph of D j on 0,. λ. ] λ. 3.33) ) = b. Fo what Lemma Let the function D = D j w) be defined by equations 3.32) and 3.33), fo 0 w < ) and define D j = b. a) D j w) w h, with equality only at w = 0 and w =. b) If λ, then D j w) inceases fom 0 to as w inceases fom 0 to b c) If λ >, then D j w) 0 fo 0 w w, and D j w) inceases fom 0 to w to. b, as w inceases fom, in which w is the unique zeo of the expession in 3.13) in Poposition 3.4. Poof. Poof of a. D j w) w h if and only if w f jw) w, 1 f j w)

22 which is equivalent to f j w) = 0, which only occus when w = 0, o f j w) 0 and w. It follows that D j w) w h holds fo 0 w, with equality only at the endpoints. This esult confims ou hypothesis that the line w = hd lies in the jump egion. To pove pats b) and c), we will use the fact that D j w) is popotional to the following: D jw) 1 h + h f jw) + h f jw) 1 λ. 3.34) [ ] Poof of b. If λ, then D j w) is inceasing on 0, if and only if 1 h + h x + h x1 λ 0, fo 0 x 1. This inequality holds because the left side equals 0 when x = 1, and the left side deceases on [0, 1]. Poof of c. If λ >, fist note that D j w) has a unique zeo at w = w because D j w) = 0 if and only if the expession in 3.13) equals 0. Because D j 0) < 0, we conclude that D jw) 0 fo [ ] 0 w w. Also, note that f j w) inceases on 0, and equals x when w = w. It follows that [ ] D j w) is inceasing on w, if and only if 1 h + h x + h x1 λ 0, fo x x 1, which is equivalent to f 3 x) 0 fo x x 1, which we know is tue fom Lemma Fom Lemma 3.13, we see that thee ae two cases to conside: λ and λ >, as thee wee fo the poblem in Section 3.1. In the next two popositions, we pove that φ given in 3.29) and 3.31), patched togethe along D = D j w), satisfies the conditions of Lemma 3.8. In the fist, we conside λ ; in the second, λ >. Poposition Suppose λ. On R b = {w, D) : 0 D < b w, 0 w pobability of eaching the bequest goal befoe uining is given by φw, D) = 1 w hd h b D) )w, if 0 D Dj w),, if D j w) < D b w. The associated optimal insuance puchasing stategy is as follows: }, the maximum 3.35) a) If 0 D D j w), then do not buy additional insuance until wealth eaches the safe level, at b which time, buy additional insuance of D. b) If D j w) < D b w, then immediately buy additional insuance of b w + D) and theeafte Poof. continually buy additional insuance to ensue that the sum of wealth and death benefit equals b. The function φ is inceasing and piecewise diffeentiable in w and D, and it equals 1 at w = Recall that we defined φ in 3.31) to be equal to 1 when w = and D = b.. Fom the definition of

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