How To Find The Optimal Stategy For Buying Life Insuance


 Lionel Turner
 7 months ago
 Views:
Transcription
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 afoementioned 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 afoementioned 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 timehomogeneous 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. SinglePemium 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 nonzeo 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 timehomogeneous 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 singlepemium 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 nonnegative, nondeceasing pocess, and ii) if wealth unde this pocess is nonnegative 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 nondeceasing 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 socalled 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 nondeceasing 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 boundayvalue 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 boundayvalue 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 boundayvalue 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 nonnegative. 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 nondeceasing, 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 onesided 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 continuouslypaid 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 timehomogeneous 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 nonnegative 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 nondeceasing, 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 onesided 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 nonnegative on [0, x ]. c) The function f 3 on [0, 1] defined by is nonpositive 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 nonnegative 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 nonnegative 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 nonpositive 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 nonpositive 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 nondeceasing, 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 buynoinsuance 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 eenfocing 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 nonnegative and nondeceasing 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 nondeceasing, 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 onesided 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 nondeceasing 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 boundayvalue 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
Chapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationExperimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival
Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuoustime model of expeimentation
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationCONCEPT OF TIME AND VALUE OFMONEY. Simple and Compound interest
CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationDefinitions and terminology
I love the Case & Fai textbook but it is out of date with how monetay policy woks today. Please use this handout to supplement the chapte on monetay policy. The textbook assumes that the Fedeal Reseve
More informationEfficient Redundancy Techniques for Latency Reduction in Cloud Systems
Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationModeling and Verifying a Price Model for Congestion Control in Computer Networks Using PROMELA/SPIN
Modeling and Veifying a Pice Model fo Congestion Contol in Compute Netwoks Using PROMELA/SPIN Clement Yuen and Wei Tjioe Depatment of Compute Science Univesity of Toonto 1 King s College Road, Toonto,
More informationThe Predictive Power of Dividend Yields for Stock Returns: Risk Pricing or Mispricing?
The Pedictive Powe of Dividend Yields fo Stock Retuns: Risk Picing o Mispicing? Glenn Boyle Depatment of Economics and Finance Univesity of Cantebuy Yanhui Li Depatment of Economics and Finance Univesity
More informationControlling the Money Supply: Bond Purchases in the Open Market
Money Supply By the Bank of Canada and Inteest Rate Detemination Open Opeations and Monetay Tansmission Mechanism The Cental Bank conducts monetay policy Bank of Canada is Canada's cental bank supevises
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk  Risk to shaeholdes esulting fom the use of debt. Financial Leveage  Incease in the vaiability of shaeholde etuns that
More informationIgnorance is not bliss when it comes to knowing credit score
NET GAIN Scoing points fo you financial futue AS SEEN IN USA TODAY SEPTEMBER 28, 2004 Ignoance is not bliss when it comes to knowing cedit scoe By Sanda Block USA TODAY Fom Alabama comes eassuing news
More informationA Capacitated Commodity Trading Model with Market Power
A Capacitated Commodity Tading Model with Maket Powe Victo MatínezdeAlbéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationFinancial Planning and Riskreturn profiles
Financial Planning and Risketun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: 201016 Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM Financial Planning and Risketun
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationLiquidity and Insurance for the Unemployed
Liquidity and Insuance fo the Unemployed Robet Shime Univesity of Chicago and NBER shime@uchicago.edu Iván Wening MIT, NBER and UTDT iwening@mit.edu Fist Daft: July 15, 2003 This Vesion: Septembe 22, 2005
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationFI3300 Corporate Finance
Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multipeiod multicf timevalueofmoney poblems: Geneal case Pepetuity
More informationCollege Enrollment, Dropouts and Option Value of Education
College Enollment, Dopouts and Option Value of Education Ozdagli, Ali Tachte, Nicholas y Febuay 5, 2008 Abstact Psychic costs ae the most impotant component of the papes that ae tying to match empiical
More informationSaving and Investing for Early Retirement: A Theoretical Analysis
Saving and Investing fo Ealy Retiement: A Theoetical Analysis Emmanuel Fahi MIT Stavos Panageas Whaton Fist Vesion: Mach, 23 This Vesion: Januay, 25 E. Fahi: MIT Depatment of Economics, 5 Memoial Dive,
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationPersonal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit () Total Private Saving Rate (S Private /Y) 12/18/2009
1 Pesonal Saving Rate (S Households /Y) 2 SAVING AND INVESTMENT 16.0 14.0 12.0 10.0 80 8.0 6.0 4.0 2.0 0.02.04.0 1959 1961 1967 1969 1975 1977 1983 1985 1991 1993 1999 2001 2007 2009 Pivate Saving Rate
More informationDefine What Type of Trader Are you?
Define What Type of Tade Ae you? Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 1 Disclaime and Risk Wanings Tading any financial maket involves isk. The content of this
More informationChris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment
Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationFirstmark Credit Union Commercial Loan Department
Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and
More informationOptimal Capital Structure with Endogenous Bankruptcy:
Univesity of Pisa Ph.D. Pogam in Mathematics fo Economic Decisions Leonado Fibonacci School cotutelle with Institut de Mathématique de Toulouse Ph.D. Dissetation Optimal Capital Stuctue with Endogenous
More informationTHE CARLO ALBERTO NOTEBOOKS
THE CARLO ALBERTO NOTEBOOKS Meanvaiance inefficiency of CRRA and CARA utility functions fo potfolio selection in defined contibution pension schemes Woking Pape No. 108 Mach 2009 Revised, Septembe 2009)
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationCapital Investment and Liquidity Management with collateralized debt.
TSE 54 Novembe 14 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee, Stéphane Villeneuve and Xavie Wain 7 Capital Investment and Liquidity Management with collatealized debt.
More informationExam #1 Review Answers
xam #1 Review Answes 1. Given the following pobability distibution, calculate the expected etun, vaiance and standad deviation fo Secuity J. State Pob (R) 1 0.2 10% 2 0.6 15 3 0.2 20 xpected etun = 0.2*10%
More informationLiquidity and Insurance for the Unemployed*
Fedeal Reseve Bank of Minneapolis Reseach Depatment Staff Repot 366 Decembe 2005 Liquidity and Insuance fo the Unemployed* Robet Shime Univesity of Chicago and National Bueau of Economic Reseach Iván Wening
More informationMoney Market Funds Intermediation and Bank Instability
Fedeal Reseve Bank of New Yok Staff Repots Money Maket Funds Intemediation and Bank Instability Maco Cipiani Antoine Matin Buno M. Paigi Staff Repot No. 599 Febuay 013 Revised May 013 This pape pesents
More informationAn application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty
An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty YinYann Chen * and HsiaoYao Fan Depatment of Industial Management, National Fomosa Univesity,
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationYARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH
nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav
More informationTrading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract
Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi email: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,
More informationPromised LeadTime Contracts Under Asymmetric Information
OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3364X eissn 15265463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised LeadTime Contacts Unde Asymmetic Infomation Holly
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More informationAn Analysis of Manufacturer Benefits under Vendor Managed Systems
An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More information9:6.4 Sample Questions/Requests for Managing Underwriter Candidates
9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated
More informationUncertain Version Control in Open Collaborative Editing of TreeStructured Documents
Uncetain Vesion Contol in Open Collaboative Editing of TeeStuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecompaistech.f Talel Abdessalem
More informationBasic Financial Mathematics
Financial Engineeing and Computations Basic Financial Mathematics Dai, TianShy Outline Time Value of Money Annuities Amotization Yields Bonds Time Value of Money PV + n = FV (1 + FV: futue value = PV
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationInstructions to help you complete your enrollment form for HPHC's Medicare Supplemental Plan
Instuctions to help you complete you enollment fom fo HPHC's Medicae Supplemental Plan Thank you fo applying fo membeship to HPHC s Medicae Supplement plan. Pio to submitting you enollment fom fo pocessing,
More informationContingent capital with repeated interconversion between debt and equity
Contingent capital with epeated inteconvesion between debt and equity Zhaojun Yang 1, Zhiming Zhao School of Finance and Statistics, Hunan Univesity, Changsha 410079, China Abstact We develop a new type
More informationNBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING?
NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING? Daia Bunes David Neumak Michelle J. White Woking Pape 16932 http://www.nbe.og/papes/w16932
More informationImproving Network Security Via CyberInsurance A Market Analysis
1 Impoving Netwok Secuity Via CybeInsuance A Maket Analysis RANJAN PAL, LEANA GOLUBCHIK, KONSTANTINOS PSOUNIS Univesity of Southen Califonia PAN HUI Hong Kong Univesity of Science and Technology Recent
More informationA THEORY OF NET DEBT AND TRANSFERABLE HUMAN CAPITAL
A THEORY OF NET DEBT AND TRANSFERABLE HUMAN CAPITAL Bat M. Lambecht Lancaste Univesity Management School Gzegoz Pawlina Lancaste Univesity Management School Abstact Taditional theoies of capital stuctue
More informationEffect of Contention Window on the Performance of IEEE 802.11 WLANs
Effect of Contention Window on the Pefomance of IEEE 82.11 WLANs Yunli Chen and Dhama P. Agawal Cente fo Distibuted and Mobile Computing, Depatment of ECECS Univesity of Cincinnati, OH 452213 {ychen,
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationPatent renewals and R&D incentives
RAND Jounal of Economics Vol. 30, No., Summe 999 pp. 97 3 Patent enewals and R&D incentives Fancesca Conelli* and Mak Schankeman** In a model with moal hazad and asymmetic infomation, we show that it can
More informationGESTÃO FINANCEIRA II PROBLEM SET 1  SOLUTIONS
GESTÃO FINANCEIRA II PROBLEM SET 1  SOLUTIONS (FROM BERK AND DEMARZO S CORPORATE FINANCE ) LICENCIATURA UNDERGRADUATE COURSE 1 ST SEMESTER 20102011 Chapte 1 The Copoation 113. What is the diffeence
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new highspeed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationSupply chain information sharing in a macro prediction market
Decision Suppot Systems 42 (2006) 944 958 www.elsevie.com/locate/dss Supply chain infomation shaing in a maco pediction maket Zhiling Guo a,, Fang Fang b, Andew B. Whinston c a Depatment of Infomation
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationAn Epidemic Model of Mobile Phone Virus
An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationOptimal Peer Selection in a FreeMarket PeerResource Economy
Optimal Pee Selection in a FeeMaket PeeResouce Economy Micah Adle, Rakesh Kuma, Keith Ross, Dan Rubenstein, David Tune and David D Yao Dept of Compute Science Univesity of Massachusetts Amhest, MA; Email:
More informationElectricity transmission network optimization model of supply and demand the case in Taiwan electricity transmission system
Electicity tansmission netwok optimization model of supply and demand the case in Taiwan electicity tansmission system MiaoSheng Chen a ChienLiang Wang b,c, ShengChuan Wang d,e a Taichung Banch Gaduate
More informationLoad Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationLoad Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO  1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationLeft and RightBrain Preferences Profile
Left and RightBain Pefeences Pofile God gave man a total bain, and He expects us to pesent both sides of ou bains back to Him so that He can use them unde the diection of His Holy Spiit as He so desies
More informationSaturated and weakly saturated hypergraphs
Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 67 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B
More informationOverencryption: Management of Access Control Evolution on Outsourced Data
Oveencyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI  Univesità di Milano 26013 Cema  Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM  Univesità
More information9.5 Amortization. Objectives
9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations!
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationIntertemporal Macroeconomics
Intetempoal Macoeconomics Genot Doppelhofe* May 2009 Fothcoming in J. McCombie and N. Allington (eds.), Cambidge Essays in Applied Economics, Cambidge UP This chapte eviews models of intetempoal choice
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationYIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE
YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1  P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More information30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research
RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 2945) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This
More informationDo Bonds Span the Fixed Income Markets? Theory and Evidence for Unspanned Stochastic Volatility
Do Bonds Span the Fied Income Makets? Theoy and Evidence fo Unspanned Stochastic olatility PIERRE COLLINDUFRESNE and ROBERT S. GOLDSTEIN July, 00 ABSTRACT Most tem stuctue models assume bond makets ae
More informationHEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING
U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 22863540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available
More information