Securitization of Senior Life Settlements: Managing Interest Rate Risk with a Planned Duration Class


 Adam Jacobs
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1 1 Securiizaion of Senior Life Selemens: Managing Ineres Rae Risk wih a Planned Duraion Class Carlos E. Oriz Arcadia Universiy Deparmen of Mahemaics and Compuer Science Charles A. Sone Brooklyn College, Ciy Universiy of New York Deparmen of Economics Anne Zissu Temple Universiy Deparmen of Finance Using duraion o measure ineres rae risk for securiies such as MBS, callable bonds and securiies backed by life insurance policies is problemaic because for hese securiies he iming of cash flows is uncerain. In order o measure duraion for securiies wih embedded opions like callable bonds and MBS, cash flow iming mus be assumed or modeled. In he case of senior life selemens, duraion is only a useful summary of ineres rae risk if he esimaed life of he insured is accurae. I is precisely because he life of he insured is uncerain ha he duraion of a pool of senior life selemen conracs will no offer a meaningful summary of ineres rae risk. In his paper we illusrae how a pool of senior life selemen conracs can be funded wih a capial srucure ha is composed of wo classes of securiies; one which has a duraion ha is insulaed from variaions in he life of he insured around he esimaed life expecancy and he oher wih a duraion which is highly sensiive o variaions in he life of he insured around he esimaed life expecancy. We name he securiy class wih a sable duraion he Planned Duraion Class (PDC) while he class wih he unsable duraion is called he suppor duraion class. Securiizaion of Senior Life Selemens: Managing Ineres Rae Risk wih a Planned Duraion Class Senior life selemens are life insurance conracs ha have been purchased from senior ciizens by invesors ineresed in he value offered by he deah benefi. In reurn for assuming he sream of insurance premiums necessary o keep he policy acive, he purchaser becomes he beneficiary of he policy. Pools of senior life selemens have been funded on he capial markes wih issues of fixed income securiies ha are backed by he
2 2 fuure deah benefis of he life insurance policies 1. The vehicle ha funds he pool of life selemen conracs mus issue securiies o fund he purchase price of he conracs and he expeced premiums or have access o a line of credi or oher source of liquidiy o fund he fuure premiums. In his paper we illusrae how a pool of senior life selemen conracs can be funded wih a capial srucure ha is composed of wo classes of securiies whose duraion is alered from he duraion of he underlying pool of insurance conracs. One class is consruced so ha is duraion is insulaed from variaions in he moraliy raes of he insured. We accomplish his by calculaing he firs derivaive of he Macaulay duraion wih respec o a change in he ime he pool of insurance conracs is ousanding above or below he expeced life (LE). Flucuaions in he ime he pool is ousanding is a measuremen of longeviy risk; he risk ha he insured lives beyond or shor of some expeced value. By fixing his derivaive a zero, we are able o find he yield/le combinaions for which duraion is a sable measure of ineres rae risk across premium and deah benefi combinaions. The resuling marix is he basis for our design of various classes of securiies wih differen exposures o ineres rae risk. By defaul he creaion of a class of securiies ha has a sable duraion necessiaes he creaion of an accompanying securiy class ha has an excepionally unsable duraion. The key o he success of his capial srucure is ha sufficien ineres rae risk can be leveraged ono he unsable class and ha his levered class can be funded a a yield ha does no erode he savings garnered by financing he porion of he pool of life selemen conracs wih he sable duraion class. The acual lifespan of he insured whose policies compose a life selemen porfolio will vary from wha is projeced because of inaccurae esimaes of life expecancies and negaive and posiive shocks o he moraliy ables such as epidemics, medical coss and he approval of new pharmaceuicals and reamens. Variaions in acual life spans around expeced life hrow off he accuracy of duraion as a measuremen of ineres rae risk making i difficul for invesors o manage heir porfolios wihin chosen ranges of ineres rae risk and immunizaion sraegies. Longeviy risk is he chance ha an insured lives beyond he expeced moraliy dae. As he marke for morgagebacked securiies has proven, here is value in he abiliy o disill and securiize various dimensions of risk. The mos obvious case is he marke for principal and ineres only srips derived from morgage passhrough securiies. In his case he risk ha prepaymens are slower han expeced is allocaed o he principal only class and he risk ha prepaymen rae are faser han expeced are allocaed o he ineres only class. Fund and asse/liabiliy managers frequenly use duraion as a meric o summarize he ineres rae risk of porfolios of fixed income securiies. Porfolios can be managed o arge duraion ranges, and ineres rae risk of a firm can be moderaed by managing he duraion gap beween asses and liabiliies. Managers will rebalance asses and or liabiliies when argeed duraions become misaligned. The duraion of a securiy is he weighed average 1 The Liferade Fund (2006) and he Senior Life Selemen AsseBacked Securiizaion Bond (SLS ASB 2006) are examples of funds ha purchase life selemen conracs.
3 3 ime ha he value of he securiy is reurned o he invesor. This duraion measure can be modified so ha i is price elasiciy wih respec o yield. As a price elasiciy, duraion measures how sensiive he price of he securiy is relaive o changes in he yield of he securiy. When duraion is modified o measure ineres rae risk, i is referred o as modified duraion. Any measure of ineres rae risk for a pool of senior life selemen conracs mus be based on he expeced moraliy rae of he pool of insured. Deviaions from he expeced moraliy rae will change he iming and magniude of cash flows o and from he pool of conracs. When he acual moraliy rae is slower han he expeced rae, he value of he premiums ha mus be paid o keep he policies in force increases and he receip of fixed deah benefis are pushed furher ino he fuure. A clear difference beween he complexiies of measuring ineres rae risk of securiies wih embedded opions and senior life selemens is ha people do no exercise an opion o die. Our objecive is o design a capial srucure which is less cosly han one in which all classes of securiies issued o fund a pool of senior life selemens share on a pro raa basis all premium paymens and all deah benefis. We do his by creaing one class of securiies ha has a sable duraion, e.g. predicable ineres rae risk ha is independen of flucuaions in moraliy raes. This class of securiies is he unlevered class or Planned Duraion Class (PDC). The naural resul of creaing an unlevered class is he creaion of a class of securiies ha is levered wih respec o longeviy risk. We call his class he levered class or Suppor Duraion Class (SDC). In his ranching process he longeviy risk embedded in he pool of life selemen conracs is sripped ou and reallocaed o one class of securiies and away from anoher class of securiies. The success of his ype of ranching is he abiliy o place he levered class a a yield ha does no offse he savings generaed by he unlevered class relaive o a single securiy srucure. This ranching of risk is a he core of he secondary morgage marke. I is sandard for issues of collaeralized morgage obligaions (CMOs) o have cerain classes of securiies ha are insulaed from prepaymen risk, and he insulaing classes or leveraged classes ha finance he prepaymen risk ha has been shifed away from he insulaed class. In a CMO ransacion he class ha is insulaed from prepaymen risk is called he planned amorizaion class or PAC and he class ha is leveraged wih respec o prepaymen risk is called he suppor class. The efficiency and liquidiy of he marke for CMOs depends on he abiliy of invesmen bankers o place he riskier suppor classes a yields ha make he overall CMO ransacion valuable o he issuer.
4 4 An invesor who buys an ineres in a pool of life selemen conracs is exposed o he following risks: 1) credi risk 2) ineres rae risk, and mos significanly 3) longeviy or moraliy risk. I is he enanglemen of he ineres rae risk and longeviy risk essenially makes he soring, ranking and selecion of life selemen invesmens based on duraion inaccurae. Ye here is value in being able o fi life selemen invesmens ino specified duraion ranges. Our soluion o his problem will give invesors his abiliy o selec and compare life selemen conracs based on duraion. In 2004 he firs securiies backed by a pool of senior life selemen conracs was srucured and issued. I was a $63 Million class A senior life selemensecuriizaion backed by $195 million in face value of life insurance policies, issued by Tarryown Second, LLC. There have been several oher public securiizaions of pools of senior life selemens, and many privae deals. I has been esimaed by Conning & Co. ha in he nex en years, he life selemens marke may grow o over $125 billion. If he marke for assebacked securiies collaeralized by senior life selemens is o grow o is full poenial, he measuremens of risks specific o his class of securiies mus be developed and refined. We find he condiions for which he Macaulay duraion of a life selemen conrac is no affeced by changes in he insured s life above or below ha of his life expecancy. Once we calculae he combinaions of discoun raes and life expecancies for which duraion remains a sable measure of ineres rae risk for a life selemen conrac, we use his informaion o srucure a class of senior life selemen backed securiies whose ineres rae risk can be summarized using he sandard measure of duraion. Of course by deleveraging one class of securiies, we are lef wih a highly leveraged class of securiies. For his smaller bu more volaile class wih respec o deviaions of he acual life of a pool of insured from he life expecancy, duraion becomes even more unreliable as a measure of ineres rae risk. Jus as he mos volaile classes of collaeralized morgage obligaions wih respec o prepaymen risk mus be placed wih invesors who specialize in esimaing and assuming prepaymen risk, our soluion o he duraion problem for senior life selemens relies on he exisence of group of invesors who would be willing o ake posiions in securiies ha are leveraged wih respec o longeviy risk. I is imporan o keep in mind ha an unexpeced increase in longeviy perhaps due o a new pharmaceuical ha is approved will reduce he value of a pool of life selemens as he pool of insured lives longer and pushes ou he dae ha deah benefis are received.. The risk cus boh ways, an exreme hea wave or flu epidemic would reduce he average life of a life selemen pool increasing he yield on he life selemen conracs. 3 J.M. Keynes (1936) inroduced he concep of a bond s price elasiciy wih respec o ineres raes. J.R. Hicks (1938) research focused on ha same elasiciy concep. I was Macaulay (1938) ha expanded Keynes s and Hick s work by developing he duraion risk meric, which has since aken on his name, Macaulay duraion. While he Macaulay duraion is calculaed for parallel shis in a fla yield curve, oher variaions of duraion have been developed o accoun for nonparallel shifs and for securiies wih embedded opions. Macaulay and modified duraion (which is simply he Macaulay duraion divided by 1+y), are widely used merics by porfolio and asse/liabiliy managers.
5 5 I is no only shocks o he moraliy ables ha lead o changes in he value of a life selemen conrac, simple errors in esimaing someone s life span will also lead o deviaions in acual value from expeced value. For he leveraged class ineres rae risk becomes quie unpredicable as even sligh differences in acual life from life expecancy creae large changes in he value of he life selemen conrac. I should be noed however, ha in order o achieve an invesmengrade raing, a securiized pool of senior life selemens mus have a group of selers ha is diversified across diseases. In he following secion of he paper we derive he changes in he duraion of a pool of senior life selemen conracs wih respec o changes in he average life of he insured whose conracs compose he life selemen pool. We show ha for a se of combinaions of discoun raes and life expecancies, duraion is a sable consisen measure of ineres rae risk for life selemen conracs and securiies backed by senior life selemens. When he life of he pool of insured people deviaes from he expeced life, duraion sill offers an accurae measure of ineres rae risk We demonsrae our resuls wih numerical examples. Macaulay Duraion and Is Sensiiviy o Shifs in Life Expecancy Tables The value of an individual senior life selemen is equal o he presen value of he premia p paid periodically during he life of he senior life seler, plus he discouned value of he benefi B received a deah of he seler. P = p[1/(1+y) 1 + 1/(1+y) /(1+y) n ] + B/(1+y) n (1) Where p sands for he insurance premium paid each year firs by he original owner of he policy and hen by he life selemen company, B is he deah benefi a he ime of deah of he life seler, y is he discoun rae and P is he presen value of he life selemen conrac. The general formula for he Macaulay Duraion, D, of a fixed income securiy is D = (i)cfi/(1+y) i /P 1=i Where i is he ime a which he cash flow is paid, y is he discoun rae, P is he price of he securiy a he ime he duraion is compued, and CF i represens he cash flow a ime i. is he ime when he seler dies, and is unknown. The Macaulay duraion can be rearranged so ha i measures he percenage change in a securiy s price over he percenage change in yield, he price elasiciy of he securiy. 3 The cash flows o and from a life selemen conrac are he yearly premia p and he deah benefi B received a he ime of he insured s deah, which we denoe. Equaion (2) is he calculaion of he Macaulay duraion D for a life selemen conrac. The presen value of each cash flow is muliplied by he ime a which i is paid, i, where i runs
6 6 from year 1 o year, being he ime of deah so he premia cease and he deah benefi is received. The sum of he presen values, muliplied by he ime a which hey are received, is divided by he presen value of he life selemen conrac, P. D = [ p(i)/(1+y) i B/(1+y) ]/P (2) i=1 Changes in Duraion for shifs in life expecancy Invesors in morgagebacked securiies are exposed o he risk ha he prepaymen rae on he underlying pool of morgages will be above or below he rae hey use o price he morgagebacked securiy. Prepaymens affec he iming and magniude of cash flows generaed by morgagebacked securiies. For example when prepaymens accelerae, principal is prepaid faser so ha he oal ineres paid declines, a change in magniude, and he principal is colleced faser, a change in iming. For MBS he cash flows generaed by a pool of morgages are always posiive. Invesors in pools of life selemen conracs are exposed o longeviy risk he risk ha he pool of insured lives longer han he expecaion upon which he pool of conracs were priced. Unlike a pool of morgages which generae posiive cash flows composed of ineres and principal paymens, he underlying pool of life selemens conracs generaes a sream of negaive cash flows, he insurance premiums and a sream of posiive cash flows, he deah benefis. In he case of a pool of senior life selemens, he iming, magniude and direcion of cash flows are affeced by he longeviy of underlying insured. The change in he direcion of he cash flows from negaive o posiive for a pool of senior life selemens is he reason ha under cerain condiions, a deviaion from life expecancy of he selers does no affec he Macaulay duraion of he pool. In Appendix A we calculae he firs derivaive of he Macaulay duraion wih respec o he change in ime above or below he expeced ime of deah, and in Appendix B we find he condiions for which he derivaive is equal o zero. Fixing his derivaive a zero is he consrain we use o locae he yields of he life selemen conracs a which deviaions from life expecancy have no impac on he duraion measuremen. This calculaion is done for various yield/life expecancy combinaions. As a maer of simplifying he derivaive calculaion in Appendix A, we se a = (1)/(1+y) and equaion (2) hen becomes: D =[( pi(a) i ) B(a) ]/P (2) i=1 Condiions for a reliable Senior Life Selemen s Macaulay Duraion In Appendix B we develop he condiions for which he Macaulay duraion of a pool of senior life selemens is no affeced by changes in he life of he insured above or below life expecancies compued a he ime he insurance conracs were sold. We use equaion (13) from Appendix A, o calculae he condiions for which D = 0. This is expressed in equaion (14).
7 7 * = [1/ln(1+y)] + [p(1+y)/y(pby)] eq. 14 Equaion (14) shows he consrain beween a seler s life expecancy (we call i * in he equaion) he premium p, he deah benefi B and he yield y of he corresponding policy, in order o obain a life selemen wih a duraion ha is no affeced by deviaions around LE. Carving ou he Planned Duraion Class and he Suppor Duraion Class Exhibi 1 shows a ypical block of senior life selemens ha could be securiized. I has a oal principal value of $82 million, a weighed average α of 4.4% (we define α as he raio of premium p o deah benefi B, α = p/b), and a weighed average LE of I is sandard for he indusry of senior life selemens o quoe life insurance policies in erms of he premium o deah benefi raio, i.e. a 4% policy may corresponds o a premium of $4000 and a deah benefi of $100,000. Exhibi 1 Principal Amoun 10,000,000 15,000,000 5,000,000 2,000,000 1,000,000 3,000,000 6,000,000 8,000,000 1,000,000 4,000,000 10,000,000 7,000,000 10,000,000 Annual Premium 500, , , , , , , , , , , , , LE We nex find he premia and deah benefis ha saisfy equaion (14) for differen yield/le combinaions. We have assumed an upward yield curve so ha yields do increase wih life expecancy in eq. (14). We assume a yield of 5.25% for a LE of 2, a yield of 5.50% for a LE of 3 and coninue o increase he yield by 25bps for each LE increase by one addiional year, as shown in Exhibi 2. When we plug a 5.25% yield, a LE of 2, and a deah benefi B of $1,100,000 in eq. (14), and solve for premium p, we obain $400,500 as shown in Exhibi 2. We solve for premia, using several oher combinaion of yields, LEs and deah benefis, all represened in Exhibi 2.
8 8 Exhibi 2 LE y 5.25% 5.50% 5.75% 6% P * 400, , , ,000 B * 1,100,000 1, , , Exhibi p 450, , , ,275, B 12,000,000 16,000,000 18,000,000 27,000,000 Exhibi 3 summarizes exhibi 1 across LEs: i adds all premia generaed by life selemens wih a LE of 2 and all corresponding deah benefis. There are only wo life selemens wih LE of 2, and he corresponding premia are $200,000 and $250,000, a oal of $450,000, shown in exhibi 3 in he second column. The corresponding deah benefis are $5 million and $7 million, a oal of $12 million. In each column we add he premia and deah benefis for each LEs going from 2 o 5. Exhibi 4: Planned Duraion Class PDC p/p * B/B * Exhibi 4 shows he raio of he opimal premium p * (from Exhibi 2) over he oal premia across LEs ranging from 2 o 5, compued in Exhibi 3, and he corresponding raio of he opimal B * (compued in Exhibi 2) over he oal deah benefis compued in Exhibi 3. If we look a he LE of 5, we can creae 8.79 PDCs wih an annual premium of $145,000 each, and a corresponding deah benefi of $1,100,000 (from Exhibi 2). More specifically, he planned duraion class, PDC, for he LE of 5 and a 6% yield, can be srucured by sripping 8.79 imes he amoun of $145,000 in premia (from a oal of $1,275,000 in Exhibi 3), wih a corresponding 8.79 imes $1,100,000 in deah benefis (from a oal of $27,000,000 in Exhibi 3). The suppor duraion class, CDC, is hen allocaed he difference beween he sum of he oal premia under he LE of 5, minus hose allocaed o he PDC, and he sum of he deah benefis under LE of 5 minus hose allocaed o he CDC. We show he oal amoun of premia p and corresponding deah benefis B for he planned duraion classes creaed under differen LEs in Exhibi 5 and he corresponding premia and deah benefis for he suppor duraion classes in Exhibi 6. The corresponding α is shown in he las row of Exhibis 5 and 6. I is ineresing o observe ha α is high for he planned duraion class, and goes o zero for he suppor duraion class. We can acually generalize he relaionship beween α and LE for a planned duraion class in Exhibi 7. We observe ha for a PDC wih a
9 9 high LE, a lower α is needed, whils for a PDC wih low LE, a higher α mus be consruced by sripping differen premia. The resuling suppor duraion class, SDC, is a zerocoupon bond ha pays deah benefis only (no negaive premium), a acual deah of he insured, which qualifies as mauriy of he zerocoupon bond. The SDC s Macaulay duraion is hen equal o he LE (he mauriy) if deviaion around LE is zero. Exhibi 5: Planned Duraion Class LE y (PDC)p (PDC)B (PDC)α Exhibi 6: Suppor Duraion Class LE (SDC)p (SDC)B 10,764,045 13,390,511 13,078,947 17,327,586 (SDC)α Exhibi 7 α/le curve for PDC α LE For simpliciy, we only analyzed LEs ranging from 2 o 5 in Exhibi 1, which corresponded o a oal face value of $73 million. This means ha we creaed a planned duraion class in he amoun
10 10 of 25% of $73 million and a suppor duraion class in he amoun of 75% of he $73 million face value. Conclusion The derivaion of he yield/le combinaions for which he duraion of a pool of life selemen conracs does no change as he age of he life selers deviaes from he life expecancy, opens up he possibiliy of srucuring wo classes of securiies o finance a pool of senior life selemen conracs; one class would be designed o have a sable duraion measure across a specrum of pool longeviy and he oher class would pick up he slack by being designed o have a duraion ha is very sensiive o small changes in pool longeviy. In fac he design of he second class is imposed by he design of he firs. When financing a fixed pool of asses such as senior life selemens wih various classes of assebacked securiies, he deleveraging of one class wih respec o a risk dimension, in his case longeviy risk, mus be accompanied by a leveraging of anoher class. The firs class would be srucured o address he needs of hose invesors looking for invesmens wih fairly cerain duraions, we call his class he Planned Duraion Class (PDC). This is done by carving ou cash flows from he pool of life selemen conracs ha saisfy equaion (14). Cash flows generaed by he life selemen pool bu no allocaed o he Planned Duraion Class would be direced o he Suppor Duraion Class (SDC). This can be achieved by combining seasoned wih unseasoned porfolios of senior life selemen conracs. In exchange for assuming he longeviy risk of he pool, invesors in he SDC class would be offered a higher yield. References A.M. Bes, Life Selemen Securiizaion, Sepember 1, D. Blake, A. J. G. Cairns and K. Dowd, Living Wih Moraliy: Longeviy Bonds And Oher MoraliyLinked Securiies, Presened o he Faculy of Acuaries, 16 January Cowley, Alex; Cummins, J. David, Securiizaion of Life Insurance Asses and Liabiliies, Journal of Risk and Insurance, June 2005, v. 72, iss. 2, pp Cummins David J., Securiizaion of Life Insurance Asses and Liabiliies, Wharon Dohery Neil A. and Singer Hal J. The Benefis of a Secondary Marke for Life Insurance Policies, The Wharon Financial Insiuions Cener, November 14, Dowd Kevin, Cairns Andrew J.G.and Blake David, Moraliydependen financial risk measures, Insurance: Mahemaics and Economics, Volume 38, Issue 3, Pages (15 June 2006).
11 11 Goldsein Mahew, Dying for 8%  Invesors Beware, Tavakoli Srucured Finance, Inc., Augus J. Hicks, Value and Capial: An inquiry ino some fundamenal principles of economic heory Ediion, Oxford: Clarendon Press, Keynes, J.M. The General Theory of Employmen Ineres and Money, New York: Harcour, Brace & World, Inc., Lin, Yijia and Cox, Samuel H., Securiizaion of Moraliy Risks in Life Annuiies, Journal of Risk and Insurance, June 2005, v. 72, iss. 2, pp Macaulay, F. Some Theoreical Problems Suggesed by he Movemens of Ineres Raes, Bond Yields, and Sock Prices in he Unied Saes Since New York: Naional Bureau of Economic Research, Milevsky, Moshe A., The Implied Longeviy Yield: A Noe on Developing an Index for Life Annuiies, Journal of Risk and Insurance, June 2005, v. 72, iss. 2, pp Modu Emmanuel, Life Selemen Securiizaion, A.M. Bes Repor Ocober 18, Richard Chrisine, Wih $70M Bond Deal, Wall S. Manages o Securiize Deah, The Wall Sree Journal, April 30, Sone Charles A. and Zissu Anne, Securiizaion of Senior Life Selemens: Managing Exension Risk, The Journal of Derivaives, Spring Sone Charles A. and Zissu Anne, Risk Managemen Risk Measuremen Leer from he Ediors, The Financier, Vol.3, No. 4 & 5, Taglich Brohers, Inc. Research Repor on Life Parners Holdings, Inc. December Weil Roman L., "Macaulay's Duraion: An Appreciaion." Journal of Business, 1973, 46(4), pp APPENDIX A The nex sep is o isolae from eq. (2) he erm, ha we call f(): f() = pi(a) i i=1 In he nex secion we find an expression for f().
12 12 Expression of f() Recall ha for every real number b and naural number k<n, he following formula holds: b i = (b n+1 b k )/(b1) (3) i=k We can rewrie f() as f() = pi(a) i = p i(a) i = (4) i=1 i=1 p(a+a 2 +a 3 + a + (5) a 2 +a 3 + a + (6) a 3 + a + (7) + (8) a ) (9) By applying formula (3) o every single line of he previous segmens (fragmens (5), (6), ec.) we ge ha f() = pi(a) i = p i(a) i = i=1 i=1 p[(a +1 a 1 )/(a1) + (a +1 a 2 )/(a1) + (a +1 a 3 )/(a1) + + (a +1 a )/(a1)] The above expression can be simplified ino f() = pi(a)i = p i(a) i = i=1 i=1 p[a +1 (a + a a )]/(a1) Applying formula (3) again we ge ha f() = pi(a) i = p i(a) i = p[a +1 (a +1 a)/(a1)]/(a1) i=1 i=1 [p/(a1)][(a +2 a +1 a +1 +a]/(a1) = [p/(a1) 2 ][a +2 (+1)a +1 +a] In summary, we proved ha f() = [p/(a1) 2 ][a +2 (+1)a +1 +a] where a=1/(1+y) and p, y, are consan wih respec o.
13 13 Expression for D We can add now he new expression of f() o ge he final expression for D: D =[( pi(a) i ) B(a) ]/P = i=1 [f() B(a) ]/P = {(p/(a1) 2 [a +2 (+1)a +1 + a]) B(a) }/P = (1/P){a [a 2 p/(a1) 2 ap/(a1) 2 B] a [ap/(a1) 2 ] + ap/(a1) 2 }= (1/P){a [ap/(a1) B] a [ap/(a1) 2 ] + ap/(a1) 2 } In summary, we ge ha D = (1/P){a [ap/(a1)  B] a [ap/(a1) 2 ] + ap/(a1) 2 } (10) For he purpose of having an expression of D using he original parameers, we can now replace a by 1/(1+y) in eq. (10) o obain ha D = (1/P){a [ap/(a1)  B] a [ap/(a1) 2 ] + ap/(a1) 2 } = (1/P){[1/(1+y) ][[p/(1+y)]/[(1/(1+y)1] B] (1/(1+y)) [p/(1+y)]/[(1/(1+y)) 1] 2 + [(p/(1+y)]/[(1/(1+y)) 1] 2 To conclude, we have he following close expression for he funcion D: D = {[p(y) 1 B]/(1+y)  p/[y 2 (1+y) 1 ] + p(1+y)/y 2 }/P (11) The Derivaive of Duraion Wih Respec o Time Noe ha he expression of D in eq.(10) is of he form D = {a [C(a1) B] a C + C}/P = (1/P){a [C(a1) B] a C + C} (12) Where a, B, P are consans and C is a consan defined as
14 14 C = ap/(a1) 2 Now we ake he derivaive of duraion as expressed in equaion (12) wih respec o. The soluion is given is equaion 13. D = (1/P){a [C(a1) B] + a [C(a1) B] [ln(a)] a C[ln(a)]} Where C and a are he consans previously specified. We simplify he las expression o arrive a equaion (13). D = (a /P){[C(a1) B] + [C(a1) B] [ln(a)] C[ln(a)]} = D = (a /P){[[C(a1) B] ln(a)] + [C(a1) B C[ln(a)]]} (13) APPENDIX B Noe ha (a /P) can never be equal o 0 because a is always posiive (1/1+y). This means ha he derivaive of duraion wih respec o ime () equals zero (D = 0) when (C(a1) B)(ln(a)) + C(a1) B C(ln(a)) = 0 This is a simple linear equaion ha we solve for. = [((C(a1) B) + C(ln(a))]/[(C(a1) B)(ln(a))] = [1/ln(a)] + C/[C(a1) B] Now we replace C wih (ap)/(a1) 2 o arrive a D = 0 when = [1/ln(a)] + [ap/(a1) 2 ]/{[ap/(a1) ] B]} This previous equaion can be simplified o : = [1/ln(a)] + [ap/(a1) 2 ]/{[ap(a1) B(a1) 2 ]/(a1) 2 } = [1/ln(a)] + ap/[ap(a1) B(a1) 2 ] Finally, we replace a by is value of 1/(1+y) o show ha D =0 when
15 * = [1/ln(1+y)] + [p(1+y)/y(pby)] eq
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