Securitized Senior Life Settlements Macauley Duration and Longevity Risk

Transcription

1 1 Securiized Senior Life Selemens Macauley Duraion and Longeviy Risk 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 In his paper we illusrae ha Macaulay duraion is an inaccurae meric of ineres rae risk for fixed income securiies ha are backed by pools of senior life selemen conracs. The inaccuracy is due o he fac ha he iming of cash flows from senior life selemen insrumens, boh he life insurance premiums and he deah benefis, are a funcion of how long he associaed pool of insured acually lives. Deviaions in acual life from life expecancy make saemens abou he ineres rae risk of a securiy backed by senior life selemens conjecural. I is only possible o summarize he ineres rae risk of a pool of senior life selemens if he saemen is modified o accoun for he uncerainy of cash flow iming and magniude. We calculae he relaionship beween duraion and changes in he iming of cash flows o and from senior life selemen conracs o illusrae he weakness of using duraion o quanify ineres rae risk for life selemen securiies. Once we have illusraed he weakness of using he duraion meric for his class of securiies we offer a soluion o he problem of no being able o summarize ineres rae risk wih duraion. Our approach is similar o wha has been done in he marke for collaeralized morgage obligaions. We demonsrae ha i is possible o carve ou a class of securiies from a pool of senior life selemens ha will have a sable duraion across a wide range of weighed average lives of a pool of senior life selemen conracs. Key words: Senior life selemens, longeviy risk, moraliy ables, duraion.

2 2 Inroducion and Moivaions 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 selemen-securiizaion backed by $195 million in face value of life insurance policies, issued by Tarryown Second, LLC. If he marke for asse-backed 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. Senior life selemens are conracs ha ransfer a life insurance policy from he insured o a financial eniy. The financial eniy purchases he policy a a discoun from he owner. The discoun will be a funcion of he presen value of he premiums due and he deah benefi boh of which depend upon he life expecancy of he insured and he yield curve. Once he life insurance policy is ransferred from he owner o he financial eniy ha has bough i, he purchaser of he policy, he life selemen company, is liable for he premium paymens and becomes he beneficiary. Upon he deah of he insured he life selemen company receives he deah benefi. The value of he fixed sream of premium paymens and fuure deah benefis are exposed o ineres rae risk. If he deah dae of he insured were known wih cerainy, duraion would be a useful measure of ineres rae risk like i is for any fixed income securiy whose cash flows are no ied o embedded opions such as morgage backed securiies, callable bonds and bank deposis or o exernal facors such naural disasers or credi evens. The erm senior in senior life selemen conracs refers o he age of hose people who life selemen companies will buy insurance conracs from. In he curren marke he life expecancy of he supply of life insurance polices o he life selemen marke is no more han welve years. I is he demand for living benefis ha is driving he supply of life insurance policies o he life selemen marke and i is he search for invesmen value ha is driving he demand for life insurance polices by life selemen companies. As of February 2005 here was approximaely $12.7 rillion of life policies in he U.S, of which approximaely 10% for insured ha are a leas 70 years old. Each year $1.5 rillion in life policies lapses or is surrendered. The marke for senior life selemen conracs was esimaed o have amouned o beween $6 billion and $8 billion in 2004, wih a projecion of $45 billion in face value by 2007 (source: We do no discuss he reasons people choose o disconinue a life insurance policy. For our purposes he ineresing poin is ha life selemen companies can offer owners of life insurance more value han he savings in premiums hey achieve by leing a policy lapse. The value of a life selemen conrac is based on he life expecancy of he seler which in urn is a funcion of he age and healh condiion of he seler. The valuaion of a life selemen is achieved by discouning he premia paid over he esablished seler s life expecancy and he benefi o be received a he ime he seler dies. If a seler lives above life expecancy, premia need o be paid over a longer period, and i akes longer o receive he deah benefi. Longeviy reduces he value of senior life selemens.

3 3 Longeviy risk is he key variable in he evaluaion of life selemen conracs Life expecancy ables are always changing. Life expecancy in developed counries has been increasing seadily over ime, alhough no uniformly across differen age ranges, as suggesed by Renshaw, Haberman, and Hazoupoulos (1996). Lin and Cox (2005) compue he percenage change in he presen values of annuiy paymens under differen simulaed moraliy shocks. From heir numerical applicaions we can see how sensiive securiies wih longeviy risk can be o deviaions from he life expecancy on which premiums were se.. We should noe ha longeviy risk is he risk of living longer han expeced and moraliy risk is he risk of dieing sooner han expeced. Obviously hese risks are he wo sides of he same coin. We conribue o Lin and Cox s research by compuing he Macaulay duraion s sensiiviy o longeviy risk. Shifs in life expecancy ables disurb he reliabiliy of he Macaulay duraion. Wihou duraion as a useful meric of ineres rae risk, invesmens in senior life selemen conracs can no be readily gauged o oher fixed income securiies. Wihou duraion as a reliable measure of ineres rae risk senior life selemen conracs will have o be discouned a a higher rae o compensae for his uncerainy. This loss of value will consrain he growh of he marke. We find he condiions for which he Macaulay duraion of a life selemen conrac is no affeced by changes in he insured person s life above or below ha of his life expecancy. Once we calculae he combinaions of discoun raes and deviaions in life from life expecancy 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 on he acual life of a pool of insured from he life expecancy, he pool duraion becomes even more unreliable as a measure of ines rae risk. Jus as he mos volaile classes of collaeralized morgage obligaions wih respec o prepaymen risk mus be placed wih fund managers and risk managers 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 life exension 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. One class is srucured so ha is duraion is a relaively sable measure of ineres rae risk while he second smaller class whose design is consrained by he design of he firs class has ineres rae risk ha can no be measured by duraion because i is highly leveraged wih respec o changes in he life of he insured.

4 4 In he following secion of he paper we derive he changes in he duraion of senior life selemen conracs wih respec o changes in he average life of he pool of insured who have sold heir policies in he life selemen marke. We show ha for a se of combinaions of discoun raes and deviaions of he acual life of he insured from his life expecancy, duraion is a sable consisen measure of ineres rae risk for life selemen conracs and securiies backed by senior life selemens. We demonsrae our resuls wih numerical examples. Modeling framework: 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. 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 non-parallel 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 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 deah, a ime. 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 from year 1 o year, being he ime he premia sop being paid and when 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 a Senior Life Selemen s Duraion for shifs in life expecancy Invesors in morgage-backed 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 morgage-backed 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 selemens conracs are exposed o longeviy risk he risk ha he pool of insured live 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 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 his secion we calculae he firs derivaive of he Macaulay duraion wih respec o he change in ime above or below he expeced deah, and find he condiions for which he derivaive is equal o zero. As a maer of simplifying he derivaive calculaion we se a = (1)/(1+y) and equaion (2) hen becomes: D =[( pi(a) i ) B(a) ]/P i=1 The nex sep is o isolae he erm, ha we call f(): f() = pi(a) i i=1

6 6 In he nex secion we find an expression for f(). 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 )/(b-1) (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 )/(a-1) + (a +1 -a 2 )/(a-1) + (a +1 -a 3 )/(a-1) + + (a +1 -a )/(a-1)] 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 )]/(a-1) Applying formula (3) again we ge ha f() = pi(a) i = p i(a) i = p[a +1 (a +1 -a)/(a-1)]/(a-1) i=1 i=1 [p/(a-1)][(a +2 a +1 a +1 +a]/(a-1) = [p/(a-1) 2 ][a +2 (+1)a +1 +a] In summary, we proved ha f() = [p/(a-1) 2 ][a +2 (+1)a +1 +a]

7 7 where a=1/(1+y) and p, y, are consan wih respec o. 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/(a-1) 2 [a +2 (+1)a +1 + a]) B(a) }/P = (1/P){a [a 2 p/(a-1) 2 ap/(a-1) 2 B] a [ap/(a-1) 2 ] + ap/(a-1) 2 }= (1/P){a [ap/(a-1) B] a [ap/(a-1) 2 ] + ap/(a-1) 2 } In summary, we ge ha D = (1/P){a [ap/(a-1) - B] a [ap/(a-1) 2 ] + ap/(a-1) 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/(a-1) - B] a [ap/(a-1) 2 ] + ap/(a-1) 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(a-1) B] a C + C}/P = (1/P){a [C(a-1) B] a C + C} (12)

8 8 Where a, B, P are consans and C is a consan defined as C = ap/(a-1) 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(a-1) B] + a [C(a-1) 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(a-1) B] + [C(a-1) B] [ln(a)] C[ln(a)]} = D = (a /P){[[C(a-1) B] ln(a)] + [C(a-1) B C[ln(a)]]} (13) Shifs in Senior Life Selemens Life Expecancies Daa in exhibi 1 describes a ypical disribuion of life expecancies for insurance policies ha are eligible for sale ino he senior life selemen marke. Exhibi 1 Typical Disribuion of Available Life Expecancies % of Insured Life Expecancy (LE) in LE Caegory LE<= 36 monhs 1 36 monhs<le<=72 monhs monhs<le<=108 monhs monhs<le<=144 monhs* monhs<le<=180 monhs monhs<le<=216 monhs 8 LE>=216 monhs 2 *As a pracical maer, he life expecancies ha are found mos commonly in life selemen ransacions are normally 12 years or less. Source: A.M. Bes, Sepember We use he disribuion from exhibi 1 o consruc a hypoheical pool of senior life selemen conracs and examine how he duraion of his pool changes wih respec o deviaions in he acual life of he insured from heir expeced life. We assume ha he pool is composed of 100 life selemen conracs (m 0 ), he number of selers from he pool dying a each poin in ime (d i ) is compued over a oal of n periods (n = 225) as follows in Exhibi 2:

9 9 Exhibi 2 d 1,.., d 36 = 1/36 = d 37,.., d 72 = 12/(72-36) =.3333 d 73,.., d 108 = 30/(108-72) = d 109,.., d 144 = 30/( ) = d 145,.., d 180 = 17/( ) = d 181,.., d 216 = 8/( ) = d 217,.., d 225 = 2/( ) = Exhibi 2 is based on he daa from exhibi 1. The rae of deah based on he Typical Disribuion of Life Expecancies for senior life selemen conracs (which we call scenario 1 hroughou he paper) is graphed in Exhibi 3. Exhibi 3: Deah rae over ime based on a Typical Disribuion of Life Expecancies for Senior Life Selemen Conracs Deah rae over ime rae monhs Exhibi 4 graphs he deah rae over ime under differen scenarios. Scenario 1 is he base case, wih Typical Disribuion of Life Expecancies ; scenario 2 shifs he disribuion of he deah rae in scenario 1 by welve monhs (life exension by welve monhs relaive o he base case). Each subsequen scenario shifs/exends he disribuion of deah rae by anoher welve monhs.

10 10 Exhibi 4 Deah rae over ime rae monhs D rae(sc1) D rae (sc2) D rae (sc3) D rae (sc4) Condiions for a reliable Senior Life Selemen s Macaulay Duraion We now develop he condiions for which he Macaulay duraion of a senior life selemen, or a pool of senior life selemens, is no affeced by changes in he life of he insured above or below is life expecancy compued a he ime he insurance conrac was sold. The Macaulay duraion of a pool of senior life selemens is he weighed average duraion of each life selemen. Using equaion (13), we can now find he condiions for which D = 0, ha is he condiions for which he duraion does no change when a seler lives above or below life expecancy. 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(a-1) B)(ln(a)) + C(a-1) B C(ln(a)) = 0 This is a simple linear equaion ha we solve for. = [(-(C(a-1) B) + C(ln(a))]/[(C(a-1) B)(ln(a))] = [-1/ln(a)] + C/[C(a-1) B] Now we replace C wih (ap)/(a-1) 2 o arrive a D = 0 when

11 11 = [-1/ln(a)] + [ap/(a-1) 2 ]/{[ap/(a-1) ] B]} This previous equaion can be simplified o : = [-1/ln(a)] + [ap/(a-1) 2 ]/{[ap(a-1) B(a-1) 2 ]/(a-1) 2 } = [-1/ln(a)] + ap/[ap(a-1) B(a-1) 2 ] Finally, we replace a by is value of 1/(1+y) o show ha D =0 when = [1/ln(1+y)] + [p(1+y)/y(-p-by)] eq. 14 Numerical Applicaions Using a premium p of $4,000 and a deah benefi B of $250,000, wih equaion (1) we obain he value of a senior life selemens across life expecancies beween 1 and 11 years, and across yields comprised beween 1% and 15%. The values are displayed in exhibi 5. Exhibi 5 y\ $243, $237, $230, $224, $218, $212, $206, $200, $194, $188, $182, $241, $232, $224, $215, $207, $199, $191, $184, $176, $169, $161, $238, $227, $217, $207, $197, $187, $178, $169, $160, $151, $143, $236, $223, $211, $199, $187, $176, $165, $155, $145, $136, $127, $234, $219, $205, $191, $178, $166, $154, $143, $132, $122, $112, $232, $215, $199, $184, $169, $156, $143, $132, $120, $110, $100, $229, $211, $193, $177, $161, $147, $134, $121, $109, $98, $88, $227, $207, $188, $170, $154, $139, $125, $112, $100, $88, $78, $225, $203, $182, $164, $146, $131, $116, $103, $91, $79, $69, $223, $199, $177, $158, $140, $123, $108, $95, $82, $71, $61, $221, $196, $173, $152, $133, $116, $101, $87, $75, $64, $54, $219, $192, $168, $146, $127, $110, $94, $81, $68, $57, $48, $217, $189, $163, $141, $121, $104, $88, $74, $62, $51, $42, $215, $185, $159, $136, $116, $98, $82, $69, $57, $46, $37, $213, $182, $155, $131, $110, $92, $77, $63, $51, $41, $32,800.96

12 12 In exhibi 6 we graph he values of he senior life selemen ha were calculaed in exhibi 5. We observe ha value is a decreasing funcion of yield and he decline in value is seeper for longer life expecancies. Exhibi 6 $300, $250, $200, $150, $100, $50, $0.00 Value of Life Selemens for differen life expecancies yield =1 =2 =3 =4 =5 =6 =7 =8 =9 =10 =11 In exhibi 7 we graph he changes in he value of life selemen conracs for differen iniial yield scenarios, from 5% o 10%, when life expecancy moves from 1 o 11 years. We observe ha he higher he life expecancy of a senior life seler is, he lower he value of he life selemen becomes. Of course his is because premia would have o be paid over a longer period of ime before he deah benefi is paid. The inverse relaionship beween life expecancy and he value of he life selemen conrac is accenuaed for higher yields.

13 13 Exhibi 7 Value of Life Selemens for differen yields $250, $200, $150, $100, $50, $ ime y=5% y=6% y=7% y=8% y=9% y=10% Exhibi 8 combines exhibis 6 and 7 in a 3-D framework. We can see how he value of a life selemen is simulaneously negaively affeced by an increase in yield and an increase in life expecancy. This is capured in he downward slope of he plane in value/ime/yield space. Value is more sensiive o increases in yields for higher life expecancies (he eas edge of he plane is seeper han he wes edge), and i is more sensiive o increases in life above iniial life expecancy for higher yields (he back edge of he plane has a seeper slope han he fron edge). Exhibi 8 $250, $200, $150, $100, $50, % $0.00 7% yield % ime 11 We previously derived he condiions for which D = 0. This was he resul of eq. (14). We resae his formula below.

14 14 * = [1/ln(1+y)] + [p(1+y)/y(p-by)] eq. (14) We call * he specific life expecancy ha for an iniial yield, he Macaulay duraion remains he same for changes in he insured s life above or below *. We calculae * using a premium p equal o $4000 and a deah benefi B equal o $250,000, for yields beween 1% and 15%. The resuls of his calculaion are displayed in exhibi 9. Exhibi 9 y * 1% % % % % % % % % % % % % % % The daa from exhibi 9 can be inerpreed as hose life expecancy/yield combinaions used o value life selemen conracs for which duraion is a sable measure of ineres rae risk, ha is when D =0, for life selers living above or below life expecancy. Exhibi 10 is he graph of he * for yields ranging from 4% o 15%.

15 15 Exhibi 10 * % 2% 4% 6% 8% 10% 12% 14% 16% yield For example, a life selemen conrac wih a life expecancy of 8.98 years, as highlighed in exhibi 9 (we round i o 9 for compuaional purpose) valued a a marke rae (or yield) of 10%, will have a Macaulay duraion of There is always he chance ha he life seler lives 10 years, one year above he seler s life expecancy, (an 11% change in ). Wih an unchanged yield, in our example, 10%, he Macaulay duraion would no be affeced by his life exension, and would remain a Again using exhibi 9 we observe he duraion measure is robus when he life selemen conrac is valued over 8 years, one year below life expecancy. Duraion remains unchanged a This means ha an invesor purchasing securiized senior life selemens wih a life expecancy of *, can rely on he compued Macaulay duraion as a measure of ineres risk knowing ha he insured may acually live longer or shorer han he expecancy on which he original valuaion was based. We now illusrae ha for a life expecancy oher han *, calculaions of duraion will be unsable for deviaions in life from he expeced life of he insured. Using same premium $4,000 and deah benefi, $250,000 as in he las example and a life expecancy of 5 years, wih curren marke raes of 5%, we obain duraion of If he seler lives 6 years insead of he expeced 5 years, he duraion goes o If on he oher hand he dies one year earlier han expeced, afer 4 years insead of 5 years, he duraion changes from -5.2 o In Exhibi 12 we graph he price/yield relaionship for hree differen life expecancies across a range of yields. The daa is presened in exhibi 11. The middle curve is he plo of a 9-year expeced life across yields ranging from 6% o 13%. The life selemen conracs are discouned a a yield of 10%, iniial poin (highlighed in exhibi 11) on he curve. The lowes curve is drawn for an exension of life by one year and he highes curve illusraes wha happens o he price/yield relaionship when life is shorened by one year. The curves shif parallel o one anoher. This parallel shif indicaes ha he duraion of he life selemen conracs does no

16 16 change and is sill equal o The slope of he angen o he price/yield curve along he 10% yield remains consan. The slope of he angen o he price/yield curve a an iniial poin is compued by aking he firs derivaive of he life selemen price wih respec o yield. When he firs derivaive is muliplied by (1+y)/P we obain he Macaulay duraion in equaion (2). Exhibi 11 y/ 8 years 9 years 10 years 0.06 $132, $120, $110, $121, $109, $98, $112, $100, $88, $103, $91, $79, $95, $82, $71, $87, $75, $64, $81, $68, $57, $74, $62, $51, $69, $57, $46, Exhibi 12 price/yield relaionship E()=9 $140, $120, $100, $80, $60, $40, $20, $ yield Exhibi 14 is he same space as exhibi 12 bu he price/yield relaionships are ploed for an expeced life of five years across yields ranging from 1% o 9%. The iniial poin (highlighed in exhibi 13) along he curve is a a yield of 5%. When he life of he insured pool deviaes by one year above or below he iniial five-year life expecancy, he new curves have a angen o he 5% yield wih differen slopes. When he life span increases from five o six years, he curve falls and

17 17 roaes so ha he new slope of he angen o he curve a he yield of 5% is higher, and he iniial duraion of -5.2 changes o When he life span of he pool of insured falls from five o four years he price/yield curve shifs upwards and he slope decreases. There is a decline in he duraion of he pool of life selemens conracs from -5.2 o Exhibi 13 y/ 4 years 5 years 6 years 0.01 $224, $218, $212, $215, $207, $199, $207, $197, $187, $199, $187, $176, $191, $178, $166, $184, $169, $156, $177, $161, $147, $170, $154, $139, $164, $146, $131, Exhibi 14 price/yield relaionship E()=5 $250, $200, $150, $100, $50, $ yield

18 18 Exhibi 15 $95, $90, $85, $80, $75, $70, yield ime We consruc Exhibi 15 by carving ou from exhibi 8 he yield/ime combinaions for which he duraion calculaion for he life selemen conracs is immune o changes in he life of he insured person above or below he life expecancy used o iniially value he conrac, and by graphing he combinaions agains he appropriae value. Conclusion The derivaion of he ime/yield 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 was 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 asse-backed 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 Sure Duraion Class (SDC). 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 Sure Duraion Class would be direced o he Duraion Companion Class (DCC). 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 DCC class would be offered a higher yield.

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