INSIUE OF EONOI SUDIES Faculy of social sciences of hales Univesiy Fixed Income Secuiies Lecue s Noes No. 3 ouse: Financial ake Insumens II eache: Oldřich Dědek
I. BOND ONRA bonds ae capial make insumens and as such have mauiies in excess of one yea saigh bond (plain vanilla bond, bulle bond pays a egula (usually semi-annual fixed coupon ove a fixed peiod o mauiy (edempion wih he eun of pincipal (pa value, nominal value on he mauiy dae bonds may have moe inicae flow paens and can be classified accoding o many cieia. lassificaion of bonds fequency of coupon paymens - convenional (saigh, plain vanilla, bulle bond pays egula coupons quaely, semi-annually o annually - eo-coupon bond do no pay coupons a all, hey ae sold a a deep discoun o pa value and all ewad fom holding he bond comes in he fom of capial gain - income bond makes coupon paymen only if he income geneaed by he issuing fim is sufficien sie of coupon paymens - convenional bond pays a fixed coupon (deemined as a pecenage of is pa value ove he whole life of he bond - vaiable coupon bond links is coupon o some economic vaiable such as he eail pice index (index-linked bond, cuen make inees ae (floaing ae noe - collaealised bond deives is coupons fom a given package of undelying asses (ABS, asse-backed secuiies
edempion dae - convenional bond has one edempion dae a mauiy (five-yea bond, en-yea bond, ec. - double-daed bond has a ange of possible edempion daes - callable bond has an opion feaue ha gives he issue he igh o deemine he acual dae of edempion - puable bond has an opion feaue ha gives he holde he igh o deemine he acual dae of edempion - consol (pepeual bond, pepeuiy have no edempion dae a all and inees on i will be paid indefiniely - conveible bond has an opion feaue ha gives he holde he igh o conve he bond a mauiy ino ohe ypes of bonds o ino equiy issue of he bond - govenmen bond is issued by a soveeign govenmen in ode o finance and manage public deb (easuies in USA, gils in UK - municipal bond is issue by a local auhoiy - copoae bond is issued by a pivae company (senio vs. junio deb, secued vs. unsecued deb, fixed vs. floaing chage deb cuency of denominaion - domesic bond is issued by a esiden issue in he domesic cuency - foeign bond is issued by a non-esiden issue in he domesic cuency Samuai (Japan, Yankee (USA, Bulldog (UK, aado (Spain, Kiwi (New Zealand, Alpine (Swieland - euobond is issued and aded in a non-esiden cuency (ohe han seling in he UK, ohe han euo in he Euoone counies, ec. many innovaions: dual cuency bond: coupon paymen ae in one cuency and he edempion poceeds ae in anohe cuency change bonds: coupons ae fis paid in one cuency and hen in anohe defeed coupon bond: hee is a delay in he paymen of he fis coupon muliple-coupon bond: coupons change ove he life of he bond in a pedeemined manne missing coupon bond: coupon paymen is missed wheneve a dividend paymen on he issue s shaes is missed 3
efacable bond: aached wih call and pu opions bull o bea bond: pincipal on edempion depends on how a sock index has pefomed defaul o cedi isk i is usually assessed in he fom of cedi aing (Sandad & oo's, oody's, Fich he lowe he cedi aing, he geae he isk of defaul and he highe he isk pemium gades of he bond: invesmen gades, non-invesmen (speculaive gades (junk bonds, high-yielding bonds. Fai picing of saigh bonds fai pice of a bond is given by he discouned pesen value of he cash flow seam using he make-deemined discoun ae fo a bond of his mauiy and isk class... fai pice of he bond...nominal value (pa value, pincipal of he bond c... coupon ae (in % annual fixed coupon paymen ( numbe of yeas o mauiy make-deemined discoun ae c a annual discouning of annual coupon paymens: c ( ( c ( ( Invese and convex elaionship beween he bond pice and he discoun ae ice Discoun ae 4
5 fai pice of a pa bond when he coupon ae is equal o he discoun ae he pice of he bond will be equal o is pa value c bonds wih a coupon ae geae han he discoun ae ae piced below is pa value and ae said o be selling a a discoun c < > bonds wih a coupon ae smalle han he discoun ae ae piced above is pa value and ae said o be selling a a pemium c > < fai pice of a pepeuiy c fai pice of a bond when valuaion dae diffes fom coupon paymen dae he bond s cash flow is discouned back o he dae of he nex coupon paymen and hen he oal sum is discouned o he pesen day ( ( ( n n c c c c c ( ( (... 360 / 360 / n numbe of days o he nex coupon dae b semi-annual discouning of semi-annual coupon paymens: / / / K c semi-annual discouning of annual coupon paymens: 4 K
d annual discouning of semi-annual coupon paymens: / / / K / ( ( (.3 lean and diy pice full (diy, goss pice of a bond is he pice ha is acually paid fo he bond i should be equal o he pesen value of all fuue cash flows obained by he holde of he bond clean pice is he quoed pice of a bond ha disegads he inees accued fom he las coupon paymen dae full pice clean pice accued inees X X X 3 3 clean pice i coupon paymen dae X i ex-dividend dae i ansacion dae (he bond has been sold o a new owne beween he coupon paymen dae and he nex ex-dividend dae he bond is aded cum dividend cum dividend means ha he issue will mail he enie coupon o a new bondholde (he will be egiseed as he owne of he bond he buye of he bond mus compensae he selle fo giving up he enie nex coupon even hough he selle held he bond fo pa of he peiod since he las coupon paymen full pice clean pice c (full pice > clean pice 360 beween he ex-dividend dae and he nex coupon paymen dae he bond is aded ex dividend 6
ex dividend means ha he issue will mail all he coupon o an oiginal bondholde (he may sill be egiseed as he owne of he bond he selle of he bond mus compensae he buye fo giving up all of he nex coupon even hough he buye will be holding he bond fo pa of he peiod since he las coupon paymen full pice clean pice c (full pice < clean pice 360 A easuy bond pays 9 % coupon annually. On June he bond has 63 days o he nex coupon paymen and hee ae 97 days since he las coupon paymen (assuming 360 days in a yea. Since he nex coupon paymen he bond will have 5 yeas o mauiy. he cuen make yield fo he bond is 8 %. full pice 00 0.09 00 0.09.99.474 63 / 360 5 5 (.08 0.08.08 (.08.036 97 accued inees 0.09 00 7. 45 360 clean pice.474 7.45 04. 05 Wha ae he calculaions if he cuen make yield fo he bond is 9 % (equal o he coupon ae? full pice 00 0.09 00 0.09.0900 07.368 63 / 360 5 5 (.09 0.09.09 (.09.05 97 accued inees 0.09 00 7. 45 360 clean pice 07.368 7.45 99. 94 he clean pice of he bond is no exacly equal o he noional amoun despie he fac ha he discoun ae is equal o he coupon ae. his is because accued inees is calculaed on a simple inees basis and pice on a compound inees basis. 7
.4 Yield measues on bonds because of complicaed paens of cash flows he bonds ae compaed in ems of yields (ae of eun eaned fom invesing in holding he bond a yield o mauiy (Y, edempion yield, inenal ae of eun Y is he mos fequenly used measue of he eun fom holding he bond Y is a discoun ae a which he pesen value of he discouned cash flows is equal o he cuen diy pice of he bond Y is he soluion of he equaion (has o be found hough numeical ieaion using a compue calculao c Y ( Y ( assumpions: annual discouning of annual coupon paymens, he pesen dae is he coupon paymen dae shocomings of Y: i Y is a measue of a bond yield ha is based on he implici assumpion ha each coupon can be einvesed a a consan Y ae (his ae of inees is consan fo all fuue ime peiods Y of a bond compleely ignoes he einvesmen isk (as opposed o he Y of a ime deposi wih compounding inees eaanged fai pice equaion: Y ( Y ( Y... ( Y nominao: eminal value of cash flows fom holding a bond unde assumpion ha all coupons ae einvesed a an Y ae fo he emaining life of he bond denominao: bond pice equal o iniial value of he invesmen ii Y is based on he implici assumpion ha bond is held o mauiy 8
b holding peiod yield HY is he aveage yield ealied duing he holding (invesmen peiod, aking ino accoun he puchasing and selling pices of he bond as well as changes in he ollove aes a which coupon paymens can be einvesed HY is he soluion of he equaion B ( HY ( ( K S B... buying pice of he bond S selling pice of he bond ollove (einvesmen ae disadvanage: HY used as an ex ane measue of he bond s aaciveness equies making assumpions abou ollove aes ha ae unceain A bond is puchased on a coupon paymen dae fo 96.50 and sold exacly wo yeas lae fo 0.30. he annual coupon is 8.7 and is paid semi-annually. he ollove aes fo he fis hee coupons ae 0.00, 0.5 and 0.40 % especively. he holding peiod yield can be found by solving he equaion HY 96.50 5.7 HY 96.50 4 / 4 3 8.7 0.0 0.0 4.4 0.30 5.7 0.093 9.3 % 0.04 0.30 c cuen yield (fla yield, unning yield, inees yield, income yield Y is he simples measue of he ae of eun on a bond annual coupon Y pice of he bond moe appopiae is he diy pice of he bond pice which is he puchasing pice bu he clean pice does no esul in he saw-ooh paen of he Y 9
he ecipocal value of he Y indicaes a numbe of coupon paymens ha will fully epay he puchasing pice of he bond disadvanage: Y does no ake ino accoun capial gains o losses esuling fom he diffeences beween he cuen pice of he bond and is mauiy value (moe suiable fo long-daed bonds wih small capial gain/loss elemen d simple yield o mauiy assumpion is made ha he capial gain/loss occus evenly ove he emaining life o he bond 00 - bond pice SY Y yeas o mauiy bond pice A bond pays an annual coupon of 8.75 and is cuen clean pice is 95.30. he bond has 4 yeas o mauiy. 8.75 Y 95.30 9.8 % 00 95.30 SY Y 9.8.3 0.4 % 4 95.30 e money make yield yields fo bonds ha have only one emaining coupon o be paid a mauiy ae someimes quoed on a basis compaable o money make insumens (simple inees, acual numbe of days capial make convenions (compound inees n ( 365 / n / 365 money make convenions (simple inees 365 n 365 n 0
II. ANALYSIS OF HE YIELD URVE yield cuve (also called em sucue of inees aes is he elaionship beween a paicula yield measue and a bond s mauiy only bonds fom a homogeneous goup should be included (fom he same isk class o wih he same degee of liquidiy Y is he mos fequen measue used in consucing yield cuves yield cuve is a plo of he bonds yield o mauiy agains hei em o mauiy Shapes of yield cuve Y ising Y declining (eveed Y fla Y Y wih a hump ime o mauiy empiical yield cuve is a plo of he Y of exising bonds agains hei em o mauiy poblems: - Y measue applied on coupon-beaing bonds suffes fom einvesmen isk - Y cuve does no disinguish beween he diffeen paymen paens of lowcoupon and high-coupon bonds wih he same mauiy (hey may have diffeen Ys because hey may diffe in ems of hei aaciveness fo invesos. Zeo yield cuve eo-coupon bonds (pue discoun bonds ae bonds ha make no inees paymen and he only cash flow is he pice paid and he pincipal amoun eceived a mauiy
eo aes (spo aes ae Ys eaned on eo-coupon bonds eo aes ae he annual inees aes a which he discouned value of he pincipal amoun equals he bond s pice eo ae of -yea eo-coupon bond ( / a clea advanage of eo-coupon yields is ha hey avoid compleely he quesion of einvesmen isk (because of his ceainy invesos may accep a slighly lowe oveall yield eo yield cuve (spo yield cuve is a plo of eo aes of eo-coupon bonds agains hei em o mauiy hee may be no complee se of eo-coupon bonds available fo he consucion he eo yield cuve, bu hee mus be heoeical eo aes which ae consisen wih he inees aes obseved in he make boosapping is he analyical mehod of building up a seies of consisen eo yields fom coupon-beaing yields pacical counepas of boosapping: - bond sipping (unbundling is a echnique of financial engineeing ha beaks down he coupon-beaing bond ino coupons and pincipal componens as sepaae secuiies ha ae aded sepaaely - synheic eo-coupon bond is a echnique of financial engineeing ha combines a seies of coupon-beaing bonds in such a way ha all he cash flows ae eo excep fo he fis and he las Bond sipping oupon bond Z Z Z 3 Z
deivaion of he eo yield cuve, ae yields o mauiy of obseved coupon-beaing bonds (he bonds ae,..., assumed o be pa bonds bu any seies of bonds can be used fo boosapping,,..., ae eo yields of heoeical eo-coupon bonds consisen wih he given se of coupon-beaing yields eo ae fo one-yea mauiy one-yea bond is aleady by assumpion a eo-coupon bond, heefoe eo ae fo wo-yea mauiy mehod of bond sipping pesen value of wo-yea coupon-beaing pa bond (wih pincipal : V ( sum of pesen values of eo-coupon secuiies sipped fom he coupon-beaing bond: V ( no-abiage condiion implies ha boh values mus be equal one can hen solve fo unknown vaiable V V ( eo ae fo wo-yea mauiy mehod of synheic eo-coupon bond puchasing one wo-yea pa coupon-beaing bond geneaes he cash flow 0 : : : issuing one-yea bonds in nominal value equal o he discouned coupon of he woyea bond geneaes he cash flow: 0 : : ( : 0 combined cashflow geneaes he cashflow paen of a wo-yea eo-coupon bond 3
0 : : 0 : ( ae of eun of he synheic wo-yea eo-coupon bond is given by he equaion i can be demonsaed ha he obained expession is equivalen o he pevious expession ( eo ae fo -yea mauiy in a simila fashion he boosapping pocess woks fowad ieaively fo all ohe mauiies wih aleady deemined eo aes,,..., he eo ae can be deemined fom he equaion ( ( pacical obsacle of boosapping: i is no geneally possible o find a seies of exising bonds wih he convenien mauiies one mus inepolae beween exising yields in ode o esablish aes fo all mauiies (assumpions mus be made abou fiing a cuve along exising poins Suppose ha a seies of bonds ae cuenly piced as follows: Bond ice oupon auiy ( (% (yeas A 90.9 0 B 97.4 9 85.6 5 3 D 04.65 3 4 4
. Deeminaion of one-yea eo ae: 00 0 % 90.9. Deeminaion of wo-yea eo ae: a bond sipping 97.4 9 ( 0.0 ( 09 89.3 b synheic eo-coupon bond 09 / 0.5 % Bond -yea -yea oal bond bond F Quaniy 0.09 x 0-97.4 8.8-89.3 9-9 0 09 0 09 / 09 0.5 % 89.3 3. Deeminaion of hee-yea eo ae: a bond sipping 85.6 5 ( 0.0 3 5 ( 0.05 05 76.6 b synheic eo-coupon bond / 3 05 (.08 % Bond 3-yea -yea -yea oal bond bond bond F Quaniy 0.0459 0.0459 x 0-85.6 4.47 4.7-76.6 5-0.4-4.59 0 5-5 0 0 3 05 0 0 05 3 3 5
/ 3 05 3.08 % 76.6 4. Deeminaion of fou-yea eo ae: a bond sipping 04.65 3 ( 0.0 4 3 ( 0.05 3 7.70 b synheic eo-coupon bond / 4.66 % 3 ( 0.08 3 3 ( Bond 4-yea 3-yea -yea -yea oal bond bond bond bond F Quaniy 0.38 0.36 0.36 x 0-04.65 0.56.07 0.33-7.69 3-0.6 -.0 -.36 0 3-0.6 -.38 0 0 3 3-3 0 0 0 4 3 0 0 0 3 4 4 / 4 3 4.66 % 7.69. Implied fowad yield cuves fowad-fowad ae is he inees ae on a cash boowing o lending ha sas on one fuue dae and ends on anohe fuue dae f p inees ae fo he peiod ha sas in yeas ime and ends in (p yeas ime (inees ae fo p yeas peiod beginning in yeas ime yield o mauiy of a p-yea eo-coupon bond ha will be issued in - yeas ime building up synheic fowad-fowad aes fom eo aes boowing fo yeas (cash inflow of a 0 a geneaes a cash ouflow ( a ime (epaymen of boowed funds 6
lending fo p yeas (cash ouflow of a 0 a p geneaes a cash inflow ( a ime p (len funds ae euned back (boowing (lending ( ( p p p ne posiion of he wo cash ansacions is lending a he yea fo implied fuue lending ae can be found fom he equaion p yeas ( f p f p p ( p ( p ( p ( p / p implied fowad aes ae he inees aes ha ae consisen wih a given eo-coupon yield cuve (hey ae calculaed on he basis of his cuve inconsisen fowad aes indicae he exisence of an abiage oppouniy Suppose ha he following eo aes wee exaced fom a given sucue of make bond yields (he same as in pevious example: auiy 3 4 Zeo ae 0.00 0.5.08.66 ( 0.05 -yea ae saing in yea s ime.04 % 0.0 3 ( 0.08 -yea ae saing in yeas ime. % ( 0.05 4 ( 0.66 -yea ae saing in yeas ime.8 % ( 0.05 / 7
Suppose ha he -yea ae saing in yea s ime is.5 % ahe hen.04 %. heefoe his inees ae is no consisen wih given sucue of eo yields and he following abiage oppouniy exiss: ake a -yea loan, inves he funds ino -yea deposi, afe one yea einves he poceeds ino anohe -yea deposi, epay he loan. final cash inflow 00.0.5. 65 final cash ouflow 00.05. 5 abiage pofi.65.5. 50 fowad yield cuve is a plo of implied fowad aes agains em o mauiy fo a given fuue saing dae fowad yield cuve in -yea ime:, f, K f ( values f 3, fowad yield cuve in -yeas ime:, f, K f ( values f 3 4,................................................................ fowad yield cuve in (--yeas ime: f ( value cuen yield cuve can be defined as he fowad yield cuve saing now f, f, K f 0 0, 0 Family of implied fowad yield cuves fowad yield cuve in yeas ime fowad yield cuve in yea s ime eo yield cuve 3 - - 8
building up eo aes fom fowad-fowad aes eo aes can be deemined as he geomeic mean of he chain of appopiae fowadfowad aes (using convenion 0 f ( ( ( ( 0 f ( ( ( f f ( f KK ( KKKKKKKKKKKKKKKK f f ( f he -yea inees ae is now 0 %. he make expecs -yea inees ae o ise o.05 % afe one yea and o.8 % afe a fuhe yea. Wha ae he cuen -yea and 3-yea eo-coupon yields consisen wih hese expecaions? [( f ( ] /..05 0.5 % 0 f 3 [( f ( f ( ] /..05.8.07 % 3 0 f 3 expeced changes in inees aes if he eo ae cuve is upwad sloping ( < hen he make expecs he fuue inees aes o ise poof fo all fuue -yea aes ( < ( ( ( f <, < f < < f fo all f poof fo all nex yea s inees aes ( ( ( f ( ( f > ( ( ( < f > ( f fo all similaly if he eo ae cuve is downwad sloping ( > hen he make expecs fuue inees aes o fall 9
.3 a yield cuve pa yield is he yield o mauiy of a bond ha is piced a o nea pa (pice of he bond is equal o is nominal value pa yield is equal o he coupon ae fo such bond (since Y fo bonds piced a pa is equal o he coupon ae pa yields consisen wih a given sucue of eo aes can be found by solving he equaion c c ( ( d d whee d ( d is he discoun faco fo he -yea eo-coupon bond wih he Y of Deemine pa yields consisen wih he following sucue of eo-coupon yields? auiy 3 4 Zeo ae 0.00 0.5.08.66 Discoun faco 0.90909 0.88688 0.7963 0.6439 d d c 0.00 %, c 0.49 % d d d d 3 d 4 c 3.00 %, c 4.5 % d d d d d d d 3 3 4 pa yield cuve is a plo of he yield o mauiy agains em o mauiy fo bonds piced a pa if he eo ae cuve is upwad sloping ( < hen he pa yield cuve lies below he eo ae cuve if he eo ae cuve is downwad sloping ( > hen he pa yield cuve lies above he eo ae cuve 0
poof fo he upwad sloping eo cuve d ( ( ( c < d ( ( ( osiion of pa, spo and fowad yield cuves fowad spo pa pa spo fowad.4 icing of floaing ae noes FRN (floaes ae ypes of bonds (capial make secuiies whose coupon paymens ae vaiable and elaed o a given sho-em make ae (i.e. 3-monh LIBOR, 6-monh LIBOR coupons of he FRN ae fixed fo a given peiod and ae ese on he coupon ese daes fai pice of FRN expeced cash flow (, f, f 3,..., f discouned a eo aes of inees of coesponding mauiies FRN ae geneally issued a pa f f 3 f f
fai pice of FRN on he coupon ese day elaionship beween eo and fowad aes: ( ( ( f ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( f f FRN K he floae is sold a pa (fo is nominal value.5 Inflaion-indexed bond inflaion-indexed (inflaion-linked bond is one whose cash flows ae linked o movemens in a specific pice index (i.e. I wih he aim of poviding poecion fo eal value of invesmen bonds ae issued wih a ceain eal ae of eun (i.e..5 % while a nominal ae of will depend on he ealied fuue pah of he pice index coupon paymen coupon ae pincipal I 0 I epaymen of pincipal pincipal I 0 I I I I,, 0 ae values of he pice index i when he bond is issued; ii when he bond pays egula coupons iii when he bond maues inflaion-linked bonds offe only impefec poecion agains eosion of puchasing powe - he weighed composiion of a geneal pice index does no coincide wih he collecion of goods consumed by invesos who wan o be poeced agains inflaion - saisical epoing and pocession lags (up o monhs mean ha hee is always a delay beween he elevan ime peiod fo which an index is compued and he dae on which ha numbe is published
- calculaion of accued inees equies o fix he nominal value of he pice index a he beginning of he six-monh coupon peiod, heefoe coupon o pincipal paymens ae based on a leas a six-monh old pice index nominal ae of eun (Y on an inflaion-linked bond depends on an unceain foecas of he inflaion ae π ( I I ( I I ( ( I I 0 ( 0 ( π ( π ( (... ( ( ( π... ( 0 Fishe equaion ( ( ρ ( π nominal yield, ρ eal yield, π inflaion ae eal ae of eun (Y on an inflaion-linked bond ( π ( π ( ( ( π ( π ( ρ( π ( ρ ( π ( ρ ( ρ ( ( π... ( (... ( ρ ( ( π... ( ρ ( π beak-even inflaion is he inflaion ha makes he money yield on an inflaion-linked bond equal o he yield on a convenional bond of he same mauiy beak-even inflaion can be deemined using he Fishe equaion Suppose ha he yield o mauiy on a convenional easuy is 8.3 % and ha he inflaionlinked easuy of he same mauiy is.5 %. oupons ae paid semi-annually. Deemine he beak-even inflaion. π ρ 0.083 0.05.058 3
A an inflaion of 5.8 % he inflaion-linked bond is equally aacive as he convenional bond. if he expeced ae of inflaion is highe han he beak-even inflaion hen invesos will pefe he index-linked bonds o convenional ones excess demand fo hese bonds will evenually pess down he yield of inflaion poeced bond o he level of convenional ones (and vice vesa in efficien makes he beak-even inflaion hus eveals he inflaion ha is expeced by make paicipans.6 heoies of he yield cuve main ask of hese heoies is o explain wha deemines he posiion and he shape of he yield cuve expecaion hypohesis agues ha a cuen long-em inees aes ae elaed o expeced fuue sho-em inees aes (in accodance wih he heoy behind he fowad yield cuve ( ( ( ( f f ( f KK ( ( 0 f f b implied fowad inees aes ae he bes indicao of expeced fuue inees aes f E p ( p f p cuen p-yea fowad ae saing in yeas ime E ( p... fuue p-yea spo ae expeced a he yea he make believes ha a ime he p-yea inees ae will have a value of f p cuen inees aes ae adjused o he expeced values of fuue inees aes he cuen one-yea ae is 6.5 % and he make is expecing he one-yea ae in a yea s ime o be 7.5 %. he expecaion hypohesis mainains ha he cuen wo-yea ae will be.065.075 0.07 7 % 4
If he make eassess he inees ae expecaion fo he second yea fom 7.5 % o 8.0 %, he cuen wo-yea spo ae changes immediaely o a value of.065.08 0.074 7.4 % componens of he nominal inees ae nominal inees ae eal inees ae inflaion pemium liquidiy pemium isk pemium eal inees ae is given by he ae of ime pefeences o by he willingness of he individuals o fogo cuen consumpions in eun fo addiional consumpion in he fuue peiods eal inees ae is he pice paid fo he willingness o ansfe esouces ove ime and equaes he supply of funds fo hose willing o lend wih he demand fo funds fo hose wishing o boow inflaion pemium is he pice paid o lendes fo he eosion of he puchasing powe of povided funds ha will be caused by an expeced inflaion inflaion pemium allows o buy in he fuue he same bundle of goods ha could be bough when money wee povided Fishe equaion: ( nominal inees ae ( eal inees ae ( inflaion ae ( ( ρ ( π & ρ π liquidiy pemium is elaed o he effec of he duaion of lending and boowing liquidiy pefeence heoy: mos lendes pefe o lend fo sho peiods because sho-em lending can be moe easily conveed o cash wihou he isk of losing capial value; in ode o induce lendes o povide long-em funding hey have o be offeed a liquidiy pemium on long money mos boowes pefe o boow fo long peiods because sho-em boowing has o be olled ove and hee is he isk ha his can be done on unfavouable ems; boowes ae willing o pay a liquidiy pemium on long money 5
pefeences of boh lendes and boowes hus combine o give he upwad-sloping yield cuve (even if he eal inees ae and he inflaion ae ae no expeced o change ove ime liquidiy pemium is likely o incease wih he em o mauiy bu a a deceasing way (he cuve iniially ise seeply and hen flaens ou segmenaion (pefeed habia heoy: financial makes ae segmened by mauiy ange and hee ae no spillove effecs beween each make segmen examples: pension funds ae acive in long-em bonds, banks ae moe ineesed in sho-em secuiies, specific ole of govenmen policy in secuing public seco boowing and managing public deb, cenal bank policies in conolling money supply, ec. he shape of he yield cuve is deemined by supply and demand condiions in each make segmen wihou efeence o condiions in ohe segmens isk pemium is elaed o he isk ha he issue of a bond will defaul on his obligaions specific isk esuls fom he evens ha ae diecly linked o a paicula issue (incompeen decisions made by manages, insufficien sales evenue o finance opeaing cos, inadequae collaeal in he even of insolvency, ec. make isk esuls fom he dependence of fim s eanings on make condiions 6
III. EASUREEN OF INERES RAE RISK 3. Risks associaed wih invesing in bonds inees ae isk (pice isk, make isk is he isk of a capial loss fo an inveso who may have o sell he bond befoe he mauiy dae (as inees aes ise he pice of bonds will fall his isk is no of concen fo an inveso who plans o hold a bond o mauiy einvesmen isk is he isk ha ineim coupons of a bond will have o be einvesed a a lowe ae such ha he inveso will ean a lowe yield han he saed yield a he ime he bond is puchased defaul isk (cedi isk is he isk ha he issue of a bond will defaul on is conacual paymens of inees ae and/o pincipal he isk of defaul is gauged by qualiy aings assigned by aing companies (o cedi eseach saff of financial fims inflaion isk is he isk ha he eun ealied fom invesing in a bond will no be sufficien o offse he loss in puchasing powe due o inflaion his isk is no of concen fo adjusable o floaing ae bonds call isk is he isk ha he issue may execise he conacual igh o eie all o pa of he issue befoe he mauiy dae disadvanages of call povisions fom he inveso s pespecive: - cash flow paen of he bond is no known wih ceainy - poceeds fom eiing he bond will have o be einvesed a a lowe inees ae because bonds ae called when inees aes have dopped - capial appeciaion poenial of he bond may be educed because of a given call pice sipulaed in a bond conac foeign exchange isk (cuency isk occus when he issue pomises o make paymens in a foeign cuency so he cash flow will depend on he foeign-exchange ae a he ime he cash flow is eceived liquidiy isk (makeabiliy isk involves he ease wih which an issue can be sold a o nea he pevailing make pice he geae he deale spead (diffeence beween he bid pice a which he issue can be sold and he ask pice a which he issue can be puchased he geae he liquidiy isk 7
3. Duaion duaion is defined as he weighed aveage mauiy of a bond using he elaive discouned cash flows in each peiod as weighs he measue can be undesood as he aveage ime aken o eceive he cash flows on a given bond he measuing uni of duaion is a yea fomula fo acaulay duaion (D D w / ( /( ( ( he fai pice equaion ensues ha he sum of weighs equals one w ( ( duaion can be compued using he analyical fomula D ( ( ( ( fomula fo modified duaion (D acaulay duaion D m A bond wih a yield o mauiy of 8 % and a coupon of 5 % paid annually has five yeas o mauiy. alculae he duaion of he bond. D 5.08 5 88.0 5 88.0 5.08 5.08 5.08 05.08 5 0.08.08 00.08 3 4 5 5 5.08.08 3.08 6.08.08 0.08 5 5 0.08.08 3 4.08 4 00 88.0 5.08 5 5.08 5 00 88.0 4.5 5.08 he bond has acaulay duaion of 4.5 yeas. odified duaion is 4.8 yeas. 5 88.0 8
popeies of duaion a duaion is always less han (o equal o mauiy D explanaion: some weigh is given o he cash flows in he ealy yeas of he bond s life ha helps o bing fowad he aveage ime a which he cash flows ae eceived duaion equals o mauiy fo eo-coupon bonds duaion of pepeuiy (, D b duaion deceases as coupon inceases (and vice vesa explanaion: as coupon ises hen moe of he elaive weigh of he cash flows is ansfeed o he pesen dae c duaion deceases as yield o mauiy inceases explanaion: as yield ises hen he pesen value of all fuue cash flows falls bu he pesen value of he moe disan cash flows falls elaively moe han ha of he neae cash flows d duaion is he measue of inees ae isk (elasiciy of he bond pice wih espec o one plus he yield o mauiy poof: d d d ( ( ( ( ( d / d ( /( d( ( ( 9 D he lowe he duaion, he less esponsive is he bonds value o he inees ae flucuaions duaion can be egaded as a fis-ode appoximaion of a change in he bond s value wih espec o he change in he yield (i measue he slope of he pesen-value pofile & D D m
Using he duaion measue esimae a new pice of he bond fom he pevious example if yields on five-yea bonds ae expeced o ise fom 8 o 0 %? (0.0 0.08 & D 88.0 4.5 7.35 0.08 A bond pice is expeced o fall fom 88.0 o (88.0 7.35 80.67. A diec calculaion based on he fai pice fomula shows ha a a yield of 0 % an exac new pice would be 8.05. Duaion hus oveesimaes a ue fall of he bond s pice. esen-value pofile and duaion ln 0 ln D 0 ln( ln( logaihmic deivaion d / D d( /( dln dln( he geae is he change in he bond s pice he geae is he appoximaion eo e immuniaion popey of duaion when inees aes ise he inveso is exposed o wo offseing effecs accumulaed value of einvesed coupons will be geae bond s pice (equal o discouned value of emaining cash flows will be smalle when inees aes fall he wo effecs (called einvesmen isk and pice isk move in he opposie diecion 30
3 immuniaion popey means ha if he bond is valued a he duaion dae han he wo effecs exacly offse each ohe poof: - fai pice of he bond ( ( ( ( ( ( K - value of accumulaed coupons a he duaion dae (einvesed coupon componen D D D ( ( ( ( R K - deivaion wih espec o gives ( ( R D d d D D - value of he bond s pice a he duaion dae (he capial gain componen D D D D ( ( ( ( ( G K - deivaion wih espec o gives ( ( ( ( ( G D D D D D d d - he assumpion ha einvesmen and pice isks offse each ohe leads o he equaion D D d d d d ( ( ( ( ( ( G R 0 - i can be shown ha he above equaion is consisen wih he analyical fomula fo duaion
3 f duaion of a pofolio of bonds is he weighed aveage of duaions of individual bonds using elaive make values of bonds as weighs ( ( ( ( ( ( ( ( ( B D n n n B D n n n n n n n n n n n n n n n n n B B D 3.3 onvexiy convexiy can be egaded as a second-ode measue (quadaic appoximaion of inees ae isk (as opposed o duaion which can be egaded as a fis-ode measue (linea appoximaion of inees ae isk convexiy measues he cuvaue of he pesen-value pofile (as opposed o duaion which measues he slope of he pesen-value pofile fai pice of a bond is a funcion ( f second-ode aylo expansion of he funcion gives ( ( d d D d d d d & K convexiy of he bond is called he expession d d K ( ( ( ( Using he duaion and convexiy measues esimae a new pice of he bond fom he pevious example if yields on five-yea bonds ae expeced o ise fom 8 o 0 %?.40.08 6 5 88.0 00.08 6 5.08 5 4.08 4 3.08 3.08 88.0 5 7 7 6 5 4 3 K
88.0 & 7.35.4 (0. 0.08 7.35 0.39 6.96 A bond pice is expeced o fall fom 88.0 o (88.0 6.96 8.06. By compaing he esimaed new pice wih he exac new pice 8.05 we see ha he convexiy em subsanially impoved he esimaion. Desiable popey of convexiy ln A B B A ln( wo bonds A and B ae ading a he same pice, a he same yield o mauiy and hey have he same duaion (hey have he same slope of pesen-value pofile bond B is moe convex han he bond A highe convexiy is desiable because - if yields ise hen he pice of B falls by less han he pice of A - if yields fall hen he pice of B ises by moe han he pice of A his siuaion is no likely o happen because invesos will be eady o pay o obain highe convexiy he bond wih he highe convexiy would sell a a highe pice and heefoe wih a lowe yield 33
IV. REO 4. lassical epo epo is a sho fo sale and epuchase ageemen wheeby he wo ansacions involved ae deal as a package ansaced unde one ageemen he fis deal is a sale of a secuiy he second deal is a evesal of he fis deal Secuiy ash Selle/lende Buye/boowe ash inees Secuiy evese epo is he opposie o he epo iniial puchase of secuiies followed by a subsequen sale (epo o one pay is a evese epo o he ohe pay legal vs. economic eamen of epo legal eamen: owneship of he secuiy does pass fom he secuiy s selle o he secuiy s buye fo he peiod of he epo consequences: if he secuiy s selle defauls on cash paymen in he evesal deal hen he counepay can keep he secuiy (he secuiy is excluded fom bankupcy poceedings economic eamen: epo is a ype of secued lending in which he cash lende holds he secuiy only as collaeal and is ewaded by being paid a egula inees (deemined by a epo ae consequences: if a coupon is payable on he secuiy duing he epo em hen he secuiy s holde will eceive i bu he is obliged o make compensaing paymen o he secuiy s selle 34
simila deals buy/sell-back is a deal in which he wo legs of he classical epo, alhough deal simulaneously, ae eaed as wo sepaae ansacions ahe han one secuiy lending is a deal in which he lende of a special secuiy lends he secuiy fo a fee and is secued agains defaul by he boowe by aking collaeal also in he fom of secuiies eminology epo eminology is based on he secuiies side of he deal no on he cash side lende, selle in he epo is he pay who lends/sells secuiies a he ouse and epuchases hem lae boowe, buye in he epo (called also inveso is he pay who boows/buys secuiies (and hus invess he cash a he ouse and sells hem lae meaning of bid and offe aes: 5.0 5.30 money make 5.0 is he bid ae a which he deale is willing o ake deposi (he bids fo cash 5.30 is he offe ae a which he deale is willing o make loan (he offes cash epo make 5.0 is he offe ae because he deale offes secuiies agains cash a 5.0 % (he effecively boows cash a ha ae 5.30 is he bid ae because he deale bids fo secuiies agains cash a 5.30 % (he effecively lends cash a ha ae cash flows in he epo cash-diven epo means ha he deal is based on a paicula amoun of cash o be boowed and he exac naue of he collaeal is no impoan secuiy-diven epo means ha he deal is iniiaed by he need o boow a paicula amoun of secuiy which is called he special (he moe special he secuiy, he lowe he epo ae A secuiy-diven epo ansacion is based on he following paamees: uency denominaion: euo Repo em: 8 days 35
Repo ae: 4.0 % (A/360 convenion Special secuiy: easuy bond wih a coupon of 8.5 %, acual clean pice of 08.95 and accued inees fo days Requied nominal value of he special: 60 mil EUR ash flow a he beginning of he epo deal: accued inees 8.5. 6EUR 360 full pice 08.95.6. 57 EUR.57 cash paid 60000000 6694600 EUR 00 Special in nominal value of 60 000 000 mil EUR Buye Selle ash 66 94 600 EUR ash flow a he end of he epo deal: 8 inees 6694600 0.04 0866 EUR 360 pincipal 6694600 EUR cash paid 0866 6694600 6750866 EUR Special in nominal value of 60 000 000 mil EUR Buye Selle ash 67 50 866 EUR magining a iniial magin has he fom of a haicu paid cash make value of collaeal ( ± haicu posiive haicu (i.e. % he collaeal value is lowe han he cash loan (he loan is unde-collaealised negaive haicu (i.e. - % he collaeal value is highe han he cash loan (he loan is ove-collaealised 36
he amoun of haicu depends on he elaive cediwohiness of he wo paies b making o make he buye ecalculaes he value of he collaeal coninually in ode o ensue ha i is of adequae value if he value of he collaeal has fallen he buye may ake a magin call equiing he selle o ansfe moe collaeal (in ems of moe secuiies o in cash if he value of he collaeal ises he selle can make a magin call equiing he buye o eun some of he collaeal vaiaion magin is he amoun of collaeal ansfeed beween he selle and he buye in esponse o a magin call subsiuion a epo deal may include he igh fo he secuiy s selle o change he exac secuiy used as collaeal duing he peiod of he epo mauiy he epo peiod is geneally fom one day (ovenigh epo of seveal monhs em epo he peiod is fixed and ageed in advance open epo eihe pay may call fo he epo o be eminaed a any ime (equiing some noice, he epo is olled ove each day cedi isk consideaions double secuiy of he epo sucue is povided by - cedi sanding of he counepay - cedi sanding of he issue of he collaeal epo aes a which i is possible o boow ae lowe hen he aes fo an unsecued inebank loan cusody epo cusody epo is a epo which involves anohe oganisaion o ac as cusodian fo he collaeal (a cusodian holds he secuiy in a sepaae accoun on he buye s behalf cusodian pefoms a couple of duies (making o make, daily epos, subsiuions and ohes he buye canno o use he secuiy in ohe epo deal 37
4. Applicaions of epo i funding a long posiion in a bond moives: - deale has bullish expecaions and heefoe wishes o puchase bonds - deale buys a bond which his counepay inends o buy - deale coves a failed puchase when he inended o buy and sell he same bond sucue of a deal: he deale boows cash fo puchasing he bond hough he epo ade in which he bond seves as he collaeal feasibiliy of he deal is ensued by he selemen pocess in which all paymens of cash o ansfes of a paicula secuiy ae need on any dae acoss all he ansacions ousanding beween he deale and he cleaing sysem uchasing bond Bond make aying cash Deale Lending bond as collaeal Repo deal Boowing cash ii coveing a sho posiion in a bond moives: - deale has beaish expecaions and heefoe wishes o sell bonds sho - deale sells a bond which his counepay inends o sell - deale coves a failed sale when he inended o buy and sell he same bond sucue of a deal: he deale boows a secuiy in a evese epo and obains he cash collaeal by selling he bond in he make Selling bond Bond make Receiving cash Deale Boowing bond as special Revese epo deal Lending cash iii yield enhancemen fund manage may sell a special secuiy fom his pofolio in a epo ansacion and eplace his special secuiy by geneal collaeal in a evese epo ansacion boh special and geneal collaeal have he same cediwohiness so cedi isk of he pofolio emains unchanged 38
pofi made is he diffeence beween he wo epo aes (he special allows o boow a a lowe ae in compaison wih he lending ae Selling special Bond make Boowing a 7 % Deale Boowing geneal collaeal Revese epo deal Lending a 7.5 % iv leveaging a bond pofolio exising holding of bonds can be used as collaeal o epo in moe cash ha can be invesed in new secuiies he leveaging pocess can be epeaed as fa as pudence and magin equiemens allow Invesmen 00 Sep Bonds 00 Bond make ash 00 Bonds 00 Repo ash 98 Sep Bonds 98 Bond make ash 98 Bonds 98 Bonds 96 Repo Sep 3 Bond make ash 96 ash 96 Bonds 96 Bonds 94 Repo Sep 4 Bond make ash 94 ash 94 oiginal invesmen 00 $ exposiion in he bond make 388 $ heoeical exposiion a a % haicu 00 / 0.0 5000 $ if he bond make collapses magin calls will be made in each epo along he whole chain and heavy losses will esul (fund manage is uged o sell bonds a depessed pices 39
v cash-and-cay ansacion in long-em inees ae fuues pocess of deliveing a D bond o he fuues conac is based on a epo deal (in ode o boow he money o buy he bond he deale obains he bond hoough a epo ansacion; he can epo he bond ou implied epo ae is he beak-even ae a which he cash-and-cay ansacion gives a eo esul (no pofi and no loss vi liquidiy managemen epo opeaions is a majo ool of moneay policy conduced by cenal banks cenal bank adds cash o he banking sysem by buying secuiies fom he banks in open make opeaion called also injecing epos cenal bank dains cash fom he banking sysem by selling secuiies o he banks in open make opeaion called also wihdawing epos epo ae a which epo ansacions ae conduced is he mechanism hough which changes in inees ae levels ae signalled o he make 40
V. ORGAGE LOANS AND ORGAGE-BAKED SEURIIES mogage make is a collecion of makes ha include a pimay (oiginaion make whee funds ae boowed fo financing of housing and a seconday make in which mogages ade mogage insumens ae elaed o he pimay mogage make and epesen vaious ypes of mogage designs ha inemediae housing financing mogage-backed secuiies (BS ae elaed o he seconday mogage make and epesen vaious ypes of secuiies ha deive is cash flow fom an undelying pool of mogages secuiiaion is a echniques of financial engineeing in which individual asses ae pooled and used as collaeal fo he issuance as well as souce of cash flow of deivaive secuiies 5. adiional mogage adiional (level-paymen, plain-vanilla mogage is a mogage in which he boowe pays a fixed inees ae (he mogage ae and epays a pa of he oiginal mogage balance (he pincipal in equal insalmens ove an ageed peiod of ime (em of he mogage in such a way ha a he end of he em he loan has been fully epaid (he mogage is fully amoied A A A H 0 balloon mogage is a mogage in which paymens ae sufficien o pay only inees and no paymens ae made o educe he oiginal balance ha mus be paid a he end of he em of he mogage (poblem wih efinancing he balance, disasous consequences duing he Gea Depession 4
mahemaics of a adiional mogage H 0 oiginal mogage balance H.. mogage balance a he end of peiod A egula mogage paymen (insalmen em of mogage mogage ae fai value of a L is a pesen value of all fuue insalmens discouned a a given mogage ae discouned pesen value should equal he oiginal mogage balance H 0 A ( A ( amoun of a egula mogage paymen (annuiy: ( A H 0 H a 0 (, a is he annuiy faco A mogage of 00000 $ was exended fo 30 yeas a an annual mogage ae of 9.5 %. he fequency of mogage paymens is monhly. Deemine he value of he monhly mogage paymen. H 0 00000, 30 360, 0.095/ 0.007967. 360 0.007967 (.007967 A 00000 840.85$ 360 (.007967 decomposiion of an insalmen ino inees paymen and pincipal epaymen in each peiod a fixed mogage paymen consiss of inees paid on an ousanding mogage balance and epaymen of a poion of he ousanding mogage balance A A H ( H H he fomula fo deemining he emaining mogage balance (a he end of he peiod H ( ( ( H 0 H ( H 0 ( ( 4
he fomula fo deemining he inees paymen (in he peiod H H 0 [( ( ( ] he fomula fo deemining he scheduled pincipal epaymen (in he peiod H H H 0 ( ( epaymen calenda of adiional mogage loan calculaion of he cash flow of an L can be aanged in a able called paymen calenda annuiy amoun appoach:. alculae he value of a egula fixed insalmen using he annuiy fomula and inse he esul ino all cells in he column oal insalmen. alculae he inees paymen (he numbe in he Beginning balance is muliplied by he mogage ae 3. alculae he pincipal epaymen (he numbe in Inees paymen is subaced fom he numbe in oal insalmen 4. alculae he ending balance (he numbe in incipal epaymen is subaced fom he numbe in Beginning balance 5. Use he numbe in Ending balance as he nex peiod s numbe in Beginning balance 6. Repea he sequence of seps fom hough 5 fo he subsequen peiod ake ou he paymen calenda fo a mogage wih a pincipal of 00000 $, em o mauiy of 0 yeas and an annual mogage ae of 3 %. Fequency of insalmen is yealy. 0 0.03 (.03 oal insalmen 00000 3443. 4 0.03 Yea Beginning balance Regula insalmen Inees paymen incipal epaymen Unpaid balance 00000,00 3443,4 6000,00 7443,4 9556,86 9556,86 3443,4 5776,7 7666,44 84890,4 3 84890,4 3443,4 5546,7 7896,43 76993,99 4 76993,99 3443,4 5309,8 833,3 68860,67 5 68860,67 3443,4 5065,8 8377,3 60483,35 43
6 60483,35 3443,4 484,50 868,64 5854,7 7 5854,7 3443,4 4555,64 8887,50 4967, 8 4967, 3443,4 489,0 954,3 3383,08 9 3383,08 3443,4 404,39 948,75 4384,34 0 4384,34 3443,4 373,53 97,6 467,7 467,7 3443,4 3440,8 000,96 04669,76 04669,76 3443,4 340,09 0303,05 94366,7 3 94366,7 3443,4 83,00 06,4 83754,58 4 83754,58 3443,4 5,64 0930,50 784,07 5 784,07 3443,4 84,7 58,4 6565,65 6 6565,65 3443,4 846,97 596,7 49969,48 7 49969,48 3443,4 499,08 944,06 3805,4 8 3805,4 3443,4 40,76 30,38 573,04 9 573,04 3443,4 77,69 67,45 305,59 0 305,59 3443,4 39,55 305,59 0,00 Sum 6886,83 6886,83 00000,00 chaaceisic feaues of paymen calenda: - he poion of he egula paymen applied o inees declines (because he mogage balance is educed wih each insalmen - he poion applied o educing he mogage balance inceases (because he insalmen is fixed and he inees componen falls - he mogage is compleely paid off (self-amoied a he end of he mogage em annuiy faco appoach: a slighly modified appoach o making ou he paymen calenda is based on he elaionship: A H 0 a H a K H a H a0 annuiy faco appoach can be inepeed as if he ousanding mogage is pepaid a he end of each peiod by means of aking a new mogage whose mauiy is shoe by one peiod and whose pincipal equals he mogage s ousanding balance sequence of seps:. alculae he annuiy faco fo he beginning balance of he mogage using he especive fomula 44
. alculae he oal insalmen (he numbe in he Beginning balance is muliplied by he numbe in he Annuiy faco 3. Decompose he oal insalmen ino an inees paymen and pincipal epaymen and calculae he ending balance in he same way as in he annuiy level appoach 4. Use he numbe in Ending balance as he nex peiod s numbe in Beginning balance 5. Repea he sequence of seps fo he subsequen peiod ake ou he paymen calenda fo he mogage fom he pevious example (pincipal 00000 $, em o mauiy 0 yeas, annual mogage ae 3 %, yealy fequency of paymen using he annuiy faco appoach. Yea Beginning balance Annuiy faco Regula insalmen Inees paymen incipal epaymen Unpaid balance 00000,00 0,0676 3443,4 6000,00 7443,4 9556,86 9556,86 0,06984 3443,4 5776,7 7666,44 84890,4 3 84890,4 0,07709 3443,4 5546,7 7896,43 76993,99 4 76993,99 0,075953 3443,4 5309,8 833,3 68860,67 5 68860,67 0,0796 3443,4 5065,8 8377,3 60483,35 6 60483,35 0,083767 3443,4 484,50 868,64 5854,7 7 5854,7 0,08856 3443,4 4555,64 8887,50 4967, 8 4967, 0,094030 3443,4 489,0 954,3 3383,08 9 3383,08 0,0046 3443,4 404,39 948,75 4384,34 0 4384,34 0,08077 3443,4 373,53 97,6 467,7 467,7 0,73 3443,4 3440,8 000,96 04669,76 04669,76 0,8434 3443,4 340,09 0303,05 94366,7 3 94366,7 0,4456 3443,4 83,00 06,4 83754,58 4 83754,58 0,60506 3443,4 5,64 0930,50 784,07 5 784,07 0,84598 3443,4 84,7 58,4 6565,65 6 6565,65 0,8355 3443,4 846,97 596,7 49969,48 7 49969,48 0,6907 3443,4 499,08 944,06 3805,4 8 3805,4 0,353530 3443,4 40,76 30,38 573,04 9 573,04 0,56 3443,4 77,69 67,45 305,59 0 305,59,030000 3443,4 39,55 305,59 0,00 Sum 6886,83 6886,83 00000,00 5. epaymens hee componens of he mogage cashflow: - inees paymen 45
- scheduled pincipal epaymen - pepaymens ae called paymens in excess of he egulaly scheduled pincipal easons fo pepaymens: - selling he popey (change of employmen, puchase of a moe expansive home, divoce, ec. - igh of boowe o pay off all o pa of he mogage balance a any ime (paiculaly if make aes fall below he ae in he conac - defaul on meeing mogage obligaion (he popey is epossessed, sold and poceeds ae used o pay off he mogage, fo insued mogages he insue will pay off he mogage balance - insued caasophe occus (he insuance poceeds ae used o pay off he mogage pepaymen ae is an esimae of a facion of he emaining pincipal in he pool of mogages ha is pepaid each monh he ae is based on he hisoical pepaymen expeience and on he cuen and expeced economic envionmen link beween annual Π p ( H A Π pojeced pincipal pepaymen fo peiod p pepaymen ae H.. mogage balance a he beginning of peiod given all pevious pepaymens A scheduled pincipal epaymen fo peiod pa and monhly p m pepaymen aes p m ( pa ake ou he paymen calenda fo he mogage fom he pevious example (pincipal 00000 $, em o mauiy 0 yeas, annual mogage ae 3 %, yealy fequency of paymen assuming ha he annual pepaymen ae is 6 %. 46
Yea Beginning balance Regula insalmen Inees paymen incipal epaymen epaymens Unpaid balance 00000,00 3443,4 6000,00 7443,4 553,4 8003,45 8003,45 3443,4 5430,0 803,04 0379,4 660,98 3 660,98 3443,4 4878,33 8564,8 94,77 44803,40 4 44803,40 3443,4 4344,0 9099,04 84,6 756,0 5 756,0 3443,4 386,86 966,8 7076,749 0869,07 6 0869,07 3443,4 336,07 07,07 6045, 94706,88 7 94706,88 3443,4 84, 060,94 5046,97 79058,65 8 79058,65 3443,4 37,76 07,38 4079,36 63908,03 9 63908,03 3443,4 97,4 55,90 34,98 4939,0 0 4939,0 3443,4 477,8 965,97 36,394 35036,85 35036,85 3443,4 05, 39,04 358,689 86, 86, 3443,4 638,58 804,56 508,8937 797,67 3 797,67 8,85 39,8 797,67 0 0 4 0 0 0 0 0 0 5 0 0 0 0 0 0 6 0 0 0 0 0 0 7 0 0 0 0 0 0 8 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 Sum 6959,55 3834,7 387,8 688,8 he 6 % pepaymen ae shoened he life of he mogage o only 3 yeas. he sum of scheduled pincipal epaymens and pepaymens equals 00000 $. pepaymen isk is he isk ha he homeowne will pepay he mogage a any unfavouable ime and ha he iming and amoun of pepaymens ae no known wih ceainy and ae difficul o pedic accuaely conacion isk is elaed o he siuaion when declining mogage aes end o speed up pepaymens (because homeownes have an incenive o efinance hei debs a a lowe ae and invesos ae foced o einves pepaymens a lowe make inees aes exension isk is elaed o he siuaion when ising mogage aes end o slow down pepaymens (because homeownes lose an incenive o efinance hei debs a a lowe ae and invesos canno einves moe cash a highe make inees aes mismach isk occus when speeding up and slowing down pepaymens disub he balanced posiion beween asses and liabiliies in ems of hei duaions 47
5.3 Inflaion-adjused mogage designs poblems caused by high inflaion in he adiional level-paymen mogage: eosion poblem: subsanial inflaion eodes he epaymens of pincipal in ems of puchasing powe (i.e. by he wenieh yea he eal value of he iniial insalmen will be down o some 5 % a a 0 % inflaion pe yea if he eosion is no made up by sufficienly high mogage aes ha peseve equied eal inees aes hen lendes will be discouaged o povide adequae funding fo he mogage make Naue of he il poblem 50 Insalmen 00 50 00 50 0 0 3 4 5 6 7 8 9 0 Inflaion il poblem: if high inflaion is addessed by high mogage aes in ode o leave eal aes lagely unchanged hen he level-paymen mogage may lead o high iniial insalmens high iniial paymens end o foeclose home owneship o lage segmens of he populaion o foce boowes o scale down hei demands a gaduaed-paymen mogage G plan consiss in eplacing he level-paymen seam wih one ha gows a he ae of inflaion in such way ha he gaduaed seam has he same pesen value as he undelying level-paymen seam ρ equied eal ae of inees (mogage ae in he absence of inflaion π ae of inflaion mogage ae, ( ρ ( π 48
G 0.. iniial value of gaduaed paymen, G G 0 ( π iniial valueg 0 can be found fom he equaion H 0 G ( π G ( π 0 0 G 0 ( ( ρ ( π ( ρ G 0 is he insalmen ha would be paid in he absence of inflaion ompae paymen calendas beween a 0-yea level-paymen mogage and a 0-yea gaduaed-paymen mogage wih beginning balances of 00000 $ unde assumpion ha a mogage bank equies a eal inees ae of 3 % and hee is a seady inflaion of 8 %. aymen calenda fo he level-paymen mogage mogage ae ( 0.03 ( 0.08 0. 4 0 0.4.4 level paymen 00000 550. 38 0.4 Yea Beginning balance Regula insalmen Inees paymen incipal epaymen Unpaid balance 00000,00 550,38 480,00 3030,38 96969,6 96969,6 550,38 39,39 337,00 93598,6 3 93598,6 550,38 760,49 3749,90 89848,7 4 89848,7 550,38 339,00 47,39 85677,34 5 85677,34 550,38 0870,3 4640,5 8037,09 6 8037,09 550,38 0348,57 56,8 75875,8 7 75875,8 550,38 9768,38 574,00 7033,8 8 7033,8 550,38 9,98 6387,40 63745,88 9 63745,88 550,38 8405,04 705,35 56640,53 0 56640,53 550,38 7606,40 7903,99 48736,54 48736,54 550,38 677,99 879,39 39944,5 39944,5 550,38 579,7 9780,66 3063,49 3 3063,49 550,38 4630,38 0880,0 983,48 4 983,48 550,38 3407,46 0,9 0780,57 5 0780,57 550,38 047,0 3463,9 9377,8 6 9377,8 550,38 0533,8 4976,56 78740,7 7 78740,7 550,38 8850,46 6659,9 6080,80 8 6080,80 550,38 6977,88 853,50 43548,30 9 43548,30 550,38 4894,83 065,55 93,74 0 93,74 550,38 577,64 93,74 0,00 Sum 5007,64 3007,64 00000,00 49
aymen calenda fo he gaduaed-paymen mogage (gowing a 8 % pe yea mogage ae ( 0.03 ( 0.08 0. 4 0 0.03.03 iniial value of gaduaed paymen 00000 3443. 4 0.03 Yea Beginning balance Regula insalmen Inees paymen incipal epaymen Unpaid balance 00000,00 458,59 480,00-796,4 0796,4 0796,4 5680,08 3374,86-7694,78 5656,9 3 5656,9 6934,49 439,76-7305,7 96,46 4 96,46 889,5 5060,87-677,6 9733,08 5 9733,08 975,39 58,00-6069,6 3580,69 6 3580,69 33,58 6504, -57,65 40974,34 7 40974,34 3039,8 7085,5-4046,33 4500,67 8 4500,67 488,3 7540,3-658,0 47678,68 9 47678,68 687,90 7839,08-966,8 48644,86 0 48644,86 90,73 7947,68 075,05 47569,8 47569,8 3344,55 786,85 357,7 4405,0 4405,0 3385, 743,46 640,66 3763,44 3 3763,44 36560,9 6709,77 9850,5 7780,93 4 7780,93 39485, 560,58 388,53 3898,40 5 3898,40 4643,9 404,8 860,74 9596,66 6 9596,66 46055,43 95,34 404,09 79,57 7 79,57 49739,87 94,05 30497,8 40694,75 8 40694,75 5379,06 584,09 37904,97 0789,79 9 0789,79 5806,58 553,57 46463,0 5636,78 0 5636,78 6657,9 633,3 5636,78 0,00 Sum 664399,33 0,00 00000,00 chaaceisic feaues of G paymen calenda: - in compaison wih he L he G insalmens ae subsanially lowe a he beginning of he mogage life bu hey ae subsanially highe a he end of he mogage life poblem fo people whose income may no keep up wih inflaion (he las insalmen is ove 4.3 imes lage han he fis insalmen mos G plans heefoe use modified vesions in which he nominal paymen gows duing a poion of he life of he conac and heeafe levels off - negaive amoiaion is a siuaion in which he peiodic paymen is insufficien o cove accued inees and hus he ousanding mogage balance inceases 50
he debo insead of epaying is geing deepe and deepe in deb as a consequence of high and pesisen inflaion negaive amoiaion may aise he mogage balance o an amoun ha exceeds he make value of he popey encouaging he boowe o defaul b pice-level-adjused mogage LA plan is designed o be fixed in puchasing powe ems and he effecive insalmen and he unpaid balance ae adjused by an inflaion coecion faco ems of he conac: - he eal inees ae (he mogage ae ha could be expeced if hee wee no inflaion in he economy, abou 3-4 % - he inflaion coecion faco (equal o an incease in he seleced pice index in a given peiod - ohe ems of he loan (mauiy, iniial amoun, ec. sequence of seps:. Wok ou he paymen calenda fo an undelying eal mogage inees ae (using he annuiy amoun appoach. uliply he eal insalmen (pevailing in he absence of inflaion by an appopiae inflaion coecion faco 3. uliply he eal unpaid balance by he same coecion faco ake ou he paymen calenda fo a LA plan wih a pincipal of 00000 $ and em o mauiy of 0 yeas unde assumpion ha he bank chages a 3 % eal mogage ae. esisen consume inflaion is expeced o say a a level o 8 % pe yea. Yea Real beginning balance Real insalmen Real unpaid balance ice index Regula insalmen Effecive unpaid balance incipal epaymen 00000,00 3443,4 9556,86,0800 458,59 0796,4-796,4 9556,86 3443,4 84890,4,664 5680,08 5656,9-7694,78 3 84890,4 3443,4 76993,99,597 6934,49 96,46-7305,7 5
4 76993,99 3443,4 68860,67,3605 889,5 9733,08-677,6 5 68860,67 3443,4 60483,35,4693 975,39 3580,69-6069,6 6 60483,35 3443,4 5854,7,5869 33,58 40974,34-57,65 7 5854,7 3443,4 4967,,738 3039,8 4500,67-4046,33 8 4967, 3443,4 3383,08,8509 488,3 47678,68-658,0 9 3383,08 3443,4 4384,34,9990 687,90 48644,86-966,8 0 4384,34 3443,4 467,7,589 90,73 47569,8 075,05 467,7 3443,4 04669,76,336 3344,55 4405,0 357,7 04669,76 3443,4 94366,7,58 3385, 3763,44 640,66 3 94366,7 3443,4 83754,58,796 36560,9 7780,93 9850,5 4 83754,58 3443,4 784,07,937 39485, 3898,40 388,53 5 784,07 3443,4 6565,65 3,7 4643,9 9596,66 860,74 6 6565,65 3443,4 49969,48 3,459 46055,43 79,57 404,09 7 49969,48 3443,4 3805,4 3,7000 49739,87 40694,75 30497,8 8 3805,4 3443,4 573,04 3,9960 5379,06 0789,79 37904,97 9 573,04 3443,4 305,59 4,357 5806,58 5636,78 46463,0 0 305,59 3443,4 0,00 4,660 6657,9 0,00 5636,78 Sum 6886,83 664399,33 00000,00 chaaceisic feaues of LA paymen calenda: - effec of negaive amoiaion means ha he ending balance may keep gowing fo a subsanial pa of he life of he mogage - effec of gowing amoun of insalmen means ha paymens sa low and hen aise smoohly a he ae of inflaion (in his case he seam is equal o he G plan, in ealiy he inflaion coecion faco mios he acual developmen of pice index - he mogage loan is fully amoied by is consucion (because i is fully amoied in eal ems c dual-ae mogage DR plan combines a fixed ae which deemines how much he boowe will effecively pay in egula insalmens and a floaing ae which deemines effecive eun on a mogage loan ems of he conac: - he paymen ae which is a ae of inees fixed fo he life of he mogage loan - he effecive ae which changes peiodically on he basis of an ageed-upon efeence sho-em ae - ohe ems of he loan (mauiy, iniial amoun, ec. sequence of seps: 5
. On he basis of he fixed paymen ae calculae he amoun of insalmen using he annuiy faco appoach. On he basis of he vaiable ae calculae he amoun he inees componen in he insalmen 3. alculae he amoun of he epaid pincipal and he amoun of unpaid balance 4. Use he amoun of unpaid balance as he amoun of subsequen peiod s beginning balance 5. Repea he sequence of seps fo he subsequen peiod ake ou he paymen calenda fo a DR plan wih a pincipal of 00000 $ and em o mauiy of 0 yeas unde assumpion ha he bank chages a 3 % paymen ae (equal o he eal ae pevailing in he absence of inflaion and a % effecive ae (coesponding o a nominal ae composed of a 3 % eal ae and an 8 % seady inflaion ae. Yea Beginning balance Annuiy faco Regula insalmen Effecive ae Inees paymen incipal epaymen Unpaid balance 00000,00 0,0676 3443,4 0, 000,00-8556,86 08556,86 08556,86 0,06984 4560,6 0, 94,5-838,09 6937,95 3 6937,95 0,07709 5773,8 0, 3863,7-8089,90 507,85 4 507,85 0,075953 709,43 0, 4753,06-766,63 3689,48 5 3689,48 0,0796 854,6 0, 5595,84-707,4 39760,7 6 39760,7 0,083767 0083,94 0, 6373,68-689,74 46050,46 7 46050,46 0,08856 78,95 0, 7065,55-583,60 5334,06 8 5334,06 0,094030 363,83 0, 7646,75-403,9 55347,98 9 55347,98 0,0046 565,79 0, 8088,8-435,49 57783,47 0 57783,47 0,08077 7860,58 0, 8356,8-495,60 5879,07 5879,07 0,73 3078,9 0, 840,70 867,49 564,58 564,58 0,8434 393,93 0, 805,7 476,65 5684,93 3 5684,93 0,4456 35854,3 0, 7685,34 868,78 4356,4 4 4356,4 0,60506 39085,89 0, 6786,78 99, 37,03 5 37,03 0,84598 468,09 0, 5433,87 748, 3968,8 6 3968,8 0,8355 467,07 0, 3536,57 384,50 90784,3 7 90784,3 0,6907 536,4 0, 0986,7 30339,87 60444,45 8 60444,45 0,353530 567,99 0, 7648,89 39073,0 37,36 9 37,36 0,56 6349,99 0, 3350,85 50079,4 79, 0 79,,030000 73430,99 0, 784,4 65588,84 5703,38 Sum 670867,08 476570,46 9496,6 53
he final insalmen of 73430.99 $ mus be opped up by he emaining value of unpaid balance of 5703.38 $. his addiional paymen makes he oal sum of pincipal epaymens equal o 00000 $. 5.4 ismach-adjused mogage designs mismach poblem aises when mogages being vey long-em asses ae lagely financed by funds of a sho-em naue (he esul is highly speculaive aciviy of boowing sho and lending vey long in he pesence of ising inees aes lending insiuions may become echnically insolven because he make value of is asses (discouned a he cuen high aes will be insufficien o cove is liabiliies he mismach poblem can by solved by edesigning he adiional mogage o poduce an asse ha would mach he sho-em make aes and hus he cos of liabiliies adjusable ae mogage is a conac in which he mogage ae is ese peiodically in accodance wih some appopiaely chosen benchmak index a he ese dae he new conac ae is equal o he benchmak index plus a spead possible eseing daes: evey monh, six monhs, one yea, wo yeas, five yeas possible benchmak indexes: - make-deemined sho-em inees ae - cos of funding of mogage oiginaos (weighed aveage inees ae cos paid by oiginaos fo hei liabiliies AR design does no addess he il poblem because he ciical ealy paymens will be oughly as high as hey ae wih a adiional mogage and will make a subsanial jump evey ime he mogage ae is adjused caps in ARs caps ae limis ha ae mean o poec he boowe agains shap unfavouable changes in paymen condiions a ese daes peiodic ae cap limis he change in he conac ae a he ese dae (i.e. he ae may no incease by moe hen % pe annum peiodic paymen cap limis he change in he monhly insalmen a he ese dae 54
lifeime cap is an uppe limi on he conac ae ha could be chaged ove he life of he loan and is expessed in ems of he iniial ae (i.e. if he iniial mogage ae is 7 % and he lifeime cap is 5 % hen he maximum inees ae he lende can chage ove he life of he loan is % cap is an opion ha he lende has effecively sold o he homeowne some of he expeced loss fo mogage oiginao o invesos due o caps will be ecouped by a mogage aes o ohe less favouable ems floos in ARs floos ae lowe limis on changes in inees ae o paymen amoun a ese days floos can be of peiodic as well as lifeime naue floo is an opion ha he homeowne has effecively sold o he lende hybid mogage is a conac ha shaes feaues of fixed-ae and adjusable-ae mogages conveible adjusable ae mogage is such AR ha can be conveed ino a fixedae mogage conveible (educible fixed ae mogage can have he mogage ae loweed if aes fall by some pedeemined level he boowe has he choice of conveing and he lende chages a nominal fee fo convesion balloon (ese mogage is a conac in which he boowe is given long-em financing bu a specified fuue daes he conac ae is enegoiaed he conac is effecively a sho-em loan in which he lende agees o povide financing fo he emainde of he em of he mogage 5.5 ogage-backed secuiies mogage secuiiaion compises vaious echniques of financial engineeing in which individual mogages ae pooled and used as souce of cash flow (collaeal fo he ceaion of new secuiies called mogage-backed secuiies (BS 55
Building blocks of secuiiaion in he mogage make Oiginao Sucue Invesos ogage ogage BS Inveso ogage ool ogage BS BS ogage 3 ogage 3 BS 3 BS 4 ice ice ogage X ogage X BS Y Inveso Z Fee Guaanees Insuance agency edi enhancemen a passhough secuiies passhough secuiy is a financial insumen ha ediecs he cashflow fom an undelying pool of secuiied mogages o invesos on a po aa basis advanages of passhoughs: - by holding passhoughs he inveso holds a divesified pofolio of mogages ha educes mos unsysemaic isk and leaves only sysemic isk - liquidiy of passhoughs is consideably bee han ha of an individual mogage (by selling a passhough he inveso can dispose of all loans simulaneously, hee is no need o dispose of loans one by one An invesmen bank puchased a pool of 500 mogage loans each of i in an amoun of 00000 $. Agains his pool he bank issued 5000 passhoughs. A million dolla invesmen can hus buy: 56
a 0 individual mogages [000000:00000] giving he igh o a cash flow fom hese mogages (subsanial unsysemaic isk b 00 passhoughs [000000:(50000000:5000] giving he igh o % [00:5000] of oal cash flow fom he pool of all 500 mogages (well divesified pofolio agency passhoughs ae passhough secuiies issued o guaaneed by govenmen o govenmen-sponsoed insiuions USA: Fedeal Naional ogage Associaion (Fannie ae Fedeal Home Loan ogage opoaion (Feddie ac Govenmen Naional ogage Associaion (Ginnie ae convenional (pivae label passhoughs ae issued by pivae conduis wihou any govenmen guaanees; hey ae aed by commecial aing agencies foms of cedi enhancemen: - copoae guaanees (he issue uses is own cedi aing o back he secuiy - lee of cedi (a guaanee povided by a financial insiuion - insuance (policy povided by an insuance company - ove-collaealiaion (pojeced cash flow fom un undelying mogage pool exceeds liabiliies fom BS - senio/subodinaed sucues (mogage pool is paiioned ino senio and subodinaed pa, senio pa is sold o invesos as convenional passhoughs and subodinaed pa can be eihe eained o sold o invesos willing o accep he geae defaul isk, he amoun of subodinaed pa o senio pa deemines cedi aing of passhoughs b collaealied mogage obligaions (O Os ae bonds backed by he cash flow fom an undelying pool of mogages o an undelying pool of passhoughs (Os backed by ohe Os ae called O squae a O belongs o a O class (anche wihin a given O sucue fo which a given se of ules indicaes how he pincipal fom he undelying pool is o be disibued among he holdes of bonds of individual O classes (i.e. sequenialpay, A bonds, accual bonds, floaing-ae bonds 57
c sipped mogage-backed secuiies sips ae ceaed by aleing he disibuion of pincipal and inees fom a po aa disibuion o an unequal disibuion IO (inees only class absobs all of he inees fom an undelying pool O (pincipal only class absobs he enie pincipal fom an undelying pool popeies of IOs and Os pice of a O moves in he opposie diecion as he change in inees aes (a O has a posiive duaion - mogage aes fall pepaymens speed up and O holdes ecove moe quickly a given eun (equal o nominal amoun of a mogage pool minus puchasing O pice his effec is senghened by he cashflow discouned a a lowe discoun ae - mogage aes ise pepaymens slow down and i is aking longe o ecove pincipal epaymens his effec is senghened by he cashflow discouned a a highe discoun ae pice of an IO end o move in he same diecion as he change in inees aes (an IO has a negaive duaion - mogage aes decline pepaymens acceleae, ousanding pincipal declines fase heeby loweing he base fo calculaing he inees his effec is usually songe hen he offseing effec of he cashflow discouned a a lowe discoun ae - mogage aes incease pepaymens deceleae ha impoves he base fo calculaing he inees his effec is usually songe hen he offseing effec of he cashflow discouned a a highe discoun ae 58
VI. ONEY ARKE SEURIIES money make secuiies ae sho-em financial insumens wih mauiies of less han a yea capial make secuiies have mauiies in excess of one yea 6. oney make convenions a day-coun convenions as a geneal ule he calculaion of inees akes accoun of he exac numbe of days beween paymens vaiaions aise in he convenions used fo he numbe of days in he base yea A/365 assumes a 365-day yea A/360 assumes a 360-day yea he yield on a deposi quoed on A/360 basis is 0.5 %. Wha is he equivalen yield expessed on A/365 basis? A 365 / 365 A / 360 0.5.04 0.65 % 360 an amoun of 00 $ deposied fo 0 days a / 360 yields R 00 360 A 0 0 00 0.05 360 360 A / an amoun of 00 $ deposied fo 0 days a / 365 yields R 00 365 A 0 0 00 0.065 365 365 A / 00.58 00.58 b efeence aes offe ae is he ae a which one bank can empoaily lend funds o anohe bank LIBOR (London Inebank Offe Rae is a se of offe aes fo vaious ems deemined in he London inebank make LIBOR aes ae quoed in a numbe of majo cuencies and fo a numbe of diffeen ems (anging fom ovenigh o one yea many money make inees aes ae se wih efeence o LIBOR aes 59
simila aes in ohe money makes: EURIBOR (Euoone, RIBOR (ague, NYBOR (New Yok, IBOR (okyo, ec. bid ae is he ae a which one bank acceps funds fom anohe bank LIBID, RIBID, NYBID, IBID and so on ae bid aes ha ae counepas o offe aes in especive naional money makes c ypes of quoaions quoaion on a yield basis on he mauiy day he inveso eceives he nominal value of he insumen plus an amoun of inees calculaed accoding o he fomula fo simple inees nominal (pa value of he insumen quoed inees ae numbe of days o mauiy inees 365 mauiy value 365 effecive ae of inees is he annualied inees ae which would give he same compound eun as he ae we ae compaing 365 effecive ae e 365 quoaion on a discoun basis on he mauiy day he inveso eceives he pa value and no explici inees which is howeve efleced in he diffeence beween he discouned issue pice and he pa value nominal (pa value of he insumen d quoed inees ae numbe of days o mauiy discoun issuing pice d 365 d 365 equivalen yield q d 365 365 d d 365 60
he equivalen yield is always geae han he discoun ae d q d < q q 365 c implied fowad money make aes inees paiy equaion: f 365 365 365 appoximae fomula fo implied fowad aes (neglecing he em wih he poduc of aes f & sandad popey of a money make yield cuve fo expeced changes in inees aes upwad sloping yield cuve indicaes ha he make expecs fuue inees aes o ise < < f downwad sloping yield cuve indicaes ha he make expecs fuue inees aes o fall > > f 6. Examples of money make insumens a money make deposi is an insumen which pays a fixed inees (quoed on yield basis and has a fixed em he inees and pincipal ae paid in one lump sum on he mauiy day he deposi canno be liquidaed befoe mauiy A money make deposi of mil USD has been opened on 5 Januay a an inees ae of 9.5 % fo one monh. 3 mauiy value 000000 0.095 007856. 6 365 365 / 3 3 effecive ae of inees 0.095 0.0965 9.65 % 365 6
b negoiable ceificae of deposi caies a fixed inees (quoed on yield basis and has a fixed em Ds canno be wihdawn befoe mauiy bu hey can be aded in a seconday make (hey ae negoiable some DS have mauiy in excess of one yea I S issuance puchase sale mauiy puchasing pice selling pice holding peiod yield h S S I c 365 365 I c 365 S S 365 365 c coupon ae S S 365 S 365 365 S, S yield on he D a puchase o a sale I numbe of days beween issue and mauiy, S numbe of days beween puchase and mauiy o sale and mauiy onside an inveso who puchases a 9-day D (wih a coupon of 0 % and nominal value of 00 wih 50 days o mauiy a a yield of 0 % and sells 30 days lae a a yield of 0 %. 9 00 0. 365 puchasing pice 0. 08 50 0. 365 6
9 00 0. 365 selling pice S 0. 935 0 0. 365 50 0. 365 holding peiod yield 365 h 0.0995 9.95 % 0 30 S 365 Even hough he yield has no changed fom 0 % ove he peiod, he holding peiod eun is less han 0 %. his is he esul of compounding: he D is piced o include accued inees bu he inees is no paid unil mauiy. c easuy bill is a sho-em govenmen secuiy sold on he basis of a discoun o he pa value on mauiy he holde eceives he pa value of he bill A 9-day 00 $ easuy bill is issued a a discoun of 0 %. 9 issue pice 00 0. 97. 5 365 0. equivalen yield q 0.06 0.6 % 9 0. 365 ohe secuiies quoed on a discoun basis bills of exchange (commecial bills ae issued by pivae companies agains he sale of goods; hey ae used o finance ade in he sho em banke s accepance is a wien pomise issued by he boowe o he bank o epay boowed funds; i can be negoiable and can be sold in a seconday make; if he boowe defauls he inveso has legal ecouse o he bank ha made he fis accepance commecial pape efes o unsecued pomissoy noes issued by lage copoaions; he noes ae no backed by any collaeal bu hey ely on he high cedi aing of he issuing copoaion 63