Two Explicit Formulae for Yield

Transcription

1 Two Explc Fomulae fo Yeld Muhamed Boogovac, Ph.D. Peseed a Eepse Rsk Maageme Smposum Soce of cuaes Chcago, IL pl 3 6, 006 Copgh 006 b he Soce of cuaes. ll ghs eseved b he Soce of cuaes. Pemsso s gaed o make bef exceps fo a publshed evew. Pemsso s also gaed o make lmed umbes of copes of ems hs moogaph fo pesoal, eal, classoom o ohe sucoal use, o codo ha he foegog copgh oce s used so as o gve easoable oce of he Soceʹs copgh. Ths cose fo fee lmed copg whou po cose of he Soce does o exed o makg copes fo geeal dsbuo, fo advesg o pomooal puposes, fo cluso ew collecve woks o fo esale.

2 bsac Hsocall, has alwas bee a challegg mahemacal ask o ge a explc fomula fo he soluo of a polomal equao of he degee, whe >. s we kow he eld ae of a pofolo s usuall a mplc soluo of he algebac equao of degee geae ha,.e., usuall cao be calculaed explcl b meas of a fe umbe of facos ad adcals. Theefoe, he eld of a pofolo of fxed come secues ma be esmaed b meas of eave mehods such as Newo s mehod, fo example. I hs pape we gve wo explc fomulae ha esmae he eld of a pofolo of fxed come secues ha ae o ees sesve. The eld s explcl esmaed ems of he dscoug ees aes of he secues ha compse he pofolo.

3 . Ioduco, Defos, Ideas Le deoe a veco of cash flows: 0,,,, a he mes 0 0,,,,, whee ca be posve, egave o zeo. Eve asaco ad pofolo ca be epeseed as a sees of cash flows. Theefoe, we wll use he wods asaco ad pofolo echageabl. The ae of ees s called a eld ae f sasfes he followg equao: P : v 0, 0 whee v, see 5.3 Kellso 99. The eld ae s cucal fo coec assessme of asacos. Le us meo ha we wll o esc ouselves o polomals he equao hs pape,.e., mes,,,,, ca be a eal umbes, cludg he mos fequel used, facos of he ea. well kow esul fom algeba s ha he geeal suao hee s o explc fomula o solve he equao, eve f s ol a polomal. The ol avalable mehods geeal ae eave. Hece, o explc fomula fo eld exss ha woks he geeal suao. We should be movaed o fd a explc fomula o esmae eld, eve f hee s ve sophscaed sofwae ha esmaes eld b eave mehods. People suggle all aeas of scece o fd explc soluos of algebac equaos eve hough umecal soluos ae avalable. The easo fo ha s ha a explc fomula explas elaos bewee he ems, whch cobue o ou udesadg of a gve poblem. I pacula, fom he fomulae ha we pese hs pape, we lea abou elaos bewee eld ad dscou aes of he gve pofolo, whch s mpossble o see fom a sofwae based o umecal mehods. The dffeece bewee kowg a umecal esul ad kowg a fomula s he same as he dffeece bewee kowg a fac ad deducg devg a fac. We eed boh. The oexsece of a fomula fo explc soluo of he equao s jus he fs poblem wh he eld ae. The secod majo poblem heed fom algeba ases fom he fac ha somemes hee s moe ha oe eal umbe soluo o equao, ad somemes hee ae o eal umbe soluos a all. Wose e, we ca cosuc asacos fo boh cases mulple eal umbe soluos, ad o eal umbe soluos. I Kellso 99, Pomslow 980, Techoew, e al. 965 ad Jea 968, we ca fd some examples of such asacos, ad 3

4 vaous aemps o coecl assess he asacos wh ehe mulple eld aes, o wh o eld aes. Example. Ths s Example 5.3 fom Kellso 99. asaco s obseved whee a peso boows oe u fo oe ea a ae 8 pece ad leds he same amou fo oe ea a ae j0%. The eld ae, as a soluo of he equao does o exs hs asaco. Example. Ths s Example 5. Kellso 99. Cosde a asaco whch a peso makes pames of $00 mmedael ad $3 a he ed of eas exchage fo a pame eu of $30 a he ed of oe ea. Equao fo hs pofolo s 00 30v 3v 0 whch asfoms o , whee deoes he ukow eld ae. These quadac equaos have he followg wo soluos: v ad v,.e.,.. 0. ad 0. Example 3. Ths s Example 5.4. Kellso 99. The cash flow s 0,, 00, 00, 0. The equao s 00 00v 0v 0 whch asfoms o00 0. Ths equao does o have a eal soluo fo. Vaous auhos ed o def ecessa ad suffce codos ha he coeffces of should sasf ha would guaaee a uque eal umbe soluo of, a uque eld of he coespodg asaco. Howeve, eve codo o coeffces of educes he se of avalable asacos. Fo example, a cocep of smple pojec s defed as a asaco whch he sgs of all s ae dffee fom ha of 0. The exsece of he uque eld ae has bee pove fo smple pojecs Techoew, e al The dffcules emege wh he osmple pojecs. I ode o sud osmple pojecs, Techoew, e al. 965 oduce wo dffee ees aes, ledg ad boowg aes. Jea 968 poved he exsece of a uque eld ae fo he asacos whe ehe fs o las flows ae egave. The dffcules of deemg eld emege fo asacos wh egave mddle flows. I Chape 5 of Kellso 99 ad Becke 988 some moe geeal suffce codos ha cash flows mus sasf wee gve ode o have a uque soluo of. Oe such suffce codo s ha he ousadg vesme balace s posve a all mes,,,. 4

5 Ths was a ve bef llusao of he ma deas ad useful classcal esuls. Those esuls ae ve mpoa, bu ode o be acheved, ve sog escos o he se of avalable asacos had o be made, cludg ha he mes ae assumed o be aual umbes,.e., ha he equao s a polomal. Thee s o eed o go o deals of a of he above esuls, because we appoached he poblem of eld ve dffeel b seg assumpo of abage fee make ahe ha makg assumpos o he coeffces of he equao as was he case all he above papes. eesg dea came fom Pomslow 980. Pomslow sas: Idvdual mus assess a asaco b usg quaes ha deped o hs pacula ccumsaces ad ha ae depede of he asaco self, as opposed o usg eld aes, whch deped solel o he asaco ad ae depede of a pacula dvdual. I ou pape, as a pacula ccumsace of a veso, we see he cue dscou aes of he pofolo. No assumpos abou he coeffces of he equao ae made advace hs pape, ulke mos of he ohe eseach abou he eld, e.g., Becke 988, Kellso 99, Pomslow 980, Techoew, e al. 965 ad Jea 968. The dea s o assume ohe, ma suaos moe aual, escos o he sees of asacos, he escos cosse wh he o abage pcple. The assumpos ha he dscou aes ae gve ad ha we ae he abage fee make esc he se of avalable asacos of he equao a aual wa. I fac, we esc ou eseach o he se of asacos such ha have he coeffce 0 equal o he value of he pofolo,.e., s he pese value of he cash flows, whee each s dscoued wh he coespodg dscou ae of he pofolo. Ths meas ha we follow he advce of Pomslow 980 o o look a he asacos as solaed eves. The fac ha he eld ae s affeced b cue make aes s ulzed. The goal s o fd a explc esmae of eld ha depeds o cue ees aes,.e., o he dscou aes of he zeos he pofolo of ees. Fo ha pupose we wll lm ouselves o pofolos of fxed come secues whch ae o ees sesve. Ths meas ha all secues a obseved pofolo P have fxed cash flows. Le us deoe he se of dscou aes of he pofolo P b R,,,. These ma be dscou aes of a sps o zeo coupo bods, adjused fo a spead ove spo aes. These dscou aes do o eed o be spo aes, eve hough he ae spo aes f we deal wh Teasu secues ad f we assume holdg hem ul mau. Le me also epea ha we assume ha he pofolo P s a pa of a abage fee make M. Ude hese assumpos, Example shows ha eld s fe, f a sk spead s o cluded o cosdeao,.e. ha case s a fee luch. I hs pape, wo fomulae, 0 ad 7, whch esmae eld of a legmae avalable 5

6 abage fee make M asaco P, ems of he gve dscou aes R,,, of P wll be pove. These fomulae ae mpoa esuls because he ae explc ad heefoe eable us o avod compuao b eave echques, ad addo, gve us a ew elao bewee eld ad dscou aes of he gve pofolo. We wll be able o esmae eld of legmae asacos mauall, ad o ceae a smple wokshee o calculae eld, as demosaed he umecal examples lae. The fomulae 0 ad 7 esmae eld ae of he abage fee asaco defed b he cash flow,,,, a mes,,, ad he coespodg dscou aes,,,. We do o eed o kow he cash flow 0 a 0, he value of he pofolo, as a pu fo he fomulae 0 ad 7. Ths s sc wh he fac ha 0 s uquel deemed he abage fee make M. zeo coupo bod maug a me s a bod ha pas s face value a ad o coupo po. Eve pofolo of fxed come secues ha ae o ees sesve ca be cosdeed as a colleco of zeo coupo bods zeos. Spo aes gve us he eld o mau of he assocaed skless zeo coupo bods. The fomula 0 s a lea appoxmao of eld, ad he fomula 7 akes o accou covex as well.. Devao of he Resuls. Lea ppoxmao Le us cosde a se,,,, >, of zeo coupo bods wh face amous of $, wh maues a mes: T,,,, ad a veco of coespodg dscou aes: R,,,. I ode o smplf oao we wll somemes def a u zeo coupo bod wh s pce a 0. I hs oao,,,... Le us deoe vecos R, as usuall b: Xx, x,, x R Le us cosde he pofolo P cossg of zeo coupo bods wh cash flows a mes,.e., P{ 0,,,, }. The he pese value of pofolo P s gve b he followg fomula 0 PV P 6

7 Noce ha eve bod s a specal case of such a pofolo, whee he fs cash flows ae he coupo pames, ad Face value coupo pame. Ths meas ha eve pofolo of fxed come secues ca be boke dow o a se of zeos P{ 0,,,, }. Example 4. I ode o demosae he above oao, le us cosde pofolo P ha cosss of wo bods. The bod maues eas. Coupo ae s 7 pece, ad spead s 0. pece. The bod B maues 3 eas, has coupo ae 8 pece, ad we assume o spead fo B. Noce ha we do o eed o kow he spo aes he make M; we ol eed o kow he fal, adjused dscou aes,, 3, 4, 5. The pofolo P, boke o buldg blocks, has he followg cash flows 7, 07, 8, 8, 08, adjused dscou aes R 0.07, 0.07, , 0.08 ad he coespodg mes T,,,, 3. /// I he above oao eld sasfes he equao: 3 The followg holds 0 4 Le us oduce a fuco of vaables x: X x, x,..., x x 5 Le us assume hee ha,,, ae suffcel close o each ohe. Ths s a ealsc assumpo because usuall he dscou aes of smla duaos ae close o each ohe. The assumpo ha he dscou aes ae suffcel close s o ve escve. Fo example, o mae how seep he eld cuve s, he po,,, s jus oe po R. Whehe o o a lage eghbohood some absolue us s suffcel small fo ou pupose depeds o he shape of he mulvaable suface of he fuco x,x,..., x aoud he po R,,, R. 7

8 8 Le us appl he mulvaable Talo fomula of he fs ode o he fuco X he eghbohood of he gve po of dscou aes R,,,. The follows ha x x x x,...,, 6 B he defo of he eld of pofolo P ad b defo of he fuco gve b equao 5 follows: 0,...,, 7 If we subsue Y,,, fom he eghbohood of R fo Xx, x,, x he equao 6 we ge:,...,, 8 Fom 3, 7, ad 8 we ge: 0 9 Solvg hs equao fo gves: 0 Ths s he lea fomula ha esmaes eld explcl ems of gve dscou aes. The above esuls ca be summazed he followg poposo: Poposo. ssume we have a pofolo of fxed come secues P a abage fee make M. s we kow we ca cosde as a se of zeos,.e.,

9 P { 0,,,, }, whee ae u zeos gve b, wh he cue dscou aes R,,, of he pofolo P, a mes 0 0,,,,, especvel. The he eld of he pofolo ca be esmaed b he fomula 0. Le us oce ha he abage fee assumpo asfoms o a assumpo o he coeffces of he equao,.e., s assumed ha he cash flow a he me 0, 0 s pese value of he cash flows,,,, wh espec o he cue dscou aes R,,,. No classcal esul uses such assumpos egadg he coeffces of he equao. Coolla. The eld of pofolo P wh dscou aes R,,, ad cash flows 0,,,, a he mes 0 0,,,,, a abage fee make M s he weghed aveage of he dscou aes. The weghs of he dscou aes ae equal o w Remak: I Babbel ad Mell 996, o pages 4, has bee pove ha YTM ca be appoxmaed as a weghed aveage of he spo aes ad ha he weghs appled o spo aes ae he dolla duaos of he especve zeos elave o he dolla duao of he ee pofolo. I he poof of ha saeme he pofolo fuco P defed b 3 Babbel ad Mell 996 was cosdeed as a fuco of oe vaable ʺʺ eld ad s P devave wh espec o was deoed b. lso, boh zeo coupo bods C ad Cm wee defed b he elao 4 as fucos of oe vaable,, ad Talo fomula fo fucos of oe vaable was used aoud he eld fo boh fucos C ad Cm. The fomula 8 was pove fo pofolo of ol wo zeo coupo bods. s C s m P C m 8 9

10 If we expad ha fomula o zeos ad f we use ou oao, ha he fomula 8 becomes w w s, whee weghs ae gve as aos of dolla duaos b fomulae, ad S s, s,, s s he se of he spo aes of he pofolo of sk fee fxed come secues. Ths meas ha ou fomula 0 s a efg of he fomula 8 fom Babbel ad Mell 996 because ou weghs ae calculaed b meas of he exac dscou ae each me po, whle weghs of 8 ae all calculaed a he eld ae ha has e o be deemed. The ohe advaage of fomula 0 ove 8 s ha he eld s o oe sde of 0, lef sde, ad all dscou aes ae o he gh sde, whle he fomula 8 has eld o boh sdes. Ou muldmesoal appoach, ad Talo developme aoud he po R,,,, o aoud, eabled ha sepaao whch he eabled us o ge a secod ode appoxmao of eld, ou fomulae 7, ems of he dscou aes. If we waed o use fomula 8 o esmae eld, we would have aga o use a eave mehod, because s o boh sdes of 8 ad cao be solved explcl ems of he spo aes s, whle he fomula 0 s a explc expesso of eld ems of he dscou aes of he pofolo.. Secod Ode ppoxmao If, sead of 6, we use he mulvaable Talo fomula of he secod ode, we wll ge a bee esmae of eld ha 0. I ode o smplf fuhe devaos, le us oduce he followg oao fo fs ad secod devaves of he fucos : D D,,,,,. 0

11 The secod ode Talo fomula of he mulvaable fuco X he eghbohood of R ca be we as x D x D X Fom, ad 3 we have: Y 3 Subsug Y fo X, ad applg 3 we ge 0 D D 4 fe some devaos we ge: 0 D D D D D 5 Le us oduce oao fo coeffces of he equao 5: D E 6.a D D F 6.b D D G, 6.c

12 If F 4EG 0, he he fomula F ± F 4EG, 7 E gves a bee esmae of eld ha he lea appoxmao 0. We have o pck up he soluo of 7 whch s close o he dscou aes, because he devaos of he fomulae wee doe ude he assumpo ha ha Y,,, s he eghbohood of R,,,. Example 4 coued: The fomula 0 gves 0.07* 7 * 0.07* *8 * 0.08*8 3* 0.08* *07 8 *8 3* I s eas o vef ha he pese value fuco akes value P whch s ve close o zeo. Ths meas ha eve he lea appoxmao of eld s ve good. The fomula 7 gves , ad P , paccall zeo. Ths calculao s doe b a speadshee, see he esuls below. If we appoxmae eld of he pofolo b he weghed aveage of elds 0.07, 0.08, whee weghs ae values of bods; , ad B 00, he we have vg , ad P , whch s a wose esul eve ha ou lea appoxmao 0. Mau Spo Cash Tmes Raes Flows Yeas s * *D *D*S *D *D *S *D *S Sums , Fomula P Fomula 7 E 75 F 548 G P P 8.33 vg Pvg

13 The soluo of 7 does o eed o be eld because s o he eghbohood of he dscou aes 0.07, ad 0.08, ad s o eld, as he polomal akes value P , o close o zeo. 3. Numecal Examples Example coued: Le us assume some ealsc spo aes he make M, e.g., s0.033%, s % ad as befoe 30, 3, he Ths meas ha he asaco fom he Example s o allowed he abage fee make M whch s0.03, ad s The ol allowed asaco wh gve cash flows 30, 3 s descbed b v 3v 0, ahe ha b00 30v 3v 0. The coespodg equao ems of eld became , ahe ha The fomula 0 gves he uque esmae fo eld. 30 * * * , 30 3 * whch s a lea esmae. The fomula 7 gves much bee esmae The secod soluo of 7, s o a soluo because does o sasf he codo o be close o he dscou aes R,. The equao ha descbes he allowed asaco, , has wo exac soluos, ad he ae , The fomula 7 cel appoxmaed, he soluo ha s close o spo aes of he make M. Example 3 coued: I ode o be able o use ou fomulae fo he cash flow, 00, 0, he dscou aes eed o be ve close as assumed befoe he poof of he fomulae, fo example 0.06, Le us show wha happes hs case

14 Ths meas ha he asaco fom Example 3 s o legmae he abage fee make M wh dscou aes 0.06, 0.06 a duaos of ea, ad eas, especvel. The ol allowed asaco wh gve cash flows 00, 0 s descbed b v 0v 0, ahe ha b00 00v 0v 0. The equao ems of eld becomes , ae ha The equao 7 gves he eld The ohe soluo fom eed o o be eld because s fa fom he dscou aes 0.06, The polomal akes value P , whch s pe close o zeo, eve hough hs example epeses a exeme suao. Because hee s NO eal soluo of he equao wh such cash flows, evebod would have o adjus some of he coeffces of he equao ode o ge some esmae of he eld. The queso ases: How o adjus he coeffces ode o ge some esmae of eld such case? The fomulae of hs pape offe oe aswe. The esmae he eld whou a adjusmes of he cash flows,,,, whch meas ha ol he coeffce 0 has o be mplcl adjused. I hs example, as well as Example, he assumpo ha he dscou aes have o be suffcel close plas a bg ole. If we ae o happ wh he esmae, ha s because he dscou aes, ae o close eough. The esmaes would be bee f he dscou aes wee close. Coolla of fomulae 0 ad 7. I case of a asaco ha does o have a eal umbe soluo fo eld we mgh be able o fd a esmae of eld fo he pofolo ha has he same cash flows,,,. The esmae of eld wll deped o he cue dscou aes,,,. The he cash flow 0 s adjused o be he pese value of cash flows,,,, dscoued b,,,, especvel. The fal esul s he legmae asaco, a appoxmao of he ogal asaco, wh he eld ae whch s appoxmaed b 0 ad 7. Example 5: Calculae eld of he pofolo ha cosss of 0 zeo coupo bods, wh Mau Tmes, T 0.5; ;.5; ; 4; 6; 7; 8; 9; 0, 4

15 Dscou Raes, R 0.03; 0.035; 0.035; ; 0.04; 0.045; 0.045; ; 0.05; 0.055, Cash Flows, 4000;,000,000; 50,000; 400,000; 700,000; 85,000;,000,000; 00,000; 00,000;,000,000 Vecos T, R, ad ae gve colums, 3,ad 4, especvel, of he followg able: Mau Spo Cash Tmes Raes Flows Yeas * *D *D* *D *D * *D * 0 0 0,30, , ,94,93 57, ,000, ,53 938,037 30,486,87,00 59,053, , ,47 344,097,043 83,5 9,090, , ,607 76,35 6,863,07,375 77,677, , ,363,30,396 9,056,064,403 44,576 7, , ,6 38,099 6,97,558,938 08,755 4, ,000, ,88 4,9,96,503 37,68,648,695,79 76, , ,987 56,870 5,06 4,56,809 5,03 0, , ,46 55,5 7,66 5,6,4 63,06 3, ,000, ,43 5,549,05 305,0 57,857,967 3,8,88 75,00 Sums,30,7 6,389,094 34,053 48,309,96,68,833 50,60 Fomula Fomula 7 E 4,54,958 F 9,070,96 G 389, Ths was a example of a osmple asaco, as defed Techoew, e al. 965 ad a asaco wh a mddle egave flow as defed Jea 968; o a eas asaco o deeme eld accodg o Techoew, e al. 965 ad Jea 968. The fomula 0 gves , whle fomula 7 gves bee esmae, Noe ha we have facoal mes could be a eal umbes ad ha hee ae hee chages of he sgs he sequece of cash flows. The secod soluo of s o eld. I s eas o vef ha does o sasf equao of hs example. We dd o expec ha o be he case, because s oo fa fom dscou aes of he make hs example. 5

16 4. Cocluso Yeld ae s mpoa o vesos fo assessg asacos. I hs pape we esmae he eld ae b meas of quaes ha deped o he make he cue ees dscou aes of he zeos ha compse he pofolo, as opposed o calculag eld aes solel b meas of he cash flows, depedel of a pacula evome. We deal wh ees aes o calculae eld, ahe ha dealg ol wh he coeffces of he equao. We lm ouselves o he pofolo P, whch cosss of fxed come secues ha ae o eessesve a abage fee make M. aual assumpo fo he dscou aes of he pofolo s ulzed; ha he ae suffcel close o each ohe. Ude hese assumpos we wee able o pove wo explc esmaes fo eld ems of dscou aes he pofolo; fomula 0 s a lea appoxmao of eld, ad fomula 7 uses secod devaves covex as well. The appoxmaos 0 ad 7 appoxmae he eld ha s closes o he dscou aes of he pofolo P. 6

17 Refeeces Babbel, D.F., ad Mell, C.B Valuao of Iees Sesve Facal Isumes. SO Moogaph M FI96. Becke, D.N geealzed pofs eleased model fo he measueme of eu o vesme fo lfe suace. Tasacos of he Soce of cuaes 40: Fabozz, F.J., ed Hadbook of Fxed Icome Secues. Sxh Edo. Columbus, OH: Iw/McGaw Hll. Jea, W.H O mulple aes of eu. Joual of Face 3: Kellso, S.G. 99. Theo of Iees. Secod Edo. Chcago: Rchad D. Iw, Ic. Luebege, D.G Ivesme Scece. New Yok: Oxfod Uves Pess. Paje, H.H., e al Facal Ecoomcs: Wh pplcaos o Ivesmes, Isuace ad Pesos. Schaumbug, IL: cuaal Foudao. Pomslow, S.D ew appoach o he heo of ees. Tasacos of he Soce of cuaes 3: Techoew, D., Robcheck,.., ad Moalbao, M aalss of cea fo vesme ad facg decsos ude cea. Maageme Scece Novembe: