hte 9 Bod d Note Vlutio d Relted Iteest Rte Fouls witte fo Ecooics 04 Ficil Ecooics by Pofesso Gy R. Evs Fist editio 2008, this editio Octobe 28, 203 Gy R. Evs The iy uose of this docuet is to show d justify vlutio fouls fo bills, bods d otes. The objective hee is to udestd how ices d/o yields (ech deteied by the othe) e deteied i the vst secody ket fo these secuities. The ehsis i this docuet is o the icig of U.S. Tesuy secuities but the geel fouls show ly to equivlet coecil bills, otes, d bods. I. Suleety d Relevt Ifotio () Most bods d otes (heefte I will use the te bods to efe to otes d bods), icludig ll issued by the U.S. Tesuy, y couo iteest eithe qutely o seithe th ully. U.S. Tesuy bods y iteest sei-ully. (2) Bills (with tuities of oe ye o less by defiitio) e discouted (sold t less th vlue) d e the edeeed t vlue. This is exlied i oe detil i Sectio II below. (3) Bods c be sold i secody ket, just like stocks. Becuse thei couo iteest te is fixed d y ot eflect the yields of equivlet secuities t the tie they e esold, they will be esold eithe t discout (below ) o eiu (bove ) so tht thei effective yields will eflect cuet ket tes t the tie they e esold. Geelly, if iteest tes hve ise sice the tie of issue, the bod will sell t discout. If iteest tes hve flle the bod will sell t eiu. The eso fo this will be show below. (3) Whe bods with t lest oe couo yet eiig e esold i the secody ket, iteest ccued to the evious owe sice the lst couo yet is dded to the fil ice of the secuity whe it is sold. Whe you buy the bod o the secody ket, if you buy betwee couo iteest yet dtes, which will lost lwys be the cse, you will y () the sk ice of the bod d (b) ccued iteest sice the lst couo yet. These two vlues will be sued d eflected i the fil ice of the bod. It is iott to udestd tht o bod ice quottios, oly the fist t, the sk ice, is icluded. You, the bod buye, will the e the etie couo yet o the bod whe it is id, eve if you uchsed the bod the dy befoe the yet dte. A detiled exle of this colicted oit will be ovided i sectio V. below. II. Discoutig Bills By defiitio, bills, such s the U.S. Tesuy Bills tht tue i 4, 3, o 26 weeks, tue i less th oe ye fo the dte of issue. Theefoe they do ot y couo iteest, which is to
Bod Vlutio - Pge 2 sy tht o sete iteest yet is de to the ivesto. Isted they e sold t discout, ice tht is less th, thei edetio vlue. Iteest eed fo bill is ilicit i the citl gi elized by buyig the bill t discout d edeeig it t. Although bills (d bods) e sold i outs tht e ultiles of $,000, thei ices e quoted i uits of 00, which is clled the vlue. Fo exle, bill with edetio vlue of $0,000 cuetly quoted t 98.72 (discout) hs cuet vlue of $9,872. Whe the bill is edeeed it will be edeeed t fo $0,000. Theefoe, lthough o iteest yet ws eve de fo this bill, iteest is ilicit i the citl gi elized. The foul fo deteiig the ulized discout yield of bill whe the discouted ice is kow is show hee:. 00 ice 52 yield ice weeks to tuity Usig the exle bove d ssuig the bill to be ewly-issued U.S. Tesuy Bill with ice of 99.02 d 3 week tuity, the discout yield will equl 00 99.02 52 2. yield 0. 03959 99.02 3 The foul fo deteiig the ice of bill with y give ube of weeks to tuity (wt) whe the discout yield is kow (d cofiig equtio 2) is 3. ice 00 wt yield 52 00 99.02 3 0.03959 52 The tie-djustet coefficiet i the fouls bove ssue tht the bill i questio hs dutio exessed i weeks. If the dutio is exessed i oths, the the coefficiet is equl to the ube of oths dutio divided by 2 d if exessed i dys, the coefficiet is equl to the ube of dys dutio divided by 365. Fo exle, bill tuig i 24 dys iced t 99.745 would be hve yield of 00 99.745 365 4. yield 0. 0389 99.745 24
Bod Vlutio - Pge 3 III. ooudig Bsics I ode to udestd the colicted bod vlutio foul, we will build the exltio i stes. We will stt by eviewig the bsic fouls fo clcultig cooud iteest. This will be esy to do i questio d swe fot. Q: If I ivest $0 t ul te of 8% fo oe ye, wht will y ivestet be woth t the ed of the ye: A: $ 0.08 $0. 80 Q: Wht if I leve y iteest eed i the ccout d let it ccue iteest fo two oe yes? Wht will y ivestet be woth t the ed of the thid ye? 3 A: $0.08 $2. 60 Fo this we geelize iteest-cooudig foul, ssuig costt iteest te ove ube of yes d ssuig the ccued iteest (iteest eed ove the yes) is left i the ccout. 5. f f = futue vlue of ivestet, = eset vlue of ivestet, = the ul iteest te, d = ube of yes of the ivestet. IV. Peset Vlue Bsics Now let us chge the oiettio soewht. I the seies bove, we wted to kow the futue vlues of ivestets lced tody. Suose we ow wt to isted kow the eset vlue of soe kow csh yet tht will be id i the futue. Suose, fo exle, tht soeoe hs give you cotct oisig to y exctly $0,000 i 0 yes. Wht is tht cotct woth tody? Q: Is t tht cotct woth $0,000? A: No, clely ot. If you wee give the otio of ccetig $0,000 tody vesus $0,000 i 0 yes, would you ccet the foe o the ltte? lely you would ccet the foe, if fo o othe eso th the fct tht you could ccet $0,000 tody d ivest
Bod Vlutio - Pge 4 it t 8% (o the evilig iteest te) fo 0 yes d, usig foul (5) bove, ed u with $2,590. So $0,000 to be id 0 yes i the futue is woth lot less th $0,000 to be id tody. Q: So how would I clculte the eset vlue of $0,000 to be id 0 yes fo ow? A: Give the evilig ket iteest te o 0 ye ivestets (usully this would be estite - let s ssue 8% fo this exle) we hve to sk elted questio: If I c exect to e 8% e ye ove the ext 0 yes d I wt to hve $0,000 t the ed of tht eiod, how uch ust I ivest tody? Wht iitil ivestet eig 8% coouded ully will leve e with $0,000 fte 0 yes? To swe this, ll we hve to do is look bck u to equtio (5) d see tht we c odify tht equtio solvig fo the eset vlue of usig the futue vlue f ove yes to get the swe: 6. f 0,000.08 0 $4,63.93 This es tht $4,63.93 ivested t 8% coouded ove 0 yes will be woth exctly $0,000 t the ed of the te yes, kig the vlue ficilly equl, so cotct offeig to y $0,000 i 0 yes would be vlued tody t $4,63.94 give ou ssutios bout iteest tes. Q: Plese clify tht lst oit. Why is the iteest te used i the clcultio beig ssued o estited? A: I soe cses the iteest te would be kow, but becuse iteest tes c chge ove tie, becuse soe ivestet ltetives e vilble tht offe diffeet tes (e.g. ivestig i Tesuy bills vs. deositig i bk), d the oble ojects ito the futue, soeties iteest-te estite ust be used. I ost cses, the evilig ket te o siil o ideticl ltetive ivestets would be used. V. A Exle of Peset Vlue Alictios - The lifoi Lottey A good exle of the eset-vlue lictio to ficil flows ove the futue c be ovided by the lifoi lottey. Wies of the lifoi lottey e ot id oe lu su whe they wi. They e id i 20 equl ul yets, the fist ight fte wiig the lottey d the lst 9 yes lte. Fo exle, if you wi $0 illio, you e id $500 thousd ight wy d the se out ech ye fo the ext 9 yes. Fo the discussio just cocluded, it is cle tht the out wo is ot elly woth $0
Bod Vlutio - Pge 5 illio becuse so uch of it is id i the futue d the lottey-wie gets o iteest. How uch, the, is it woth? Wht, i othe wods, is the eset vlue of this lottey wi? At fist glce this is clely oe difficult oble th the oble eeseted i equtio (6), which ivolved the eset vlue of oly oe yet. I this oble thee e 20 yets. The solutio hee becoes esy to gs s soo s we elize tht whe yet ste icludes oe th sigle yet, the eset vlue of those yets will be equl to the sued eset vlue of ech idividul yet. I the cse of ou lottey exle, the eset vlue of wi is equl to the eset vlue of ech of the idividul 20 ul yets, sued: 7. 500k 500k 500k Vlue 500k 500k 0 9 2 8.08.08.08.08.08 $5,30,800 Exessig the se i sutio ottio: 8. Vlue 9 500k t0 t $5,30,800 I few wods, the tue vlue of wiig $20 illio i the lifoi lottey, if we c ssue the evilig iteest te to be 8%, is $5,30,800. VI. Eleety Bod Vlutio - Sttig Poit Evlutig bod i y esects is ot uch diffeet th givig vlue to wiig the lifoi lottey. I the eyes of the fice kets ote o bod, whethe ewly issued o beig esold o the secody kets, is vey little oe th futue csh-yet ste. Fo exle, ewly-issued 30-ye bod yig iteest oce e ye is oise to y () 30 equl ul iteest yets ove 30 yes, d (b) the bod s vlue (00) i exctly 30 yes. This is uch like the lifoi lottey yet, excet the fist yet is de t the ed of the ye isted of ieditely, d i the cse of the bod, the vlue of the bod is edeeed t the ed of the 30 yes, esultig i oe lge fil yet with vlue equl to. I is iott to udestd tht i the lictio below, we e usig the foul to evlute the vlue of bod o secody ket fo bod tht ws issued i the st t tie whe iteest tes wee highe o lowe. Suose we e icig the hyotheticl bod discussed bove - 30-ye bod yig iteest oly oce ye. Assue, s lwys, tht the bod hs vlue of 00 d couo te of
Bod Vlutio - Pge 6 8% e ye. This es, of couse, tht the bod ys $8 iteest e ye (e 00). Suose dditiolly, tht 0 yes hve elsed sice the oigil issue of the bod, levig oly 20 yes to tuity (exctly). Filly, suose tht ket iteest yields o equivlet 20-ye bods e ow t 0%. Wht will be the ket vlue of this bod? Q: Befoe you swe, let e ke sue I hve this stight. The bod is beig esold with 20 yes eiig to tuity, so it ust coete with othe ficil ssets tht lso hve te yes of life eiig, icludig ew 20-ye otes. Is tht coect? A: Yes. Theefoe the effective yield ust be coetitive with these othe ficil ssets. Sice the couo te, 8%, is fixed fo the life of the bod, this bod ust sell t discout ( ice below ) to effectively ise its yield to 0% fo the uchse. Ad gi, wht is beig vlued is cshflow ste cosistig of 20 eiig iteest yets d the fil edetio vlue of the bod (00). Hee is the eleety bod foul tht will llows us to clculte the vlue of this bod: 9. 2 P 3 = eset ket vlue (sked ice). = the couo yet (out) equl to the couo te ties. = evilig iteest te o equivlet secuities. = ube of yes to tuity. Exessig the se foul i sutio ottio: 0. P i i Filly, to solve fo ou hyotheticl exle:. 20 8 i i.0.0 00 82. 97 20 Becuse iteest tes e highe th t the tie this bod ws fist issues, the bod is tdig t discout fo the ice of 82.97. This es tht if this bod ws oigilly issued i the deoitio of $0,000, it is ow woth $8,297. Redes with good th bckgoud will ecogize tht fouls (9) d (0) e geoetic
Bod Vlutio - Pge 7 seies. Theefoe the eleety bod foul c be educed to sile foul tht is esie to clculte o clculto o i coute og: 2. P The hyotheticl oble used i this sectio would be solved s 20.0 3. 8 00 82. 97 20 0.0.0 The deivtio of this foul is show i the Aedix A. Note tht the vlues fo equtios () d (3) gee, s they should. VII. Eleety Bod Vlutio - Peiodic Iteest Pyets The foul bove is just begiig oxitio. It does ot tke ito ccout the fcts tht bods usully y iteest eithe twice e ye, qutely o othly. Fo exle U.S. Tesuy Bods y iteest twice e ye. The foul fo eiodic yets is slight odifictio of foul (9) bove: P 4. 3 2 whee vibles hve the se vlues s bove excet = the ube of ties e ye tht iteest is id. = ube of eiig couo yets the th ube of eiig yes. Note tht still equls the ul vlue of the couo yet (the couo te ties ) d tht the fil te is uchged fo foul (9), give tht is equl to ties the ube of yes. The sutio ottio vesio of this equtio eeds oly sll odifictio of equtio (0):
Bod Vlutio - Pge 8 5. i i P Usig the se exle of 30-ye bod with couo te of 8%, but this tie id seiully isted of ully, with 20 yes eiig i its life t tie whe equivlet bods e yig 0% (the se ssutios, excet fo sei-ul iteest yets, used to clculte equtio () bove): 40 8 00 6. 2 83. 50 40 20.05.0 i Filly, the educed-fo equtio fo this geoetic seies is slight odifictio of equtio (2): 7. P Usig the se vlues used i equtio (6) lied to equtio (7) yields 40.05 8. 8 00 83. 50 2 20 0.05.0 Note tht the solutio vlues e the se fo equtios (6) d (8). Agi, the deivtio of this educed-fo equtio is show i the Aedix A. As exected, the swes obtied fo equtios () d (3) whe coed to equtios (6) d (8) e siil but slightly diffeet becuse iteest is beig coouded twice e ye i the ltte exle, kig tht bod woth oe. VIII. The olete Bod Vlutio Foul The fil foul fo bod vlutio fo sles o the secody ket tkes ito ccout the fct tht you seldo buy o sell bod o the exct dy couo yet is de. Usully the
Bod Vlutio - Pge 9 tsctio is de o dy i betwee couo yets. This oses two obles: () the eset vlue clcultio ust be ble to tke ito ccout fctios of couo yet eiods, d (2) sice the egisteed bod owe o the dy of couo yet lwys eceives the full iteest yet fo the evious couo eiod, soe echis ust be i lce to tsfe ceti otio of iteest ccued sice the lst couo yet to the evious owe. It is the ltte coditio tht colictes the foul the ost. As stted i the oeig sectio o I. Suleety d Relevt Ifotio the yet fo bod fte uchse will cosist of two ts: () the ket vlue (the sk ice) of the bod, the vlue of which we hve bee clcultig ll log, d (2) ccued iteest to the evious owe sice the lst couo yet. The ltte will be yet to the io bod owe fo ccued iteest sice the lst couo yet. Most iott to udestd, the bod ice quottios, such s those tht e i the Wll Steet Joul, iclude oly the fist t, the sk ice (o the bid ice, fo those sellig the bod - the quoted yield is lwys clculted fo the sk ice, d eithe icludes the ccued iteest). If you buy bod o the secody ket, you will y the quoted sk ice lus ccued iteest. Q: This is cofusig. I eed exle. A: OK. Suose you buy the bod beig used fo ou exle o Febuy 8, 2009. Suose lso tht the bod kes its couo yets o Mch 5 d Setebe 5 of evey ye d the bod tues o Setebe 5, 2029. Q: So I' i betwee couo yets. Thee is little oe th oth befoe the ext. Is tht you oit? A: Yes. O settleet dte of Febuy 8, it will bee 46 dys sice the lst couo yet. The ext will be i oly 36 dys. Yet o tht dy you will eceive the full couo yet fo hlf ye. Q: So techiclly I owe the evious owe 46 dys of ccued iteest, which will be icluded i the fil ice. Is thee foul fo clcultig the ccued iteest? A: Yes, d tht foul will be icluded i the fil foul fo bod vlutio. Hee is the foul fo ccued iteest usig ou exle: 46 9. AI 4 $3. 2 2 82 whee AI = ccued iteest
= the ul couo yet (couo te ties ) = ccued dys, o the ube of dys sice the lst couo yet = the ube of dys i the full couo eiod. Fo exle, ssuig tht the bod hd 8% couo, Bod Vlutio - Pge 0 At this oit it is obbly best to show the colete bod vlutio foul, the exli it. Reebeig tht this is the foul fo clcultig sk ice, hee it is: 20. whee i i P 365 = ket vlue, the quoted sk ice of the bod, = the ul couo yet, equl to the couo te ties, P = 00 = the evilig ul ket yield, exessed s sk yield o yield-to-tuity, = the ube of couo yets e ye, = the ube of eiig couo yets, = the ube of dys i this couo eiod (betwee 8 d 84, use 82 if ukow), = the ube of dys betwee the lst couo yet d the settleet dy. The fist te eesets the eset vlue of ll eiig couo yets, the secod te eesets the eset vlue of the edetio t the tie of tuity, d the thid te, which is subtcted fo the fist two, eesets the ccued iteest tht is owed to the io owe. Also otice tht the exoets i the fist two tes e o loge whole ubes, but fctios eesetig ccutely the ube of dys to ech couo yet d tuity. Let us ly this foul to ou hyotheticl bod discussed bove. Hee is eide of the ssutios de: Puchse dte: Febuy 8, 2009 (46 dys sice lst couo yet, 36 to ext), Next couo dte: Mch 5, 2009 (the fist of 42 couos eiig), Redetio dte: Setebe 5, 2029 (fo d lst couo), ouo te/out: 8% yieldig $4 e couo yet, Peset ket te (sk yield): 0%. Kowig tht the fctio
Bod Vlutio - Pge 46 82 0.8022 hee is the clcultio fo the ket vlue (sk ice) which would be quoted: 2. Silifyig this we get 22. 42 8 00 8 0. 8022 i0.8022 20.50.099 2 i.0.0 2 2 00 42 4 0 20. 60 i i.8022.05.0 3.2 83. 3 This bod would be quoted with sk ice of $83.3. Accued iteest would equl $3.2. To buy this bod would cost $86.52 e 00 (), the su of equtios (22) d (9). To buy $0,000 "woth" of this bod would cost $8,652 of which $32 is ccued iteest. The couo-yet cooet is, oce gi, geoetic seies tht c be educed to uch sile equtio tht is esie to solve with clculto, i Excel, o with coute og. Hee is the educed-fo equtio: 23. / P 365 Usig the se vlues tht wee used fo the geoetic seies vesio of this i equtio (2), whe lied to the educed-fo equtio (23),
Bod Vlutio - Pge 2 24. 42.05 4 00 20.5 0. 099 0.05.0 0.8022.05 4 0.8022 83. 3 Deivtio of equtio (23) is show i Aedix B. Q: this foul be used to solve fo the ket yield () the th the ket ice? Suose I wted to y 86 fo this bod. ould I use this foul to clculte the effective ket yield? A: Yes d o. You c't diectly use this foul to solve fo (too y oots!) but you c use this foul d coo itetive techique (equiig coute og o dedicted clculto chi) to covege to solutio. Ay busiess clculto, Excel, o coute og tht c solve fo equtio (20) o (23) bove c lso, give ket vlue, solve fo s ukow. APPENDI A: Reduced-fo Vesio of the Eleety Bod Foul Afte itoducig equtios (9) d (0) i the i text, both wee idetified s geoetic seies tht c be educed to sile equtio, which ws itoduced t equtio (2). Show below e coies of equtios (9) d (2). 9. P 3 2 2. P The uose of this edix is to show how equtio (9) c be coveted to equtio (2). If we let eeset the vlue of the couo yet d V eeset the eset discouted vlue of the couo iteest yets oly d substitute fo ( + ) the the fist hlf of the bod vlue foul c be eeseted s
Bod Vlutio - Pge 3 A. This c be coveted to stdd geoetic seies equtio by ultilyig both sides ties. This will esult i A2. A stdd geoetic seies of the fo bove c be educed to oe eleety equtio by ultilyig both sides of the equtio ties ( ) which esults i A3. To ke this oe elegt ultily both sides of the equtio ties A4. Solvig fo esults i A5. Substitutig ( + ) bck ito equtio (5) fo yields the fot t of the educed fo equtio s show below V 2 V V V V 2
Bod Vlutio - Pge 4 A6. Reebeig tht is equl to the couo te ties, if the foul fo discoutig the eset vlue of the bod t edetio is dded to equtio (A6), the educed fo equtio is ideticl to equtio (2) d is colete. APPENDI B: Reduced-fo Vesio of the olex Bod Foul Equtio (20), eoduced below, icludes cooet, the eset vlue clcultio of the couo iteest yets, tht c be educed to sile equtio (23), lso eoduced below. 20. 23. As i Aedix A, defie V to eeset the eset discouted vlue of the couo iteest yets oly, let equl ( + /), let equl /, d let equl V/. Give these ssutios, the eset vlue of the couo iteest yet fo equtio (20) c be witte B. 3 2 Multily both sides of the equtio ties, which gives us ou stdd geoetic seies, V P 365 / P i i 365
Bod Vlutio - Pge 5 B2. 2 To silify, ultily both sides of the equtio ties ( - ) B3. To futhe silify, ultily both sides of the equtio ties (the lst two stes could hve bee doe i oe ste, but it is esie to see how we e gettig to ou esults by doig it i two stes) B4. At this oit it is useful to substitute fo d V to see whee this is goig B5. V Regig d solvig fo V (the eset discouted vlue of the couo yets loe) yields B6. V / If we ow dd the eset discouted vlue of the edetio vlue d the subtctio fo ccued iteest, equtio B6 becoes equtio (23) bove, which, desite its visul colexity, is esy to og ito Excel, coute og, o clculto. The elegce of equtio B6 is esie to see if we ssue oly oe couo iteest yet e
Bod Vlutio - Pge 6 ye, which ilies tht = d tht is equl to the ube of eiig yes i the life of the bod. Equtio B6 becoes B7. V / Gy R. Evs, 2008-203. This teil y be used without eissio of the utho fo eductiol uoses oly. The use of this teil fo coecil uoses without eissio of the utho is ohibited.