Yield to Maturity Is Always Received as Promised

Transcription

1 JOURAL OF ECOOMICS AD FIACE EDUCATIO Volume 7 umber Summer Yield o Mauriy Is Always Received as Promised Richard J. Cebula and Bill Z. Yang 2 ABSTRACT This noe commens on a misconcepion ha yield o mauriy from holding a coupon bond unil mauriy is only promised, bu no really received, unless coupon paymens are reinvesed a he same rae as he (original) yield o mauriy. I shows ha yield o mauriy is always earned no maer how coupon paymens are allocaed spen or reinvesed a any rae. I illuminaes ha he realized compounding yield in fac measures he yield o mauriy from a combinaion of wo invesmens raher from simply holding he bond iself unil mauriy. Inroducion Yield o mauriy (YTM hereafer) is he sandard measure of he oal rae of reurn of he bond over is life. This ineres rae is ofen viewed as a measure of he average rae of reurn ha will be earned on a bond if i is bough now and held unil mauriy (Bodie, e al, 2002, p. 426). And i is considered he mos accurae measure of ineres rae (Mishkin, 2004, p. 64). Unforunaely, due o a fac ha yield o mauriy will equal he rae of reurn realized over he life of he bond if all coupons are reinvesed a an ineres rae equal o he bond s yield o mauriy (Bodie, e al, 2002, p. 429), YTM has been widely misinerpreed as he rue rae of reurn an invesor would received by holding he securiy unil is mauriy if each ineres paymen is reinvesed a he yield o mauriy (Srong, 2004, p.70, ialic original). Similar inerpreaions can be also found in, o name a few, Reilly and Brown (997, pp ), Madura (998, p. 27), and Fabozzi and Modigliani (2002, p. 364). This noe poins ou ha he above-menioned common reamen in many exbooks urns ou o be a fallacy. The ruh is ha YTM on a (coupon) bond is always received regardless of how coupon paymens are re-invesed, provided ha he bond is held unil mauriy wihou defaul. I addresses a basic quesion in bond heory: beween YTM and realized compounding yield (RCY hereafer), which concep measures he rue rae of reurn from holding a coupon bond unil mauriy? I is well acceped ha YTM measures he rae of reurn from holding a bond unil mauriy for boh coupon bond and zero-coupon bond as well. By definiion, he YTM received from holding a bond is independen of how coupon paymens are allocaed, as long as hey are paid on ime as conraced. By comparing he iniial invesmen and he final value accumulaed over he invesmen horizon, on he oher hand, RCY on a bond measures he rae of reurn from an accoun (or rus) ha holds he bond and he ineress paid. Of course, i depends on how coupon paymens are reinvesed. We demonsrae ha he RCY acually measures he YTM from a combined invesmen - holding a coupon bond plus an addiional periodic invesmen wih each coupon paymen received. o surprisingly, YTM and RCY would be normally unequal; RCY equals YTM if and only if coupon paymens are reinvesed a he same rae as he iniial YTM. However, his conclusion should no be inerpreed as he yield o mauriy is acually received only if coupon paymens are reinvesed a he yield o mauriy. Yield o Mauriy vs. Realized Compounding Yield Richard J. Cebula, Shirley and Philip Solomons Eminen Scholar Chair and Professor of Economics, Armsrong Alanic Sae Universiy, Savannah, GA 349, Richard.cebula@armsrong.edu 2 Bill Z. Yang, Associae Professor of Economics, Georgia Souhern Universiy, Saesboro, GA , billyang@georgiasouhern.edu

2 JOURAL OF ECOOMICS AD FIACE EDUCATIO Volume 7 umber Summer Yield o mauriy (YTM) of a coupon bond is defined as he soluion for variable y from he following equaion C ( y) F y = ( ), () where P is he purchase price of he bond, C is he periodical coupon paymen, F is he face value and is he erm o mauriy. The YTM measures he heoreic annual rae of reurn from his invesmen, provided he invesor holds i unil mauriy and receives C per period as well as F a mauriy as conraced. Tha is, he YTM is compleely deermined by he cash flows paid and received by he invesor over he invesmen horizon. By definiion, nohing has been assumed regarding how coupon paymens are allocaed reinvesed a a specific rae or simply spen when received. The only implici assumpion on coupon paymens (and he par value) is ha hey are received on ime as promised, i.e., no defaul. Why do so many auhors emphasize ha he YTM is acually received only if he coupon paymens are reinvesed a he same rae as YTM? I sems from misinerpreing anoher measure of (annual) rae of reurn realized compound yield (RCY), which is formally defined as follows: RCY = - V (2) P where funds iniially invesed (or iniial purchase price), V = curren value accumulaed from he invesmen a he end of period. oe ha RCY is deermined exclusively by he iniial invesmen and he final value accumulaed from he invesmen wihou specifying he cash (in- or ou-) flows on he invesmen during he invesmen horizon. Solving for P, we can rewrie (2) as V ( RCY ). (3) I implies immediaely from equaion (3) ha if an invesor holds o mauriy a zero-coupon bond ha pays cash in-flows only a mauriy, obviously, RCY = YTM. Proposiion. For a zero-coupon bond, RCY = YTM. If an invesor holds a coupon bond ha pays cash in-flows periodically unil mauriy, he value accumulaed from all cash in-flows a he end of invesmen horizon, V, depends on wheher coupon paymens are spen or reinvesed, and a wha rae if reinvesed. 3 Hence, RCY may or may no equal he YTM as calculaed a he ime of purchase. To illuminae how he saemen is incorrecly reached ha YTM is acually received only if coupon paymens are reinvesed a YTM, we firs show how RCY is linked wih YTM from holding a coupon bond. For he purpose of exposiion, we examine hree differen invesmens: ) buying and holding a coupon bond unil mauriy; 2) (re)invesing every coupon paymen whenever received; 3) holding a porfolio ha combines hese wo invesmens. Coupon bond -P C C C C CF When a coupon paymen is received and simply spen raher han reinvesed, i can be inerpreed as being reinvesed a a rae of - 00%.

3 JOURAL OF ECOOMICS AD FIACE EDUCATIO Volume 7 umber Summer Reinvesing Coupon paymens ( y ) C -C -C -C -C r = Combinaion of wo invesmens = 0 -P ( y ) C F r Figure : Time lines of hree differen invesmens Figure above describes he ime lines wih cash flows of each of he hree invesmens. Formally, le YTM denoe he YTM on holding he coupon bond unil mauriy. By definiion, YTM is deermined in he following equaion C F. (') = ( YTM ) ( YTM) Clearly, YTM is enirely deermined by parameers P, C, F and, and independen of how C s are allocaed simply spen or reinvesed a any rae. To inves an amoun equal o he coupon paymen periodically when hey are received a a rae of y r is anoher invesmen. The YTM from such a separae invesmen, denoed as YTM 2, is deermined by is cash ou-flows and in-flows as follows 0 = = Rewriing (4) as C ( YTM ) 2 ( YTM ) 2 = C( y ). (4) r = ( ) C YTM = 2 = C ( ), (4 ) y r we can obain ha ha YTM 2 = y r. 4 Holding he coupon bond unil mauriy and reinvesing all coupon paymens a a rae of y r when received, as a maer of fac, combines he above wo invesmens. Le YTM 2 denoe he YTM on his combined invesmen. From he ime lines in Figure, i implies ha i is deermined in he following equaion (5): 0 ( yr ) C F =. (5) ( YTM ) 2 4 For he uniqueness of soluion o such an equaion, see, for example, Theorem 6.2(d) on Descares Rule of sign, in Henrici (974, p. 422).

4 JOURAL OF ECOOMICS AD FIACE EDUCATIO Volume 7 umber Summer oe ha his combined invesmen is like a zero-coupon bond ha does no generaes any cash in-flows unil mauriy. Explicily solving for YTM 2 from equaion (5), we obain YTM 2 = - ( yr ) C F = 0. (6) P Comparing equaion (6) wih equaion (2), we have YTM 2 = RCY wih all coupon paymens being reinvesed a a rae of y r. Hence, RCY is essenially he YTM on he combined invesmen ha holds he coupon bond and reinvess is coupon paymens when received. Then, how is RCY (= YTM 2 ) relaed o YTM? If y r = YTM, hen YTM also solves equaion (5), since YTM solves equaion (). oe from (6) ha RCY is an increasing funcion of y r. The uniqueness of soluion of equaion () and he monooniciy of RCY in y r imply ha RCY = YTM if and only if y r = YTM. This is quie inuiive. By srucure, RCY (= YTM 2 ) measures he annual rae of reurn from he combinaion of he firs wo invesmens. Tha is, RCY is a weighed average of YTM and YTM 2. Therefore, when YTM = YTM 2, heir average, RCY, mus be equal o boh of hem. We summarize he oucomes from above analysis in he following: Proposiion 2. For an invesor who holds a coupon bond unil mauriy, (i) (ii) YTM as defined in equaion () measures he annual rae of reurn acually received by he bond invesor, regardless of how coupon paymens are re-invesed, ie., independen of y r. RCY = YTM 2 measures he yield o mauriy from a combined invesmen of holding he bond unil mauriy plus reinvesing coupon paymens a a rae of y r. (iii) RCY (<) YTM if and only if y r (<) YTM. The Roo of he Fallacy The resuls summarized in Proposiion 2 are quie clear and inuiive. However, resul (i) has been ignored, hough i can be easily seen from he definiion of YTM per se; oucome (ii) has been probably overlooked and i is no recognized ha RCY is in fac he YTM of a porfolio/rus; and he wors, conclusion (iii) has been misinerpreed as YTM (from holding he coupon bond unil mauriy) is acually received only if RCY = YTM. This misinerpreaion simply ells sudens and readers ha if a coupon bond is purchased a a price of P, he YTM as calculaed from Equaion () is only a promise o he bond holder a he ime of purchase, bu may no be really received, unless P( YTM) can be accumulaed a he end of period by reinvesing all coupon paymens a he same rae as he iniial YTM. Implicily, he concep behind such a misinerpreaion is ha he RCY, raher han he YTM, should be he measure of rae of reurn on holding a bond per se o mauriy. Do bond invesors have o reinves all coupon paymens a he same rae as YTM o earn such an annual rae of reurn by holding i unil mauriy? Le s look a a simple example. Two bond invesors, A and B, boh have bough he same coupon bond a he face value and hold i unil mauriy. When receiving he coupon paymen, invesor A always deposis he same amoun of money as he coupon paymens in a savings accoun ha happens o offer he same ineres rae as he coupon rae, whereas invesor B has o rely on he coupon paymens as a source of income o pay bills. A he ime when he bond is maure, invesor A has accumulaed much more wealh due o reinvesing coupon paymens and he power of compounding, bu invesor B only has he face value redeemed. So, wha annual rae of reurn does each of wo invesors earn from holding he bond? Should we measure i wih YTM or RCY? In his example, he wo invesors have earned he same YTM bu differen RCY. For invesor A, RCY = YTM, while for invesor B, RCY = 0. Hence, if RCY is used o measure he annual rae of reurn acually earned from holding he bond, hen invesor A has earned an annual rae of reurn ha is equal o

5 JOURAL OF ECOOMICS AD FIACE EDUCATIO Volume 7 umber Summer YTM, and invesor B has a zero rae of reurn! If one ells inves B ha he has earned nohing from holding he bond, he would eiher laugh or ge very confused. By invesing he amoun of face value (F), invesor B has received coupon paymen (C) year in year ou unil mauriy. Obviously, every year he has earned a rae of reurn of C/F (= YTM in his example) while holding he bond. Indeed, invesor A would have accumulaed much more wealh han invesor B over such an invesmen horizon. I is, however, no because he reinvesmen makes A s coupon bond yield higher han B s bond does, bu because he former has addiional funds periodically invesed. To summarize, RCY does no measure he rae of reurn earned from holding he bond per se. Raher, i measures he rae of reurn earned from wo invesmens holding he bond plus saving a consan amoun every period while holding he bond. When claiming YTM is acually earned only if coupon paymens are reinvesed a YTM, one migh have confused beween YTM 2 (= RCY) and YTM ; in fac, he former measures he rae of reurn from a porfolio or rus ha holds he bond as well as is coupon paymens, while he laer measures he rae of reurn from holding he bond per se no maer how coupon paymens are reinvesed. Confusion beween he wo invesmens may be he roo of he fallacy. Yield o Mauriy or Realized Compounding Yield: A pracical discussion As analyzed above, we ariculae ha YTM and RCY are wo differen and imporan mehods o measure he rae of reurn from invesmens. Basically, YTM measures he rae of reurn from holding a bond no maer how coupon paymens are disposed, while RCY measures he rae of reurn from holding he bond plus re-invesing coupon paymens when received. Hence, in pracice invesors wih differen purposes for invesmen may wan o use differen measure when making an invesmen decision. Specifically, if one decides o spend he coupon paymens when received, hen YTM should be used o measure he rae of reurn from holding a coupon bond. If insead, one plans o save every penny for he fuure and likes o know his rae of reurn from he invesmen porfolio, hen he should consider RCY as he measure of he rae of reurn from he (combined) invesmens since hey include re-invesmen as well. Moreover, if one is mosly concerned abou reinvesmen risk, hen zero-coupon bond is a beer vehicle (han coupon bond) in invesmen since RCY = YTM for zero-coupon bond and here is no re-invesmen risk a all. References Bodie, Zvi, Alex Kane, and Alan J. Marcus, Invesmens, 5 h ediion, McGraw-Hill Irwin Fabozzi, Frank, J. and Franco Modigliani, Capial markes Insiuions and Insrumens, 3 rd ediion, Prenice Hall Henrici, Peer, 974. Applied and compuaional complex analysis, vol., John Wiley & Sons, Inc. Madura, Jeff, 998. Financial markes and insiuions, 4 h ediion, Souh Wesern College Publishing Mishkin, Frederic S., The economics of money, banking and financial markes, 7 h ediion, Pearson Addison Wesley Reilly, Frank K., and Keih C Brown, 997. Invesmen analysis and porfolio managemen, 5 h ediion, Dryden Srong, Rober A., Pracical invesmen managemen, 3 rd ediion, Thomas Souhwesern