The yield curve, and spot and forward interest rates Moorad Choudhry

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1 he yield curve, and spo and forward ineres raes Moorad Choudhry In his primer we consider he zero-coupon or spo ineres rae and he forward rae. We also look a he yield curve. Invesors consider a bond yield and he general marke yield curve when underaking analysis o deermine if he bond is worh buying; his is a form of wha is known as relaive value analysis. All invesors will have a specific risk/reward profile ha hey are comforable wih, and a bond s yield relaive o is perceived risk will influence he decision o buy (or sell) i. We consider he differen ypes of yield curve, before considering a specific curve, he zero-coupon or spo yield curve. Yield curve consrucion iself requires some formidable mahemaics and is ouside he scope of his book; we consider here he basic echniques only. Ineresed readers who wish o sudy he opic furher may wish o refer o he auhor s book Analysing and Inerpreing he Yield Curve. B. HE YIELD CURVE We have already considered he main measure of reurn associaed wih holding bonds, he yield o mauriy or redempion yield. Much of he analysis and pricing aciviy ha akes place in he bond markes revolves around he yield curve. he yield curve describes he relaionship beween a paricular redempion yield and a bond s mauriy. Ploing he yields of bonds along he erm srucure will give us our yield curve. I is imporan ha only bonds from he same class of issuer or wih he same degree of liquidiy be used when ploing he yield curve; for example a curve may be consruced for gils or for AA-raed serling Eurobonds, bu no a mixure of boh. In his secion we will consider he yield o mauriy yield curve as well as oher ypes of yield curve ha may be consruced. Laer in his chaper we will consider how o derive spo and forward yields from a curren redempion yield curve. C. Yield o mauriy yield curve he mos commonly occurring yield curve is he yield o mauriy yield curve. he equaion used o calculae he yield o mauriy was shown in Chaper. he curve iself is consruced by ploing he yield o mauriy agains he erm o mauriy for a group of bonds of he same class. hree differen examples are shown a Figure 2.. Bonds used in consrucing he curve will only rarely have an exac number of whole years o redempion; however i is ofen common o see yields ploed agains whole years on he x-axis. Figure 2.2 shows he Bloomberg page IYC for four governmen yield curves as a 2 December 2005; hese are he US, UK, German and Ialian sovereign bond yield curves.

2 From figure 2.2 noe he yield spread differenial beween German and Ialian bonds. Alhough boh he bonds are denominaed in euros and, according o he European Cenral Bank (ECB) are viewed as equivalen for collaeral purposes (implying idenical credi qualiy), he higher yield for Ialian governmen bonds proves ha he marke views hem as higher credi risk compared o German governmen bonds. Yield % Negaive Posiive Humped Years o mauriy Fig 2. Yield o mauriy yield curves Figure 2.2 Bloomberg page IYC showing hree governmen bond yield curves as a 2 December 2005 Bloomberg L.P. Used wih permission. Visi he main weakness of he yield o mauriy yield curve sems from he un-real world naure of he assumpions behind he yield calculaion. his includes he assumpion of a consan rae for coupons during he bond s life a he redempion yield level. Since marke raes will flucuae over ime, i will no be possible o achieve his (a feaure known as reinvesmen risk). Only zero-coupon bondholders avoid reinvesmen risk as no coupon is paid during he life of a zero-coupon bond. Neverheless he yield o mauriy curve is he mos commonly encounered in markes. For he reasons we have discussed he marke ofen uses oher ypes of yield curve for analysis when he yield o mauriy yield curve is deemed unsuiable. Moorad Choudhry 200,

3 C. he par yield curve he par yield curve is no usually encounered in secondary marke rading, however i is ofen consruced for use by corporae financiers and ohers in he new issues or primary marke. he par yield curve plos yield o mauriy agains erm o mauriy for curren bonds rading a par. he par yield is herefore equal o he coupon rae for bonds priced a par or near o par, as he yield o mauriy for bonds priced exacly a par is equal o he coupon rae. hose involved in he primary marke will use a par yield curve o deermine he required coupon for a new bond ha is o be issued a par. As an example consider for insance ha par yields on one-year, wo-year and hree-year bonds are 5 per cen, 5.25 per cen and 5.75 per cen respecively. his implies ha a new wo-year bond would require a coupon of 5.25 per cen if i were o be issued a par; for a hree-year bond wih annual coupons rading a par, he following equaliy would be rue : = ( ) ( ) 2 3. his demonsraes ha he yield o mauriy and he coupon are idenical when a bond is priced in he marke a par. he par yield curve can be derived direcly from bond yields when bonds are rading a or near par. If bonds in he marke are rading subsanially away from par hen he resuling curve will be disored. I is hen necessary o derive i by ieraion from he spo yield curve. C. he zero-coupon (or spo) yield curve he zero-coupon (or spo) yield curve plos zero-coupon yields (or spo yields) agains erm o mauriy. In he firs insance if here is a liquid zero-coupon bond marke we can plo he yields from hese bonds if we wish o consruc his curve. However i is no necessary o have a se of zero-coupon bonds in order o consruc his curve, as we can derive i from a coupon or par yield curve; in fac in many markes where no zero-coupon bonds are raded, a spo yield curve is derived from he convenional yield o mauriy yield curve. his of course would be a heoreical zero-coupon (spo) yield curve, as opposed o he marke spo curve ha can be consruced from yields of acual zerocoupon bonds rading in he marke. he zero-coupon yield curve is also known as he erm srucure of ineres raes. Spo yields mus comply wih equaion 4., his equaion assumes annual coupon paymens and ha he calculaion is carried ou on a coupon dae so ha accrued ineres is zero. Moorad Choudhry 200,

4 P d = C M + = ( + rs ) ( + rs ) = Cx D + Mx D = (4.) where rs is he spo or zero-coupon yield on a bond wih years o mauriy D /( + rs ) = he corresponding discoun facor In 4., rs is he curren one-year spo yield, rs 2 he curren wo-year spo yield, and so on. heoreically he spo yield for a paricular erm o mauriy is he same as he yield on a zero-coupon bond of he same mauriy, which is why spo yields are also known as zero-coupon yields. his las is an imporan resul. Spo yields can be derived from par yields and he mahemaics behind his are considered in he nex secion. As wih he yield o redempion yield curve he spo yield curve is commonly used in he marke. I is viewed as he rue erm srucure of ineres raes because here is no reinvesmen risk involved; he saed yield is equal o he acual annual reurn. ha is, he yield on a zero-coupon bond of n years mauriy is regarded as he rue n-year ineres rae. Because he observed governmen bond redempion yield curve is no considered o be he rue ineres rae, analyss ofen consruc a heoreical spo yield curve. Essenially his is done by breaking down each coupon bond ino a series of zero-coupon issues. For example, 00 nominal of a 0 per cen wo-year bond is considered equivalen o 0 nominal of a one-year zero-coupon bond and 0 nominal of a wo-year zero-coupon bond. Le us assume ha in he marke here are 30 bonds all paying annual coupons. he firs bond has a mauriy of one year, he second bond of wo years, and so on ou o hiry years. We know he price of each of hese bonds, and we wish o deermine wha he prices imply abou he marke s esimae of fuure ineres raes. We naurally expec ineres raes o vary over ime, bu ha all paymens being made on he same dae are valued using he same rae. For he one-year bond we know is curren price and he amoun of he paymen (comprised of one coupon paymen and he redempion proceeds) we will receive a he end of he year; herefore we can calculae he ineres rae for he firs year : assume he one-year bond has a coupon of 0 per cen. If we inves 00 oday we will receive 0 in one year s ime, hence he rae of ineres is apparen and is 0 per cen. For he wo-year bond we use his ineres rae o calculae he fuure value of is curren price in one year s ime : his is how much we would receive if we had invesed he same amoun in he one-year bond. However he wo-year bond pays a coupon a he end of he firs year; if we subrac his amoun from he fuure value of he curren price, Moorad Choudhry 200,

5 he ne amoun is wha we should be giving up in one year in reurn for he one remaining paymen. From hese numbers we can calculae he ineres rae in year wo. Assume ha he wo-year bond pays a coupon of 8 per cen and is priced a If he was invesed a he rae we calculaed for he one-year bond (0 per cen), i would accumulae in one year, made up of he 95 invesmen and coupon ineres of On he paymen dae in one year s ime, he one-year bond maures and he woyear bond pays a coupon of 8 per cen. If everyone expeced ha a his ime he wo-year bond would be priced a more han (which is minus 8.00), hen no invesor would buy he one-year bond, since i would be more advanageous o buy he wo-year bond and sell i afer one year for a greaer reurn. Similarly if he price was less han no invesor would buy he wo-year bond, as i would be cheaper o buy he shorer bond and hen buy he longer-daed bond wih he proceeds received when he one-year bond maures. herefore he wo-year bond mus be priced a exacly in 2 monhs ime. For his o grow o (he mauriy proceeds from he wo-year bond, comprising he redempion paymen and coupon ineres), he ineres rae in year wo mus be.92 per cen. We can check his using he presen value formula covered earlier. A hese wo ineres raes, he wo bonds are said o be in equilibrium. his is an imporan resul and shows ha here can be no arbirage opporuniy along he yield curve; using ineres raes available oday he reurn from buying he wo-year bond mus equal he reurn from buying he one-year bond and rolling over he proceeds (or reinvesing) for anoher year. his is he known as he breakeven principle. Using he price and coupon of he hree-year bond we can calculae he ineres rae in year hree in precisely he same way. Using each of he bonds in urn, we can link ogeher he implied one-year raes for each year up o he mauriy of he longes-daed bond. he process is known as boo-srapping. he average of he raes over a given period is he spo yield for ha erm : in he example given above, he rae in year one is 0 per cen, and in year wo is.92 per cen. An invesmen of 00 a hese raes would grow o 23.. his gives a oal percenage increase of 23. per cen over wo years, or 0.956% per annum (he average rae is no obained by simply dividing 23. by 2, bu - using our presen value relaionship again - by calculaing he square roo of plus he ineres rae and hen subracing from his number). hus he one-year yield is 0 per cen and he wo-year yield is per cen. In real-world markes i is no necessarily as sraighforward as his; for insance on some daes here may be several bonds mauring, wih differen coupons, and on some daes here may be no bonds mauring. I is mos unlikely ha here will be a regular spacing of redempions exacly one year apar. For his reason i is common for praciioners o use a sofware model o calculae he se of implied forward raes which bes fis he marke prices of he bonds ha do exis in he marke. For insance if here are several one-year bonds, each of heir prices may imply a slighly differen rae of ineres. We will choose he rae which gives he smalles average price error. In pracice all bonds are used o find he rae in year one, all bonds wih a erm longer han one year are used o calculae he rae in year wo, and so on. he zero-coupon curve can also be calculaed direcly from Moorad Choudhry 200,

6 he par yield curve using a mehod similar o ha described above; in his case he bonds would be priced a par (00.00) and heir coupons se o he par yield values. he zero-coupon yield curve is ideal o use when deriving implied forward raes. I is also he bes curve o use when deermining he relaive value, wheher cheap or dear, of bonds rading in he marke, and when pricing new issues, irrespecive of heir coupons. However i is no an accurae indicaor of average marke yields because mos bonds are no zero-coupon bonds. Zero-coupon curve arihmeic Having inroduced he concep of he zero-coupon curve in he previous paragraph, we can now illusrae he mahemaics involved. When deriving spo yields from par yields, one views a convenional bond as being made up of an annuiy, which is he sream of coupon paymens, and a zero-coupon bond, which provides he repaymen of principal. o derive he raes we can use (4.), seing P d = M = 00 and C = rp, shown below. 00 = rp x D + 00 x D = = rp x A + 00 x D (4.2) where rp is he par yield for a erm o mauriy of years, where he discoun facor D is he fair price of a zero-coupon bond wih a par value of and a erm o mauriy of years, and where A = D = A + D (4.3) = is he fair price of an annuiy of per year for years (wih A 0 = 0 by convenion). Subsiuing 4.3 ino 4.2 and re-arranging will give us he expression below for he -year discoun facor. D rp A = x + rp (4.4) In (4.) we are discouning he -year cash flow (comprising he coupon paymen and/or principal repaymen) by he corresponding -year spo yield. In oher words rs is he ime-weighed rae of reurn on a -year bond. hus as we said in he previous secion he spo yield curve is he correc mehod for pricing or valuing any cash flow, including an irregular cash flow, because i uses he appropriae discoun facors. his conrass wih Moorad Choudhry 200,

7 he yield-o-mauriy procedure discussed earlier, which discouns all cash flows by he same yield o mauriy. 4.5 he forward yield curve he forward (or forward-forward) yield curve is a plo of forward raes agains erm o mauriy. Forward raes saisfy expression (4.5) below. P d C C = ( rf ) ( rf )( rf ) ( + rf )...( + rf ) M = (4.5) where C M + = ( + i rf i) ( + i rf i) i= i= in rf is he implici forward rae (or forward-forward rae) on a one-year bond mauring year Comparing (4.) and (4.2) we can see ha he spo yield is he geomeric mean of he forward raes, as shown below. ( rs ) = ( + 0rf )( + rf 2 )...( + rf ) + (4.6) his implies he following relaionship beween spo and forward raes : ( rf ) + = ( + rs ) ( + rs ) D = D (4.7) Moorad Choudhry 200,

8 C. heories of he yield curve As we can observe by analysing yield curves in differen markes a any ime, a yield curve can be one of four basic shapes, which are : normal : in which yields are a average levels and he curve slopes genly upwards as mauriy increases; upward sloping (or posiive or rising) : in which yields are a hisorically low levels, wih long raes subsanially greaer han shor raes; downward sloping (or invered or negaive) : in which yield levels are very high by hisorical sandards, bu long-erm yields are significanly lower han shor raes; humped : where yields are high wih he curve rising o a peak in he medium-erm mauriy area, and hen sloping downwards a longer mauriies. Various explanaions have been pu forward o explain he shape of he yield curve a any one ime, which we can now consider. Unbiased or pure expecaions hypohesis If shor-erm ineres raes are expeced o rise, hen longer yields should be higher han shorer ones o reflec his. If his were no he case, invesors would only buy he shorerdaed bonds and roll over he invesmen when hey maured. Likewise if raes are expeced o fall hen longer yields should be lower han shor yields. he expecaions hypohesis saes ha he long-erm ineres rae is a geomeric average of expeced fuure shor-erm raes. his was in fac he heory ha was used o derive he forward yield curve in (4.5) and (4.6) previously. his gives us : ( rs ) ( rs )( rf ) ( rf ) or + = (4.0) ( ) ( ) ( + rs = + rs + rf ) (4.) where rs is he spo yield on a -year bond and - rf is he implied one-year rae years ahead. For example if he curren one-year rae is rs = 6.5% and he marke is expecing he one-year rae in a year s ime o be rf 2 = 7.5%, hen he marke is expecing a 00 invesmen in wo one-year bonds o yield : 00 (.065)(.075) = 4.49 afer wo years. o be equivalen o his an invesmen in a wo-year bond has o yield he same amoun, implying ha he curren wo-year rae is rs 2 = 7%, as shown below. Moorad Choudhry 200,

9 00 (.07) 2 = 4.49 his resul mus be so, o ensure no arbirage opporuniies exis in he marke and in fac we showed as much, earlier in he chaper when we considered forward raes. A rising yield curve is herefore explained by invesors expecing shor-erm ineres raes o rise, ha is rf 2 >rs 2. A falling yield curve is explained by invesors expecing shorerm raes o be lower in he fuure. A humped yield curve is explained by invesors expecing shor-erm ineres raes o rise and long-erm raes o fall. Expecaions, or views on he fuure direcion of he marke, are a funcion of he expeced rae of inflaion. If he marke expecs inflaionary pressures in he fuure, he yield curve will be posiively shaped, while if inflaion expecaions are inclined owards disinflaion, hen he yield curve will be negaive. Liquidiy preference heory Inuiively we can see ha longer mauriy invesmens are more risky han shorer ones. An invesor lending money for a five-year erm will usually demand a higher rae of ineres han if he were o lend he same cusomer money for a five-week erm. his is because he borrower may no be able o repay he loan over he longer ime period as he may for insance, have gone bankrup in ha period. For his reason longer-daed yields should be higher han shor-daed yields. We can consider his heory in erms of inflaion expecaions as well. Where inflaion is expeced o remain roughly sable over ime, he marke would anicipae a posiive yield curve. However he expecaions hypohesis canno by iself explain his phenomenon, as under sable inflaionary condiions one would expec a fla yield curve. he risk inheren in longer-daed invesmens, or he liquidiy preference heory, seeks o explain a posiive shaped curve. Generally borrowers prefer o borrow over as long a erm as possible, while lenders will wish o lend over as shor a erm as possible. herefore, as we firs saed, lenders have o be compensaed for lending over he longer erm; his compensaion is considered a premium for a loss in liquidiy for he lender. he premium is increased he furher he invesor lends across he erm srucure, so ha he longesdaed invesmens will, all else being equal, have he highes yield. Segmenaion Hypohesis he capial markes are made up of a wide variey of users, each wih differen requiremens. Cerain classes of invesors will prefer dealing a he shor-end of he yield curve, while ohers will concenrae on he longer end of he marke. he segmened markes heory suggess ha aciviy is concenraed in cerain specific areas of he marke, and ha here are no iner-relaionships beween hese pars of he marke; he relaive amouns of funds invesed in each of he mauriy specrum causes differenials in supply and demand, which resuls in humps in he yield curve. ha is, he shape of he yield curve is deermined by supply and demand for cerain specific mauriy invesmens, each of which has no reference o any oher par of he curve. Moorad Choudhry 200,

10 For example banks and building socieies concenrae a large par of heir aciviy a he shor end of he curve, as par of daily cash managemen (known as asse and liabiliy managemen) and for regulaory purposes (known as liquidiy requiremens). Fund managers such as pension funds and insurance companies however are acive a he long end of he marke. Few insiuional invesors however have any preference for mediumdaed bonds. his behaviour on he par of invesors will lead o high prices (low yields) a boh he shor and long ends of he yield curve and lower prices (higher yields) in he middle of he erm srucure. Furher views on he yield curve As one migh expec here are oher facors ha affec he shape of he yield curve. For insance shor-erm ineres raes are grealy influenced by he availabiliy of funds in he money marke. he slope of he yield curve (usually defined as he 0-year yield minus he hree-monh ineres raes) is also a measure of he degree of ighness of governmen moneary policy. A low, upward sloping curve is ofen hough o be a sign ha an environmen of cheap money, due o a more loose moneary policy, is o be followed by a period of higher inflaion and higher bond yields. Equally a high downward sloping curve is aken o mean ha a siuaion of igh credi, due o more sric moneary policy, will resul in falling inflaion and lower bond yields. Invered yield curves have ofen preceded recessions; for insance he Economis in an aricle from April 998 remarked ha in he Unied Saes every recession since 955 bar one has been preceded by a negaive yield curve. he analysis is he same: if invesors expec a recession hey also expec inflaion o fall, so he yields on long-erm bonds will fall relaive o shor-erm bonds. here is significan informaion conen in he yield curve, and economiss and bond analyss will consider he shape of he curve as par of heir policy making and invesmen advice. he shape of pars of he curve, wheher he shor-end or long-end, as well ha of he enire curve, can serve as useful predicors of fuure marke condiions. As par of an analysis i is also worhwhile considering he yield curves across several differen markes and currencies. For insance he ineres-rae swap curve, and is posiion relaive o ha of he governmen bond yield curve, is also regularly analysed for is informaion conen. In developed counry economies he swap marke is invariably as liquid as he governmen bond marke, if no more liquid, and so i is common o see he swap curve analysed when making predicions abou say, he fuure level of shor-erm ineres raes. Governmen policy will influence he shape and level of he yield curve, including policy on public secor borrowing, deb managemen and open-marke operaions. he markes percepion of he size of public secor deb will influence bond yields; for insance an increase in he level of deb can lead o an increase in bond yields across he mauriy range. Open-marke operaions, which refers o he daily operaion by he Bank of England o conrol he level of he money supply (o which end he Bank purchases shorerm bills and also engages in repo dealing), can have a number of effecs. In he shorerm i can il he yield curve boh upwards and downwards; longer erm, changes in he level of he base rae will affec yield levels. An anicipaed rise in base raes can lead o a Moorad Choudhry 200,

11 drop in prices for shor-erm bonds, whose yields will be expeced o rise; his can lead o a emporary invered curve. Finally deb managemen policy will influence he yield curve. (In he Unied Kingdom his is now he responsibiliy of he Deb Managemen Office.) Much governmen deb is rolled over as i maures, bu he mauriy of he replacemen deb can have a significan influence on he yield curve in he form of humps in he marke segmen in which he deb is placed, if he deb is priced by he marke a a relaively low price and hence high yield. B. SPO AND FORWARD RAES: Spo Raes and boo-srapping Par, spo and forward raes have a close mahemaical relaionship. Here we explain and derive hese differen ineres raes and explain heir applicaion in he markes. Noe ha spo ineres raes are also called zero-coupon raes, because hey are he ineres raes ha would be applicable o a zero-coupon bond. he wo erms are used synonymously, however sricly speaking hey are no exacly similar. Zero-coupon bonds are acual marke insrumens, and he yield on zero-coupon bonds can be observed in he marke. A spo rae is a purely heoreical consruc, and so canno acually be observed direcly. For our purposes hough, we will use he erms synonymously. A par yield is he yield-o-mauriy on a bond ha is rading a par. his means ha he yield is equal o he bond s coupon level. A zero-coupon bond is a bond which has no coupons, and herefore only one cash flow, he redempion paymen on mauriy. I is herefore a discoun insrumen, as i is issued a a discoun o par and redeemed a par. he yield on a zero-coupon bond can be viewed as a rue yield, a he ime ha is i purchased, if he paper is held o mauriy. his is because no reinvesmen of coupons is involved and so here are no inerim cash flows vulnerable o a change in ineres raes. Zero-coupon yields are he key deerminan of value in he capial markes, and hey are calculaed and quoed for every major currency. Zero-coupon raes can be used o value any cash flow ha occurs a a fuure dae. Where zero-coupon bonds are raded he yield on a zero-coupon bond of a paricular mauriy is he zero-coupon rae for ha mauriy. No all deb capial rading environmens possess a liquid marke in zero-coupon bonds. However i is no necessary o have zero-coupon bonds in order o calculae zero-coupon raes. I is possible o calculae zero-coupon raes from a range of marke raes and prices, including coupon bond yields, ineres-rae fuures and currency deposis. We illusrae shorly he close mahemaical relaionship beween par, zero-coupon and forward raes. We also illusrae how he boo-srapping echnique could be used o calculae spo and forward raes from bond redempion yields. In addiion, once he discoun facors are known, any of hese raes can be calculaed. he relaionship beween he hree raes allows he markes o price ineres-rae swap and FRA raes, as a swap rae is he weighed arihmeic average of forward raes for he erm in quesion. Moorad Choudhry 200, 2008

12 Discoun Facors and he Discoun Funcion I is possible o deermine a se of discoun facors from marke ineres raes. A discoun facor is a number in he range zero o one which can be used o obain he presen value of some fuure value. We have PV = () where d x FV PV FV d is he presen value of he fuure cash flow occurring a ime is he fuure cash flow occurring a ime is he discoun facor for cash flows occurring a ime Discoun facors can be calculaed mos easily from zero-coupon raes; equaions 2 and 3 apply o zero-coupon raes for periods up o one year and over one year respecively. d d = ( + rs ) = ( + rs ) (2) (3) where d rs is he discoun facor for cash flows occurring a ime is he zero-coupon rae for he period o ime is he ime from he value dae o ime, expressed in years and fracions of a year Individual zero-coupon raes allow discoun facors o be calculaed a specific poins along he mauriy erm srucure. As cash flows may occur a any ime in he fuure, and no necessarily a convenien imes like in hree monhs or one year, discoun facors ofen need o be calculaed for every possible dae in he fuure. he complee se of discoun facors is called he discoun funcion. Implied Spo and Forward Raes In his secion we describe how o obain zero-coupon and forward ineres raes from he yields available from coupon bonds, using a mehod known as boo-srapping. In a governmen bond marke such as ha for US reasuries or UK gils, he bonds are considered o be defaul-free. he raes from a governmen bond yield curve describe he Moorad Choudhry 200,

13 risk-free raes of reurn available in he marke oday, however hey also imply (risk-free) raes of reurn for fuure ime periods. hese implied fuure raes, known as implied forward raes, or simply forward raes, can be derived from a given spo yield curve using boo-srapping. his erm reflecs he fac ha each calculaed spo rae is used o deermine he nex period spo rae, in successive seps. able shows an hypoheical benchmark gil yield curve for value as a 7 December he observed yields of he benchmark bonds ha compose he curve are displayed in he las column. All raes are annualised and assume semi-annual compounding. he bonds all pay on he same coupon daes of 7 June and 7 December, and as he value dae is a coupon dae, here is no accrued ineres on any of he bonds. he clean and diry prices for each bond are idenical. Bond erm o mauriy (years) Coupon Mauriy dae Price able Hypoheical UK governmen bond yields as a 7 December 2000 Gross Redempion Yield 4% reasury % 07-Jun % 5% reasury 200 5% 07-Dec % 6% reasury % 07-Jun % 7% reasury % 07-Dec % 8% reasury % 07-Jun % 9% reasury % 07-Dec % he gross redempion yield or yield-o-mauriy of a coupon bond describes he single rae ha presen-values he sum of all is fuure cash flows o is curren price. I is essenially he inernal rae of reurn of he se of cash flows ha make up he bond. his yield measure suffers from a fundamenal weakness in ha each cash-flow is presenvalued a he same rae, an unrealisic assumpion in anyhing oher han a fla yield curve environmen. So he yield o mauriy is an anicipaed measure of he reurn ha can be expeced from holding he bond from purchase unil mauriy. In pracice i will only be achieved under he following condiions: he bond is purchased on issue; all he coupons paid hroughou he bond s life are re-invesed a he same yield o mauriy a which he bond was purchased; he bond is held unil mauriy. Benchmark gils pay coupon on a semi-annual basis on 7 June and 7 December each year. Moorad Choudhry 200,

14 In p racice hese condiions will no be fulfilled, and so he yield o mauriy of a bond is no a rue ineres rae for ha bond s mauriy period. he bonds in able pay semi-annual coupons on 7 June and 7 December and have he same ime period - six monhs - beween 7 December 2000, heir valuaion dae and 7 June 200, heir nex coupon dae. However since each issue carries a differen yield, he nex six-monh coupon paymen for each bond is presen-valued a a differen rae. In oher words, he six-monh bond presen-values is six-monh coupon paymen a is 4% yield o mauriy, he one-year a 5%, and so on. Because each of hese issues uses a differen rae o presen-value a cash flow occurring a he same fuure poin in ime, i is unclear which of he raes should be regarded as he rue ineres rae or benchmark rae for he six-monh period from 7 December 2000 o 7 June 200. his problem is repeaed for all oher mauriies. For he purposes of valuaion and analysis however, we require a se of rue ineres raes, and so hese mus be derived from he redempion yields ha we can observe from he benchmark bonds rading in he marke. hese raes we designae as rs i, where rs i is he implied spo rae or zero-coupon rae for he erm beginning on 7 December 2000 and ending a he end of period i. We begin calculaing implied spo raes by noing ha he six-monh bond conains only one fuure cash flow, he final coupon paymen and he redempion paymen on mauriy. his means ha i is in effec rading as a zero-coupon bond, as here is only one cash flow lef for his bond, is final paymen. Since his cash flow s presen value, fuure value and mauriy erm are known, he unique ineres rae ha relaes hese quaniies can be solved using he compound ineres equaion (4) below. ( nm) rs FV = PV + i m rs = ( nm) i m FV PV (4) where FV PV rs i m n is he fuure value is he presen value is he implied i-period spo rae is he number of ineres periods per year is he number of years in he erm he fir s rae o be solved is referred o as he implied six-monh spo rae and is he rue ineres rae for he six-monh erm beginning on 2 January and ending on 2 July Moorad Choudhry 200,

15 Equaion (4) relaes a cash flow s presen value and fuure value in erms of an associaed ineres rae, compounding convenion and ime period. Of course if we re-arrange i, we may use i o solve for an implied spo rae. For he six-monh bond he final cash flow on mauriy is 02, comprised of he 2 coupon paymen and he 00 (par) redempion amoun. So we have for he firs erm, i =, FV = 02, PV = 00, n = 0.5 years and m = 2. his allows us o calculae he spo rae as follows : nm) rs i = m ( FV/PV ) ( 0.5 x 2) rs = 2 02/ 00 rs rs = = 4.000% ( ) hus he implied six-monh spo rae or zero-coupon rae is equal o 4 per cen. 2 We now need o deermine he implied one-year spo rae for he erm from 7 December 2000 o 7 June 200. We noe ha he one-year issue has a 5% coupon and conains wo fuure cash flows : a 2.50 six-monh coupon paymen on 7 June 200 and a one-year coupon and principal paymen on 7 December 200. Since he firs cash flow occurs on 7 June - six monhs from now - i mus be presen-valued a he 4 per cen six-monh spo rae esablished above. Once his presen value is deermined, i may be subraced from he 00 oal presen value (is curren price) of he one-year issue o obain he presen value of he one-year coupon and cash flow. Again we hen have a single cash flow wih a known presen value, fuure value and erm. he rae ha equaes hese quaniies is he implied one-year spo rae. From equaion (4) he presen value of he six-monh 2.50 coupon paymen of he one-year benchmark bond, discouned a he implied six-monh spo rae, is : (0.5 x 2) PV 6-mo cash flow, -yr bond = 2.50/( /2) = he presen value of he one-year coupon and principal paymen is found by subracing he presen value of he six-monh cash flow, deermined above, from he oal presen value (curren price) of he issue : PV -yr cash flow, -yr bond = = he implied one-year spo rae is hen deermined by using he presen value of he one-year cash flow deermined above : (5) 2 Of course inuiively we could have concluded ha he six-monh spo rae was 4 per cen, wihou he need o apply he arihmeic, as we had already assumed ha he six-monh bond was a quasi-zero-coupon bond. Moorad Choudhry 200,

16 rs ( ) ( x2) = / = = % he implied.5 year spo rae is solv ed in he same way: PV 6-mo cash flow,.5-yr bond = 3.00 / ( / 2) (0.5x2) = PV -yr cash flow,.5-yr bond = 3.00 / ( / 2 = PV.5-yr cash flow,.5-yr bond = = x 2) rs3 = 2 ( 03 / ) = = % Exending he same process for he wo-year bond, we calculae he implied wo-year spo rae rs 4 o be per cen. he implied 2.5-year and hree-year spo raes rs 5 and rs6 are per cen per cen respecively. ) (x2) he ineres raes rs, rs 2, rs 3, rs 4, rs 5 and rs 6 describe he rue zero-coupon ineres raes for he six-monh, one-year,.5-year, wo-year, 2.5-year and hree-year erms ha begin on 7 December 2000 and end on 7 June 200, 7 December 200, 7 June 2002, 7 December 2002, 7 June 2003 and 7 December 2003 respecively. hey are also called implied spo raes because hey have been calculaed from redempion yields observed in he marke from he benchmark governmen bonds ha were lised in able. Noe ha he one-,.5-, wo-year, 2.5-year and hree-year implied spo raes are progressively greaer han he corresponding redempion yields for hese erms. his is an imporan resul, and occurs whenever he yield curve is posiively sloped. he reason for his is ha he presen values of a bond s shorer-daed cash flows are discouned a raes ha are lower han he bond s redempion yield; his generaes higher presen values ha, when subraced from he curren price of he bond, produce a lower presen value for he final cash flow. his lower presen value implies a spo rae ha is greaer han he issue s yield. In an invered yield curve environmen we observe he opposie resul, ha is implied raes ha lie below he corresponding redempion yields. If he redempion yield curve is fla, he implied spo raes will be equal o he corresponding redempion yields. Once we have calculaed he spo or zero-coupon raes for he six-monh, one-year,.5-year, wo-year, 2.5-year and hree-year erms, we can deermine he rae of reurn ha is implied by he yield curve for he sequence of six-monh periods beginning on 7 Moorad Choudhry 200,

17 December 2000, 7 June 200, 7 December 200, 7 June 2002 and 7 December hese period raes are referred o as implied forward raes or forward-forward raes and we denoe hese as rf i, where rf i is he implied six-monh forward ineres rae oday for he ih period. Since he implied six-monh zero-coupon rae (spo rae) describes he reurn for a erm ha coincides precisely wih he firs of he series of six-monh periods, his rae describes he risk-free rae of reurn for he firs six-monh period. I is herefore equal o he firs period spo rae. hus we have rf = rs = 4.0 per cen, where rf is he risk-free forward rae for he firs six-monh period beginning a period. We show now how he risk-free raes for he second, hird, fourh, fifh and sixh six-monh periods, designaed rf 2, rf 3, rf 4, rf 5 and rf 6 respecively may be solved from he implied spo raes. he benchmark rae for he second semi-annual period rf 2 is referred o as he one-period forward six-monh rae, because i goes ino effec one six-monh period from now ( oneperiod forward ) and remains in effec for six monhs ( six-monh rae ). I is herefore he six-monh rae in six monhs ime, and is also referred o as he 6-monh forwardforward rae. his rae, in conjuncion wih he rae from he firs period rf, mus provide reurns ha mach hose generaed by he implied one-year spo rae for he enire oneyear erm. In oher words, one pound invesed for six monhs from 7 December 2000 o 7 June 200 a he firs period s benchmark rae of 4 per cen and hen reinvesed for anoher six monhs from 7 June 200 o 7 December 200 a he second period s (as ye unknown) implied forward rae mus enjoy he same reurns as one pound invesed for one year from 7 December 2000 o 7 December 200 a he implied one-year spo rae of per cen. his reflecs he law of no-arbirage. A momen s hough will convince us ha his mus be so. If his were no he case, we migh observe an ineres rae environmen in which he reurn received by an invesor over any given erm would depend on wheher an invesmen is made a he sar period for he enire mauriy erm or over a succession of periods wihin he whole erm and reinvesed. If here were any discrepancies beween he reurns received from each approach, here would exis an unrealisic arbirage opporuniy, in which invesmens for a given erm carrying a lower reurn migh be sold shor agains he simulaneous purchase of invesmens for he same period carrying a higher reurn, hereby locking in a risk-free, cos-free profi. herefore forward ineres raes mus be calculaed so ha hey are arbirage-free. Forward raes are no herefore a predicion of wha spo ineres raes are likely o be in he fuure, raher a mahemaically derived se of ineres raes ha reflec he curren spo erm srucure and he rules of no-arbirage. Excellen mahemaical explanaions of he no-arbirage propery of ineres-rae markes are conained in Ingersoll (987), Jarrow (996), and Rober Shiller (990) among ohers. he exisence of a no-arbirage marke of course makes i sraighforward o calculae forward raes; we know ha he reurn from an invesmen made over a period mus equal he reurn made from invesing in a shorer period and successively reinvesing o a maching erm. If we know he reurn over he shorer period, we are lef wih only one unknown, he full-period forward rae, which is hen easily calculaed. In our example, Moorad Choudhry 200,

18 having esablished he rae for he firs six-monh period, he rae for he second sixmonh period - he one-period forward six-monh rae - is deermined below. he fuure value of invesed a rf, he period forward rae, a he end of he firs sixmonh period is calculaed as follows : ( 0.5 x 2) rf FV = = + 2 = he fuure value of a he end of he one-year erm, invesed a he implied benchmark one-year spo rae is deermined as follows : ( x2) rs2 FV2 = = + 2 = he implied benchmark one-period forward rae rf 2 is he rae ha equaes he value of FV (.02) on 7 June 200 o FV 2 ( ) on 7 December 200. From equaion (4) we have : FV ( ) 2 rf2 = x 2 FV = 2.02 = = % In oher words invesed from 7 December o 7 June a 4.0 per cen (he implied forward rae for he firs period) and hen reinvesed from 7 June o 7 December 200 a per cen (he implied forward rae for he second period) would accumulae he same reurns as invesed from 7 December 2000 o 7 December 200 a per cen (he implied one-year spo rae). he rae for he hird six-monh period - he wo-period forward six-monh ineres rae may be calculaed in he same way: Moorad Choudhry 200,

19 FV 2 = (.5 x 2) FV 3 = x ( + rs 3 / 2) = x ( / 2) 3 = FV 2 ( 0.5 2) 3 rf 3 = FV 4 = = = 8.25% In he same way he hree-period forward six-monh rae rf 4 is calculaed o be per cen. he res of he resuls are shown in able 2. We say one-period forward rae because i is he forward rae ha applies o he six-monh period. he resuls of he implied spo (zero-coupon) and forward rae calculaions along wih he given redempion yield curve are illusraed graphically in Figure. he simple boosrapping mehodology can be applied using a spreadshee for acual marke redempion yields. However in pracice we will no have a se of bonds wih exac and/or equal periods o mauriy and coupons falling on he same dae. Nor will hey all be priced convenienly a par. In designing a spreadshee spo rae calculaor herefore, he coupon rae and mauriy dae is enered as sanding daa and usually inerpolaion is used when calculaing he spo raes for bonds wih uneven mauriy daes. A spo curve model ha uses his approach in conjuncion wih he boo-srapping mehod is available for downloading a Marke praciioners usually use discoun facors o exrac spo and forward raes from marke prices. For an accoun of his mehod, see Choudhry e al (200), chaper 9. erm o mauriy Yield o mauriy Implied spo rae able 2 Implied spo and forward raes Implied oneperiod forward rae % % % % % % % % 8.25% % % % % % % % % % Moorad Choudhry 200,

20 6.0% 4.0% Ineres rae % 2.0% 0.0% 8.0% 6.0% 4.0% Par yield Spo rae Forward rae 2.0% erm o mauriy Figure Par, spo and forward yield curves Examples Example Consider he following spo yields : -year 0% 2-year 2% Assume ha a bank s clien wishes o lock in oday he cos of borrowing -year funds in one year s ime. he soluion for he bank (and he mechanism o enable he bank o quoe a price o he clien) involves raising -year funds a 0% and invesing he proceeds for wo years a 2%. As we observed above, he no-arbirage principle means ha he same reurn mus be generaed from boh fixed rae and reinvesmen sraegies. Using he following formula: 2 ( rs ) = ( + rs )( rf ) rf = 2 ( + y2 ) ( + y ) he relevan forward rae is calculaed o be 4.04 per cen. Example 2 Moorad Choudhry 200,