Fixed Income Analysis: Securities, Pricing, and Risk Management

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1 Fixed Income Analysis: Securiies, Pricing, and Risk Managemen Claus Munk This version: January 23, 2003 Deparmen of Accouning and Finance, Universiy of Souhern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. Phone: Fax: Inerne homepage: hp:// cmu

2 Conens Preface viii 1 Basic ineres rae markes, conceps, and relaions Inroducion Markes for bonds and ineres raes Discoun facors and zero-coupon bonds Zero-coupon raes and forward raes Annual compounding Compounding over oher discree periods LIBOR raes Coninuous compounding Differen ways o represen he erm srucure of ineres raes Deermining he zero-coupon yield curve: Boosrapping Deermining he zero-coupon yield curve: Parameerized forms Cubic splines The Nelson-Siegel parameerizaion Addiional remarks on yield curve esimaion Exercises Fixed income securiies Inroducion Floaing rae bonds Forwards on bonds Ineres rae forwards forward rae agreemens Fuures on bonds Ineres rae fuures Eurodollar fuures Opions on bonds Opions on zero-coupon bonds Opions on coupon bonds Caps, floors, and collars Caps Floors Collars Exoic caps and floors Swaps and swapions i

3 Conens ii Swaps Swapions Exoic swap insrumens Exercises Sochasic processes and sochasic calculus Probabiliy spaces Sochasic processes Differen ypes of sochasic processes Basic conceps Markov processes and maringales Coninuous or disconinuous pahs Brownian moions Diffusion processes Iô processes Sochasic inegrals Definiion and properies of sochasic inegrals Leibniz rule for sochasic inegrals Iô s Lemma Imporan diffusion processes Geomeric Brownian moions Ornsein-Uhlenbeck processes Square roo processes Muli-dimensional processes Change of probabiliy measure Exercises Asse pricing and erm srucures: discree-ime models Inroducion A one-period model Asses, porfolios, and arbirage Invesors Sae-price vecors and deflaors Risk-neural probabiliies Redundan asses Complee markes Equilibrium and represenaive agens in complee markes A muli-period, discree-ime model Asses, rading sraegies, and arbirage Invesors Sae-price vecors and deflaors Risk-neural probabiliy measures Redundan asses Complee markes

4 Conens iii Equilibrium and represenaive agens in complee markes Discree-ime, finie-sae models of he erm srucure Concluding remarks Exercises Asse pricing and erm srucures: an inroducion o coninuous-ime models Inroducion Asse pricing in coninuous-ime models Asses, rading sraegies, and arbirage Invesors Sae-price deflaors Risk-neural probabiliy measures From no arbirage o sae-price deflaors and risk-neural measures Marke prices of risk Complee vs. incomplee markes Exension o inermediae dividends Equilibrium and represenaive agens in complee markes Oher probabiliy measures convenien for pricing A zero-coupon bond as he numeraire forward maringale measures An annuiy as he numeraire swap maringale measures A general pricing formula for European opions Forward prices and fuures prices Forward prices Fuures prices A comparison of forward prices and fuures prices American-syle derivaives Diffusion models and he fundamenal parial differenial equaion One-facor diffusion models Muli-facor diffusion models The Black-Scholes-Meron model and Black s varian The Black-Scholes-Meron model Black s model Problems in applying Black s model o fixed income securiies An overview of coninuous-ime erm srucure models Overall caegorizaion Some frequenly applied models Exercises The Economics of he Term Srucure of Ineres Raes Inroducion Real ineres raes and aggregae consumpion Real ineres raes and aggregae producion Equilibrium ineres rae models Producion-based models

5 Conens iv Consumpion-based models Real and nominal ineres raes and erm srucures Real and nominal asse pricing No real effecs of inflaion A model wih real effecs of money The expecaion hypohesis Versions of he pure expecaion hypohesis The pure expecaion hypohesis and equilibrium The weak expecaion hypohesis Liquidiy preference, marke segmenaion, and preferred habias Concluding remarks Exercises One-facor diffusion models Inroducion Affine models Bond prices, zero-coupon raes, and forward raes Forwards and fuures European opions on coupon bonds: Jamshidian s rick Meron s model The shor rae process Bond pricing The yield curve Forwards and fuures Opion pricing Vasicek s model The shor rae process Bond pricing The yield curve Forwards and fuures Opion pricing The Cox-Ingersoll-Ross model The shor rae process Bond pricing The yield curve Forwards and fuures Opion pricing Non-affine models Parameer esimaion and empirical ess Concluding remarks Exercises

6 Conens v 8 Muli-facor diffusion models Wha is wrong wih one-facor models? Muli-facor diffusion models of he erm srucure Muli-facor affine diffusion models Two-facor affine diffusion models n-facor affine diffusion models European opions on coupon bonds Muli-facor Gaussian diffusion models General analysis A specific example: he wo-facor Vasicek model Muli-facor CIR models General analysis A specific example: he Longsaff-Schwarz model Oher muli-facor diffusion models Models wih sochasic consumer prices Models wih sochasic long-erm level and volailiy A model wih a shor and a long rae Key rae models Quadraic models Final remarks Calibraion of diffusion models Inroducion Time inhomogeneous affine models The Ho-Lee model exended Meron) The Hull-Whie model exended Vasicek) The exended CIR model Calibraion o oher marke daa Iniial and fuure erm srucures in calibraed models Calibraed non-affine models Is a calibraed one-facor model jus as good as a muli-facor model? Final remarks Exercises Heah-Jarrow-Moron models Inroducion Basic assumpions Bond price dynamics and he drif resricion Three well-known special cases The Ho-Lee exended Meron) model The Hull-Whie exended Vasicek) model The exended CIR model Gaussian HJM models Diffusion represenaions of HJM models

7 Conens vi On he use of numerical echniques for diffusion and non-diffusion models In which HJM models does he shor rae follow a diffusion process? A wo-facor diffusion represenaion of a one-facor HJM model HJM-models wih forward-rae dependen volailiies Concluding remarks Marke models Inroducion General LIBOR marke models Model descripion The dynamics of all forward raes under he same probabiliy measure Consisen pricing The lognormal LIBOR marke model Model descripion The pricing of oher securiies Alernaive LIBOR marke models Swap marke models Furher remarks Exercises The measuremen and managemen of ineres rae risk Inroducion Tradiional measures of ineres rae risk Macaulay duraion and convexiy The Fisher-Weil duraion and convexiy The no-arbirage principle and parallel shifs of he yield curve Risk measures in one-facor diffusion models Definiions and relaions Compuaion of he risk measures in affine models A comparison wih radiional duraions Immunizaion Consrucion of immunizaion sraegies An experimenal comparison of immunizaion sraegies Risk measures in muli-facor diffusion models Facor duraions, convexiies, and ime value One-dimensional risk measures in muli-facor models Duraion-based pricing of opions on bonds The general idea A mahemaical analysis of he approximaion The accuracy of he approximaion in he Longsaff-Schwarz model Alernaive measures of ineres rae risk Morgage-backed securiies 299

8 Conens vii 14 Credi risky securiies Sochasic ineres raes and he pricing of sock and currency derivaives Inroducion Sock opions General analysis Deerminisic volailiies Opions on forwards and fuures Forward and fuures prices Opions on forwards Opions on fuures Currency derivaives Currency forwards A model for he exchange rae Currency fuures Currency opions Alernaive exchange rae models Final remarks Numerical echniques 315 A Resuls on he lognormal disribuion 316 References 319

9 Preface This book provides an inroducion o he markes for fixed-income securiies and he models and mehods ha are used o analyze such securiies. The class of fixed-income securiies covers securiies where he issuer promises one or several fixed, predeermined paymens a given poins in ime. This is he case for sandard deposi arrangemens and bonds. However, several relaed securiies wih paymens ha are ied o he developmen in some paricular index, ineres rae, or asse price are ypically also ermed fixed-income securiies. In he broades sense of he erm, he many differen ineres rae and bond derivaives are also considered fixed-income producs. Maybe a more descripive erm for his broad class of securiies is ineres rae securiies, since he values of hese financial conracs are derived from curren ineres raes and expecaions and uncerainy abou fuure ineres raes. The key concep in he analysis of fixed-income securiies is he erm srucure of ineres raes, which is loosely defined as he dependence beween ineres raes and mauriies. The ouline of his book is as follows. The firs wo chapers deal wih he mos common fixedincome securiies, fix much of he noaion and erminology, and discuss basic relaions beween key conceps. The main par of he book discusses models of he evoluion of he erm srucure of ineres raes over ime. Chaper 3 inroduces much of he mahemaics needed for developing and analyzing modern dynamic models of ineres raes. In Chapers 4 and 5 we review some of he imporan general resuls on asse pricing. In paricular, we define and relae he key conceps of arbirage, sae-price deflaors, and risk-neural probabiliy measures. The connecion o marke compleeness and individual invesors behavior is also addressed, jus as he implicaions of he general asse pricing heory for he modeling of he erm srucure are discussed. Chaper 6 applies he general asse pricing ools o explore he economics of he erm srucure of ineres raes. For example we discuss he relaion beween he erm srucure of ineres raes and macro-economic variables such as aggregae consumpion, producion, and inflaion. We will also review some of he radiional hypoheses on he shape of he yield curve, e.g. he expecaion hypoheses. Chapers 7 o 12 develops models for he pricing of fixed income securiies and he managemen of ineres rae risk. Chaper 7 goes hrough so-called one-facor models. This ype of models was he firs o be applied in he lieraure and daes back a leas o The one-facor models of Vasicek and Cox, Ingersoll, and Ross are sill frequenly applied boh in pracice and in academic research. Chaper 8 explores muli-facor models which have several advanages over one-facor models, bu are also more complicaed o analyze and apply. In Chaper 9 we discuss how oneand muli-facor models can be exended o be consisen wih curren marke informaion, such as bond prices and volailiies. Chaper 10 inroduces and analyzes so-called Heah-Jarrow-Moron viii

10 Preface ix models, which are characerized by aking he curren marke erm srucure of ineres raes as given and hen modeling he evoluion of he enire erm srucure in an arbirage-free way. We will explore he relaion beween hese models and he facor models sudied in earlier chapers. Ye anoher class of models is he subjec of Chaper 11. These marke models are designed for he pricing and hedging of specific producs ha are raded on a large scale in he inernaional markes, namely caps, floors, and swapions. Chaper 12 discusses how he differen ineres rae models can be applied for ineres rae risk managemen. The subjec of Chaper 13 only some references are lised in he curren version) is how o consruc models for he pricing and risk managemen of morgage-backed securiies. The main concern is how o adjus he models sudied in earlier chapers o ake he prepaymen opions involved in morgages ino accoun. In Chaper 14 only some references are lised in he curren version) we discuss he pricing of corporae bonds and oher fixed-income securiies where he defaul risk of he issuer canno be ignored. Chaper 15 focuses on he consequences which sochasic variaions in ineres raes have for he valuaion of securiies wih paymens ha are no direcly relaed o ineres raes, such as sock opions and currency opions. Finally, Chaper 16 only some references are lised in he curren version) describes several numerical echniques ha can be applied in cases where explici pricing and hedging formulas are no available. Syle... Prerequisies... There are several oher books ha cover much of he same maerial or focus on paricular elemens discussed in his book. Books emphasizing descripions of markes and producs: Fabozzi 2000), van Horne 2001). Books emphasizing modern ineres rae modeling: James and Webber 2000), Pelsser 2000), Rebonao 1996),... I appreciae commens and correcions from Lennar Damgaard, Hans Frimor, Mee Hansen, Frank Emil Jensen, Kasper Larsen, Moren Mosegaard, Per Plonikoff, and oher people. I also appreciae he excellen secrearial assisance of Lene Holbæk.

11 Chaper 1 Basic ineres rae markes, conceps, and relaions 1.1 Inroducion The value of an asse equals he value of is fuure cash flow. Tradiionally, he value of a bond is compued by discouning all is fuure paymens using he same ineres rae, namely he yield of he bond. If we have wo bonds wih he same paymen daes bu differen yields, we will herefore discoun he paymens from he bonds wih differen ineres raes. This is clearly illogical. The presen value of a given paymen a a given fuure poin in ime is independen of which asse he paymen sems from. All sure paymens a he same dae should be discouned wih he same rae. On he oher hand, here is no reason o discoun paymens a differen daes wih he same discoun rae. The ineres rae on a loan will normally depend on he mauriy of he loan, and on he bond markes here will ofen be differences beween he yields on shor-erm bonds and long-erm bonds. The erm srucure of ineres raes is he relaion beween ineres raes and heir mauriy. In his chaper we firs provide a brief inroducion o he mos basic markes for borrowing and lending and discuss differen ypes of ineres raes and discoun facors as well as he relaion beween hem. We will briefly inroduce some heories of he level and shape of he erm srucure of ineres raes. Finally, we will discuss how informaion abou he erm srucure can be exraced from marke daa. 1.2 Markes for bonds and ineres raes This secion gives a brief descripion of markes for bonds and oher deb conracs. A bond is simply a radable loan agreemen. Mos bonds ha are issued and subsequenly raded a eiher organized exchanges or over-he-couner OTC) represen medium- or long-erm loans wih mauriies in excess of one year and ofen beween 10 and 30 years. Some shor-erm bonds are also issued and raded, bu much of he shor-erm borrowing aciviy akes place in he so-called money marke, where large financial insiuions including he cenral bank) and large corporaions form several ypes of loan agreemens wih mauriies ranging from a few hours up o around one year. These agreemens are usually no raded afer he original conracing. The ineres raes se in he money marke direcly affec he ineres raes ha banks offer and charge heir commercial 1

12 1.2 Markes for bonds and ineres raes 2 and household cusomers. Small invesors may paricipae in he money markes hrough money marke funds. Below, we inroduce he mos imporan ypes of bonds ha are raded. More deails on bond markes can be found in e.g. Fabozzi 2000). We also look a some of he deb conracs available in he money marke. In Chaper 2 we will discuss many oher ineres rae relaed securiies, such as fuures and opions on bonds and ineres raes. The issuer of he bond he debor or borrower) issues a conrac in which he is obligaed o pay cerain paymens a cerain fuure poins in ime. Typically, a bond issue consiss of a series of idenical bonds. The simples possible bond is a zero-coupon bond, which is a bond promising a single paymen a a given fuure dae, he mauriy dae of he bond, and no oher paymens. Bonds which promise more han one paymen when issued are referred o as coupon bonds. Typically, he paymens of coupon bonds follow a regular schedule so ha he paymens occur a regular inervals quarerly, semi-annually, or annually) and he size of each of he paymens is deermined by he face value, he coupon rae, and he amorizaion principle of he bond. The face value is also known as he par value or principal of he bond, and he coupon rae is also called he nominal rae or saed ineres rae. Mos coupon bonds are so-called bulle bonds or sraigh-coupon bonds where all he paymens before he final paymen are equal o he produc of he coupon rae and he face value. The final paymen a he mauriy dae is he sum of he same ineres rae paymen and he face value. Oher bonds are so-called annuiy bonds, which are consruced so ha he oal paymen is equal for all paymen daes. Each paymen is he sum of an ineres paymen and a parial repaymen of he face value. The ousanding deb and he ineres paymen are gradually decreasing over he life of an annuiy, so ha he repaymen increases over ime. Some bonds are so-called serial bonds where he face value is paid back in equal insalmens. The paymen a a given paymen dae is hen he sum of he insalmen and he ineres rae on he ousanding deb. The ineres rae paymens, and hence he oal paymens, will herefore decrease over he life of he bond. Finally, few bonds are perpeuiies or consols ha las forever and only pay ineres. The face value of a perpeuiy is never repaid. Mos coupon bonds have a fixed coupon rae, bu a small minoriy of bonds have coupon raes ha are rese periodically over he life of he bond. Such bonds are called floaing rae bonds. Typically, he coupon rae effecive for he paymen a he end of one period is se a he beginning of he period a he curren marke ineres rae for ha period, e.g. o he 6-monh ineres rae for a floaing rae bond wih semi-annual paymens. Bond markes can be divided ino he naional markes of differen counries and he inernaional marke also known as he Eurobond marke). The larges naional bond markes are hose of he U.S., Japan, Germany, Ialy, and France followed by oher Wesern European counries and Ausralia. In he naional marke of a counry, primarily bonds issued by domesic issuers are raded, bu ofen some bonds issued by cerain foreign governmens or corporaions or inernaional associaions are also raded. The bonds issued in a given naional marke mus comply wih he regulaion of ha paricular counry. Bonds issued in he less regulaed Eurobond marke are usually underwrien by an inernaional syndicae and offered o invesors in several counries simulaneously. Many Eurobonds are lised on one naional exchange, ofen in Luxembourg or London, bu mos of he rading in hese bonds akes place OTC over-he-couner). Oher Eurobonds are issued as a privae placemen wih financial insiuions. Mos Eurobonds are issued

13 1.2 Markes for bonds and ineres raes 3 in U.S. dollars Eurodollar bonds ), he common European currency Euro, Pound Serling, Swiss francs, or Japanese yen. Divided according o he ype of issuer, naional bond markes have wo or hree major caegories: governmen-relaed) bonds, corporae bonds, and in some counries morgage-backed bonds. In addiion, some bonds issued by inernaional insiuions or foreign governmens are ofen raded. Eurobonds are ypically issued by inernaional insiuions, governmens, or large corporaions. In mos naional bond markes, he major par of bond rading is in governmen bonds, which are simply bonds issued by he governmen o finance and refinance he public deb. In mos counries, such bonds can be considered o be free of defaul risk, and ineres raes in he governmen bond marke are hen a benchmark agains which he ineres raes on oher bonds are measured. However, in some economically and poliically unsable counries, he defaul risk on governmen bonds canno be ignored. In he U.S., governmen bonds are issued by he Deparmen of he Treasury and called Treasury securiies. These securiies are divided ino hree caegories: bills, noes, and bonds. Treasury bills or simply T-bills) are shor-erm securiies ha maure in one year or less from heir issue dae. T-bills are zero-coupon bonds since hey have a single paymen equal o he face value. Treasury noes and bonds are coupon-bearing bulle bonds wih semi-annual paymens. The only difference beween noes and bonds is he ime-omauriy when firs issued. Treasury noes are issued wih a ime-o-mauriy of 1-10 years, while Treasury bonds maure in more han 10 years and up o 30 years from heir issue dae. Treasury sells wo ypes of noes and bonds, fixed-principal and inflaion-indexed. The fixed-principal ype promises given dollar paymens in he fuure, whereas he dollar paymens of he inflaion-indexed ype are adjused o reflec inflaion in consumer prices. 1 Finally, he U.S. Treasury also issue socalled savings bonds o individuals and cerain organizaions, bu hese bonds are no subsequenly radable. While Treasury noes and bonds are issued as coupon bonds, he Treasury Deparmen inroduced he so-called STRIPS program in 1985 ha les invesors hold and rade he individual ineres and principal componens of mos Treasury noes and bonds as separae securiies. 2 These separae securiies, which are usually referred o as STRIPs, are zero-coupon bonds. Marke paricipans creae STRIPs by separaing he ineres and principal pars of a Treasury noe or bond. For example, a 10-year Treasury noe consiss of 20 semi-annual ineres paymens and a principal paymen payable a mauriy. When his securiy is sripped, each of he 20 ineres paymens and he principal paymen become separae securiies and can be held and ransferred separaely. 3 In some counries including he U.S., bonds issued by various public insiuions, e.g. uiliy companies, railway companies, expor suppor funds, ec., are backed by he governmen, so ha he defaul risk on such bonds is he risk ha he governmen defauls. In addiion, some bonds are issued by governmen-sponsored eniies creaed o faciliae borrowing and reduce borrowing 1 The principal value of an inflaion-indexed noe or bond is adjused before each paymen dae according o he change in he consumer price index. Since he semi-annual ineres paymens are compued as he produc of he fixed coupon rae and he curren principal, all he paymens of an inflaion-indexed noe or bond are inflaionadjused. 2 STRIPS is shor for Separae Trading of Regisered Ineres and Principal of Securiies. 3 More informaion on Treasury securiies can be found on he homepage of he Bureau of he Public Deb a he Deparmen of he Treasury, see

14 1.2 Markes for bonds and ineres raes 4 coss for e.g. farmers, homeowners, and sudens. However, hese bonds are ypically no backed by he governmen and are herefore exposed o he risk of defaul of he issuing organizaion. Bonds may also be issued by local governmens. In he U.S. such bonds are known as municipal bonds. In some counries, corporaions will radiionally borrow funds by issuing bonds, so-called corporae bonds. This is he case in he U.S., where here is a large marke for such bonds. In oher counries, e.g. Germany, corporaions end o borrow funds hrough bank loans, so ha he marke for corporae bonds is very limied. For corporae bonds, invesors canno ignore he possibiliy ha he issuer defauls and canno mee he obligaions represened by he bonds. Bond invesors can eiher perform heir own analysis of he crediworhiness of he issuer or rely on he analysis of professional raing agencies such as Moody s Invesors Service or Sandard & Poor s Corporaion. These agencies designae leer codes o bond issuers boh in he U.S. and in oher counries. Invesors will ypically rea bonds wih he same raing as having nearly) he same defaul risk. Due o he defaul risk, corporae bonds are raded a lower prices han similar defaul-free) governmen bonds. The managemen of he issuing corporaion can effecively ransfer wealh from bond-holders o equiy-holders, e.g. by increasing dividends, aking on more risky invesmen projecs, or issuing new bonds wih he same or even higher prioriy in case of defaul. Corporae bonds are ofen issued wih bond covenans or bond indenures ha resric managemen from implemening such acions. U.S. corporae bonds are ypically issued wih mauriies of years and are ofen callable bonds, so ha he issuer has he righ o buy back he bonds on cerain erms a given poins in ime and for a given price). Some corporae bonds are converible bonds meaning ha he bondholders may conver he bonds ino socks of he issuing corporaion on predeermined erms. Alhough mos corporae bonds are lised on a naional exchange, much of he rading in hese bonds is in he OTC marke. Morgage-backed bonds consiue a large par of some bond markes, e.g. in he U.S., Swizerland, and Denmark. A morgage is a loan ha can parly) finance he borrower s purchase of a given real esae propery, which is hen used as collaeral for he loan. Morgages can be residenial family houses, aparmens, ec.) or non-residenial corporaions, farms, ec.). The issuer of he loan he lender) is a financial insiuion. Typical morgages have a mauriy beween 15 and 30 years and are annuiies in he sense ha he oal scheduled paymen ineres plus repaymen) a all paymen daes are idenical. Fixed-rae morgages have a fixed ineres rae, while adjusable-rae morgages have an ineres rae which is rese periodically according o some reference rae. A characerisic feaure of mos morgages is he prepaymen opion. A any paymen dae in he life of he loan, he borrower has he righ o pay off all or par of he ousanding deb. This can occur due o a sale of he underlying real esae propery, bu can also occur afer a drop in marke ineres raes, since he borrower hen have he chance o ge a cheaper loan. Morgages are pooled eiher by he issuers or oher insiuions, who hen issue morgage-backed securiies ha have an ownership ineres in a given pool of morgage loans. The mos common ype of morgage-backed securiies is he so-called pass-hrough, where he pooling insiuion simply collecs he paymens from borrowers wih loans in a given pool and passes hrough he cash flow o invesors less some servicing and guaraneeing fees. Many pass-hroughs have

15 1.3 Discoun facors and zero-coupon bonds 5 paymen schemes equal o he paymen schemes of bonds, e.g. pass-hroughs issued on he basis of a pool of fixed-rae annuiy morgage loans have a paymen schedule equal o ha of annuiy bond. However, when borrowers in he pool prepay heir morgage, hese prepaymens are also passed hrough o he securiy-holders, so ha heir paymens will be differen from annuiies. In general, owners of pass-hrough securiies mus ake ino accoun he risk ha he morgage borrowers in he pool defaul on heir loans. In he U.S. mos pass-hroughs are issued by hree organizaions ha guaranee he paymens o he securiies even if borrowers defaul. These organizaions are he Governmen Naional Morgage Associaion called Ginnie Mae ), he Federal Home Loan Morgage Corporaion Freddie Mac ), and he Federal Naional Morgage Associaion Fannie Mae ). Ginnie Mae pass-hroughs are even guaraneed by he U.S. governmen, bu he securiies issued by he wo oher insiuions are also considered virually free of defaul risk. Finally, le us ake a brief look a some of he deb conracs made in he money marke. While we focus on he U.S. marke, similar conracs exis in many oher counries and many of he conracs are also made in he Euromarke. The deb conracs in he money marke are mainly zero-coupon loans, which have a single repaymen dae. Financial insiuions borrow large amouns over shor periods from each oher by issuing cerificaes of deposi, also known in he marke as CDs. In he Euromarke deposis are negoiaed for various erms and currencies, bu mos deposis are in U.S. dollars and for a period of one, hree, or six monhs. Ineres raes se on deposis a he London inerbank marke are called LIBOR raes LIBOR is shor for London Inerbank offered rae). To manage very shor-erm liquidiy, financial insiuions ofen agree on overnigh loans, socalled federal funds. The ineres rae charged on such loans is called he Fed funds rae. The Federal Reserve has a arge Fed funds rae and buys and sells securiies in open marke operaions o manage he liquidiy in he marke, hereby also affecing he Fed funds rae. Banks may obain emporary credi direcly from he Federal Reserve a he so-called discoun window. The ineres rae charged by he Fed on such credi is called he federal discoun rae, bu since such borrowing is quie uncommon nowadays, he federal discoun rae serves more as a signaling device for he arges of he Federal Reserve. Large corporaions, boh financial corporaions and ohers, ofen borrow shor-erm by issuing so-called commercial papers. Anoher sandard money marke conrac is a repurchase agreemen or simply repo. One pary of his conrac sells a cerain asse, e.g. a shor-erm Treasury bill, o he oher pary and promises o buy back ha asse a a given fuure dae a he marke price a ha dae. A repo is effecively a collaeralized loan, where he underlying asse serves as collaeral. As cenral banks in oher counries, he Federal Reserve in he U.S. paricipaes acively in he repo marke o implemen heir moneary policy. The ineres rae on repos is called he repo rae. 1.3 Discoun facors and zero-coupon bonds We will assume hroughou ha he face value is equal o 1 dollar) unless saed oherwise. Suppose ha a some dae a zero-coupon bond wih mauriy T is raded in he financial markes a a price of B T. This price reflecs he marke discoun facor for sure ime T paymens. If many zero-coupon bonds wih differen mauriies are raded, we can form he funcion T B T, which we call he marke) discoun funcion prevailing a ime. Noe ha B = 1, since

16 1.4 Zero-coupon raes and forward raes 6 he value of geing 1 dollar righ away is 1 dollar, of course. Presumably, all invesors will prefer geing 1 dollar a some ime T raher han a a laer ime S. Therefore, he discoun funcion mus be decreasing, i.e. 1.1) 1 B T B S 0, T < S. An example of an esimaed marke discoun funcion is shown in Figure 1.1 on page 17. Nex, consider a coupon bond wih paymen daes 1, 2,..., n, where we assume wihou loss of generaliy ha 1 < 2 < < n. The paymen a dae i is denoed by Y i. Such a coupon bond can be seen as a porfolio of zero-coupon bonds, namely a porfolio of Y 1 zero-coupon bonds mauring a 1, Y 2 zero-coupon bonds mauring a 2, ec. If all hese zero-coupon bonds are raded in he marke, he price of he coupon bond a any ime mus be 1.2) B = i> Y i B i, where he sum is over all fuure paymen daes of he coupon bond. If his relaion does no hold, here will be a clear arbirage opporuniy in he marke. Example 1.1 Consider a bulle bond wih a face value of 100, a coupon rae of 7%, annual paymens, and exacly hree years o mauriy. Suppose zero-coupon bonds are raded wih face values of 1 dollar and ime-o-mauriy of 1, 2, and 3 years, respecively. Assume ha he prices of hese zero-coupon bonds are B +1 = 0.94, B +2 = 0.90, and B +3 = According o 1.2), he price of he bulle bond mus hen be B = = If he price is lower han , riskfree profis can be locked in by buying he bulle bond and selling 7 one-year, 7 wo-year, and 107 hree-year zero-coupon bonds. If he price of he bulle bond is higher han , sell he bulle bond and buy 7 one-year, 7 wo-year, and 107 hree-year zero-coupon bonds. If no all he relevan zero-coupon bonds are raded, we canno jusify he relaion 1.2) as a resul of he no-arbirage principle. Sill, i is a valuable relaion. Suppose ha an invesor has deermined from privae or macro economic informaion) a discoun funcion showing he value she aribues o paymens a differen fuure poins in ime. Then she can value all sure cash flows in a consisen way by subsiuing ha discoun funcion ino 1.2). The marke prices of all bonds reflec a marke discoun funcion, which is he resul of he supply and demand for he bonds of all marke paricipans. We can hink of he marke discoun funcion as a very complex average of he individual discoun funcions of he marke paricipans. In mos markes only few zero-coupon bonds are raded, so ha informaion abou he discoun funcion mus be inferred from marke prices of coupon bonds. We discuss ways of doing ha in Secions Zero-coupon raes and forward raes Alhough discoun facors provide full informaion abou how o discoun amouns back and forh, i is prey hard o relae o a 5-year discoun facor of I is far easier o relae o he

17 1.4 Zero-coupon raes and forward raes 7 informaion ha he five-year ineres rae is 5%. Ineres raes are always quoed on an annual basis, i.e. as some percenage per year. However, o apply and assess he magniude of an ineres rae, we also need o know he compounding frequency of ha rae. More frequen compounding of a given ineres rae per year resuls in higher effecive ineres raes. Furhermore, we need o know a which ime he ineres rae is se or observed and for which period of ime he ineres rae applies. Spo raes applies o a period beginning a he ime he rae is se, whereas forward raes applies o a fuure period of ime. The precise definiions follow below Annual compounding Given he price B T a ime on a zero-coupon bond mauring a ime T, he relevan discoun rae beween ime and ime T is he yield on he zero-coupon bond, he so-called zero-coupon rae or spo rae for dae T. Le ŷ T hen have he following relaionship: 1.3) B T = 1 + ŷ T T ) ) or 1.4) ŷ T = B T ) 1/T ) 1. denoe his rae compued using annual compounding. We The zero-coupon raes as a funcion of mauriy is called he zero-coupon yield curve or simply he yield curve. I is one way o express he erm srucure of ineres raes. An example of a zero-coupon yield curve is shown in Figure 1.2 on page 17. While a zero-coupon or spo rae reflecs he price on a loan beween oday and a given fuure dae, a forward rae reflecs he price on a loan beween wo fuure daes. The annually compounded relevan forward rae a ime for he period beween ime T and ime S is denoed by ˆf T,S. Here, we have T < S. This is he rae, which is appropriae a ime for discouning beween ime T and S. We can hink of discouning from ime S back o ime by firs discouning from ime S o ime T and hen discouning from ime T o ime. We mus herefore have ha 1.5) 1 + ŷ S ) S ) = 1 + ŷ T ) T ) 1 + ˆf T,S ) S T ), from which we find ha ˆf T,S T )/S T ) = 1 + ŷt ) 1 + ŷ S 1. ) S )/S T ) We can also wrie 1.5) in erms of zero-coupon bond prices as ) S T ) 1.6) B S = B T T,S 1 + ˆf, so ha he forward rae is given by 1.7) ˆf T,S = Noe ha since B = 1, we have B T B S ) 1/S T ) 1. ˆf,S = B B S ) 1/S ) 1 = B S ) 1/S ) 1 = ŷ S, i.e. he forward rae for a period saring oday equals he zero-coupon rae or spo rae for he same period.

18 1.4 Zero-coupon raes and forward raes Compounding over oher discree periods LIBOR raes In pracice, many ineres raes are quoed using semi-annually, quarerly, or monhly compounding. An ineres rae or R per year compounded m imes a year, corresponds o a discoun facor of 1 + R/m) m over a year. The annually compounded ineres rae ha corresponds o an ineres rae of R compounded m imes a year is 1 + R/m) m 1. This is someimes called he effecive ineres rae corresponding o he nominal ineres rae R. As discussed earlier, ineres raes are se for loans wih various mauriies and currencies a he inernaional money markes, he mos commonly used being he LIBOR raes ha are fixed in London. Tradiionally, hese raes are quoed using a compounding period equal o he mauriy of he ineres rae. If, for example, he hree-monh ineres rae is l he presen value of one dollar paid hree monhs from now is Conversely, he hree-monh rae is More generally, he relaions are B = l = ) B T = and l ) 1 B l T T ) l T = 1 1 T B T ) 1.. per year, i means ha T,T +0.5 Similarly, a six-monh forward rae of L valid for he period [T, T + 0.5] means ha so ha B T +0.5 = B T L T,T +0.5 ) 1, T,T +0.5 L = 1 ) B T 0.5 B T More generally, 1.9) L T,S = 1 B T S T B S ) 1. Alhough such spo and forward raes are quoed on many differen money markes, we shall use he erm spo/forward) LIBOR rae for all such money marke ineres raes compued wih discree compounding Coninuous compounding Increasing he compounding frequency m, he effecive annual reurn of one dollar invesed a he ineres rae R per year increases o e R, due o he mahemaical resul saying ha 1 + m) R m = e R. lim m

19 1.4 Zero-coupon raes and forward raes 9 A nominal, coninuously compounded ineres rae R is equivalen o an annually compounded ineres rae of e R 1 which is bigger han R). Similarly, he zero-coupon bond price B T is relaed o he coninuously compounded zerocoupon rae y T by 1.10) B T = e yt T ), so ha 1.11) y T = 1 T ln BT. The funcion T y T is also a zero-coupon yield curve ha conains exacly he same informaion as he discoun funcion T B T and also he same informaion as he annually compounded yield curve T ŷ T. We have he following relaion beween he coninuously compounded and he annually compounded zero-coupon raes: y T = ln1 + ŷ T ). If f T,S denoes he coninuously compounded forward rae prevailing a ime for he period beween T and S, we mus have ha B S = B T T,S f e S T ), in analogy wih 1.6). Consequenly, 1.12) f T,S = ln BS ln B T S T Using 1.10), we ge he following relaion beween zero-coupon raes and forward raes under coninuous compounding: 1.13) f T,S = ys S ) y T T ). S T In he following chapers, we shall ofen focus on forward raes for fuure periods of infiniesimal lengh. The forward rae for an infiniesimal period saring a ime T is simply referred o as he forward rae for ime T and is defined as f T = lim S T f T,S. The funcion T f T is called he erm srucure of forward raes. Leing S T in he expression 1.12), we ge. 1.14) f T = ln BT T = BT / T B T, assuming ha he discoun funcion T B T is differeniable. Conversely, 1.15) B T = e T f u du. Applying 1.13), he relaion beween he infiniesimal forward rae and he spo raes can be wrien as 1.16) f T = [yt T )] T = y T + yt T T )

20 1.5 Deermining he zero-coupon yield curve: Boosrapping 10 under he assumpion of a differeniable erm srucure of spo raes T y T. The forward rae reflecs he slope of he zero-coupon yield curve. In paricular, he forward rae f T and he zerocoupon rae y T will coincide if and only if he zero-coupon yield curve has a horizonal angen a T. Conversely, we see from 1.15) and 1.10) ha 1.17) y T = 1 T f u du, T i.e. he zero-coupon rae is an average of he forward raes Differen ways o represen he erm srucure of ineres raes I is imporan o realize ha discoun facors, spo raes, and forward raes wih any compounding frequency) are perfecly equivalen ways of expressing he same informaion. If a complee yield curve of, say, quarerly compounded spo raes is given, we can compue he discoun funcion and spo raes and forward raes for any given period and wih any given compounding frequency. If a complee erm srucure of forward raes is known, we can compue discoun funcions and spo raes, ec. Academics frequenly apply coninuous compounding since he mahemaics involved in many relevan compuaions is more elegan when exponenials are used. There are even more ways of represening he erm srucure of ineres raes. Since mos bonds are bulle bonds, many raders and analyss are used o hinking in erms of yields of bulle bonds raher han in erms of discoun facors or zero-coupon raes. The par yield for a given mauriy is he coupon rae ha causes a bulle bond of he given mauriy o have a price equal o is face value. Again we have o fix he coupon period of he bond. U.S. reasury bonds ypically have semi-annual coupons which are herefore ofen used when compuing par yields. Given a discoun funcion T B T, he n-year par yield is he value of c ha solves he equaion 2n i=1 c B 2) +0.5i + B +n = 1. I reflecs he curren marke ineres rae for an n-year bulle bond. The par yield is closely relaed o he so-called swap rae, which is a key concep in he swap markes, cf. Secion Deermining he zero-coupon yield curve: Boosrapping In many bond markes only very few zero-coupon bonds are issued and raded. All bonds issued as coupon bonds will evenually become a zero-coupon bond afer heir nex-o-las paymen dae.) Usually, such zero-coupon bonds have a very shor mauriy. To obain knowledge of he marke zero-coupon yields for longer mauriies,we have o exrac informaion from he prices of raded coupon bonds. In some markes i is possible o consruc some longer-erm zero-coupon bonds by forming porfolios of raded coupon bonds. Marke prices of hese synheical zero-coupon bonds and he associaed zero-coupon yields can hen be derived. Example 1.2 Consider a marke where wo bulle bonds are raded, a 10% bond expiring in one year and a 5% bond expiring in wo years. Boh have annual paymens and a face value of 100. The one-year bond has he paymen srucure of a zero-coupon bond: 110 dollars in one year and nohing a all oher poins in ime. A share of 1/110 of his bond corresponds exacly o a

21 1.5 Deermining he zero-coupon yield curve: Boosrapping 11 zero-coupon bond paying one dollar in a year. If he price of he one-year bulle bond is 100, he one-year discoun facor is given by B +1 = The wo-year bond provides paymens of 5 dollars in one year and 105 dollars in wo years. Hence, i can be seen as a porfolio of five one-year zero-coupon bonds and 105 wo-year zero-coupon bonds, all wih a face value of one dollar. The price of he wo-year bulle bond is herefore cf. 1.2). Isolaing B +2, we ge B 2 = 5B B +2, 1.18) B +2 = B B+1. If for example he price of he wo-year bulle bond is 90, he wo-year discoun facor will be B +2 = From 1.18) we see ha we can consruc a wo-year zero-coupon bond as a porfolio of 1/105 unis of he wo-year bulle bond and 5/105 unis of he one-year zero-coupon bond. This is equivalen o a porfolio of 1/105 unis of he wo-year bulle bond and 5/ ) unis of he one-year bulle bond. Given he discoun facors, zero-coupon raes and forward raes can be calculaed as shown in Secion The example above can easily be generalized o more periods. Suppose we have M bonds wih mauriies of 1, 2,..., M periods, respecively, one paymen dae each period and idenical paymen dae. Then we can consruc successively zero-coupon bonds for each of hese mauriies and hence compue he marke discoun facors B +1, B +2,..., B +M. Firs, B +1 is compued using he shores bond. Then, B +2 is compued using he nex-o-shores bond and he already compued value of B +1, ec. Given he discoun facors B +1, B +2,..., B +M, we can compue he zero-coupon ineres raes and hence he zero-coupon yield curve up o ime + M for he M seleced mauriies). This approach is called boosrapping or yield curve sripping. Boosrapping also applies o he case where he mauriies of he M bonds are no all differen and regularly increasing as above. As long as he M bonds ogeher have a mos M differen paymen daes and each bond has a mos one paymen dae, where none of he bonds provide paymens, hen we can consruc zero-coupon bonds for each of hese paymen daes and compue he associaed discoun facors and raes. Le us denoe he paymen of bond i i = 1,..., M) a ime +j j = 1,..., M) by Y ij. Some of hese paymens may well be zero, e.g. if he bond maures before ime + M. Le B i denoe he price of bond i. From 1.2) we have ha he discoun facors, B +2,..., B +M B ) mus saisfy he sysem of equaions B 1 Y 11 Y Y 1M B 2 Y 21 Y Y 2M. = Y M1 Y M2... Y MM B M B +1 B +2.. B +M.

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