Analyzing the Term Structure of Credit Spreads on Corporate Bonds over Treasury Using the Extended- Nelson-Siegel Model

Transcription

1 Maser Thesis 30 ECTS Analyzing he Term Srucure of Credi Spreads on Corporae Bonds over Treasury Using he Exended- Nelson-Siegel Model Maser Thesis in Mahemaics wih Specializaion in Financial Engineering Auhors: Helena Frisk Vicoria Hallin Supervisor: Consuling Senior Lecurer Jan Röman Supervisor Examinaor Jan Röman Dimirii Silvesrov Keywords: Nelson-Siegel model, corporae bonds, governmen bonds, credi spreads, defaul risk, bond raing, invesmen grade bonds, speculaive grade bonds, spo rae, curren yield, forward rae, yield o mauriy, zero-coupon bond, accrued ineres, duraion, floaing rae noe, forward rae agreemen, bulle bond, selemen dae.

2 Acknowledgemens: We graefully acknowledge he valuable help wih daa, from SEB Capial Markes in Sockholm and Sandard & Poor s corporaion in London. Furher, we would like o hank our supervisor, Consuling Senior Lecurer Jan Röman a he Mälardalen Universiy Deparmen of Mahemaics and Physics for his help and commens. Finally, we would like o hank everyone ha was presen a our presenaion of his paper for heir suiable commens. Any errors are our own.

3 ABSTRACT In his paper we esimae he erm srucure of credi spreads for differen raing caegories wih an exension of he Nelson-Siegel model. In our sudy we analyse wheher credi spread changes depends on bond characerisics such as raing and mauriy. Our resuls indicae ha credi spread changes especially depend on raing and even if he bond is plus raed, minus raed or fla raed. The credi spread changes also depend on mauriy bu o a much less significan exen. From our analysis we can see ha he lower he raing caegory and he longer he mauriy of he bond he higher credi spread. Tha means ha bonds wih lower raings are more affeced han bonds wih higher raing. Furher, he resuls from sudying he spreads for differen invesmen grade raing caegories indicae ha credi spreads on plus raed bonds have significanly lower credi spreads han minus raed bonds and ha resul is in line wih our firs sudy. This paper is wrien as a maser hesis in mahemaics wih specializaion in financial engineering a Mälardalen Universiy deparmen of mahemaics and physics and in cooperaion wih Mälardalen Universiy School of economics. The paper includes 30 ECTS and corresponds o a school erm. This hesis is he finishing elemen for a maser degree in mahemaics wih specializaion in financial engineering and will give an insigh in he erm srucure of credi spreads on corporae bonds over reasury using he Exended- Nelson-Siegel Model. 2

4 CONTENTS ABSTRACT INTRODUCTION BACKGROUND PURPOSE PROBLEM SPECIFICATION AND DISCUSSION DELIMITATION TARGET GROUP THEORETICAL FRAMEWORK AN INTRODUCTION TO DEBT SECURITIES Governmen Bonds Municipal Bonds Morgage Bonds Fannie Maes Ginnie Maes Freddie Mac Corporae Bonds The Huge Aracion in Corporae Bonds Aracive Yields Dependable Income Safey Diversiy Markeabiliy Basic Terms of Corporae Bonds Mauriy Srucure Fixed Rae -, Floaing Rae- and Zero Coupon Bonds Accrued Ineres Rae Day Coun Convenion E/ Ac/ Ac/Ac NL/ Ineres Rae Risk The Risk Facors Tha Corporae Bond Invesors Face Raing Caegories Invesmen Grade Speculaive Grade Duraion and Convexiy The Governmen Bond and Corporae Bond Yield Curve Exemplifying Credi Spreads Defaul Risk Tax Premium Risk Premium Liquidiy Explaining he Yield Curve Yield o Mauriy and Zero-Coupon Raes THE NELSON-SIEGEL MODEL EMPIRICAL ANALYSIS PRACTICAL METHODOLOGY Finding he Credi Spreads wih he Exended-Nelson-Siegel Model Daa Descripion The Model

5 4 RESULT RESULT ANALYSIS ANALYSIS OF THE RESULTS CONCLUSION CONCLUSION REFERENCE LIST LITERATURE ELECTRONIC SOURCES WORKING PAPERS DEFINITIONS APPENDIX 4

6 INTRODUCTION This chaper conains an inroducion o he subjec. A background is presened, followed by a discussion abou he purpose of he hesis and will follow wih problem specificaion and discussion and delimiaion. Finally he assigned arge group is described. 5

7 . Background In lae years he corporae bond marke has become more liquid and wider. The number of corporae bonds has increased rapidly over he years. Due o his fac i has become more imporan o sudy he markes of insrumens wih defaul risk, such as corporae bonds. There is a broad lieraure on credi risk models bu here is lile empirical esing of he models. However, we believe ha here are several reasons o sudy he behaviour of credi spreads for differen raing caegories. In his paper we analyse he erm srucure of credi spread changes on invesmen grade Swedish corporae bonds. We have used a large se of daa from 2004 o 2007 of Swedish corporae bonds and a large se of Swedish governmen bonds for our sudy. The erm srucure of credi spreads is esimaed as he difference in erm srucure of spo raes on Swedish corporae bonds and Swedish governmen bonds. We look a credi spreads for several mauriies and our sudy begin wih esimaing he erm srucure of credi spreads for invesmen grade raing caegories: BBB, BB, A and AA. Furher, we coninue wih dividing he resuls beween plus raed, minus raed and neurally raed bonds e.g. AA divides ino AA, AA- and AA+. Principally, governmen bonds are used o esimae he risk-free erm srucure of spo raes and he corporae bonds are used o esimae he risky erm srucure of spo raes. We have divided his paper ino 5 pars: In he firs secion we will give a shor inroducion o deb securiies and he risk hey carry. The second secion gives an overview of he mehodology o exrac spo raes using he Exended Nelson-Siegel model and he four measures of fi. In he hird secion, we firs presen he daa and he esimaion resuls of he erm srucure of credi spreads. In he fourh secion, we analyze he main deerminans of credi spread changes for differen raing caegories and years o mauriy. Evenually, he las secion concludes. 6

8 .2 Purpose The purpose of his paper is o examine he erm srucure of credi spreads which is esimaed as he difference in erm srucure of spo raes on Swedish corporae bonds and Swedish governmen bonds and es wheher bond characerisics such as raing and mauriy has any influence on he credi spread changes..3 Problem specificaion and discussion We will in paricular examine if here is a risk premium in corporae bond spreads and, if so, why i exiss. We will exrac he erm srucure of spo raes for differen raing caegories using he Exended-Nelson-Siegel model. The erm srucure of credi spreads is esimaed as he difference in erm srucure of spo raes on Swedish corporae bonds and Swedish governmen bonds. The Exended-Nelson-Siegel model is used by many cenral banks. Sweden used he Exended-Nelson-Siegel model unil 200 bu is sill calculaing he Nelson-Siegel parameers o have a backup mehod. Insrumens wih high liquidiy and low credi risk should be used when esimaing he erm srucure. Tha is one of he reasons why many counries use governmen bonds. The daa Sweden uses for hese calculaion are he benchmark governmen bonds from 2-0 years, Treasury bills wih mauriies closes o 3,6,9,2 monhs and he repo rae. They are calculaing i on a daily basis and send i once a week o he Bank for Inernaional Selemens. From our analysis we expec ha he yields on corporae bonds are higher han he yields on governmen bonds because governmen bonds are assumed o be risk-free. Corporae bonds wih lower credi raing han AA are more risky han a governmen bond and should because of ha reason generae a higher reurn. Our sudy did no resul as we expeced. We have found ha he erm srucure of spo raes on Swedish corporae bonds are boh above and below he erm srucure of spo raes on Swedish governmen bonds and here is no academic evidence ha basis poins can be added o an AAA raed bond o find he yield for a bond wih anoher credi rae. BIS Papers NO 25,Zero-coupon yield curves: echnical documenaion, p 27 7

9 .4 Delimiaion As his hesis was o represen 30 ECTS credis, he sudy is limied o es he Exended- Nelson-Siegel Model. Moreover, we limied he selecion of daa o only a careful seleced se of corporae bonds beween he years 2004 o 2007 and our sudy include invesmen grade corporae bonds and corporaions which have a hisory in credi raing a Sandard & Poor s..5 Targe group Our proposed arge group is sudens sudying financial mahemaics on advanced level and has read up o Swedish D level a Universiy. Excellen knowledge in saisics and economics are assumed. 8

10 2 THEORETICAL FRAMEWORK In his secion, we will explain bonds and he differen ypes of bonds ha exis and he risk hey carry. Furher, we will examine credi spreads and he yield curve. We will also explain why we use spo raes when measuring credi spreads. Moreover, we will discuss possible explanaions and evaluae he imporance of credi spreads. Finally we will horoughly explain he Nelson-Siegel model and he exended version. 9

11 2. An Inroducion o Deb Securiies Bonds are securiies ha represen a loan. Bonds can be issued by a Sae or a corporaion. They give rise o regular paymen of coupons, which consiue he ineres on he loan, and redempion of he securiy a mauriy. The cash flow is herefore known in advance. Bonds represen an invesmen ha is less risky han equiies, bu also less lucraive over he long erm. A corporae bond is more risky han a reasury bond and he risk on corporae bonds can be measured in several ways. The credi risk, evaluaed according o he qualiy of he issuer, which is measured by he raing sysem. The raing, which is made in public, conribues o an efficien marke and allows he reurn on he bond o be linked o is risk. The marke risk, or he ineres rae risk, is analyzed as a funcion of he opporuniy cos and represened by he difference beween he reurn ensured by he bond and he reurn of he marke for an equivalen mauriy. Bonds are grouped ogeher according o mauriy or he qualiy of he issuer. There are hree main ypes of bonds:. Governmen bonds 2. Municipal bonds 3. Morgages 4. Corporae bonds 2.. Governmen Bonds Governmen bonds represen he borrowing of he governmen. Fixed income securiies are classified according o he lengh of ime before mauriy. These are markeable securiies ha are known collecively as Treasuries and he hree main caegories are:. Treasury Bills - deb securiies mauring in less han one year. 2. Treasury Noes - deb securiies mauring in one o 0 years. 3. Treasury Bonds - deb securiies mauring in more han 0 years. All deb issued by he governmen are regarded as exremely safe. Like companies, counries can defaul on paymens and one should keep in mind ha he deb of many developing counries does carry subsanial risk. Governmen bonds are he simples o 0

12 value; hey pay ineres a a fixed rae and have a saed principal. However, here is one complicaing facor; some bonds he governmen can force he invesor o give up a a saed price a he governmen s discreion i.e. callable bonds. However, his is only rue for bonds ha are issued before Municipal Bonds Municipal bonds are he bonds nex in urn in erms of risk afer governmen bonds. These are deb obligaions of saes, ciies, and sae or ciy auhoriies. These bonds have defaul risk, bu even if i can happen ciies don' go bankrup ha ofen. The major advanage o a municipal bond is ha he reurns are free from federal ax bu also because of hese ax savings, he yield on municipal bonds are usually lower han he yield of axable bonds Morgage Bonds Morgages are deb obligaions backed by real esae. Mos ypes of morgage bonds are issued by morgage companies, banks, and credi unions. There are differen ypes of morgage bonds. These include Ginnie Maes, Fannie Mae and Freddie Macs. Furher, we will discuss hese ypes and look a how hey compare o each oher Fannie Maes Fannie Maes go is sar during he Grea Depression, when Congress creaed he Federal Naional Morgage Associaion in 938 o help he counry lifing iself ou of his depression, and did so by creaing such agencies. This governmen program called FNMA or Fannie Mae for shor was creaed as a company who would help low, moderae and middle income families realize wha has been referred o as he "American Dream", home ownership. In 960, Fannie became parially separaed from he governmen and on 968, he company wen public and was lised on he NYSE which would allow hem o reach a broader secion of he populaion. Ye, some governmen connecions remain, and Congress coninues o provide he charer under which he company operaes bu provides no financial backing. This governmen connecion allows he company o operae under he guise of being federally backed, giving borrowers and lenders alike added securiy in he process. Since 999 Fannie Mae is a publicly raded company and is main consideraion is o produce a profi. Fannie Maes do differ from Ginnie Maes and while he backing is only 2 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p. 503

13 perceived in Fannie Mae, i is real in he case of a Ginnie Mae backed securiy or morgage Ginnie Maes Governmen Naional Morgage Associaion (GNMA) are morgages backed by a pool of morgages and provides a link beween capial markes (he lenders) and he Federal Housing markes. This makes Morgage backed Securiies more aracive o invesors, such as pension funds and he like. Besides providing a higher reurn han Treasury noes and having he governmen's backing agains defaul, Ginnie Maes have anoher advanage: They are highly liquid and can be resold on he secondary marke. Ginnie Maes have a very large minimum invesmen fee, bu one can purchase shares in Ginnie Mae muual funds for less. The main difference beween his company and Fannie Mae lies in he governmen backing. Furher, Freddie Mac and Fannie Mae creae morgage pools ha are somewha larger han hose of Ginnie Mae, and invesors look o hem o also provide a relaively high reurn compared wih oher governmen securiies Freddie Mac Freddie Mac is also a governmenal creaion. This company is also a buyer of morgages on he secondary marke. Like Fannie Mae, Freddie Mac provides money o lenders o coninue heir effors a selling morgages. This keeps he marke liquid, which is jus anoher way of saying ha i will remain funded. Remaining funded means ha money will be here i lend, and he ineres raes are kep lower as a resul. Money supply is he key elemen here. Freddie Mac, like Fannie Mae, is a publicly raded company and has been for hiry years Ibid 5 Ibid 2

14 2..4 Corporae Bonds A company can issue bonds jus as i can issue sock. Corporae bonds represen a loan and are issued by privae or public corporaions. The hree differen ypes of corporae bonds are:. Shor-erm noes mauring in less han five years 2. Inermediae-erm noes/bonds mauring in five o welve years 3. Long-erm bonds mauring in more han welve years The funds ha companies raise from selling bonds are used for a variey of purposes, from purchasing equipmen o building faciliies o expanding he business. The corporae bond marke is large and liquid and mos corporae bonds are raded Over- The-Couner (OTC). This marke does no exis in a specific locaion, i is made up of bond dealers and brokers from around he counry who rade deb securiies over he phone or elecronically. The OTC marke is much larger han he exchange markes, and he vas majoriy of bond ransacions ake place in his marke. Invesors in corporae bonds include large financial insiuions, such as endowmens, pension funds, muual funds, insurance companies and banks. Individuals, from very wealhy o people of modes means, also inves in corporae bonds because of he many aracions hese securiies offer. 6 When an invesor is invesing in a corporae bond he invesor is lending money o he corporaion ha has issued he bond. In reurn he corporaion promises o reurn your money paid for he bond on a, for he bond specified mauriy dae. Unil he mauriy dae he bond pays he invesor a saed rae of ineres, usually semi-annually. The ineres paymens you receive from corporae bonds are axable. However, unlike socks a bond does no give you an ownership ineres in he issuing corporaion. 7 Corporae bonds are characerized by higher yields because here is a higher risk of a company defauling han he governmen. The upside is ha hey can also be he mos 6 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p Ibid p

15 rewarding fixed income invesmens because of he risk he invesor mus ake on. Corporae bonds are backed by he credi of he issuing company. I is he company s abiliy o earn money and mee he obligaions of he deb issue ha deermines he bonds defaul risk. The higher he qualiy, he lower he ineres rae he invesor receives. Generally corporae bonds are divided ino invesmen grade and speculaive grade. Alhough many differen ypes of opion feaures can be presen on corporae bonds, callabiliy, sinking funds, and converibiliy are he mos common. A call provision on a bond gives he issuing company he righ o force he bondholder o sell he bond back o he company a a paricular price. The price is known and may vary over ime. Callable bonds mus offer a higher reurn o compensae he holder for a disadvanageous call since he righ o call res wih he company. The sinking fund reiremen is inended o preven he company from having o make one large paymen. The company generally has he righ of purchasing he bonds in he open marke or calling hem back o mee he sinking fund. Because of his he invesor will require a higher reurn. However, he presence of a sinking fund lowers he risk ha he company will defaul on he enire issue of bonds or perhaps any of i. Finally, converible bonds are bonds ha can be exchanged for anoher securiy, usually common equiy The Huge Aracion in Corporae Bonds The reason o he huge aracion in corporae bonds is because of a variey of reasons and some of he mos imporan are:. Aracive Yields 2. Dependable Income 3. Safey 4. Diversiy 5. Markeabiliy 8 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p

16 Aracive Yields Corporae bonds mos ofen offer higher yields han comparable governmen bonds. However, his higher yield is accompanied by higher risks Dependable Income Invesors who like a seady income from heir invesmens meanwhile preserving heir principal do ofen include corporae bonds in heir porfolios Safey Corporae bonds are raed and evaluaed based on heir credi hisory and abiliy o repay heir obligaions. The beer abiliy hey have shown o do so he higher he raing and he safer he invesmen for he invesor. We will laer on in his paper explain how corporae bonds are raed Diversiy A corporae bond provides he invesor o choose from a variey of secors, srucures and credi-qualiy characerisics o mee he objecives of he invesor Markeabiliy If an invesor comes o he poin ha he/she wan o sell a bond before mauriy, in mos insances he/she can do so easily and quickly because of he liquidiy of he marke. There are five main classificaions of issuers represening various secors ha issue corporae bonds:. Public Uiliies 2. Transporaion Companies 3. Indusrial Corporaions 4. Financial Services Companies 5. Conglomeraes 5

17 Basic Terms of Corporae Bonds Earlier in his paper we have already menioned mauriy bu here are some oher basic erms of bonds ha are of imporance o cover:. Mauriy 2. Srucure 3. Fixed-Rae 4. Floaing-Rae 5. Zero-Coupon 6. Accrued Ineres Mauriy The mauriy is one of he key invesmen feaures. The mauriy of he bond ells you when you should expec o ge your principal back and for how long you can expec o receive ineres paymens. However, i should be menioned ha some corporae bonds have call or redempion feaures ha can affec he dae when your principal is reurned. As saed earlier in his paper corporae bonds are jus like Treasuries divided ino hree groups:. Shor-erm noes mauring in less han five years 2. Inermediae-erm noes/bonds mauring in five o welve years 3. Long-erm bonds mauring in more han welve years Srucure However, if you do consider buying a corporae bond i is imporan o know abou a bond s srucure. When dealing wih radiional deb securiies he invesor lends he issuer a specified amoun of money for a specified ime. In reurn for his he invesor receives fixed paymens of ineres on a regular basis for he life of he bond wih he oal principal reurned a mauriy Fixed Rae -, Floaing Rae- and Zero Coupon Bonds For he las years he radiional fixed ineres rae has been inegraed wih a mixure of oher raes and he mos common of hese are jus Floaing Rae and Zero Coupon. 6

18 Floaing Rae bonds are bonds wih variable ineres raes ha are adjused periodically according o a specific marke rae. However, hese kinds of bonds have lower yields han hose of fixed rae securiies because such bonds offer proecion agains increases in ineres raes. 9 A Zero Coupon bond does no pay any coupons. Insead hey are sold a a deep discoun o face value and redeemed for he full face value a mauriy Accrued Ineres Rae To be able o price bonds he accrued ineres has o be calculaed. Accrued ineres is he ineres earned since he las coupon paymen. The accrued ineres rae can be calculaed as: AI 360 d = C 360 Where C is he coupon and d are he days since he las coupon. The year is assumed o have 360 days bu ha can be changed and depend on he day coun convenion. The bond can be priced wih eiher clean price or diry price and he accrued ineres is included in he diry price. To calculae he accrued ineres he day coun convenion mus be known. There are differen day coun convenions. 9 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p Ibid p

19 Day Coun Convenion 30/360 Each monh has 30 days and he year has 360 days. There are wo excepions. The monh is considered o have i acual days if he lae dae is he las day of February. The monh is considered o have is acual days when he laer dae is he 30 h or 3 s and he firs dae is no he 30 h or 3 s E/360 Each monh has 30 days and he year has 360 days. There is one excepion which is ha he monh is considered o have is acual days if he laer dae is he las day in February Ac/360 The monh is considered o have is acual days and he year is considered o have 360 days Ac/Ac The monh is considered o have is acual days and he year is considered o have is acual days NL/360 The monh is considered o have is acual days. There is one excepion, when here is a leap year February is considered o have 28 days Ineres Rae Risk Like all oher bonds, corporae bonds rise in value when ineres raes fall and fall in value when ineres raes rise. By holding a bond unil mauriy you may be less concerned abou hese price flucuaions also known as marke risk or ineres rae risk. A reason o his fac is because you will receive he face value of your bond a mauriy. This inverse relaionship beween bonds and ineres raes seem o confuse mos invesors, bu he explanaion is essenially sraighforward and wo of he mos obvious explanaions are: 2 Röman 2007 Lecure noes in Analyical Finance 2, p.4 2 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p

20 . When ineres raes rise new issuers will come o he marke wih higher yields han older securiies and because of his, he older securiies are now worh less han before and herefore heir price will decrease. 2. When ineres raes decline new issuers will come o he marke wih lower yields han he older securiies and because of his he older securiies are now worh more and herefore heir price will increase. As a resul, if you have o sell your bond before mauriy, i may be worh more or less han you paid for i. There are various hings ha will affec he ineres raes boh negaively and posiively. A rule of humb is ha ineres raes ypically climb when he economy is growing and fall during economic downurns. Furher, rising inflaion leads o rising ineres raes alhough a some poin higher raes hemselves become conribuors o higher inflaion and moderaing inflaion leads o lower ineres raes. Inflaion is one of he mos influenial forces on ineres raes The Risk Facors Tha Corporae Bond Invesors Face The risk of defaul or credi risk is he risk associaed wih any kind of credi even. I is associaed wih changes in credi qualiy which also includes downgrading or upgrading in raing. If a bond is downgraded, i will be perceived as riskier by he marke. This may resul in a decrease in bond value. Thus, even if no defaul occurs, a downgrading migh influence he value of he bond. There are hree op agencies ha deal in credi raings for he invesmen world. These are: Moody s Invesor Services, Sandard & Poor s Corporaion and Fich Invesors Service. Sandard & Poor s, Moody s and Fich are financial services companies. They provide credi raings, equiy research, risk soluions and much more. They rae bonds from invesmen grade o defaulable bonds. Each of hese agencies aims o provide a raing sysem o help invesors deermine he risk associaed wih invesing and

21 Raing Caegories In his secion we will explain he meaning of he differen raing qualiy s a company can have. We sar wih invesmen grade and coninue wih speculaive grade i.e. high-yield bonds. 4 Anyhing below hese grades is already in defaul. For our analysis we will only use invesmen grade bonds, bu ha will be explained furher on in his paper. Defaul is generally defined as one of he following:. Failure of an obligor o make imely paymen of principal and/or ineres under he conracual erms of any financial obligaion. 2. The bankrupcy filings, adminisraion, receivership, liquidaion or oher windingup or cessaion of business of an obligor. 3. The disressed or oher coercive exchange of an obligaion, where crediors were offered securiies wih diminished srucural or economic erms compared wih he exising obligaion. 4 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p

22 Invesmen Grade Fich: AAA Moody s: Aaa Sandard & Poors: AAA Bonds ha are raed like his have he highes credi qualiy and his raing denoe he lowes expecaion of credi risk. Invesmen paymens are proeced by an excepionally srong capaciy for paymen of financial commimens. This capaciy is highly unlikely o be adversely affeced by foreseeable evens. Fich: AA Moody s: Aa Sandard & Poors: AA Bonds ha are raed wih Sandard & Poor s AA are judged o have a very high credi qualiy and his raing indicaes expecaions of very low credi risk. They indicae very srong capaciy for paymen of financial commimens. This capaciy is no significanly vulnerable o foreseeable evens. They are raed lower han he bes bonds because invesmen paymens may no be proeced by an excepionally srong capaciy for paymen of financial commimens as higher raed bonds. Fich: A Moody s: A Sandard & Poors: A Bonds ha have Sandard & Poor s raing A are o be considered as upper medium-grade obligaions. This raing denoes expecaions of low credi risk. The capaciy for paymen of financial commimens is considered srong, bu elemens may be presen ha sugges a suscepibiliy o impairmen someimes in he fuure. 2

23 Fich: Moody s: Sandard & Poors: BBB Baa BBB This is he lowes invesmen grade caegory and hese raings are neiher highly proeced nor poorly secured. This is a good credi qualiy and his raing indicaes ha here are currenly expecaions of low credi risk. The capaciy for paymen of financial commimens is considered adequae bu adverse changes in circumsances and economic condiions are more likely o impair his capaciy Speculaive Grade Fich: BB Moody s: Ba Sandard & Poor s: BB Bonds ha are raed in his caegory are no invesmen grade and are judged o have speculaive elemens. I indicaes ha here is a possibiliy of credi risk developing, paricularly as he resul of adverse economic change over ime; however, business or financial alernaives may be available o allow financial commimens o be me. Fich: B Moody s: B Sandard & Poor s: B Bonds ha are raed B are highly speculaive and do generally lack characerisics of he desirable invesmen. They indicae ha significan credi risk is presen, bu a limied margin of safey remains. Financial commimens are currenly being me; however, capaciy for coninued paymen is coningen upon a susained, favourable business and economic environmen. 5 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p

24 Fich: CCC Moody s: Caa Sandard & Poor s: CCC Bonds ha are raed in his caegory are of poor sanding and defaul is a real possibiliy. These bonds may be in defaul and heir capaciy for meeing financial commimens is solely relian upon susained, favourable business or economic condiions. Fich: CC Moody s: Ca Sandard & Poor s: CC For bonds ha are raed in his caegory defaul of some kind appears probable. Fich: C Moody s: C Sandard & Poor s: C For bonds ha are raed in his caegory defaul is imminen. 6 The credi risk is also associaed wih changes in credi spreads, and he defaul even ec. The defaul risk of a bond is he possibiliy ha he bondholder does no receive he paymens saed in he conrac, such as principal amoun and ineres paymens. The credi risk of a bond arises because he level and he iming of payoffs o invesors may be uncerain. Because of his uncerainy, he corporae bonds should offer higher yields han comparable defaul free bonds, i.e. Treasuries. Consequenly, a corporae bond rades a a lower price in erms of mauriy and coupon han a corresponding governmen bond. The difference beween he yield on he risky bond and he yield of he corresponding defaul free bond is called he credi spread. In urn, Credi spread changes can be easily viewed as an excess reurn of corporae bonds over Treasuries. The credi spread will be discussed and explained laer on in his paper. 6 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p

25 2..5 Duraion and Convexiy The duraion of a bond is he weighed average of he bonds mauriy and measures he ime i akes for he bond o be profiable. The duraion of a zero-coupon bond is equal o is mauriy and he duraion for a coupon bearing bond is less han is mauriy. The duraion can also be inerpreed as he firs derivaive of he bond price wih respec o he ineres rae. Duraion or Macaulay s duraion can be expressed as or as = yi icie D P n N C i D = + n i P YTM + i= YTM Where P is he presen value of he bonds price, C is he coupon size, N is he nominal amoun and n is he number of years o mauriy. Duraion can be a risk measure and a long duraion means a higher risk because i akes longer before he bond is profiable. 7 An ineres change doesn usually resul in a linear change of he bond price as he duraion measures i. I is usually a curved funcion and he convexiy measures he curvaure of he bond price. Convexiy is he second derivaive of he bonds price wih respec o he ineres rae. The convexiy can be expressed as. 8 or as D = 2 P = 2 P y P 2 y [ C e i ] C 2 Convexiy = ( ) n + i i P YTM + 00 i i 7 Röman 2007 Lecure noes in Analyical Finance 2, p.87 8 Ibid p.90 24

26 2..6 The Governmen Bond and Corporae Bond Yield Curve Swedish governmen bonds and governmen bonds in general have almos no defaul risk and all bonds are from one issuer so he uncerainy abou dae and amoun of paymens is reduced. Also as previously saed, mos governmen bonds do no have any special provisions such as call feaures, making i possible o use only bonds wihou opions for he esimaion process. Since he qualiy of daa ses available for Swedish governmen bonds is very high, yield curve esimaion can be done wih high accuracy. However, for corporae bonds he siuaion is differen. Compared o he Swedish governmen yield curve here are several imporan new facors ha influence he corporae bond yield curve ha will arise. The mos imporan of hese facors is he defaul risk, ha is, he risk ha he issuing company is no able or perhaps no willing o complee heir obligaions o pay coupons or pay back he face value of he bond. Corporae bond holders ask for a premium o compensae for he defaul risk ha hey face. This may be significan in he case of low raed issuers. Furher, corporae bonds have a lower liquidiy so an invesor of corporae bonds also has he risk of no being able o sell a every poin in ime. Compared o governmen bonds, corporae bonds also have differen ax regulaions and a larger share of bonds wih call provisions or oher special feaures. As corporae bonds face more risk facors, invesors will ask for a premium for aking on hese risks Exemplifying Credi Spreads The yield curves for corporae bonds are significanly higher han for governmen bonds. Reasons for ha are ha observed credi spread are credi risk and a risk premium ha invesors ask for aking on his risk. This will be explained in more deail below. Wha is credi risk? Credi risk refers o he possibiliy ha a company migh defaul or ha he raing of he company worsens, so ha he bond loses value. Furher facors responsible for he spread are opions included in he bond, liquidiy, and axes. When we sudy he impac of credi risk we will sor he bonds by raings according o Sandard & Poor's corporaion. As previously saed he corporae bond yield curve consiss of he governmen bond yield curve and he credi spread curve. Therefore, he corporae bond yield curve will move more or less closely wih he governmen bond yield curve depending on how much and in which direcion credi spreads change. The credi spread can be viewed as he premium 9 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p. 288ff 25

27 ha invesors ask for because of he exra risk hey ake upon when invesing in corporae bonds and herefore he ineres rae on governmen bonds is seen as he risk less ineres rae. This secion will discuss possible explanaions and evaluaion he imporance of credi spreads. As previously explained he credi spread is defined as he difference beween he yield o mauriy on a corporae bond and he yield o mauriy on a governmen bond of he same mauriy, ha is, an alernaive for corporae bond excess loss: Credi Spread = ym CB ym GB Where ym is he yield o mauriy. The credi spreads should differ across raing classes. Furher, he credi spreads should also be posiive for each raing class because of defaul risk, ax premium, risk premium, and liquidiy ha corporae bonds come wih. These are he mos imporan facors for explaining credi spreads. All in all, hese four componens explain mos of he credi spread. The four componens are explained in more deail below: Defaul Risk I will happen ha some corporae bonds will defaul because corporaions face a significanly higher defaul risk han Treasuries, which inroduces a significan higher defaul risk for corporae bonds and herefore invesors will require a higher promised paymen o compensae for he expeced loss from defauls. For esimaing he yield curve, companies should have a comparable risk class. The yield curve can be esimaed using raing as an indicaor for defaul risk. The esimaion of he yield curve is made using only bonds of companies wih a cerain raing provided by Sandard & Poor s corporaion. When valuing liabiliies, one should also accoun for defaul risk. When invesing he amoun equivalen o discouned liabiliies, one probably would no be able o service he liabiliies because some bonds in he porfolio will defaul. The credi spread due o defaul risk is higher he lower he raing of he company is Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p. 280ff 2 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p. 288ff 26

28 Tax Premium Anoher reason for he spread is he differen ax regulaions for corporae and governmen bonds. Ineres paymens on governmen bonds are subjec only o federal axes, whereas corporae bonds are also axed a he sae level, from which governmen bonds are exemp. Because of ha he invesor will earn only ineres reduced by axes. This is why corporae bonds have a ax premium. Taxes accoun for a large par of he credi spread beween governmen bonds and corporae bonds Risk Premium Invesors should require a premium for he higher risk since he reurn on corporae bonds is riskier han he reurn on governmen bonds. The credi spreads on corporae bonds vary sysemaically wih he same facors ha affec common sock reurns. So, if invesors in common socks require compensaion for his risk, so should invesors in corporae bonds. Due o he fac ha he sysemaic marke risk can no be diversified he risk premium is a large par of he credi spread Liquidiy Liquidiy cerainly plays a role, bu he size of is effec is probably quie small. One pricing error relaed o corporae bonds is a price discoun for corporae bonds ha do no have much liquidiy. The risk of liquidiy inroduces he risk ha invesors of corporae bonds migh no be able o sell he bond a he ime hey wan o. For his exra risk hey expec a premium, which leads o a discoun on he price. Furher, he bond prices usually are averages of dealer quoes. For bonds wih a low liquidiy, dealers migh no updae heir quoes regularly because here is no much business o arac. This can resul in prices ha no longer reflec he marke prices. 24 However, one should noe ha he defaul risk, ax premium, risk premium, and liquidiy have only limied influence o explain he changes in he credi spreads. The remaining spread afer adjusing for hese facors is explained in large par by proxies for sysemaic risk facors in he sock marke. 22 Ibid p. 280ff 23 Ibid p. 280ff 24 Ibid p. 280ff 27

29 As poined ou previously we define credi spreads as he difference in yield o mauriy on a zero coupon corporae bond, he corporae bond spo rae and he yield o mauriy on a zero coupon governmen bond of he same mauriy, he governmen bond spo rae. Furher ahead in he paper when we will refer o his rae we will refer o i as he spo rae raher han he longer expression yield o mauriy on a zero coupon. However, we will sar wih defining he many definiions of raes ha exis. The rae ha mos common used rae o compare bonds is he yield o mauriy. The mehod used o calculae yield o mauriy varies across bond caegories and herefore he resuls may no be comparable. Basically, yield o mauriy is he inernal rae of reurn earned from holding a bond o mauriy. Alhough i is he mos common rae used he yield o mauriy have some disadvanages. The yield o mauriy is he reurn if all cash flows received before he horizon is invesed a he yield o mauriy a he horizon. 25 Anoher ype of rae is curren yield. Curren yield is simply he annual coupon paymen divided by he price. Curren yield is he yield you see in newspapers. The curren yield has very limied usefulness. Curren yield is no he reurn of he bond if held a mauriy and i s no he expeced reurn of he bond over he year. The curren yield on a bond ha does no pay ineres is zero. If you were o inves in a bond based of curren yield you would wih mos cerainy rejec bonds wih low coupons bu large reurns in form of capial gains. The curren yield is he coupon rae divided by he marke price of he bond. Anoher ype of ineres rae is he spo rae. The spo rae is he yield o mauriy on bonds ha pay only one cash flow o he invesor of he bond and hey have a grea imporance when valuaing bonds. A reason for ha is ha is bonds are no priced a a price equal o he presen value of heir presen value of heir cash flows discouned a he spo rae, profiable swaps will exis. A final ype of ineres raes ha we will explain here is he forward rae. The forward rae is he ineres rae on bonds where he dae when he obligaion is made and he dae he money is loaned are dissimilar. The basic reason why we will use spo raes raher han yield o mauriy on coupon deb in our sudy are ha arbirage argumens hold wih spo raes and no wih yield o mauriy. Because a risk less coupon paying bond can always be expressed as a porfolio of zeros, 25 Elon, Gruber, Brown and Goezmann 2007 Modern porfolio heory and invesmen analysis 7h ed p

30 spo raes are he raes ha mus be used o discoun cash flows on risk less coupon paying deb o preven arbirage. This is however no rue for yield o mauriy. In addiion, he yield o mauriy depends on coupon. Thus, if yield o mauriy is used o define he spread, he spread will depend on he coupon of he bond ha is picked. Finally, calculaing spread as difference in yield o mauriy on coupon-paying bonds wih he same mauriy means one is comparing bonds wih differen duraion and convexiy. The disadvanage of using spos is ha hey need o be esimaed Explaining he Yield Curve We have in our sudy seen ha while yields o mauriy on bonds of various mauriies are reasonably similar, yields do differ. Bonds wih shorer mauriies generally offer lower yields o mauriy han longer erm bonds (see graphs laer). The graphical relaionship beween yield o mauriy and he erm o mauriy is called he yield curve. The same relaionship is also known in he lieraure as erm srucure of ineres raes because i relaes yields o mauriy o he erm of each bond. Basically we can say ha he yield curve or erm srucure of ineres raes is he funcion ha associaes he level of he ineres rae wih each mauriy. To predic he fuure value of he rae insrumen o be prediced one can measure he curren raes, forecasing fuure raes and anicipaing shifs in he yield curves. The yield curve can be consiued hrough observaion of he prices of financial insrumens, by esablishing a relaionship beween he mauriies of he securiies and he levels of raes. Two alernaives exis: using yield o mauriy or using zero-coupon raes Yield o Mauriy and Zero-Coupon Raes An invesor ha is considering purchasing a bond is no quoed a promised rae of reurn. Therefore, insead he invesor mus use he mauriy dae, bond price, and coupon paymens o infer he reurn offered by he bond over is life. We can herefore define he yield o mauriy as he discoun rae ha makes he presen value of a bond s paymen equal o is price. The yield o mauriy of a bond is defined as he rae ha makes is = or quoaion price equal o he discouned sum of is fuure cash flows ( F i ) i,..., n 26 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p. 304ff 29

31 P = n F i i= ( + r) i Where n is he number of periods and i is he ineres rae period. 27 Excel provides a funcion for yield o mauriy and ha is: =Yield(selemen dae, mauriy dae, annual coupon rae, bond price, redempion, value as percen of par value, number of coupon paymens per year) The equaion above has only one unknown variable, he ineres rae, r. The bond s yield o mauriy is he bond s inernal rae of reurn, bu he use of r o define he reurn ha is acually obained for he bondholder presens a cerain number of disadvanages. In order for he rae o correspond o he reurn obained a mauriy, we mus be able o reinves he coupons a he same rae as r, which is no necessarily he case. In addiion, his definiion of he rae assumes ha he yield curve is fla and ha each bond has is own ineres rae, while we would wish o have a erm srucure for he raes wih a single rae associaed wih each mauriy. To achieve his, we urn o zero-coupon bonds. A zero-coupon securiy is a securiy ha only pays ou a single cash flow a is expiry dae. Le B be he elemenary zero-coupon securiy paying ou one euro a dae. The zerocoupon price and is mauriy rae of reurn respec he following relaionship: B = ( + r ) assuming ha he reference dae is 0. The complee se of raes, r for he differen zero-coupon mauriies allows us o define he range of zero-coupon raes. The use of zero-coupon raes allows us o consruc a yield curve ha associaes a single rae wih each mauriy. The curve ha is hereby obained is hen used as a basis for valuing rae insrumens and measuring he risks. Any bond ha 27 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p. 288ff 30

32 pays ou, F i cash flows a daes i, i =,..., n, can be modelled by a porfolio of zerocoupon securiies, wih each cash flow considered o be a zero coupon. 28 The number of zero-coupon securiies raded on he markes is no sufficien o allow a direc esimae of he yield curve using hose securiies. Several models have herefore been proposed using Treasuries wih mauriies ha allow all yield curves o be covered. These bonds are coupon bonds. In his case, each coupon is equaed o a zero-coupon securiy wih a mauriy ha corresponds o he dae on which he coupon falls. The Nelson-Siegel model allow he zero-coupon raes o be consruced heoreically. The yield curve can be buil from prices in he bond marke or from prices in he money marke. Governmen bonds are used if he yield curve is buil from bonds. Insrumens used from he money marke could be LIBOR raes, fuures and ineres swaps. The yield curve can be represened in hree differen ways. I can be represened by he yields from discoun bonds. I can also be represened by associaing he coninuously compounded spo rae 29 R (, T ) wih he discoun bond price Which gives The hird way is in erm of he forward rae [ R(, T)( T )] P(, T ) = exp [ p(, T )] ln R(, T ) = T f (, T ) = [ ln[ p(, T ) ] T Then he discoun funcion can be wrien as p ( ; T ) = exp T f (, u) du and he spo rae as 28 Bodie, Kane, Marcus 2007 Essenials of invesmens 6h ed p. 296ff 29 Jan Röman 09/0/2008, Lecure noes in Analyical Finance 2, p.? 3

33 T R (, T ) = f (, u) du T When deriving he erm srucure of ineres raes eiher he yield errors or he price errors can be minimized. For insrumens wih a shor mauriy he minimizing of price errors can lead o a large yield error. To give an equal weigh o each bond wihou consider ha he bonds have differen duraions leads o over fiing of long erm bonds wih he expense on he shorer erm bonds. A small change in he yield resuls in a small change in he price of a bond wih shor mauriy and a large change in he price of a bond wih longer mauriy. To solve his problem he bonds can be weighed by he inverse of is duraion. 30 The rading volume of a bond can decrease when i is close o is mauriy and his can resul in a large yield error because he quoed price may no reflec he price when rading ook place The Nelson-Siegel Model A parsimonious funcional descripion of he forward curve was proposed by Nelson and Siegel in 987. f s s s 0 2 exp τ τ τ (, s) β ( ) + β ( ) exp + β ( ) = However, i was exended by Svensson in 994. Tha model includes an exra polynomialexponenial erm: 32 f s s s s s exp τ τ τ τ 2 τ (, s) β ( ) + β ( ) exp + β ( ) exp + β ( ) = 2 If we inegrae he Nelson-Siegel forward rae expression above we will ge: 30 BIS Papers NO 25,Zero-coupon yield curves: echnical documenaion, pviii 3 Ibid 32 Jan Röman 09/0/2008, Lecure noes in Analyical Finance 2, p

34 33 ( ) ( ) ( ) ( ) + + = ds s s s r exp exp τ τ β τ β β Furher, if make he following change in variables we will ge: Variables: τ x s = and dx ds = τ 33 ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) + + = + + = + + = τ τ τ τ τ τ τ τ τ β τ β β τ β τ β β τ β τ β β x x x x x e e e dx e xe e dx e x dx e r This implies he spo yield curves below. Nelson and Siegel proposed a model for he yield curve, ( ) NS NS r Θ,. A model ha was exended by Svensson in 994, ( ) ENS ENS r Θ,. The Nelson-Siegel formula can be wrien:. ( ) ( ) ( ) ( ) j NS NS j m m m m m m r, 2 0, exp exp exp, ε τ τ τ β τ τ β β = Θ And he Exended-Nelson-Siegel formula can be wrien: 33 Jan Röman 09/0/2008, Lecure noes in Analyical Finance 2, p. 02

35 2. m m m exp exp exp ENS r, j exp m m τ m τ τ τ 2 τ 2 ENS τ τ ( ) m τ 2 m m, Θ = β 0 + β + β 2 exp + β 3 + ε, j ~ 2 wih ε ( ), j N 0, σ j We denoe he spo rae a ime, j { AAA, AA, A, BBB and BB} = by r, j and mauriy m of he bonds. The yield curve Θ NS = ( β, β, β τ ) and Θ NS = ( β, β, β, β, τ τ ) 0 2, 0 2 3, 2 for he Exended-Nelson-Siegel are some smooh ye flexible funcions represening he ineres raes as a funcion of mauriy m. We have implemened he model in Excel. We esimae he parameers ( β, β, β, β, τ andτ ) in he Exended-Nelson-Siegel model by minimizing he sum of squared bond price errors weighed by ( / Φ ): j Min P j P j β0, β, β2, β3, τ, τ 2 ENS [ ( β, β, β, β, τ, τ )] Φ j Where Φ equals he duraion price / ( + yield o mauriy) of he bond. The advanage of he Nelson-Siegel model is.. The model work well on ineres rae daa 2. The yield curve converges o a consan value r( ) = ( ) 3. We can inerpre he basis funcions, given by: lim β 0. ϕ = e ϕ2 = λ λ ϕ = 2 e λ λ λ e 34

36 Whereϕ denoes he parallel shif, ϕ2 he iling and ϕ3 he flexing. Where λ is a fixed coefficien ha deermines he exponenial decay of he firs and second componen in he Nelson-Siegel model. 34 As previously explained he shape and he form of he yield curve are deermined by he hree componens (four in he exended version) and heir associaed weighs in β. The firs componen β0 akes he value (consan) and can for ha reason be inerpreed as he overall level ha influences he long-run level of ineres raes. The second componen β converges o one as τ decreases o zero and converges o zero as τ approaches infiniy ( ) for a given. Because of ha his componen mosly influences shor-erm ineres raes. The hird and las componen in he Nelson-Siegel model β2 converges o zero as τ decreases o zero and as τ approaches infiniy ( ) bu is concave in τ, for a given. This componen is herefore he medium-erm componen. If he ime o mauriy m goes o infiniy, he spo rae converges o β 0. If m goes o zero, he spo rae converges o β 0 + β. To avoid negaive ineres raes, β0 and β 0 + β should be posiive. β0 can be inerpreed as he long-run ineres rae and β 0 + β as he insananeous ineres rae. This implies ha β can be inerpreed as he slope of he yield curve and he slope of he curve will be negaive if β is posiive and posiive if β is negaive. β also indicaes he speed a which he curve evolves owards is long-run rend. The hird componen in he Nelson-Siegel model β2 deermines he magniude and he direcion of he hump or hrough he yield curve. The parameer τ is a ime consan ha should be posiive in order o assure convergence o he long-erm value β 0. This parameer specifies he posiion of hump or hrough he yield curve. Given ha he firs componen is he only one ha equals one as τ approaches infiniy ( ), is resulan β 0 coefficien is usually linked wih long-erm ineres rae. 34 Jan Röman 09/0/2008, Lecure noes in Analyical Finance 2, p