Table of contents Yield to maturity between two coupon payment dates Influences on the yield to maturity: the coupon effect...

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1 able of conens. ime value of money. ime value of money..... Simple versus compound ineres..... Presen and fuure value Annuiies Coninuous discouning and compounding Bond yield measures Curren yield Yield o mauriy Yield o mauriy beween wo coupon paymen daes Influences on he yield o mauriy: he coupon effec Yield o call Oher yields Oher basic conceps Spo raes Forward raes Relaion beween spo rae and forward rae erm srucure of ineres raes Yield curves and shapes heories of erm srucures Expecaions hypohesis Liquidiy preferences Marke segmenaion and preferred habia heories Oher heories Bond price analysis Yield spread analysis ypes of spreads Deerminans of yield spreads Bond valuaion Valuaion of a zero-coupon bond Saic arbirage and valuaion of coupon bonds Srips markes Price / yield relaionship Risk measuremen Risk measuremen ools Duraion and modified duraion Definiion Inerpreaions and implici assumpions An example o illusrae he calculaion of duraion Deerminans of duraion Using duraion o approximae price changes Convexiy* Duraion and convexiy beween coupon paymen daes Impac of coupon paymens and ime lapse on duraion Resricions on using he duraion and convexiy Porfolio duraion and convexiy... 63

2 .6 Credi risk Indusry consideraions Raio analysis Credi raing and raing agencies * final level

3 . ime value of money. ime value of money he ime value of money principle saes ha a dollar oday is worh more han a dollar omorrow. his is due o he fac ha he presen is cerain, bu he fuure is no; presen consumpion is beer han deferring graificaion o he fuure. For his reason, an incenive is required o moivae us o defer graificaion. his incenive is represened by he ineres rae. Ineres raes are usually expressed on an annual basis, and express he remuneraion ha has o be paid from he borrower o he lender for he service of lending money. here are several ypes of ineres raes... Simple versus compound ineres Simple ineres assumes ha ineres does no iself earn ineres, and is calculaed by he following formula: Simple ineres = (iniial value) (ineres rae) (number of years) he iniial value is he principal amoun on which ineres is paid over a given period. Wha is he simple ineres earned on 000 CHF invesed a 7% p.a. (per annum) afer 0 years? he answer is '000 CHF = 700 CHF However in he real world ineres paymens received are reinvesed o earn more ineres in subsequen periods. Compound ineres assumes ha ineres is reinvesed; so compound ineres is simple ineres plus ineres earned on ineres. he formula o calculae compound ineres on a given iniial value over a given period is: Compound ineres = (iniial amoun) [( + ineres rae) number of years ] Wha is he compounded ineres earned on 000 CHF invesed a R=7% p.a. (per annum) afer 0 years? he answer is '000 CHF [(+0.07) 0 - ] = CHF chaper / page

4 Proof: Year Capial a begin of period Ineres [] []=[] R Capial a end of period []+[] So, we see ha, afer en years, he compound ineres is CHF, as opposed o he simple ineres of 700 CHF... Presen and fuure value he process of deermining he presen value of a fuure paymen (or receip) or series of fuure paymens (or receips) is called discouning. he compound ineres rae used for discouning cash flows is also called he discoun rae. So he presen value (or acual value) of a fuure income is given by: Fuure value Presen value = number of (+ Ineres rae) he presen value of a promised fuure cash flow is inversely relaed o boh he lengh of he invesmen period and he level of ineres raes. A financial firm offers o pay you 00'000 CHF in 0 years, if you give he firm 60'000 CHF oday. Using a 7% ineres rae, he presen value of 00'000 CHF in 0 years is: 00'000 Presen value = = 50'835CHF 0 (.07) Acceping his offer would imply paying 60'000 CHF for somehing ha is worh 50'835 CHF. You should definiely refuse! he process of finding he fuure value of he paymen (or receip) or series of paymens (or receips) using he concep of compound ineres is known as compounding. he general formula for compounding is: years Fuure value = (Presen value) ( + number of Ineres rae) years 00 CHF are deposied in a bank accoun wih a 5% annual ineres rae. Wha is he balance of he accoun a he end of he firs and he second year respecively? A he end of he firs year, we have A he end of he second year, we have 00 ( ) = 05 CHF 05 ( ) = 0.5 CHF chaper / page

5 Why no jus 0 CHF? Because we also have earned a 5% ineres on he 5 CHF paid a he end of he firs year (5 5% = 0.5). We could also have direcly saed: 00 (+0.05) = 0.5 CHF A high ineres rae environmen and long invesmen period lead o greaer accumulaion of compound ineres. Suppose ha '000 CHF were deposied in a saving accoun on s of January 934. Wha is he balance on 3s of December 000, if ineres was paid a a rae of 0.5%? here have been 67 years of compounding. he final balance is: '000 (.05) 67 = 804' CHF In he case of simple ineres, we would have a balance of only '000 + (' ) = 8'035 CHF..3 Annuiies In he special case of an annuiy, a fixed amoun of money is paid each year for a specified number of years. he presen value of his series of cash flows is given by he following formula: Presen va lue n = ( + R) = = R ( + R) n When using his formula we suppose ha he firs paymen is received in one year from now. he same financial firm as above offers o pay you CHF a he end of each year during 0 years, if you give he firm 70'000 CHF oday. Using a 7% ineres rae, he presen value of his 0-year annuiy is: 0'000 Presen value = = 70' ( 0.07) + Acceping his offer would imply paying 70'000 CHF for somehing ha is worh CHF. You should definiely accep i! he fuure value of he same series of cash-flows assumes ha all individual cash-flows are reinvesed a he same ineres rae R, and can be calculaed wih he following formula: n + = ( R) Fuure value ( + R) = = 0 R n Like for he presen value, he formula above supposes ha he firs cash flow is received in one year from now. Le us illusrae his wih a simple example. chaper / page 3

6 Wha is he fuure value of he series of coupons of a 7%, 0-year bond purchased a par ('000 CHF), if we assume ha all paymens are reinvesed a a 7% rae? he answer is: ( ) = he simple ineres in his case would have been: ' = 700 CHF 67.5 CHF (= CHF 700 CHF) is he addiional amoun earned by reinvesing he coupon paymens and earning ineres on he ineres...4 Coninuous discouning and compounding Compounding can ake place no only wih annual frequency, bu also wih higher frequency. I can be shown ha if here are m compounds per year (i.e. ineres is paid m imes per year) hen an iniial amoun N 0 invesed a an annual ineres rae R during n years becomes N R m m n 0 ( + ). When m ends o infiniy (i.e. ineres is paid a every insan), i can be shown ha his formula becomes following formula: N R n 0 e, where e.78 is he Euler number. So we can use he Fuure value = (Acual value) e ime Insananeous ineres rae Coninuous compounding will lead o a higher fuure value. As ineres is paid coninuously, here is more ineres on ineres. 00 CHF are deposied in a bank accoun wih a 5% coninuous annual ineres. Wha is he accoun balance a he end of he firs year and second year respecively, using coninuous compounding? A he end of he firs year, he balance is A he end of he second year, he balance is 00 e 0.05 = 05.3 CHF 00 e 0.05 = 0.5 CHF chaper / page 4

7 . Bond yield measures.. Curren yield he curren yield of a bond is simply he annual coupon paymen divided by he marke price of he bond (excluding accrued ineres). Curren yield = Annual coupon in USD Price in USD Since he coupon rae is generally fixed, he bond s curren yield varies inversely wih he bond s price. As he bond s price rises (declines), is curren yield falls (increases) since is coupon reurn is now a lesser (larger) amoun per USD of bond value. So, all oher facors held consan, a bond wih a higher curren yield sells a a lower price! he XYZ % bond is quoed a 98. on he Is curren yield is 3.5/98. = 3.3%. he curren yield only akes ino accoun he annual coupon income of he bond, and herefore, i is no an adequae ool o compare wo bonds. For example, he curren yield of a zero-coupon bond is zero as i pays no coupon. he curren yield of a bond priced under par will decrease as he bond approaches mauriy (he coupon is fixed, bu he price rises oward par). One should noe ha some Japanese bond dealers sill use a modified version of he curren yield, also called Japanese curren yield, defined as: Price (%) 00 Annual coupon - Remaining life Japanese curren yield = Price he Japanese curren yield considers no only he coupon paymen, bu also he capial gains/losses made by he invesor over he life of he bond. hus, a bond purchased a premium will have a lower curren yield due o capial losses suffered over is life-ime, while bonds purchased a discoun will exhibi higher curren yield due o he capial gains componen... Yield o mauriy he yield o mauriy (YM) is he discoun rae ha equaes he presen value of he bond s fuure cash flows ill mauriy wih he curren marke price of he bond. Jus afer he coupon paymen, we have : P = = = + ( + YM) ( + YM) ( + YM) ( + YM) chaper / page 5

8 where: P price of he bond (marke price) cash flow received a he end of period (coupons or repaymen) remaining life of he bond (ime o mauriy). I can also be defined as he inernal rae of reurn (IRR) of he invesmen in he bond. An invesor can buy a bond for 6.00 wih a 0% coupon, '000 CHF face value and 4 years unil mauriy. he coupon has jus been paid. Wha is he bond s yield o mauriy? he yield o mauriy YM solves he following equaion: ' = 4 3 ( + YM) ( + YM) ( + YM) ( + YM) ' 60 By ieraion wih a compuer, we find he correc answer, which is YM = 5.44%. he yield o mauriy assumes ha he bond is held o mauriy, and ha all cash flows are received as scheduled hrough final mauriy. As will be seen laer (in paragraph.4.3), he yield o mauriy should no be confused wih he oal reurn on he bond invesmen. he case of six-monhly coupons is easy o handle. Firs, calculae a six-monhly yield o mauriy YM S : P = = ( + YM ) ( + YM ) ( + YM ) ( + YM ) S = S + S where is he cash flow received a he end of period semeser (coupons or repaymen), and is he number of semesers in he remaining life of he bond (ime o mauriy). hen, conver his six-monhly yield in an annual yield YM A using one of he following formulae: on a Euromarke: S YM A = ( + YM ) S on he US or English marke: YM US A = YM by convenion, he annual yields are no compounded on hese markes (which gives a lower yield han on he Euromarkes). he following example will illusrae his concep. S chaper / page 6

9 A 0-year eurobond pays semi-annually an annual coupons of 6% and is quoed a he coupon has jus been paid. Wha is is yield o mauriy? Wha would be is yield o mauriy if he coupon was paid annually? he yield o mauriy solves: 3 0 = + ( + YM ) S ( + YM S ) ( + YM S ) he soluion is YMS =.37%. he annual corresponding yield is YM A = ( + YM S ) = 4.79% If he coupon was paid annually, we would have a yield of 4.7%, which is lower. his is predicable, as a semi-annual ineres paymen allows ineres compounding. Noe ha for a US bond, we would have.37% = 4.74%! he same mehodology can be used in he case of quarerly coupons.... Yield o mauriy beween wo coupon paymen daes Beween wo coupon paymen daes, he buyer of a bond mus pay he accrued ineres o he seller. he accrued ineres is due for he fracion f of he oal period beween wo coupon daes las coupon oday nex coupon f f one period Hence, he oal price o be paid for he bond is (in he case of an annual coupon paymen): oal price = Quoed price + f Coupon he formula o calculae he yield o mauriy mus be modified o include accrued ineres in he oal price: P+ f C = = = f f f ( + YM) ( + YM) ( + YM) ( + YM) f = ( + YM) + ( + YM) ( + YM) ( + YM) f Le us illusrae his. chaper / page 7

10 A bond wih an annual coupon of 6% is quoed a on he marke, 9 years and 3 monhs before is mauriy. Wha is is yield? We have: he effecive price paid is and he yield o mauriy solves:.5 f = 9 / = 0.75 years = ( ) ( ) ( ) ( ) = + YM YM + YM + YM he soluion is YM = 4.90% (by ieraion). While he mehodology is general, one should always keep in mind ha he way o calculae he number of days for he accrued ineres varies from one counry o anoher.... Influences on he yield o mauriy: he coupon effec he yield o mauriy of wo bonds having he same mauriy bu differen cash flows is no necessarily he same. his is called he coupon effec or coupon bias. Bond A is a wo-year 0% coupon, while bond B is a wo-year 5% coupon. he reurns on money obained for one year and wo years are R 0, = 6%, and R 0, = 7% respecively. he bonds prices are P P A B 0 0 = = = = Bu he yields o mauriy of he wo bonds are differen: P P A B 0 0 = + ( + k ) ( + k ) A A 5 05 = + ( + k ) ( + k ) B B = YM = % = YM = % B A he differences in he bond yields arise because he yield o mauriy is a complex average of he spo raes applied o one and wo years cash invesmens. In our example, bond B has a greaer fracion of is value ied o he higher wo years ineres rae. Noe ha here is no coupon effec wih zero-coupon bonds. chaper / page 8

11 If one compares he price formula of a coupon bearing bond P = = = wih he definiion of he yield o mauriy ( + R 0, ) ( + R 0, ) ( + R 0, ) ( + R 0, ) P = = = ( + YM) ( + YM) ( + YM) ( YM) i is clear ha he yield o mauriy is a complex average of he spo raes. Hence, he reader should carefully disinguish beween yield o mauriy and he spo rae R 0,. In fac, if he series of raes R 0, are increasing, one can show ha he yield o mauriy (YM) will underesimae he corresponding spo rae R 0,. R 0, R 0, YM R 0, Mauriy Figure -: Yield o mauriy versus spo rae We can illusrae his by he following example : Le us consider he following increasing spo raes: R 0, = %, R 0, = 4%, R 0,3 = 5%, R 0,4 = 5.5%, R 0,5 = 6%. If we selec hree bonds A, B, and C differing only by heir coupon raes, we can compue heir prices and heir reurns o mauriy. A B C Mauriy 5 years 5 years 5 years Annual coupon rae 0% 3% 0% Repaymen 00% 00% 00% Price Yield o mauriy 6% 5.9% 5.76% herefore, i is clear ha using coupon-paying bonds, we will underesimae he effecive spo rae for he considered mauriy (6% for 5 years in our example). he bias will increase for larger coupon raes. Source : DUMON Pierre André, 995, Les obligaions ordinaires: ypologie, procédures d émission e aspecs boursiers, HEC-Universiy of Geneva, Geneva. chaper / page 9

12 If he series of raes R 0, are decreasing, one can show ha he yield o mauriy (YM) will overesimae he corresponding spo rae (R 0, ). R 0, R 0, YM R 0, Mauriy Figure -: Yield o mauriy versus spo rae We can illusrae his by he following example: Le us consider he following decreasing spo raes: R 0, = 6%, R 0, = 4%, R 0,3 = 3%, R 0,4 =.5%, R 0,5 = %. If we selec hree bonds A, B, and C differing only by heir coupon raes, we can compue heir prices and heir yields o mauriy. A B C Mauriy 5 years 5 years 5 years Annual coupon rae 0% 3% 0% Repaymen 00% 00% 00% Price Yield o mauriy %.08%.% herefore, i is clear ha using coupon-paying bonds, we will overesimae he effecive spo rae for he considered mauriy (% for 5 years in our example). he bias will increase for larger coupon raes. In he case of a coupon-bearing bond, he spo rae (R 0, ) and he yield o mauriy (YM) are usually unequal. Bu his issue will be deal wih laer...3 Yield o call For callable bonds, he yield o call is he discoun rae YM C ha equaes he presen value of he bond s fuure cash flows received hrough he call dae o he bond s curren marke price: C P = = = ( + YM ) ( + YM ) ( + YM ) ( YM ) C C where is he cash flow received a he end of period (coupons or repaymen), and C is he remaining ime unil he call dae. I assumes ha he bond will be called, and ha all cash flows are received as scheduled hrough he call dae. C C C C chaper / page 0

13 An invesor can buy he same bond as in he previous example (0% coupon, '000 CHF face value, 4-year bond, quoed price of 6.00). he bond is callable in hree years a 03. Wha is he bond s yield o call? he yield o call YMc solves he following equaion: ( + YM ) ( + YM ) ( + YM ) 3 C C C + = 60 By successive approximaion, we find he correc answer, which is YM = 5.07%. he yield o call differs from he yield o mauriy as he discouning period is shorer (since he call dae precedes he mauriy dae), and he final cash flow is generally higher (since he call price is generally above par-value)...4 Oher yields Yield o average life he yield o average life YM AL is only used o compare bonds wih a series of principal repaymen (like sinking fund bonds, morgage backed securiies,...) and bulle bonds ha repay principal a mauriy. For simpliciy, he full principal repaymen is supposed o occur on he average life dae. he average life of a bond is he weighed average mauriy of he principal repaymen (noe ha he coupon rae plays no role in he average life, as i only considers principal repaymens): Principal paid a ime Average life in years = AL = oal principal o be repaid = 0 he yield o average life is simply he inernal rae of reurn (IRR) o he average life dae (as if he average life dae was he final mauriy dae of he bond): AL P = + = ( + YM ) ( + YM ) ( + YM ) C AL AL ( + YM ) ( + YM ) AL AL AL + = C AL AL Principal + C AL +... where C is he coupon received a he end of period (wihou repaymen), and AL is he average life of he bond. All principal repaymen are assumed o be made on he average dae. An invesor can buy a 5% coupon, '000 CHF face value, 0-year bond, quoed a a price of 0. he bond has a 90% sinker 3, wih sinking funds paymens saring a he end of he firs year, and repaying 0% of he bonds annually hrough he ninh year. All repaymens are made a par value. he average life is: 00 AL = = years he sinker percenage is he percenage of bonds reired before mauriy. chaper / page

14 he yield o average life solves he following equaion: ( + YM ) ( + YM ) ( + YM ) ( + YM ) 5 55 AL AL AL AL = 00 Noe ha he las paymen is '05 CHF because we only earn ineres (of 5 CHF) for half a year, as he average life is five years and a half. By ieraion, we find he correc answer, which is YM AL = 4.58%. In essence, our bond is roughly comparable wih a bulle bond giving a 4.58% yield o mauriy, and mauring in 5.5 years. Call-adjused yield For a callable bond, he call-adjused yield is simply he yield o mauriy a he grossed-up, non callable equivalen bond price. Assume XYZ Inc. can issue a par (= 00.00) a 8% coupon, 0-year bond, wih an 8% yield o mauriy. he bond is callable in 5 years a A he same ime, a similar long erm governmen bond yields 7%. A firs, one may hink ha he invesor receives % (= 00 basis poins) addiional yield for he credi risk of XYZ. Bu if (for example) he call opion is valued a 4.00 poins, we have: Price of he non callable bond = Price of he callable bond + Value of he call opion or Price of he non callable bond = = he yield o mauriy of our bond a par is 8% (quoed yield). Bu he yield o mauriy of he same bond if he price was is 7.4 (call-adjused yield). hus, our invesor receives only 0.4% (4 basis poins) for he credi risk of XYZ...5 Oher basic conceps hree daes are essenial while deermining any rae of ineres: he commimen dae, which is he dae a which he borrower and lender se he fixed rae on he loan. he lending dae, a which he money is o be loaned. he repaymen dae, a which he money is o be repaid. In he following discussion, we will use he concep of spo raes and forward raes which are defined below...5. Spo raes he spo rae, denoed by R 0,, is defined as he annual ineres rae received on a pure discoun securiy 4 (zero-coupon bond) mauring a ime. I is, a ime 0, he required rae of reurn o lend money from ime 0 o ime, if only one final paymen is made for boh ineres and principal. 4 Noe ha he exac definiion of a pure discoun bond is a bond ha pays CHF a mauriy. chaper / page

15 One should remember ha: he commimen dae and he lending dae are he same. spo raes are ineres raes on loans or bonds ha pay only one cash flow o he invesor. Generally, spo raes are quoed as annual raes. A bond involves an invesmen of CHF and reurns a principal of '000 CHF in exacly wo years. Wha is he wo years spo rae? Wha abou he one year spo rae? As he bond is a pure discoun bond, is reurn will be he wo-year spo rae. 000 ' = ( + R ) R = % p.a. 0 0,, he rae is expressed on an annualised basis. Nohing can be said abou he one-year spo rae...5. Forward raes he forward rae, denoed by F,h, is he rae of ineres on a bond where he commimen dae (0) and he dae he money is len () are differen. If a commimen is made oday on a wo-years loan o begin in one year (), he annualised ineres rae from year o year 3 (from year o year h) is a forward rae. One should remember ha: he commimen dae is oday, bu he lending dae differs. forward raes are ineres raes on loans or bonds ha pay only one cash flow o he invesor. Forward raes are also generally quoed as annual raes. A commimen involves a loan of CHF in one year and a principal and ineres repaymen of '000 CHF in hree years. Wha is he wo-year forward rae o begin in year one? As he bond is a pure discoun bond, is reurn will be he wo-year forward rae. 000 ' = ( + F ) he rae is expressed on an annualised basis. F = 9% p.a. 3, 3, chaper / page 3

16 ..5.3 Relaion beween spo rae and forward rae Generally, he forward rae can be calculaed as he raio of he end-of-period wealh o he beginning-of-period wealh, or as he raio of he corresponding spo raes: F, = ( + R 0, ) ( + R 0, ) he spo raes are R 0, = 6%, R 0, = 7% and R 0,3 = 7.5%. Wha is he implici one-year forward rae a he end of he firs year? A he end of he second year? he one-year forward rae a he end of he firs year can be derived by using wo consecuive spo raes. If he invesor invess 00 CHF for wo years a he curren spo rae R 0,, he will receive a he end of he second year: 00 ( ) ( ) = 4.49 CHF Anoher soluion would be o inves 00 CHF for one year a he curren spo rae R 0,, and o reinves he proceeds a he forward rae F,. Noe ha all posiions in his rading sraegy are deermined a ime 0. A he end of he firs year, he invesor will receive 00 (.06) = CHF He will reinves his amoun a he forward rae, and he should end wih he same amoun as he wo-year sraegy. Hence: and we have: ( + F, ) = 4.49 CHF Wealh posiion (end of year ) 4.49 F, = = = = 8. 0% Wealh posiion (end of year ) Similarly, we can calculae he implici one year forward rae a he end of he second year: 3 Wealh posiion (end of year 3) 00 (. 075) F 3, = = Wealh posiion (end of year ) 4.49 = = 8. 5% By consrucion, he spo rae may also be seen as he geomeric average of implici consecuive forward raes: 0, [ 0,,,3 F, ] ( + R ) = ( + R ) ( + F ) ( + F )... ( + ) chaper / page 4

17 .3 erm srucure of ineres raes.3. Yield curves and shapes he economics of ineres raes deals wih he pure price of ime (ime value of money). Awareness and appreciaion of he ineres rae-mauriy relaionship is essenial in bond managemen. he relaionship beween he yields on oherwise comparable bonds wih differen mauriies is called he erm srucure of ineres raes; is graphical depicion is known as he yield curve. 9 8 Spo raes 7 6 Spo rae Years Spo rae Year ime o Mauriy Figure -3: he erm srucure of ineres raes he problems in building he erm srucure of ineres raes are ha o avoid coupon effecs and reinvesmen risk, he erm srucure of ineres raes should be buil using only zero-coupon bonds. some raes are no available: one usually knows he,, 3, 5, and 0 year raes, bu how abou a 7.5 year rae? here are no spo raes published for non-governmen bonds, as here are very few corporae zero-coupon bonds. hus, mos people will insead use he yield curve, which plos he yield o mauriy of various bonds agains heir respecive mauriy, holding all oher facors equal. chaper / page 5

18 9 8 Yield o mauriy ime o Mauriy Figure -4: he yield curve Formally, he erm srucure deals wih he relaionship beween spo raes and ime o mauriy, whereas he yield curve deals wih yield o mauriy and ime o mauriy. Generally, boh are very similar. Bu in he analysis of mauriy-reurn relaionship, i is beer o work wih spo raes raher han yields o mauriy, as hey are (among oher hings) no conaminaed by he coupon effec. A nominal ineres rae can be decomposed ino hree basic componens: nominal rae = real ineres rae + inflaion premium + risk premium he real ineres rae is he compensaion for he invesor for deferring consumpion o a fuure period (ime value of money). he inflaion premium is inended o preserve he invesor s purchasing power over ime, and reflecs he expeced fuure inflaion level over he life span of he invesmen. he risk premium proecs he invesor agains all oher poenial negaives, including defaul risk, redempion risk, marke risk, ec. 5 5 Noe ha some of hese risks can be diversified. chaper / page 6

19 he yields level of all bonds will reflec hese hree componens. Consequenly, differen issuer secors will be ploed on differen yield curves. Low qualiy secors (lower raings) will be raded a higher yields. Yield o mauriy B BBB A AAA Governmen bonds ime o mauriy Figure -5: he yield/ime o mauriy relaionship of various raings For he same reasons, callable bonds will be raded a higher yields han similar qualiy noncallable issues. Yield o mauriy Callable bonds Non callable bonds ime o mauriy Figure -6: he yield/ime o mauriy relaionship of callable/non callable bonds Because of his, he liquidiy risk, credi risk, call risk, coupon rae, and degree of premium/discoun as well as any oher risk should be sufficienly similar beween he issues in order o build a useful yield curve. he erm srucure of ineres raes can exhibi four basic shapes: posiively sloped, negaively sloped, fla, and humped. he following figures shows hese four configuraions for illusraive purposes only. chaper / page 7

20 Spo raes posiively sloped erm srucure negaively sloped erm srucure ime o mauriy Figure -7: Basic shapes of he erm srucure: posiively and negaively sloped Spo raes humped erm srucure fla erm srucure ime o mauriy Figure -8: Basic shapes of he erm srucure: fla and humped he shor mauriy secion is mainly influenced by moneary policy, while he long mauriy segmen is more sensiive o inflaionary expecaions..3. heories of erm srucures here are hree primary heories ha ry o explain he shape of he erm srucure of ineres raes: he expecaions hypohesis, he liquidiy preference, and he marke segmenaion heory..3.. Expecaions hypohesis he expecaions heory conends ha he shape of he erm srucure reflecs he marke consensus forecas on fuure ineres raes levels. Advocaes of his heory believe ha he implici forward rae is an unbiased esimae of he fuure spo rae. chaper / page 8

21 F ~ ( h) = E R h,, A good way o undersand his heory is o assume ha invesors are risk neural, and ha hey will selec he securiies ha give hem he highes expeced reurn (whaever heir ime horizon is). Consider he following example: he yield o mauriy on a one-year pure discoun bond is 0%, and % for a wo-year pure discoun bond. Invesors expec he one-year spo rae o be 6% in one year. Wha should invesors do if hey consider one or wo years invesmen horizon? he wo years invesor can inves CHF in a wo-year bond, wih a final value of (.) (.) =.54 CHF or hold wo one-year bonds, wih an expeced final value of (.0) (.6) =.76 CHF. All wo-years invesors will wan o hold wo one-year bonds. he one year invesor can inves CHF in a one-year bond, wih a final value of (.0) =.0 CHF or hold a wo-year bond, ha hey will sell in one year, wih expeced final value of (. ) (. ) (. 6) =08. CHF So, all one year invesors will also wan o hold one-year bonds. Given his universal preference, all invesors will choose a rollover sraegy and hold only he one-year bonds; hus, prices (i.e. ineres raes) should adjus unil he expeced reurn from holding a wo-years bond is exacly he same as he expeced reurn from holding wo one-year bonds. he yield o mauriy on a one-year discoun bond is 0%, and.96% for a wo-year discoun bond. Invesors expec he one year spo rae o be 6% in one year. Wha should invesors do if hey consider one or wo years invesmen horizon? he wo years invesor can inves CHF in a wo-year bond, wih a final value of (.96) (.96) =.76 CHF or hold wo one-year bonds, wih an expeced final value of (.0) (.6) =.76 CHF wo-year invesors will be indifferen beween boh bonds. he one-year invesor can inves one franc in a one-year bond, wih a final value (afer one year) of (.0) =.0 CHF or hold a wo-year bond, ha hey will sell in one year, wih expeced final value (afer one year) of 96 (. 96) (. ) (. 6) =0. CHF So, all one year invesors will also be indifferen beween boh bonds. chaper / page 9

22 If we accep he fac ha he implici forward rae is an unbiased esimae of he fuure spo rae, fuure spo raes can be derived from spo raes, as hey are also he marginal yield beween wo spo raes; and i implies ha, if here are no ransacion coss, each bond is a perfec subsiue for any oher bond, whaever is mauriy. he expeced reurn will be he same, whaever he bond combinaion seleced by he invesor. Indeed, an invesor who has a hree year invesmen horizon could do any of he following: buy a pure zero-coupon bond ha maures a he end of he invesmen period, and hold i o mauriy ( buy and hold sraegy ). buy a shor-erm mauriy bond, and reinves regularly he proceeds ( rollover sraegy ). buy a long-erm bond, and sell i wih a loss or a gain prior o mauriy. he loss or he gain is predicable, using he forward raes. he invesor would expec he same reurn as long as one acceps he idea ha F = E( R ~ ),, he expecaions heory can explain he four differen shapes of he erm srucure of ineres raes: a posiively sloped (respecively negaively sloped) erm srucure implies ha ineres raes are expeced o rise (respecively o decrease) in he fuure, while a fla erm srucure represens a marke consensus for sable yields. Finally, a humped erm srucure shows ha marke paricipans expec a rising rae environmen over he inermediae imes o mauriy, followed by a long-erm decline in yield levels. he saemen ha implici forward raes are unbiased esimaes of fuure spo raes is based on he following assumpions: invesors have homogenous expecaions. invesors choose beween shor or long-erm bonds in order o maximise heir final expeced wealh for a given invesmen period. here are no ransacion coss. bond markes are efficien, and new informaion is insananeously refleced in bond prices. In realiy, all of hese assumpions are subjec o criicism. Expecaions heory or Expecaions heories? One should noe ha in fac, here are several differen versions of he expecaions heory. We will briefly presen hem: he naive expecaions hypohesis (or globally equal expeced holding period reurn) version saes ha expeced reurns from any sraegy for any holding period are equal. We will show laer ha his hypohesis is inernally inconsisen. chaper / page 0

23 he local expecaions version saes ha he expeced oal reurns from a longerm bond over a shor-erm invesmen horizon is he same as oday s ineres rae over his horizon; more generally if we denoe by P(,) he price a ime of a zerocoupon bond paying CHF a ime, independenly of he bond s mauriy dae, we have dp E P (, ) = r () d P (, ) = E exp r () ds r () where r is he insananeous rae of reurn a ime. he above equaion saes he common sense resul ha if here is no erm premium, hen, he discoun rae o be used a each insan is he prevailing spo rae. hus, he local expecaions version is less comprehensive han he naive expecaions hypohesis, as i refers only o oal reurns over a (shor) period beginning a he presen. he unbiased expecaions (or Malkiel s hypohesis) version saes ha he forward raes are equal o he fuure expeced spo raes, ha is, In ha case, we have: P (, ) F = E( R ~ ),, ~ ~ ~ ( R, + ) E( R+, + ) E( R+, + 3)... E( R, ) = he reurn o mauriy expecaions (or Luz s hypohesis) version saes ha he expeced reurn of holding any discoun bond up o mauriy has o be equal o he expeced reurn we would obain by rolling over a sequence of single-period bonds over he same horizon, i.e.: P (, ) = E [( + R ) ( + R ~ ) ( + R ~ )... ( )],,, 3 R ~ , his is incompaible wih he unbiased expecaions, unless he level of fuure ineres raes are muually uncorrelaed. When ineres raes are posiively correlaed over ime, because of uncerain cross producs erms, bonds wih mauriies in excess of wo periods will have higher prices under he unbiased expecaions hypohesis han under he reurn o mauriy expecaions hypohesis. he yield o mauriy version saes ha he periodic rae of reurn (or holding period yield, such as an annual reurn) from holding a bond o mauriy is equal o he expeced holding period yield from rolling over a sequence of shor erm bonds over he same horizon. his version deals wih periodic reurns, while he reurn o mauriy version is concerned wih oal reurns over he invesmen horizon. chaper / page

24 In a remarkable paper 6, Cox, Ingersoll, and Ross have shown ha he naive expecaions hypohesis canno be lierally valid if here is uncerainy abou fuure ineres raes. he remaining four versions are no equivalen or even consisen wih each oher wih uncerain ineres raes. Only he local expecaions hypohesis is consisen wih an equilibrium. All oher versions would imply ha some sraegies can earn excessive reurns, and creae some arbirage profis. Inconsisency of he naive expecaions hypohesis Why is he naive expecaions hypohesis inconsisen? Le s imagine we have a wo-period economy, and wo pure discoun bonds mauring a he end of period and a he end of period respecively. We will denoe by P( 0, ) he price a ime 0 of a zero-coupon bond paying CHF a ime. For he one year holding-period, we can buy he one year zero coupon for P( 0, ), and receive CHF a. Our reurn is cerain: R = P ( ) 0, buy he wo year zero-coupon, and sell i a. Our expeced reurn is ER ( ~ E P ~ (, ) ) = P ( 0, ) According o he naive expecaions hypohesis, we mus have P (, ) 0 ~ (, ) P ( 0, ) = E P ha we can rewrie as P ( 0, ) = (relaion A) P (, ) EP ~ [ (, ) ] 0 6 COX John C., INGERSOLL Jonahan and ROSS Sephen A., 98, A Re-Examinaion of radiional Hypohesis abou he erm Srucure of Ineres Raes, Journal of Finance, pp chaper / page

25 For he wo years holding period, we can buy he wo years zero-coupon bond for P( 0, ), and receive CHF a. Our reurn is cerain: R = P ( ) 0, buy he one year zero-coupon bond for P( 0, ), and receive CHF a ; hen, reinves in buying bond which maures a (rollover sraegy). Our expeced reurn is equal o he expeced payoff divided by he original price: ER ( ~ ) = E ~ P ( 0, ) P (, ) According o he naive expecaions hypohesis, we mus have ha we can rewrie as Bu in fac, relaion A and B are conradicory. is inconsisen wih since, by Jensen s inequaliy 7, we know ha = E ~ P ( 0, ) P ( 0, ) P (, ) P ( 0, ) = E ~ P (, ) P (, ) (relaion B) 0 P ( 0, ) = P (, ) ~ EP (, ) [ ] 0 P ( 0, ) = E ~ P (, ) P (, ) 0 ~ E ~ EP (, ) P (, ) [ ] unless P ~ (, ) is deerminisic. hus, he naive expecaions hypohesis is inernally inconsisen. 7 Jensen s inequaliy saes ha E(f(x)) is no equal o f(e(x)). chaper / page 3

26 .3.. Liquidiy preferences In he expecaions heory, we assumed ha invesors do no have any mauriy preference; in fac, here is no obvious reason o have a mauriy preference in a world wihou uncerainy, since any combinaion of bonds would provide he same reurn for he same invesmen period. Once he cerainy assumpion is relaxed, we have jus seen ha only he local expecaions hypohesis is susainable. he liquidiy preference heory assers ha invesors prefer o hold liquid securiies, liquidiy being defined as he abiliy o conver rapidly a bond ino cash, while minimising he principal loss. As he flucuaion risk in long-erm securiies is higher han for shor-erm ones, invesors will prefer shor-erm securiies 8. hus, here is a shorage of longer erm invesors. Bond A is a one year bond wih a 6% coupon. Bond B is a wo-year bond wih a 6% coupon. he one year and wo-year spo raes are equal o 6%. Wha happens if here is a sudden unexpeced rise of all spo raes o 7%? Boh bonds were priced a par (= 00.00). Afer he rise, he wo-year securiy will drop o 98.9 in price, while he one year bond will drop o So, for one percenage poin increase in yield, he wo-years securiy has a more imporan price decrease han he one year securiy. Hence, for a risk-averse invesor, he wo-year securiy seems riskier. On he oher hand, he borrowers (governmens, firms,...) prefer o issue long-erm securiies, o avoid ineres raes flucuaions consequences on heir expenses. In order o induce invesors o inves in long-erm securiies, hey will offer hem an exra risk premium (a liquidiy premium or erm premium). So, here are in fac wo facors in he observed erm srucure of ineres raes: fuure expeced shor-erm spo raes (as prediced in he expecaions hypohesis heory) a posiive liquidiy premium 8 Anoher simple reason o jusify his is he unexpeced inflaion risk: if here is an unexpeced rise in he inflaion rae, he nominal ineres raes should also rise; invesors holding shor erm bonds will be able o reinves heir money a a higher rae, while invesors holding long erm bonds will have o wai for he final reimbursemen before o ake profi of he higher ineres raes. chaper / page 4

27 Yield o mauriy implici forward raes liquidiy premium expeced yield according o he liquidiy preference heory ime o mauriy Figure -9: he liquidiy premium concep We should have a posiive difference beween implici forward raes and expeced spo raes: and ( ~ ) F, = E R, + L, (for > 0) L, + > 0 As his heory suggess a higher yield for longer mauriy issues caused by heir lower degree of liquidiy, he expeced reurn on a buy and hold sraegy has o be higher han he expeced reurn on a rollover sraegy. ( + R0, ) = ( + R0, ) ( + R, + L, ) ( + R, 3 + L, 3)... ( + R, + L, ) Furhermore, as risk increases wih ime, we should observe: L, < L,3 < L 3,4 <... < L n-,n Hence, he erm srucure of ineres raes should be (mainly) upward sloping because of he preference of invesors for liquidiy Marke segmenaion and preferred habia heories he marke segmenaion heory (or preferred habia heory) views he bond marke as a series of disinc markes ha differ by heir mauriy. Each issuer or invesor will have a preferred mauriy, and he will be sufficienly risk-averse o operae only in his desired mauriy specrum. Wihin a given mauriy range, he relaive supply and demand for funds deermines he appropriae clearing price (i.e. he appropriae ineres rae). chaper / page 5

28 Price (%) Supply of money i Demand for money Q Quaniy Figure -0: Supply and demand of money for a given mauriy hus, conrary o he liquidiy preference heory, he marke segmenaion heory leads o he resul ha he risk premium aached o bonds can be eiher posiive, negaive, or zero. F ( ~ ) = E R + Π, +, +, + and nohing can be said ex ane abou he sign of Π, +. In he marke segmenaion heory, he money is considered as a commodiy, is marke clearing price is he ineres rae, and he supply and demand of each individual segmen connec o creae he overall composie erm srucure of ineres raes. By examining flows of funds ino he marke segmens, one could - in heory - predic changes in he erm srucure of ineres raes. he marke segmenaion heory explains all four basic erm srucure of ineres raes shapes: In he case of a posiively sloped erm srucure of ineres raes, he invesors (buyers) have a preference for he shor-erm segmens of he marke, hus prices of bonds wih small mauriies are high and heir yields are low; he reverse is rue for he long-erm segmen. In he case of a negaively sloped erm srucure of ineres raes, we have he reverse case of he above scenario. If he erm srucure of ineres raes is fla, invesors have similar preferences for all segmens of he marke. finally, he humped erm srucure of ineres raes is due o differen preferences, which depend on he mauriy segmens. he major criicism of his heory is ha even if invesors have a srong mauriy preference, he effec of segmenaion on ineres raes should be offse as soon as some invesors sar considering relaive yields and allocae heir funds o anoher segmen which offers a (sufficienly) higher yield. Any invesor will ry o reduce risks by saying in is preferred habia; bu he will leave i as soon as he is given a risk premium high enough o cover he assumed risks and he cos of leaving is preferred habia. chaper / page 6

29 .3..4 Oher heories he expecaion hypohesis, liquidiy preference and marke segmenaion heory are hree non exclusive ways of hinking abou ineres raes. Bu as far as bond pricing is concerned, he mos promising heories are he sochasic process no-arbirage approaches. hey rely on he following assumpions: he erm srucure and he bond prices are relaed o some sochasic facors hese facors evolve over ime according o a paricular hypohesised sochasic process (i.e. a process wih some uncerainy) here should be no arbirage opporuniy Various models have been developed, using single or muliple facors. For example, he Ogden model (987) assumes ha he erm srucure of ineres raes is driven only by he shor erm ineres rae flucuaions, and uses he following process o describe he shor erm ineres rae variaions: dr = β (u r) d predicable componen σ r dz() 443 unpredicable componen where dr is he insananeous change in he rae, β is a speed-of-adjusmen componen, u is he average level of he rae, d is he passage of ime, dz() is a sochasic process, and σ r is he sandard deviaion of he process. In words, such an equaion says ha he change in he shor erm rae has wo componens: one is predicable (he exen o which he curren rae differs from is long-erm value, muliplied by a coefficien ha measures is rae of adjusmen o is long-erm value), and one unpredicable (he produc of he sandard deviaion of he rae, of he iniial level, and of some sochasic process, ha acs as a random generaor). Using his specificaion of he shor-erm rae, and by solving a parial differenial equaion, i is possible o find an analyical soluion (or a numerical soluion) for he bond prices, and herefore for he erm srucure of ineres raes. Of course, oher specificaions of he process followed by he shor-erm raes would lead o anoher erm srucure of ineres raes. I is also possible o use oher facors (such as he long-erm rae, he spread beween shor-erm and long-erm ineres raes,...), or more han one facors. Bu each facor sochasic process has o be carefully specified, and he addiion of facors complicaes he solving of he parial differenial equaion 9. 9 For some ineresing specificaions, see: BRENNAN Michael and SCHWARZ Eduardo S., 98, Bond Pricing and Marke Efficiency, Financial Analyss Journal, pp chaper / page 7

30 .4 Bond price analysis.4. Yield spread analysis Fixed income insrumens differ in a variey of ways (markeabiliy, ax saus, credi risk..). I is hus possible o examine he impac of he various differences in he characerisics of bonds on heir yield o mauriy. I is ofen helpful o consider one difference a a ime: for example, when he only difference is given by he mauriy, aenion is paid o he erm srucure of yields; when he bonds are equal bu heir credi risk, he aenion is focused on he risk srucure of yields and so on. In any case, he differenial in he yields of wo or more bonds is called yield spread and he analysis of he causes and consequences of hose spreads is called yield spread analysis. he yield spread of a given insrumen is usually measured agains he yield of a reasury securiy having comparable mauriy; as a maer of fac, reasury securiies are he insrumens of highes qualiy in he fixed income marke in erms of markeabiliy, credi risk and ofen of ax saus. he yield spread beween a given securiy and he corresponding reasury securiy has he naure of risk premium because i reflecs he risks (lower liquidiy, higher credi risk...) ha an invesor has o face when he invess in non-reasury securiies. Normally yield spreads are measured in erms of basis poins, where a basis poin, as previously menioned, is equal o 0.0%: so, for example, a given bond A has a yield o mauriy equal o 7% and anoher o 7.50%, he yield spread is 0.50%, ha is 50 basis poins. Moreover, one can also measure yield spread in erms of he yield level and calculae he relaive yield spread defined as or he yield raio yield bond B - yield bond A relaive yield spread = yield bond A yield raio = yield bond B yield bond A In our example, he relaive yield spread is (0.50/7) and he yield raio ypes of spreads he bond marke can be subdivided ino secors by ype of issuer (reasury, corporae, financial insiuions), credi qualiy (summarized by he raing grade), mauriy (shor, medium and long erm) and level of coupon. Each secor of he marke is characerized by a yield spread versus oher secors of he bond marke..4.. Deerminans of yield spreads Many facors may affec he yield spread; in principle, any difference in any characerisics beween wo bonds should be refleced in a yield differenial. For simpliciy, he deerminans of he yield spread are usually classified in: chaper / page 8

31 mauriy of he insrumen; crediworhiness of he issuer; embedded opions; ax saus of he insrumen; liquidiy of he securiy. As far as he firs iem is concerned he reader is referred o secion.3. he crediworhiness (probabiliy of defaul) of he issuer clearly affecs he yield: as long as here is a possibiliy of defaul, he expeced yield (which akes ino accoun he likelihood of defaul) is lower han he promised yield (which is calculaed on he promised cash flows aken a heir face value). he greaer he likelihood of defaul (defaul probabiliy) and he greaer he amoun los in case of defaul (he loss given he even of defaul), he wider he spread beween promised and expeced yield (yield spread). An example of he relaionship beween credi risk and expeced yield is depiced in Figure Figure - (risk srucure of ineres raes): higher risk bonds command higher yields o mauriy. Yield o mauriy BBB bond AAA bond Mauriy Figure -: erm srucure of differen raing grade bonds Yield spreads due o crediworhiness end o widen when he economy is likely o face a recession and narrow in boom phases; invesors in fac see a higher probabiliy of defaul and a greaer severiy of losses when he economy is no in good healh and consequenly hey assign an increasing probabiliy of defaul o lower qualiy borrowers and require a higher risk premium. When he credi risk is aken ino accoun, he yield earned by he invesor mus be high enough o accoun for defaul probabiliy and for he losses incurred in case of defaul. he following relaionship holds, where r f is he risk free rae, r p is he risky rae and (-p) is he defaul probabiliy (so ha p is he probabiliy of exac fulfillmen of he paymen obligaions). Le's assume a firs ha, in case of defaul, he bondholders do no ge any paymen. In ha case: p ( + r ) + ( p) 0 = + p r f chaper / page 9

32 hence and he premium for he credi risk is r p + r = p f ( + rf ) premium = rp rf = ( + r p If he bondholder, as i is generally rue, can ge a fracion γ of he amoun owed by he issuer, hen and he credi risk premium is p ( + r ) + γ ( p) ( + r ) = + r r p p r f ( + rf ) ( γ + p γ p) = ( + r ) As far as embedded opions are concerned, many bond issues include provisions graning eiher he bondholder or he issuer, or boh, some opions o ake some acions in his self ineres (like repaying he bond before he mauriy). Clearly, he opion benefis he pary who can choose o exercise i (he holder of he opion) i and "damages" he pary who has wrien i. he advanage of holding an opion has o be paid for by a yield differenial. For example, he call provision allows he issuer o shoren a his exclusive choice he mauriy of he bond; ha opion will be exercised only when is advanageous for he issuer himself who is able o replace a higher yield insrumen wih a lower yield one. Since each opion has o be paid for, a bond wih an embedded call provision should command a higher yield (he issuer has o pay he opion) han a similar bond wihou he call provision iself. Conversely, a puable bond (a bond ha can be resold o he issuer a a given price by he invesor a his exclusive choice) should carry a lower yield (he bondholder has o pay for he opion he holds) han a bond similar bu for he pu provision. he effec of he ax saus is quie clear: anyone looks a he ne income (from labour, from invesmens and so on). Given ha, a axable bond (like a corporae bond, for example) has o pay a higher gross yield in order o compee wih an exemp bond (like a Governmen or municipal bond, for example). he afer ax (ne) yield on a axable fixed income securiy is defined as: afer-ax (ne) yield = preax (gross) yield (- ax rae) If he yield on a given axable bond is 0% and he relevan ax rae is 30%, he afer ax yield is 0 % ( 30%) = 7% Likewise, i is possible o calculae he equivalen axable yield of a ax exemp securiy wih he following formula: equivalen axable yield = f p - ax rae f f ) ax exemp yield chaper / page 30

33 As far as he liquidiy is concerned, he greaer he expeced liquidiy, he lower he yield required by he invesors. he reason is sraighforward: when an issue is illiquid, he invesor migh experience some problems should he decide o sell he bond before he mauriy. he liquidiy of bonds has hree dimensions: markeabiliy, ha is he exisence of a broad and deep marke on a given insrumen; ime o mauriy, because a mauriy (unlike socks for example) he bond will be paid back and so "ransformed" in cash; financiabiliy, o he exen ha a given issue can be uilized as a collaeral in order o borrow funds..4. Bond valuaion Bond valuaion is based on he discouned cash flow mehod and on he equilibrium concep..4.. Valuaion of a zero-coupon bond he simples bond o consider is a zero-coupon bond which pays a single cash flow a he end of period. he price of such a bond, denoed by B 0,, is equal o he presen value of is final (and only) cash flow: B 0, = ( + k) where is he cash flow received a he end of period, and k is he appropriae discoun rae. Wha is he price oday of a zero-coupon bond ha will pay '000 CHF in exacly 5 years, assuming a discoun rae of 7%? How abou a 7-year bond, sill assuming a discoun rae of 7%? and 000 ' B 0,5 = = CHF ( 07. ) ( 07. ) ' B 0,7 = = CHF 7 In he previous example, he discoun rae was assumed o be he same regardless of he mauriy of he bond. Bu generally he discoun rae varies from mauriy o mauriy. If we denoe he annual rae of reurn demanded by a lender o lend money from ime 0 o ime (also called he spo rae) by R 0, and if only one final paymen is made owards boh ineres and principal, he price of such a zero-coupon bond will be calculaed as: B 0, = ( + R ) 0, he above formula allows us o use differen discoun raes for differen mauriies. chaper / page 3