Interest Rate Swap Pricing: A Classroom Primer

Transcription

1 Ineres Rae Swap Pricing: A Classroo Prier Parick J. Cusais, CFA, The Pennsylvania Sae Universiy - Harrisburg ABSTRACT In his paper I presen an inroducory lesson on ineres rae swaps and wo odels or ineres rae swap valuaion. I begin by oulining iporan conceps o he ineres rae swap arke. I show ha a swap can be valued based on siple presen value echniques and using iplied orward raes, wih he sae resul. I provide a basic odel or copuing an a-arke swap rae. In all hree sages, I give swap valuaion exaples ha can be applied in an undergraduae or graduae inance curriculu. INTRODUCTION The ineres rae swap arke is he larges and ases growing derivaive arke. Ineres rae swaps are iporan ools or hedgers, speculaors, and invesors. According o a survey perored by he Bank or Inernaional Seleens, he ousanding volue o ineres rae swaps was $309.6 rillion as o he end o This was an increase o 34.78% ro year-end 006. Given he size and growh o he ineres rae swap arke, i is a opic ha canno be ignored in undergraduae and graduae curriculus. Undersanding ineres rae swap pricing is criical o he undersanding o he echanics o ineres rae swaps. Soe o he ineres rae debacles o he pas were due o a isundersanding o he correc valuaion o ineres rae swaps. For hese reasons, i is iporan o inroduce sudens o basic pricing odels ha capure he key aspecs o he ineres rae swap arke. This paper presens wo relaively siple swap pricing odels ha ephasize he iporan characerisics o ineres rae swaps. The irs odel is based on basic presen value echniques. I serves as an inroducion o ineres rae swap valuaion. The second odel is ore coplex and involves iplied orward raes. I can be used o value swaps wih ore coplex srucures. I also explain he calculaion o an a-arke swap rae. This is helpul in copleing sudens undersanding o he ineres rae swap arke. 1 hp:// A good exaple o his is he Bankers Trus and Procer and Gable ineres rae swap ransacion ha was unwound in See Sih (1997). 1

2 This paper is organized as ollows. Firs, I describe he basic srucure o an ineres rae swap. Then a siple ehodology or valuing a swap based on presen value echniques is presened. I hen inroduce iplied orward raes and develop a ore accurae valuaion echnique. Finally, I presen an exaple o how o copue an a-arke swap rae. VANILLA INTEREST RATE SWAPS A plain vanilla ineres rae swap is a conrac under which wo counerparies, a loaing-rae payer and a ixed-rae payer, agree o exchange ne payens a a series o uure poins o ie. A noional principal aoun is used o calculae he aoun o he ne payens. An ineres rae swap can have a auriy or enor in excess o 30 years. The ixed swap rae is se on he pricing dae o he swap. The loaing rae is esablished or he irs payen based on arke levels on he pricing dae and rese periodically based on arke levels. The loaing rae on a plain vanilla swap is based on 3- onh LIBOR. Ineres raes are ypically rese quarerly wih ne payens ade seiannually. An ineres rae swap can be srucured wih counless variaions o his srucure. Exaple loaing rae bencharks include he coercial paper rae, he U.S. Treasury-bill rae, 1-onh LIBOR, and he SIFMA index. Payens can be calculaed and ade onhly, quarerly or a soe oher desired periodiciy. A spread o he loaing rae can also be added based on arke condiions or he needs o he swap iniiaor. In an ineres rae swap, he ixed rae payer is said o be he buyer and he loaing rae payer is said o be he seller; however, ro an econoic or hedging sandpoin he opposie is rue. The value o an ineres rae swap changes wih changes in ineres raes. Figure 1 shows he change in he value o a swap relaive o changes in ineres raes. Alhough hese appear o be sraigh lines, hey represen he acual change in he value o a hypoheical swap given a change in ineres raes. For he ixed rae payer (loaing rae receiver) he value o he swap increases when ineres raes increase. This eans ha a porolio anager ha is long bonds can eecively hedge he value o his posiion by enering ino a swap ha requires he anager o pay ixed and receive loaing. For he loaing rae payer (ixed rae receiver) he value o he swap increases when ineres raes decrease.

3 Figure 1: Value o Ineres Swap Conrac Change in Swap Value Change in Ineres Raes Pay Fixed, Receive Floaing Pay Floaing, Receive Fixed THE ZERO-COUPON YIELD CURVE The zero-coupon yield, or spo rae o ineres, is he yield-o-auriy on a bond or invesen wih only one cash low occurring on a speciic dae or auriy. U.S. Treasury Bills and U.S. Treasury STRIPS are exaples o deaul-ree zero-coupon securiies. U.S. Treasury STRIPS are oen used o consruc a represenaive risk-ree axable zero-coupon curve. STRIPS, an acrony or Separaely Traded Regisered Ineres and Principal Securiies, represen zero-coupon bonds ha are creaed by sripping ull-coupon U.S. Treasury bonds. One proble wih relying on acively raded zero-coupon bonds is ha hey ay no be raded a desired auriies and yields ay have o be inerpolaed beween auriies. For his reason, zero-coupon yields are oen calculaed ro ull coupon raes. A yield curve consruced ro deaul-ree zero-coupon bonds is known as he er srucure o ineres raes. Swap pricing is based on he zero-coupon yield curve. I is oen he case ha coupon bond prices are available, bu zero-coupon bond prices are no available. In hese circusances, i is possible o esiae zero-coupon bond prices ro ull-coupon bond prices using a procedure known as boosrapping. 3 The yields associaed wih hese esiaed zero-coupon bond prices are known as iplied zero-coupon yields. Iplied zero-coupon yields are he se o discoun raes iplied in he par yield curve ha equaes he cash lows o a ull-coupon bearing bond o hose o a se o zero-coupon 3 In pracice, swap yield curves are consruced using a cobinaion o oney raes, Eurodollar uures, and arke swap raes. For a ore deailed discussion see Cusais and Thoas (006) and Young (1997). 3

4 bonds. The heoreical iplied zero-coupon yield adjuss he ull-coupon rae or he loss or gain associaed wih he periodic reinvesen o ineres payens. To illusrae his concep, assue ha a non-callable, ull-coupon bond ha aures in n years, akes a payen o C/ in each ie period, =1,,3, n. A auriy, he ullcoupon bond also pays he $100 ace value. I he is currenly selling a par, he bond price is calculaed as 100 = n C y y n = 1 (1 + ) (1 + ). To calculae iplied zero-coupon yields using he boosrapping procedure, we replace he yield o auriy, y, wih he appropriae zero-coupon yield in each period. A se o zero-coupon yields, r, exiss ha, when used as discoun raes, will resul in he curren price o he bond. We begin wih he zero-coupon yield or he shores period o ie and calculae he iplied zero-coupon yields or subsequen periods. The zero-coupon yield or he shores ie period can be observed in he arke. The boosrapping procedure o derive an iplied zero-coupon curve is illusraed by he ollowing exaple. Exaple: Table 1 presens daa or en coupon bonds selling a par. Bond one, which aures in one period has an annualized coupon rae o.00%. The iplied one-period zerocoupon yield, r 1, is equal o 1.00%, he seiannual arke yield on bond one. Table 1: Annualized Yields on Coupon Bonds Bond Mauriy () Coupon Rae %.0.5% % % % % % % % % r The iplied zero-coupon yield or period,, is calculaed by solving he ollowing equaion: 4

5 100 = 1.15 (1.010) r. (1 + ) r Solving or, he iplied wo-period zero-coupon yield is equal o 1.135% which is an annualized yield o.507%. Using he iplied one-period and wo-period zero-coupon yields, he hree-period iplied zero-coupon yield solves he equaion: = + +. (1.010) ( ) r3 3 (1 + ) The zero-coupon yield ha solves his equaion is % which is an annualized yield o.5039%. The boosrapping procedure requires ha we solve or each subsequen zerocoupon yield and use he resuls o calculae he nex iplied zero-coupon yield. Table shows he annual iplied zero-coupon yield curve or en periods. Table : Iplied Zero-Coupon Yields Mauriy Annual Full-Coupon Yield Annual Iplied Zero-Coupon Yield Bond c r %.0000%.0.5%.507% %.5039% % % % 3.477% % % % % % % % % % 5.144% For sei-annual coupon bonds, he generalized orula is: C C n Pn = +. n = 1 r rn where P is he price o he bond, C is he annual coupon equal o c(100), and r is he zerocoupon yield or a bond auring in years. 5

6 A SIMPLE PRESENT VALUE MODEL FOR VALUING INTEREST RATE SWAPS In his secion, I begin wih a siple valuaion odel or ineres rae swaps. The odel is based on basic presen value relaionships. In his odel, swap cash lows are deined as a cobinaion o a loaing rae bond and ixed rae bond, wihou principal a auriy. 4 The value o a ixed rae bond is equal o he presen value o he expeced uure cash lows discouned a he arke yield. Siilarly, he value o he ixed leg o an ineres rae swap, PV ixed, is equal o he presen value o he ixed payens: C n PV = ixed, = 1 r + 1 where C is he ixed annual swap payen, is he nuber o payen periods per year, and r is he discoun rae a ie. The loaing leg o he swap is siilar o a loaing rae bond ha pays no principal. A loaing rae bond is always priced a is ace value on an ineres payen dae because he coupon payens adjus o arke raes in each ie period. The price o a loaing rae bond, P, is equal o he presen value o is expeced cash lows. I we deine I as he expeced loaing coupon a ie, hen he value o a loaing rae bond wih ace value F is equal o I n F P = +. n = 1 r rn The only unknowns in he pricing orula are he expeced loaing rae payens in each ie period, I. Since ineres rae swaps do no require he payen o principal, he irs er on he righ hand side o he equaion is he value o he loaing rae leg o he swap. I we rearrange he equaion above and replace P wih F (since he bond always sell or is ace value), he pricing orula can be expressed as: I n F F =. = n r + n 1 r Thereore, he value o he loaing leg o he swap, PV loaing is equal o: 4 An alernaive valuaion odel includes he principal values a auriy. Since one bond is shor and he oher is long, he principal aouns cancel. 6

7 F PV loaing = F. n rn 1 + The value o he swap or he ixed rae payer, V ixed, is: V ixed = PV loaing PV ixed, and he value o he swap or he loaing rae payer, V loaing, is: V loaing = PV ixed PV loaing. Exaple: A $10,000,000 ineres rae swap has seiannual payens based on average 3- onh LIBOR. The swap aures in 5 years and has a ixed payen rae o 4.50%. The value o he swap or boh he ixed and loaing rae payers can be calculaed using he zerocoupon discoun raes in Table 3. DF reers o he discoun acor and is calculaed as ( ) r r Table 3: Discoun Raes and Facors r DF % 1.000% % 1.15% % 1.5% % 1.507% % 1.714% % 1.870% %.08% %.70% %.405% %.57% We begin by calculaing he value o he ixed leg o he swap. The seiannual payen is $10,000,000(.045)(½) or $5,000. Thereore, he value o he ixed leg is he presen value o he srea o $5,000 payens which is equal o $,018,484. The valuaion o he ixed leg o he swap is suarized in Table 4. r 7

8 Table 4: Valuaion o he Fixed Leg o a Swap r T r DF ($5,000)(DF) % 1.000% $,77.507% 1.15% $ 0, % 1.5% $ 16, % 1.507% $ 11, % 1.714% $ 06, % 1.870% $ 01, %.08% $ 194, %.70% $ 188, %.405% $ 181, %.57% $ 174,538 Toal $,018,484 Using he equaion derived above, he value o he loaing leg o he swap is: PV loaing = $ 10,000,000 ($10,000,000)(0.7757) = $,4, Thereore, he value o he swap o he ixed rae payer is: V ixed = $,47,759 - $,018,484 = $4,75, and he value o he swap o he loaing rae payer is: V loaing = $,018,484 - $,47,759 = -$4,75. In his exaple, ineres raes have risen since he issuance o he swap--he swap has a ixed rae o 4.50% and he curren ive-year arke rae is 5.00%. For his reason, he value or he ixed rae payer has increased by $4,75 and he value or he loaing rae payer has decreased by he sae aoun. THE TERM STRUCTURE OF INTEREST RATES AND IMPLIED FORWARD RATES While siple and accurae, he valuaion ehodology described above has liiaions. Coplex swaps canno be valued wih a siple odel. For coplex swap srucures we require a ore lexible valuaion ehodology. In his secion I develop a swap valuaion odel based on iplied orward raes. The Pure Expecaions Hypohesis Several hypoheses have been developed o explain how he yield curve conveys inoraion o arke paricipans. The pure expecaions hypohesis saes ha expeced uure shor-er raes are equal o he orward raes iplied in he yield curve. One iplicaion o his hypohesis is ha he yield curve can be decoposed ino a series o 5 This value is calculaed wihou rounding he discoun acor. 8

9 expeced uure shor-er raes ha will adjus in such a way ha invesors receive equivalen expeced holding period reurns. Under pure expecaions, invesors are assued o be risk-neural. Since risk-neural invesors apply no risk-relaed discoun o he value o shor-er bonds, he shape o he yield curve is driven only by invesor expecaions. I an upward sloping yield curve prevails, invesors expec higher uure shor-er ineres raes, whereas an invered yield curve iplies expecaions o lower uure shor-er raes. This heory iplies a la yield curve when invesors expec ha shor-er raes will reain consan. The pure expecaions hypohesis saes ha he expeced average annual reurn on a long-er bond is he geoeric ean o he expeced shor-er raes. 6 For exaple, he wo-period spo rae can be hough o as he one-year spo rae and he one-year rae expeced o prevail one year hence. Since expeced shor-er raes are iplied in he yield curve, an invesor would be indieren beween holding a 0-year invesen, a series o 0 consecuive one-year invesens, or wo consecuive en-year invesens. Pure expecaions is perhaps he bes known and easies o he heories o he er srucure o quaniy and apply. For his reason, i is widely used in he capial arkes as a pricing convenion or ineres rae coningen securiies. The se o orward raes derived under pure expecaions, he iplied orward yield curve, is he basis or he valuaion o any ixed-incoe securiies. Iplied Forward Raes Coupon bonds ay be viewed as a porolio o zero-coupon bonds wih unique yields, r, or each coupon payen received a ie. As such, coupon bonds can be viewed as a series o separae bonds o dieren overlapping auriies. Consider bond wo in he above exaple. Using zero-coupon yields, he bond can be priced as: = +. (1.0100) ( ) The irs cash low is discouned a a yield o.0000% or one year and he second cash low is discouned a a yield o.507% or wo years. An alernaive view o he second cash low is ha i is invesed over wo one-year periods. Since we know he yield over he irs period, here is an iplied yield or he second period ha saisies he ollowing relaionship: = +. (1.0100) ( )(1 + 1 ) The iplied yield or he second period, 1, is he orward rae on he bond ro period 1 o period and is equal o % which is an annualized rae o.5017%. The iplied 6 Epirical evidence suggess ha iplied orward raes have been a poor predicor o uure shor-er ineres raes. Faa (1975) ound ha a naïve orecasing ehod, which used curren raes o predic uure shor-er raes, produced ore accurae orecass han one using iplied orward raes. 9

10 orward rae is siply he yield earned on a one-period bond ro period 1 o period when he invesor conracs o inves in he bond oday. Viewed in his way, long-er bonds can be considered a porolio o a one-period invesen a he prevailing spo rae o ineres and a series o orward conracs o inves in one-period bonds a raes agreed upon oday. The one-period orward ineres raes are ebedded in he price o long-er bonds and can be calculaed ro he zero-coupon yield curve using he equaion: r + + = r , where -1 is he annualized iplied orward rae or period -1 o and r is he annualized iplied zero-coupon yield or ie. Solving or -1, he iplied orward rae or period is be calculaed as r = r Alernaively, his calculaion can be expressed as: r = 1. 1 r Table 5 presens he annualized iplied orward yields associaed wih he iplied zerocoupon yields ro our exaple. 10

11 Table 5: Iplied Zero-Coupon and Iplied-Forward Yields Annual Bond Mauri y Iplied Zero- Coupon Yield Annual Iplied Forward Yields r %.0000%.0.507%.5017% % 3.011% % 4.555% % % % % % 6.79% % 7.003% % % % % An alernaive swap valuaion ehod uses iplied orward raes. The loaing rae leg is valued as he presen value o expeced cash lows using he iplied orward raes, 1 1, o calculae he expeced cash lows. Using his ehod, he value o a swap o he loaing rae payer is: C 1 n n F V = loaing, = = 1 r + 1 r and he value o a swap o he ixed rae payer is: 1 C n F n V = ixed = = 1 r + 1 r In he previous exaple, he ixed rae on he swap is 4.50% and he noional aoun o he swap is $10,000,000. Using he inoraion ro Table, we can calculae iplied orward raes. Table 6 shows he iplied orward raes and he presen values o he iplied orward raes and ixed rae payens. The presen value o he loaing rae payens is calculaed as he su o he discouned values o he iplied orward raes. The presen value o he ixed rae payens is he su o he discouned value o he.5% ixed rae payens. The value o he ixed leg o he swap is he su o he ne payens as a 11

12 percenage o noional principal. The value o he ixed leg o he swap is $4,75. The sae value is calculaed using he previous odel. 7 Table 6: Swap Valuaion Using Iplied Forward Raes r 1 Ne Payen DF PV loaing PV ixed (Fixed Leg) % 1.000% ,010,77 (13,76) 1.15% 1.51% ,317 0,00 (97,703) 3 1.5% 1.506% ,046 16,757 (71,711) %.78% ,530 11,930, %.544% ,675 06,67 7, %.653% ,398 01,331 36, % 3.361% , ,783 96, % 3.600% , ,014 11, % 3.494% ,09 181, , % 4.085% , ,538 14,30 Toal ,4,759,018,484 4,75 CALCULATING AN AT-MARKET SWAP RATE Generally, a he ie an ineres rae swap is seled, he presen value o he expeced ne payens has a value o zero. Neiher pary expecs o have zero payens in every period. I he yield curve is upward-sloping, he ixed-rae payer expecs o ake posiive swap payens in he early years and receive posiive swap payens in he laer years. I he yield curve is downward sloping, he ixed-rae payer will expec o receive posiive swap payens in he early years and ake posiive swap payens in he laer years. In a la yield curve environen, he expeced uure payens or boh he ixed and loaing rae payers are zero. In any ineres rae environen, he ixed rae ha akes he presen value o he expeced ne payens equal zero is known as he a-arke swap rae. Recall ro above ha he value o a swap is based on iplied orward raes. When solving or he a-arke swap rae, all o he valuaion inpus are known excep or he ixed swap rae. For an a arke swap, he ollowing us hold: c 1 n F n F =. = = 1 r + 1 r Solving he equaion or he a-arke swap rae, c, we ge: 7 The sligh dierence is due o rounding. 1

13 n 1 F = 1 r 1 + c = ( ) n 1 F = 1 r 1 + The a-arke swap rae is equal o he su o he presen value o he iplied loaing rae payens divided by he noional aoun ies he su o he discoun acors. This aoun is uliplied by he nuber o payen periods per year. Exaple: Using he daa ro Table 6, we calculae he a-arke swap rae. The su o he presen value o he loaing rae payens is,4,749. The su o he discoun acors is ; hereore he a-arke swap rae is [,4,749 / (8.9710*10,000,000)]() = 500,000. Thereore he a-arke swap rae is approxiaely equal o 5.00%. Table 7 suarizes he expeced cash lows on he loaing and ixed rae legs o he swap using he 5.00% a-arke swap rae. This proves he accuracy o he a-arke swap rae since he su o he ne payens (a presen value) expeced on he swap a he aarke swap rae is zero. This holds since PV ixed and PV loaing are boh equal o,4,749. SUMMARY Table 7: Calculaion o A-arke Swap Rae T PV loaing PV ixed Ne Payen (o Floaing Leg) 1 99,010 47,55 (148,515) 1,317 44,467 (1,150) 3 145,046 40,841 (95,795) 4 14,530 35,477 (0,947) 5 33,675 9,635 4, ,398 3,701 13, ,000 16,46 74, ,835 08,905 91, ,09 01,85 80, , ,931 1,97 Toal,4,759,4,759 0 In his paper, I presen wo siple odels as an inroducion o ineres rae swap pricing or sudens. The odels are ean o aciliae he undersanding o how ineres rae swaps are srucured and how ineres rae oveens aec heir value. I also show a 13

14 siple copuaion o he a-arke swap rae which urher ephasizes hese ideas. All hree exaples are ean o build a basic undersanding o ineres rae swaps. Swaps are an iporan hedging ool. As he swap arke coninues o grow i becoes an increasingly vial par o he inance curriculu. The odels presened in his paper are well-suied or use in a inancial odeling class. Swap valuaion in pracice is uch ore deailed and require any inpus ro he capial arkes. Swap raders use sophisicaed odels ha generae real-ie swap values based on live daa eeds. However, he odels oulined in his paper are accurae and provide insigh ino swap pricing and he echanics o he swap arke. REFERENCES Cusais, Parick and Marin Thoas (006) Hedging Insruens and Risk Manageen. McGraw Hill. Faa, Eugene (1975) Shor Ter Ineres Raes as Predicors o Inlaion. Aerican Econoic Review, Sih, Donald J. (1997) Aggressive Corporae Finance: A Close Look a he Procer & Gable-Bankers Trus Leveraged Swap. The Journal o Derivaives, Young, Andrew R. (1997) A Morgan Sanley Guide o Fixed Incoe Analysis, Morgan Sanley. 14