Interest Rate Parity, Money Market Basis Swaps, and Cross-Currency Basis Swaps

Transcription

1 Fixed Income Liquid Markes Research June 23 Ineres Rae Pariy, Money Marke Basis Swaps, and Cross-Currency Basis Swaps Bruce uckman, Pedro Porfirio Because he classic ineres rae pariy condiion requires defaul-free raes as inpu, he common pracice of using raes derived from swap curves is no valid. his paper derives ineres rae pariy condiions ha depend on basis swap spreads in addiion o swap raes.

2 CONACS Ediors Hua Vasan Naik Bruce Quaniaive Sraegies (Europe Vasan Naik Ma David Ineres Rae Modelling (U.S. Bruce Fei Zhou Quaniaive Research (Asia Hua Charles W. Liu Manabu FX Research Jim McCormick Alexei Anne Shrui Addiional Conacs Amiabh Arora Ganlin Marco Naldi

3 Lehman Brohers LMR Quarerly LMR Quarerly, vol. 24-Q2 Ineres Rae Pariy, Money Marke Basis Swaps, and Cross-Currency Basis Swaps Bruce uckman Pedro Porfirio Because he classic ineres rae pariy condiion requires defaul-free raes as inpu, he common pracice of using raes derived from swap curves is no valid. his paper derives ineres rae pariy condiions ha depend on basis swap spreads in addiion o swap raes. he derivaions reveal: i he relaively well acceped proposiion ha money marke basis swaps reflec he credi risk inheren in one floaing rae versus anoher; and ii he less known proposiion ha cross-currency basis swaps reflec he difference beween he credi risk embedded in he shor-erm raes of one currency versus he oher. Several empirical examples are given in suppor of his less known proposiion CLASSIC INERES RAE PARIY Ineres rae pariy is an arbirage argumen used o derive forward foreign exchange raes. o describe he classic form of his argumen, define he following variables: S : Spo exchange rae of dollars per uni of foreign currency. y : -year defaul-free dollar spo (ie, zero-coupon rae of ineres. y% : -year defaul-free foreign spo (ie, zero-coupon rae of ineres. F : Forward exchange rae of dollars per uni foreign currency for delivery in years. Now consider he following able of oday s ransacions (Figure 1. I is assumed hroughou ha here is no counerpary defaul risk in forward or swap conracs. 2 Figure 1. he arbirage argumen for classic ineres rae pariy ransacion oday($ ime ($ oday (Fgn ime (Fgn Borrow S dollars S + S ( + y 1 SellSdollars, buy foreign S +1 Inves foreign -1 ( 1+ y% Sell foreign forward + F ( 1+ y% ( 1+ y% oal 1 2 he auhors hank Fredrik Akesson, Dev Joneja, Jean Paul Paradis, and Fei Zhou for helpful commens and suggesions. Pedro Porfirio works on he FX rading Desk. Counerpary defaul risk is largely miigaed by he financial soundness of he eniies involved, by requiremens o pos collaeral based on changing marke values, and by he imposiion of credi lines. June 23 2 Please see imporan analys cerificaions a he end of his repor.

4 Lehman Brohers LMR Quarerly Reprined from he LMR Quarerly, vol. 24-Q2 Because hese ransacions neiher generae nor require cash oday, ruling ou arbirage requires ha hey neiher generae nor require cash a ime eiher. Mahemaically, ( ( S y + F + y% = (1 Solving for he forward price, F = S ( 1+ y ( 1+ y% (2 he problem wih applying his classic argumen in pracice is ha marke paricipans observe spo raes implied from swap raes, no defaul-free spo raes. Because swap raes are fixed raes ha are fair agains 3-monh LIBOR, which has a buil-in credi premium, spo raes derived from swap raes are oo high o be used in equaion (2. he following secion describes how money marke basis swaps link raes conaining differen credi premiums. 2. MONEY MARKE BASIS SWAPS A money marke basis swap is an exchange of floaing rae paymens based on one index for floaing rae paymens based on anoher index. Liquid examples include: Fed Funds vs. 3- monh LIBOR; -Bills vs. 3-monh LIBOR; 1-monh vs. 3-monh LIBOR; 1-monh vs. 6- monh LIBOR; and 3-monh vs. 6-monh LIBOR. aking one example a bi furher, on March 4, 23, a rader migh have agreed o exchange 1-monh LIBOR plus 1.5 basis poins monhly for 6-monh LIBOR semiannually for hree years. Consider an imaginary basis swap o exchange he defaul-free rae of one erm for he defaul-free rae of anoher erm, eg, o exchange he defaul-free 1-monh rae monhly for he defaul-free 3-monh rae quarerly. Appendix 1 shows ha his and all similar swaps should rade fla. Inuiively, he definiion of he erm srucure of defaul-free raes is precisely ha borrowers and lenders are indifferen beween 1-monh money rolled over a quarer and 3-monh money. Unlike hese imaginary swaps of defaul-free raes, observed money marke basis swaps exchange raes wih a buil-in credi premium. Furhermore, he credi premium buil ino a paricular rae index differs from ha buil ino anoher. For example, he credi risk of a 3- monh loan is greaer han ha of rolling over 1-monh loans for a quarer. Hence, o clear markes, 3-monh LIBOR mus be se such ha is cerain receip, ie, is receip wih no possibiliy of defaul, has greaer value han he cerain receip of 1-monh LIBOR over he corresponding quarer. Hence, in a basis swap wihou any counerpary defaul risk, 1-monh LIBOR plus a spread is fair agains 3-monh LIBOR. I is imporan o disinguish beween he pricing of loans made a he index rae and he pricing of basis swaps. he credi characerisics of Corporaion A migh be such ha i fairly borrows a 1-monh LIBOR. Similarly, he credi characerisics of Corporaion B migh be such ha i fairly borrows a 3-monh LIBOR. Neverheless, as argued in he previous paragraph, paries o a swap wih no counerpary defaul risk willingly exchange 1-monh LIBOR plus a spread versus 3-monh LIBOR. Because hese swap paries do no bear any of he credi risk buil ino he loan raes, hey look o he value of a cerain flow of one rae agains he value of a cerain flow of he oher rae. March 24 3

5 Lehman Brohers LMR Quarerly Reprined from he LMR Quarerly, vol. 24-Q2 3. CROSS-CURRENCY BASIS SWAPS A cross-currency basis swap is essenially an exchange of a floaing rae noe in one currency for a floaing rae noe in anoher currency. For example, on March 4, 23 he Canadian dollar (CAD raded for.677 US dollars (USD and a rader migh have agreed o he following cross-currency basis swap: Pay 1 CAD and receive.677 USD a iniiaion Receive 3-monh CDOR 3 plus 1 basis poins on 1 CAD and pay 3-monh USD LIBOR quarerly on.677 USD for hree years Receive 1 CAD and pay.677 USD a expiraion o undersand why 3-monh CDOR plus 1 is fair agains 3-monh USD LIBOR, i is bes o begin by considering an imaginary cross-currency basis swap exchanging a defaul-free, overnigh CAD rae for a defaul-free, overnigh USD rae. Under relaively mild assumpions, Appendix 2 proves ha his cross-currency basis swap should rade fla. Inuiively, paying 1 CAD oday, receiving he defaul-free CAD rae on 1 CAD, and receiving 1 CAD a expiraion is worh 1 CAD oday. Similarly, receiving.677 USD, paying he defaul-free USD rae on.677 dollars, and paying.677 USD a expiraion is worh.677 dollars oday. herefore, because he exchange rae is.677 USD per CAD, he exchange of hese floaing rae noes is fair oday. Unlike his imaginary swap of defaul-free raes, observed cross-currency basis swaps exchange LIBOR raes. herefore, one may hink of a cross-currency basis swap of LIBOR raes as a porfolio of hree imaginary swaps: a cross-currency basis swap of overnigh, defaul-free raes; a money marke basis swap of dollar LIBOR versus dollar, defaul-free, overnigh raes; and a money marke basis swap of foreign LIBOR versus foreign, defaulfree, overnigh raes. Viewed his way, i becomes clear ha observed cross-currency basis swap spreads arise from he difference beween local basis spreads, ha is, from he difference beween he wo erm srucures of credi spreads. In he example, if USD 3-monh LIBOR has more credi risk han 3-monh CDOR, hen, in a swap wih no defaul risk, a sream of USD LIBOR would be worh more han a sream of CDOR. herefore, CDOR plus a spread would rade fair agains USD LIBOR. 4. INERES RAE PARIY REVISIED his secion derives wo ineres rae pariy relaionships which depend on raes derived from he swap curve and on basis swap spread levels. For his purpose, define he following variables. R : -year dollar par swap rae wih fixed flows paid quarerly. R % : -year foreign par swap rae wih fixed flows paid quarerly. Y : -year dollar zero-coupon swap rae. Y % : -year foreign zero-coupon swap rae. L : 3-monh dollar LIBOR a ime. L % : 3-monh foreign LIBOR a ime. 3 See secion 5.2 for a descripion of Canadian shor-erm raes. March 24 4

6 Lehman Brohers LMR Quarerly Reprined from he LMR Quarerly, vol. 24-Q2 b : A swap of he overnigh dollar defaul-free rae is fair agains 3-monh dollar LIBOR minus b. b % : A swap of he overnigh foreign defaul-free rae is fair agains 3-monh foreign LIBOR minus b %. X : A swap of 3-monh dollar LIBOR plus X is fair agains 3-monh foreign LIBOR. he derivaion of he firs pariy relaionship, using par swaps, zero-coupon swaps, and a 3-monh cross-currency basis swap, is explained in Figure 2. Inerim cash flows are annualized. Ruling ou arbirage opporuniies implies ha he ime- dollar payoff is zero, ie: ( ( ( S 1+ X R 1+ Y X R + F 1+ Y% = (3 Solving for he forward rae, ( Y ( 1 Y% ( Y ( F = S 1 + X R Y (4 he obvious advanage of equaion (4 over he classic pariy condiion (2 is ha (4 depends on swap raes and on he cross-currency marke basis swap spread. By conras, he classic condiion depends on he generally unobservable defaul-free raes of ineres. According o he definiions of R and Y, he erm muliplying X in (4 is he presen value facor for a sream of quarerly paymens a he annual rae of X. Hence his pariy relaionship may be rewrien as: ( 1+ Y ( 1 Y where PV[ X ] gives he presen value jus described. [ ] F = S 1+ PV X + % (5 Figure 2. he ineres rae pariy argumen wih a cross-currency basis swap ransacion oday ($ Inerim ($ ime ($ Spo FX S +1 Cross-currency swap + S S ( L + X S Par swap: rec fixed/ pay floaing on SX R SX [ SX RL ] Zero-coupon swap: rec floaing on S ( + X R S ( + X R L S ( X R ( Y Zero-coupon swap: pay floaing 1 1 oday (Fgn Inerim (Fgn ime (Fgn L % +1 L % ( Y% 1+ 1 Sell foreign forward + F ( 1+ Y % ( 1+ Y % oal March 24 5

7 Lehman Brohers LMR Quarerly Reprined from he LMR Quarerly, vol. 24-Q2 he second pariy relaionship uses a cross-currency swap of overnigh, defaul-free raes and an overnigh versus 3-monh basis swap in each currency insead of a cross-currency swap of 3-monh LIBOR raes. Because cross-currency swaps of overnigh, defaul-free raes do no rade, his second relaionship is of less pracical use han (4 or (5. However, his second resul is useful for undersanding he deerminaion of he 3-monh LIBOR cross-currency basis swap spread. Define r and r% o be he overnigh, dollar and foreign defaul-free raes and consider he se of ransacions described in Figure 3. Ruling ou arbirage opporuniies requires ha S 1 S b R + Y + F + Y% b R% + Y% = (6 ( ( ( (% ( Solving for he forward rae, F = S ( ( Y% ( Y ( Y 1 b R Y ( 1+ Y% b% R% ( 1+ Y% (7 Comparing (7 wih (4 reveals ha he cross-currency basis swap spread depends on he relaive magniudes of he money marke basis spreads in heir own currencies. If he credi risk embedded in he dollar curve is relaively large, so ha b is large relaive o b %, hen X ends o be negaive and dollar LIBOR minus a spread ends o be fair agains foreign LIBOR. Conversely, if he credi risk embedded in he foreign curve is greaer, so ha b % is large relaive o b, X ends o be posiive and dollar LIBOR plus a spread ends o be fair agains foreign LIBOR. March 24 6

8 Lehman Brohers LMR Quarerly LMR Quarerly, vol. 23-Q2 Figure 3. he ineres rae pariy argumen wih money marke basis swaps oday Inerim ime oday Inerim ime ransacion ($ ($ ($ (Fgn (Fgn (Fgn Spo FX S +1 Cross-currency swap S + rs S -1 r% +1 O/N vs. 3-monh swaps ( r% + L % b % + rs L b S Par swaps on Sb Rand b% R% bs [ bs R ] L + b% ( b% R% L% Zero-coupon swaps on ( S 1 b R and 1 b% R% S ( 1 b R L S ( b R ( Y ( 1 b % R % L % ( b% R% ( Y% Sell foreign forward + F ( 1+ Y % ( b% R % ( 1+ Y % 1 ( % ( % % ( % 1+ Y + b R 1+ Y 1 oal March 24 7

9 Lehman Brohers Fixed Income Research LMR Quarerly, vol. 24-Q2 5. EMPIRICAL EVIDENCE Cross-currency basis swaps spreads are normally quoed as USD LIBOR versus he foreign currency plus or minus a spread. he heory presened in he previous secions predics ha when he credi risk embedded in foreign LIBOR raes is greaer han he credi risk embedded in USD LIBOR raes, he cross-currency basis swap spread should be negaive, ie, cerain paymens of USD LIBOR are fair agains foreign LIBOR minus a spread. Similarly, when he credi risk embedded in foreign LIBOR raes is less han ha in USD LIBOR raes, he spread should be posiive. his secion presens some empirical evidence supporing hese implicaions. 5.1 Cable versus USD Boh US and UK markes rade 1-monh versus 3-monh money marke basis swaps. Quoed versus local 3-monh raes, represenaive levels in April 23 were 1-monh USD LIBOR plus.75 basis poins and 1-monh GBP LIBOR plus 2 basis poins. hese levels srongly imply ha more credi risk is buil ino GBP LIBOR han ino USD LIBOR. herefore, heory predics ha he cross-currency basis swap spread should be negaive. his was in fac he case: a represenaive level in April 23 was -1.5 basis poins. In May 23, he money marke basis swaps moved o 1-monh USD LIBOR plus 1.5 basis poins and 1-monh GBP LIBOR plus 1.75 basis poins. he widening of he US spread and ighening of he UK spread can be a leas parially explained by a subsanial weakening of he US dollar, 45-year lows in 1-year US yields, and a srong risk in he Lehman Risk Premium Index. In any case, as prediced by heory, he narrowing of he difference beween he money marke basis swap spreads coincided wih he cross-currency basis swap spread falling from -1.5 in April o -.75 in May. 5.2 CAD versus USD he floaing rae of CAD swaps are called CDOR raes (pronounced see-dor, which are keyed off Canadian BAs (Bankers Accepances. Since BAs are collaeralized by receivables whereas LIBOR raes are unsecured bank loans, CDOR raes should reflec lower credi risk han LIBOR raes. In ha case, heory implies ha USD versus CAD cross-currency basis swap spreads should be posiive. As of May 23, he 1-year swap raded a 8 basis poins, he 2-year a 9 basis poins, and he 5- and 1-year a 1 basis poins. 5.3 Yen versus USD Given he difficulies of he Japanese economy and banking sysem over he pas several years, i is reasonable o conclude ha shor-erm Japanese raes reflec much higher credi risk han equivalen mauriy US raes. Furhermore, under his supposiion, heory predics ha he cross-currency basis swap spread should be subsanially negaive. Figure 4, ploing his spread since January 21, suppors his conenion. he declining magniude of he spread since he end of 22 may reflec opimism surrounding recen changes a he Bank of Japan. March 24 8

10 Lehman Brohers Fixed Income Research LMR Quarerly, vol. 24-Q2 Figure 4. Recen Hisory of he USD/JPY Cross-Currency Basis Swap Spread Basis Poins May-1 Jul-1 Oc-1 Jan-2 Mar-2 Jun-2 Sep-2 Dec-2 Feb-3 May-3 6. CONCLUSION his paper derives ineres rae pariy formulas ha include he effecs of non-zero crosscurrency basis swap spreads. he proposiion ha hese spreads reflec credi differences across shor-erm raes in various currencies is suppored by some evidence from recen levels of GBP/USD, CAD/USD, and Yen/USD cross-currency basis swaps. APPENDIX 1: MONEY MARKE SWAPS OF DEFAUL-FREE RAES RADE FLA he defaul-free erm srucure of raes is defined such ha for he risk-neural process of he insananeous, coninuously compounded rae r(, he price of a zero coupon bond mauring a ime is: E e r s ds ( (8 I follows ha a floaing rae noe paying he insananeous rae is worh s E r s e ds Ee r( d r( s ds ( + (9 Evaluaing he inegral in he firs expecaion erm, March 24 9

11 Lehman Brohers Fixed Income Research LMR Quarerly, vol. 24-Q2 Hence, his floaing rae noe is worh par. s r( d r( s ds E e E e + s= r( s ds r( s ds = E e + 1+ E e = 1 If paymens are based on raes of a paricular erm, he floaing rae noe is sill worh par. Le y τ be he rae of ermτ years a ime iτ. hen, he value of he noe is: i Where, by definiion, (1 ( i+ 1 τ ( r s ds r( s ds E y τ iτ e E e + (11 i 1 = E i e 1+ y iττ ( i+ 1 τ iτ r( s ds (12 Using he law of ieraed expecaions, (11 may be rewrien as: iτ ( i+ 1 τ ( ( r s ds r s ds r( s ds iτ E yi ττ e Ei e E e + (13 i Subsiuing (12 ino (13 and using he law of ieraed expecaions again, iτ r( s ds r( s ds r( s ds iτ E e 1 e E e + i ( i+ 1 τ iτ ( i+ 1 τ r( s ds r( s ds r( s ds = E e e E e + (14 i r( s ds r( s ds = 1 E e + E e = 1 Hence, a floaing rae noe paying he defaul-free rae of any erm is also worh par. Because all floaing rae noes keyed off defaul-free raes are worh par, a swap of any defaul-free rae for anoher is worh zero. March 24 1

12 Lehman Brohers Fixed Income Research LMR Quarerly, vol. 24-Q2 APPENDIX 2: CROSS-CURRENCY BASIS SWAPS OF DEFAUL-FREE RAES RADE FLA As in Appendix 1, le r( denoe he risk-neural process for he valuaion of dollar claims. Le S( be he exchange rae process of dollars per uni of foreign currency which is riskneural wih respec o he valuaion of dollar claims. By hese definiions, he mean of S( is consrained by an insananeous form of ineres rae pariy. o wi, consider using 1 dollar o buy 1 S unis of foreign currency, invesing hose proceeds a he foreign rae and, afer ime, convering back o dollars. Because he processes are risk-neural wih respec o pricing dollar claims, he presen value of his sraegy mus be 1 dollar. Mahemaically, leing he foreign currency ineres rae be r%, S + S r% r E e e 1 S = S ( r r% E = e 1 S (15 Leing approach zero, ds E = ( r r d S % (16 he dollar value of a claim in he foreign currency may be compued by convering he foreign currency cash flows o dollars, discouning a dollar ineres raes, and aking expecaions using he risk-neural process for valuing dollar claims. In he case of a floaing rae noe in he foreign currency, he value is: rudu ( rd ( ( ( + % (17 E S r e d E S e For concreeness, le he risk-neural exchange rae follow he sochasic process ds S ( r r d dw( = % +σ (18 which implies ha ( S = S e ( r( s r% ( s ds σ 2 + σw( 1 2 (19 Using (19, he firs erm of (17 may be simplified as follows: March 24 11

13 Lehman Brohers Fixed Income Research LMR Quarerly, vol. 24-Q2 2 ( ( 1 rudu rsds % 2 σ + σw ( ( ( = ( % % E S r e d S E r e d = r( s ds d % W( SE σ + σ e e d d = SE e e + e d e = = S SE e ( r % s ds r( s ds W( % σ + σ 1 σ + σw( 2 2 r% ( s ds e 2 σ + σw( 1 2 { } (2 he las equaliy holds because he las erm in he brackes of he penulimae equaliy is a maringale and, herefore, zero in expecaion. Using (19, he second erm of (17 is: SEe ( 1 2 r % s ds 2 σ + σw( (21 Combining he las equaliy of (2 wih (21 o add he wo erms of (17, he value of he foreign floaing rae noe is S dollars or 1 uni of foreign currency. From Appendix 1, he dollar floaer is worh is noional amoun; from his appendix, he foreign floaer is worh is noional amoun. Hence, a cross-currency basis swap, ie, an exchange of S noional amoun of dollar floaing rae swaps for 1 noional amoun of foreign floaing rae swaps, is worh zero. In oher words, he cross-currency basis swap rades fla. March 24 12

14 Lehman Brohers Fixed Income Research analyss produce proprieary research in conjuncion wih firm rading desks ha rade as principal in he insrumens menioned herein, and hence heir research is no independen of he proprieary ineress of he firm. he firm's ineress may conflic wih he ineress of an invesor in hose insrumens. Lehman Brohers Fixed Income Research analyss receive compensaion based in par on he firm's rading and capial markes revenues. Lehman Brohers and any affiliae may have a posiion in he insrumens or he company discussed in his repor. he views expressed in his repor accuraely reflec he personal views of Bruce uckmann, Pedro Porfirio, he primary analys(s responsible for his repor, abou he subjec securiies or issuers referred o herein, and no par of such analys(s' compensaion was, is or will be direcly or indirecly relaed o he specific recommendaions or views expressed herein. he research analyss responsible for preparing his repor receive compensaion based upon various facors, including, among oher hings, he qualiy of heir work, firm revenues, including rading and capial markes revenues, compeiive facors and clien feedback. Any repors referenced herein published afer 14 April 23 have been cerified in accordance wih Regulaion AC. o obain copies of hese repors and heir cerificaions, please conac Larry Pindyck (lpindyck@lehman.com; or Valerie Monchi (vmonchi@lehman.com; his maerial has been prepared and/or issued by Lehman Brohers Inc., member SIPC, and/or one of is affiliaes ( Lehman Brohers and has been approved by Lehman Brohers Inernaional (Europe, regulaed by he Financial Services Auhoriy, in connecion wih is disribuion in he European Economic Area. his maerial is disribued in Japan by Lehman Brohers Japan Inc., and in Hong Kong by Lehman Brohers Asia Limied. his maerial is disribued in Ausralia by Lehman Brohers Ausralia Py Limied, and in Singapore by Lehman Brohers Inc., Singapore Branch. his documen is for informaion purposes only and i should no be regarded as an offer o sell or as a soliciaion of an offer o buy he securiies or oher insrumens menioned in i. No par of his documen may be reproduced in any manner wihou he wrien permission of Lehman Brohers. We do no represen ha his informaion, including any hird pary informaion, is accurae or complee and i should no be relied upon as such. I is provided wih he undersanding ha Lehman Brohers is no acing in a fiduciary capaciy. Opinions expressed herein reflec he opinion of Lehman Brohers and are subjec o change wihou noice. he producs menioned in his documen may no be eligible for sale in some saes or counries, and hey may no be suiable for all ypes of invesors. If an invesor has any doubs abou produc suiabiliy, he should consul his Lehman Brohers represenaive. he value of and he income produced by producs may flucuae, so ha an invesor may ge back less han he invesed. Value and income may be adversely affeced by exchange raes, ineres raes, or oher facors. Pas performance is no necessarily indicaive of fuure resuls. If a produc is income producing, par of he capial invesed may be used o pay ha income. Lehman Brohers may make a marke or deal as principal in he securiies menioned in his documen or in opions, fuures, or oher derivaives based hereon. In addiion, Lehman Brohers, is shareholders, direcors, officers and/or employees, may from ime o ime have long or shor posiions in such securiies or in opions, fuures, or oher derivaive insrumens based hereon. One or more direcors, officers, and/or employees of Lehman Brohers may be a direcor of he issuer of he securiies menioned in his documen. Lehman Brohers may have managed or co-managed a public offering of securiies for any issuer menioned in his documen wihin he las hree years, or may, from ime o ime, perform invesmen banking or oher services for, or solici invesmen banking or oher business from any company menioned in his documen. 23 Lehman Brohers. All righs reserved. Addiional informaion is available on reques. Please conac a Lehman Brohers eniy in your home jurisdicion.