# Interest Rate Forwards and Swaps

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1 Interest Rate Forwards and Swaps Forward rate agreement (FRA) mxn FRA = agreement that fxes desgnated nterest rate coverng a perod of (n-m) months, startng n m months: Example: Depostor wants to fx rate on 3-month depost startng n 3 months enters 3x FRA wth bank. FRA rate s rate that causes nvestor to acheve same return through ether: -month depost, or 3-month depost, followed by renvestment of prncpal and nterest for 3 addtonal months at FRA rate Calculatng FRA rate F = { [+ R L Days L ] [+ R S Days S ] } ( Days F ) R L =3.5% 0 3 MO MO R S =3.0% F=FRA rate Where: R L R S F Days L Days S Days F = spot Lbor for long perod = spot Lbor for short perod = forward Lbor = no. of days n long perod = no. of days n short perod = no. of days n forward perod FRA Mechancs Customer does not need to place funds wth bank quotng the FRA. Customer places funds anywhere she lkes whle FRA s net settled wth bank. Example: FRA rate= 3.948%, 3m-Lbor reset=3.75% and Notonal= \$00. Interest earned on depost = \$00 x (3.75% x 9/) = \$ Under FRA, bank pays \$00 x (3.948%-3.75%) x 9/ = \$ Sum of 2 amounts gves customer \$.0022 annualzed 3.948% for a 9-day perod

2 Calculaton above assumed FRA settles at end of forward perod. In realty, FRA conventon s to settle net payment as soon as Lbor fxng takes place.e. at begnnng of forward perod. Amount of net settlement reflects tme value and thus equals Notonal (Contract rate Settlement rate) (Days (+ Settlement Rate Days ) ) FRA Arbtrage Assume on Jan, 2009 dealer quotes 3x FRA at 4.5% whle m Lbor = 3.5% and 3m Lbor = 3%. To arbtrage these rates: You borrow at 3.5% for months. Invest at 3% for 3 months. Lock n today 4.5% re-nvestment rate under 3 x FRA. Proft per \$ at maturty = \$0.003 = ( + 3% x 90/) x ( + 4.5% x 9/) ( + 3.5% x 8/) Conversely, f FRA = 3.%: nvest for months at 3.5%, borrow for 3 months at 3%, and lock n 3.% borrowng rate for 3 months n 3 months under the FRA Proft per \$ at maturty = \$ = ( + 3.5% x 8/) ( + 3% x 90/) x ( + 3.% x 9/) When bd-offer spreads are taken nto consderaton, arbtrage opportunty exsts f ether: ( + R 3m (bd) x 90/) x ( + FRA 3m (bd) x 9/) > ( + R m (offer) x 8/); or ( + R m (bd) x 8/) > ( + R 3m (offer) x 90/) x ( + FRA 3m (offer) x 9/) Man resdual rsks of arbtrage are: Rsk of default by enttes wth whom you nvest the money. Rsk of default by FRA counterparty when FRA s n your favor at settlement. Rsk that settlement payment under FRA cannot be renvested at rate rate used for dscountng under FRA net settlement formula. Interest Rate Exposure Assume on Jan, 2009 a borrower has borrowed \$00MM for 3 years at 3m Lbor flat. Lbor spot = 3% frst nterest payment = \$0.75MM. Exposure to Lbor for all subsequent perods: Lbor nterest expense and reported ncome. 2

3 Quoted FRA rates: Perod Number Contract Desgnaton Perod Start Date Number of Days n perod Rate for clent Spot Jan % 2 3x Aprl % 3 x9 July % 4 9x Oct % 5 x5 Jan % 5x8 Apr % 7 8x2 July % 8 2x24 Oct % 9 24x27 Jan 90.00% 0 27x30 Apr 9.25% 30x33 July 92.50% 33x3 Oct 92.75% Borrower could hedge each reset usng seres of FRAs Unequal and nterest expenses n each quarter. Clent needs smoother pattern of nterest outflows: nterest rate swap (IRS). Prcng: aggregate PV of CFs under IRS fxed leg (usng sngle rate) = aggregate PV of CFs pad under seres of FRAs. All settlements under IRS occur at end of nterest perods n contrast to FRA conventon Borrower 5.22% Lbor Bank Smple IRS formula F = = Days (Notonal Lbor DF Days (Notonal DF = Equaton explans why swap rate F s often descrbed as tme-weghted average of relevant Lbors. 3

4 Perod Lbor/ FRA Net payment DFs PVNP Adjusted net payment PVANP 3.00% 750, ,47,30,53,29, %,0, ,538,32,08,298, %,50, ,7,,335,598,297, %,23, ,5,088,335,598,28, %,250, ,84,93,30,53,238, %,327, ,24,53,32,08,235, %,405, ,29,78,335,598,232, %,49, ,33,027,335,598,24, %,500, ,343,54,30,53,70, %,579, ,393,8,32,08,4,978.50%,, 0.874,440,895,335,598,58,53.75%,725, ,470,940,335,598,38,890 Totals 4,728,09 4,728,09 PV of Floatng leg PV of Fxed Leg IRS formula ncludng dealer proft margn If dealer proft margn n PV terms = Proft, IRS rate F may be derved from followng equaton: = Days Days (Notonal Lbor DF) + Proft = (Notonal F = DF) f dealer s party recevng fxed; and = (Notonal Lbor Days DF) = (Notonal F Days DF ) + Proft f dealer s party recevng Lbor = 4

5 IRS formula when notonal amount changes over tme When IRS notonal decreases over tme Amortzng swap When IRS notonal ncreases over tme Accretng swap IRS rate F may be derved from followng equaton: = Days Days (Notonal Lbor DF) = (Notonal F = DF) Notonal = swap notonal at begnnng of perod Swap rate for amortzng/accretng notonal, as compared to swap rate for bullet notonal s summarzed n ths table: Lbor spot/forward curve Upward Inverted slopng Amortzng notonal Lower Hgher Accretng notonal Hgher Lower Forward-startng swap Prced usng precedng equaton by settng Notonal = 0 untl start date. If curve s upward slopng: forward-start IRS rate > spot-start IRS rate; and vce-versa f curve s nverted. Frequences and day-counts Each leg of IRS may have dfferent frequency and/or day-count. Example: Swap 3m Lbor-based 3-year bullet loan nto sem-annual fxed wth 30/ day-count. In ths case, IRS rate F s the soluton to followng equaton: Notonal Lbor Days DF = Notonal F 0.5 DF = 2 = 5

6 Lbor-n-arrears (LIA) swap Works exactly lke a regular swap n all respects, except that fxng of Lbor for a perod takes place 2 busness days before the end of that perod, nstead of 2 busness days before the begnnng. Regular Swap: LIA swap: Fxng Begnnng Settlement End Begnnng Fxng Settlement End 2 bus. days Interest Perod Interest Perod 2 bus. days Regular IRS formula apples for LIA swap where Lbor s Lbor rate that fxes 2 busness days before end of nterest perod (as opposed to begnnng n regular IRS). If curve s upward slopng rate on LIA swap fxed leg. Accurate prcng needs a small convexty adjustment, especally for long-dated swaps and n volatle rate envronments. IRS Replcaton Any IRS can be vewed as 2 bond postons: one long and one short. To receve fxed / pay floatng under IRS s _ to: Issung Lbor-based lablty and Investng proceeds n fxed-rate bond; and vce-versa -month Lbor Coupons on fxed rate bond Dealer -month Lbor Fxed Bass swap Involves exchange of floatng leg aganst another floatng leg but n another currency, and ncludes prncpal exchange at maturty. Example: suppose /\$ spot =.50 US bank wants to borrow 00MM for 5 yrs but has no access to fundng market. UK bank wants to borrow \$50MM for 5 yrs but has no access to \$ fundng market. US bank ssues \$50MM 5-yr FRN whch pays m \$ Lbor, and converts proceeds nto 00MM n spot FX market UK bank ssues 00MM 5-yr FRN whch pays m Lbor and converts proceeds nto \$50MM n spot FX market

7 In fact 2 banks can execute spot FX transactons wth each other 2 banks now enter nto swap descrbed n ths dagram: 00 MM at maturty \$50 MM bond at \$ Lbor US Bank Lbor on 00 MM sem-annually \$ Lbor on \$50 MM sem-annually UK Bank 00 MM bond at Lbor \$50 MM at maturty Ths s a Lbor bass swap that, unlke IRS, ncludes an exchange of prncpal at maturty A postve or negatve bass pont adjustment to ether leg may be necessary, dependng on demand/supply, relatve credtworthness of US Bank v. UK Bank, and other factors. Cross-currency Swap Company ssues \$50MM 3-yr 5.5% bond (S.A.) but needs to fund a UK expanson. Wth /\$ spot at.50, Company converts \$ proceeds nto, and seeks to hedge future \$ coupon and prncpal. Company would lke: Bank to make all payments requred (n \$) under the bond; n return Company makes to the bank, usually on same dates, payments n whose profle s dentcal to that of a fxed-rate sem-annual bond: 00 MM at maturty \$50 MM bond at \$ 5.5% Your Company Fxed on 00 MM sem-annually \$ 5.5% on \$50 MM sem-annually Bank \$50 MM at maturty The GBP fxed rate under the CCS, F, s derved from the followng equaton: Notonal \$ F \$ DCF \$ \$ ( DF ) + Notonal \$ \$ DF = = FXSpot Notonal F DCF j DF j + Notonal DF j= 7

8 Where: Notonal \$ and Notonal = notonals n \$ and respectvely = \$50MM and 00MM. DF \$ and DF = dscount factors for \$ and curves respectvely. DCF \$ and DCF = Day-count factor, accordng to approprate conventon, for \$ and respectvely. F \$ = US \$ fxed rate = 5.5%. FX Spot = /\$ FX spot rate =.50. Bass Pont Converson Cannot smply add or subtract equal numbers of bps to each leg (\$ and ) after prcng CCS. For example, addng 25bps to \$ fxed rate does not necessarly result n dentcal ncrease of 25 bps to leg. PVs obtaned by dscountng off 2 dfferent curves must be equal. Equvalent number of bass ponts n can be derved from ths equaton: Notonal \$ 25bps DCF \$ \$ ( DF ) + Notonal \$ \$ DF = j= = FXSpot Notonal Equv DCF DF + Notonal DF Constructng/hedgng CCS Typcally ths s done usng 3 swaps: 2 IRS and Lbor bass swap: \$50 MM at maturty \$Fxed on \$50 MM \$L on \$50 MM IRS \$ Bond at 5.50% Company \$Fxed on \$50 MM Fxed on 00 MM Bank 00 MM at maturty L+ x bps on 00 MM \$L on \$50 MM \$50 MM at maturty Lbor Bass Swap 00 MM at maturty L on 00 MM Fxed on 00 MM IRS2 If we are constructng a fxed/floatng CCS, one of the IRSs would become unnecessary. Frequences and day-count for the 2 legs do not have to match: could have \$ leg based on quarterly, 30/, and leg on m Lbor, act/. 8

9 \$ Lbor Curve \$ DFs \$ Pmts PV of \$ Pmts Days Lbor Curve PV of DFs Pmts Pmts 3.94% % % % % % % % % % % % Totals n \$ Fxed 5.50% PV Dff 0.00 Fxed 4.5% 50.2 n \$ \$ Notonal 50 FX Spot.5 Notonal 00 IRS Swap Revaluaton Fxed rate recever has MTM gan when rates : fxed payments are dscounted at dscount rates recevng fxed under a swap s one way of takng a vew on nterest rates. Fxed rate recever poston = ownng a fxed rate bond funded at Lbor gan on asset when rates whle lablty s value remans almost the same. DV0 of IRS DV0 of fxed rate bond whose prncpal, coupon, maturty and daycount/frequency match those of IRS fxed leg. IRS also has convexty changes n ts value are not lnear relatve to changes n rates 9

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