7 Swaps. Overview. I have friends in overalls whose friendship I would not swap for the favor of the kings of the world. Thomas A.

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1 7 Swaps I have frieds i overalls whose friedship I would ot swap for the favor of the kigs of the world. Thomas A. Ediso Overview Mechaics of iterest rate swaps Day cout issues (Cofirmatios skip) The comparative-advatage argumet

2 2 Ia Buckley The ature of swap rates Determiig the /swap zero rates Valuatio of iterest rate swaps Currecy swaps Valuatio of currecy swaps Credit risk Other types of swap Summary Preamble First cotracts 980s Agreemet to exchage cash flows at specified future times accordig to certai specified rules Cash flows from future value of variable: IR, FX, etc. Geeralises a forward cotract if cash settled swap forward price for market value Multiple dates Focus o swaps: Fwd 00 oz Gold $400/oz + yr plai vailla iterest rate fixed-for-fixed currecy ª Swap $40, S + yr Nature of Swaps Defiitio 7.. A swap is a agreemet to exchage cash flows at specified future times accordig to certai specified rules. Mechaics of iterest rate swaps Most commo plai vailla iterest rate swap Defiitio 7.2. A plai vailla iterest rate swap is a agreemet i which a compay agrees to pay cash flows equal to iterest at a predetermied fixed rate i retur for iterest at a floatig rate, o a otioal pricipal, for a period of time. Floatig rate typically (Chapter 4) Referece IR for loas i markets (cf. prime i markets)

3 CMFM03 Fiacial Markets 3 A Example of a Plai Vailla Iterest Rate Swap A agreemet by Microsoft to Figure receive 6-moth & pay a fixed rate of 5% per aum every 6 moths for 3 years o a otioal pricipal of $00 millio Itel 5.0 % ôøøø Microsoft Figure 7.: Iterest rate swap betwee Microsoft ad Itel Cash Flows to Microsoft See Hull 2006 Table 7., page 5 Table 7.. Cash flows (millios of dollars) to Microsoft, i a $00 millio, 3-year iterest rate swap, whe a fixed rate of 5% is paid ad is received. The et cash flow is the differece. (Igore day cout issues.) Date 6-moth (%) Floatig received Fixed paid Net cash flow Mar. 5, Sept. 5, Mar. 5, Sept. 5, Mar. 5, Sept. 5, Mar. 5, Cash Flows to Microsoft with exchage of pricipal Table 7.2. Cash flows (millios of dollars) to Microsoft, i a $00 millio, 3-year iterest rate swap, whe a fixed rate of 5% is paid ad is received. The et cash flow is the differece.

4 4 Ia Buckley Date 6-moth (%) Floatig received Fixed paid Net cash flow Mar. 5, Sept. 5, Mar. 5, Sept. 5, Mar. 5, Sept. 5, Mar. 5, Code Plot Cash flow 3 2 Cash flow Date Date HaL No exchage of pricipal HbL Exchage of pricipal Figure 7.2: Cash flows (millios of dollars) to Microsoft, i a $00 millio, 3-year iterest rate swap, whe a fixed rate of 5% is paid ad is received. The et cash flow is the differece. Variats are (a) without ad (b) with exchage of pricipal at the ed. Remark The et cash flow is the same whether or ot the pricipal is exchaged. Typically the pricipal is ot exchaged (hece otioal pricipal) However, case with exchage of pricipal gives isight Typical Uses of a Iterest Rate Swap Covertig a liability from fixed rate to floatig rate floatig rate to fixed rate Covertig a ivestmet from Figure fixed rate to floatig rate floatig rate to fixed rate

5 CMFM03 Fiacial Markets 5 Cashflow to trasform Liability Asset 5.0 % 5.2 % ôøøø ôø Itel % Itel 5.0 % ôøøø Microsoft Microsoft øøøö + 0. % 4.7 % ôø Figure 7.3: Use of a iterest rate swap to trasform a liability or a asset, without a fiacial itermediary. Cashflow to trasform Liability % 5.2 % ôøøøø ôø Itel øøøö FI 5.05 % ôøøøø øøøö Microsoft + 0. % Asset % Itel % ôøøøø øøøö FI 5.05 % ôøøøø øøøö Microsoft 4.7 % ôø Figure 7.4: Use of a iterest rate swap to trasform a liability or a asset, with a fiacial itermediary. Trasformig a liability Covertio of debt paymets Microsoft from + 0.% to Itel from 5.2% to Trasformig a asset Covertio of asset cash (i)flows Microsoft from 4.7% to Itel from - 0.2% to Fiacial itermediary Typically ears basis poits Has cotracts If oe party defaults does Ñ does ot Ñ have to hoour its agreemet with other Further detail Musiela Rutkowski (2004) Usually paymets are set i advace, paid/settled i arrears Paid/settled i advace also possible, but arrears paymets require discoutig Covetios vary

6 6 Ia Buckley Discout paymets float by float, fixed by float or fix Day cout issues (e.g. 6-mo i Table 7.) is a moey market rate, hece quoted o basis Example 7.. The first floatig paymet i Table 7. is for a rate of 4.2%. What is the correct iterest paymet if day cout covetios are take ito accout? Referece period Mar 5 to Sept 5, 2004 is 84 days Iterest eared is $00ä0 6 ä0.042ä ÅÅÅÅÅÅÅÅÅ 84 = $ based cash flows Notatio L R pricipal rate umber of days sice last paymet Note: Greek L, Lambda, to save L for rate, below. Swap cash flow expressio Cash flow = L R ÅÅÅÅÅÅÅÅÅÅÅÅÅ Å 360 (7.) Fixed rate cash flows Usually actual/365 30/360 Ofte factor 360/365 required to compare 6-mo with fixed

7 CMFM03 Fiacial Markets 7 The comparative-advatage argumet The Comparative Advatage Argumet Hull Table 7.4, page 57 AAACorp wats to borrow floatig BBBCorp wats to borrow fixed Table 7.3. Borrowig rates for two corporatios Compay Fixed Floatig AAACorp 4.0% + 0.3% BBBCorp 5.2% +.0% Key feature differece betwee rates, greater tha betwee rates 9 AAACorp = has a comparative advatage i 9 BBBCorp = markets Fiacial itermediary Agreemet No ô 4 % AAACorp 3.95 % ôøøø BBBCorp øøøö + % Yes ô 4 % AAACorp 3.93 % ôøøø FI 3.97 % ôøøø BBBCorp + % Figure 7.5: Illustratio of comparative advatage agreemet for two corporatios, without ad with a fiacial itermediary. Criticism of the Comparative Advatage Argumet 4.0% ad 5.2% rates available to AAACorp ad BBBCorp i fixed rate markets are 5-year rates The +0.3% ad +% rates available i the floatig rate market are six-moth rates BBBCorp s fixed rate depeds o spread above it borrows at i the future The Nature of Swap Rates Six-moth is short-term AA borrowig rate

8 8 Ia Buckley 5-year swap rate has risk ~ 0 six-moth loas made to AA borrowers at Leder ca eter ito swap icome from loas exchaged for 5-year swap rate Swap rates Practical defiitio Defiitio 7.3. Swap rates are the fixed rates at which fiacial istitutios offer iterest rate swap cotracts to their cliets. M&R 2004, p. 328 Theoretical defiitio Defiitio 7.4. The swap rate is that value of the fixed rate that makes the value of the swap zero at iceptio. M&R 2004, p. 479 Cf. forward iterest rate ad value of FRA Istitutios offer swap cotracts at rates with appropriate spreads above ad below the theoretical oes Determiig the /swap zero rates Risk-free rates for derivative valuatio purposes 2 mos Eurodollar futures -2 yrs Swaps 2 yrs /swap zero curve Chapter 4 par yields ow put to use! Overview of argumet Cosider a ew swap with fixed rate = swap rate Add pricipals o both sides o fial paymet date fl swap ª exchage of fixed rate ad floatig rate bods Value of bods/swaps Floatig-rate rate bod *. Swap. fl fixed-rate bod worth. fl swap rates defie par yield bods; used to bootstrap the (or /swap) zero curve * Argumet below

9 CMFM03 Fiacial Markets 9 Swap rates from bod prices Notatio S swap rate for a -year agreemet Result Propositio 7.5. The -period swap rates S ca be expressed i terms of prices of zero coupo bods P 0 by the relatioship S = -P 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ i= P 0i S = - P 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ i= P 0i (7.2) No surprises here; this was expressio for par yield from Chapter 4 Mathematica implemetatio to explore terms SwapRate@_D:= P@0, D P@0, id i= Table@8S@D, SwapRate@D<, 8, 3<D êê TableForm S@D S@2D S@3D P@0,D P@0,D P@0,2D P@0,D+P@0,2D P@0,3D P@0,D+P@0,2D+P@0,3D Proof Cosider 4-year swap S 4 S 4 2 Figure 7.6: Diagram of cashflows for fixed leg of a 4-year swap agreemet Pay to get 4 coupos of swap rate iitial ivestmet back at 4 = S 4 P 0 + S 4 P 02 + S 4 P 03 +H + S 4 L P 04 Similarly for =, 2, 3, S 4 S Bod prices from swap rates Result Propositio 7.6. The prices of zero coupo bods P 0 ca be expressed i terms of the -period swap rate S by the relatioship P 0 = - S r=0 - r k=0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ +S -k

10 0 Ia Buckley - P 0 = - S r=0 r k=0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + S -k (7.3) Mathematica implemetatio to explore terms ZeroBod@_D:= S@D +S@ kd r r=0k=0 Table@8P@0, D, ZeroBod@D<, 8, 3<D êê TableForm P@0, D S@D +S@D P@0, 2D S@2D I P@0, 3D S@3D I +S@2D + H+S@DLH+S@2DL M +S@3D + H+S@2DLH+S@3DL + H+S@DLH+S@2DLH+S@3DL M Proof The swap rate is the level of the fixed rates such that the swap has zero value at iceptio Zero value occurs whe floatig rate bod equals fixed rate bod At iceptio, value of floatig rate bod is uity (up to a commo factor of the pricipal) = H + S L P 0 = S 2 P 0 +H + S 2 L P 02 = S 3 P 0 + S 3 P 02 +H + S 3 L P 03 ª ª = S i= P 0 i + P 0 ª ª Solve iteratively i terms of P 0i P 0 = P 02 = ÅÅÅÅÅÅÅÅÅÅÅÅÅ H+S = - S ÅÅÅÅÅÅÅÅÅÅÅ L -S 2 P 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ +S +S 2 = ÅÅÅÅÅÅÅÅÅÅÅ +S 2 - ÅÅÅÅÅÅÅÅÅÅÅ S 2 +S 2 ÅÅÅÅÅÅÅÅÅÅÅ +S +S 2 - ÅÅÅÅÅÅÅÅÅÅÅ S 2 +S 2 ÅÅÅÅÅÅÅÅÅÅÅ +S = - S 2 ÅÅÅÅÅÅÅÅÅÅÅ +S 2 - ÅÅÅÅÅÅÅÅÅÅÅ S 2 +S 2 ÅÅÅÅÅÅÅÅÅÅÅ +S = - S 2 I ÅÅÅÅÅÅÅÅÅÅÅ +S 2 + ÅÅÅÅÅÅÅÅÅÅÅ +S ÅÅÅÅÅÅÅÅÅÅÅ +S 2 M = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ +S 2-S 2 +S 3 + P 03 = = - S 3 I ÅÅÅÅÅÅÅÅÅÅÅ ª ª P 0 = - S r=0 ª ª - r k=0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H+S 2 LH+S 3 + L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ +S -k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H+S LH+S 2 LH+S 3 L M FRNs are worth par after a coupo paymet Not i Hull This argumet courtesy of L.P. Hughsto. Notatio

11 CMFM03 Fiacial Markets a,b start ad ed of time iterval P ab, PHa,bL price of zero-coupo bod at a, with uit payoff at b L ab iterest rate umber of days sice last paymet Setup Cosider a 3-yr ivestmet payig a coupo, aually L 0 L 2 L Figure 7.7: Diagram of cashflows for 3-yr ivestmet payig a coupo, aually With a compoudig frequecy of m = ÅÅÅÅÅÅÅÅ, the rate ad price of a zero coupo b-a bod are related: P ab = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + L ab Hb - al (7.4) For our simple case, b - a = L ab = ÅÅÅÅÅÅÅÅÅÅÅ P ab - Argumet Each coupo paymet is discouted back to the previous paymet date usig a discout rate ÅÅÅÅÅÅÅÅÅ P 0 - ÅÅÅÅÅÅÅÅÅ P 2 - ÅÅÅÅÅÅÅÅÅ P L 0 L 2 L Figure 7.8: Diagram of cashflows for 3-yr ivestmet with coupos expressed i terms of bod prices Value of 3rd coupo at time 2 J ÅÅÅÅÅÅÅÅÅÅÅ - + N ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = P 23 + L 23 Value of 3rd ad 2d coupos at time i j ÅÅÅÅÅÅÅÅÅÅÅ - + k P 2 previous step Æy Repeat iteratively dow to t = 0 Value of 3rd, 2d & st coupos at time 0 i j ÅÅÅÅÅÅÅÅÅÅÅ - + k P 0 previous step Æy z ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = { + L 2 z ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = { + L 2

12 2 Ia Buckley If coupo has just bee paid, floatig rate ote is worth par For a alterative proof see e.g. Cuthbertso & Nitzsche (200) Fiacial Egieerig: Derivatives ad Risk Maagemet Example Example 7.2. The 6, 2 & 8-moth zero rates have bee determied to be 4%, 5% & 4.8%, respectively. The 2-year semi-aually compouded swap rate is 5%. Fid the 2-yr zero rate. Bod with semi-aual coupo of 5% sells for par 0.025äI -0.04ä ä ä.5 M +.025ä -r 2 ä2.0 = r 2 = % Valuatio of iterest rate swaps Swap ª log positio i oe bod with a short positio i aother portfolio of forward rate agreemets Valuatio i terms of bod prices Valuatio of bods Notatio fixed rate usual way floatig rate worth par immediately after ext paymet date V swap B fix B fl c i r i value of swap value of fixed rate bod value of floatig rate bod cash flow at time T i zero coupo yield for i D umber of paymets Swap value V swap = B fix - B fl (7.5)

13 CMFM03 Fiacial Markets 3 Fixed rate bod B fix = c i -r i T i + L -r T i= = c i P 0i + L P 0 i= (7.6) Value of floatig rate bod Notatio k * t * r * ext coupo paymet time to ext coupo paymet zero coupo bod discout rate t * D Table 7.4. Expressios for the value of a floatig rate bod Time, relative to coupo paymet Head Immediately after L Immediately before L+k * Time t * before HL+k * L -r* t * Uiversal swap pricig formula I your ext course o swaps Value at t of forward start payer swap V swap Ht, SL = PHt, T 0 L - S PHt, T j L - PHt, T L i= (7.7) M&R (2004) p. 477 Lueberger (998) p. 275 Cuthbertso & Nitzsche (200) p.380 Example Example 7.3. Value a swap betwee 6-mo ad 8% fixed with semi-aual compoudig, with pricipal of $00mi ad remaiig life of.25 years. rates for 3, 9 & 5 moth maturities are 0%, 0.5% & %, resp. The 6-mo rate at the last paymet date was 0.2%.

14 4 Ia Buckley k * = 0.5 ä 0.02 ä 00 = $5. mi T * =.25 See table for valuatio of B fix ad B fl V swap = B fix - B fl = $H $02.505Lä0 6 = -$4.267 mi Table 7.5. Table accompayig exercise Time B fix cash flow B fl cash flow Discout factor PV B fix cash flow PV B fl cash flow ä ä ä Total Valuatio i Terms of FRAs Each exchage of paymets i IR swap is FRA FRA values assume today s forward rates are realized Steps Example Obtai forward rates from zero rates Replace radom floatig rates by forward rates Discout ad sum Example 7.4. Value the swap from the previous example, but cosiderig it as a portfolio of FRAs. Fixed rate is 00ä0.06ä0.5 = 4.0 Floatig paymet at 3mos, already kow For remaiig floatig paymets, replace radom future rate by forward (FRA pricig trick) R F = R 2 T 2 -R T ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ T 2 -T R 3,9 = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.05ä ä0.25 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = (c.c.) 0.5 i.e..044% with semi aual compoudig Similarly for R 9,5. Table 7.6. Table accompayig exercise

15 CMFM03 Fiacial Markets 5 Time Fixed cash flow Float cash flow Net Discout factor PV et cash flow ä 0.02 ä 0.5= ä ä 0.5= ä ä ä Total Agrees Ã At iceptio, ad later, FRAs do ot have zero value Currecy swaps A Example of a Currecy Swap A agreemet to pay % o a sterlig pricipal of 0,000,000 & receive 8% o a US$ pricipal of $5,000,000 every year for 5 years IBM H$L 4.0 % ôøøøø øøøøö H L 7.0 % BP Figure 7.9: Currecy swap betwee IBM ad BP Exchage of pricipal I a iterest rate swap the pricipal is ot exchaged I a currecy swap the pricipal is usually exchaged at the begiig ad the ed of the swap s life The Cash Flows Hull Table 7.7, page 66 Fixed-for-fixed Table 7.7. Cash flows to IBM due to currecy swap cotract with BP

16 6 Ia Buckley Year $ Typical Uses of a Currecy Swap Coversio from a liability i oe currecy to a liability i aother currecy Coversio from a ivestmet i oe currecy to a ivestmet i aother currecy Comparative Advatage Argumets for Currecy Swaps Hull Table 7.8, page 67 Geeral Motors wats to borrow AUD Qatas wats to borrow USD Table 7.8. Rates for borrowig available to two compaies i two currecies. Compay USD AUD GM 5 % 2.6 % Quatas 7 % 3.0 % GM 9 Quatas = has a comparative advatage i the 9 = market Differece betwee 9 USD = rates is 9 AUD = Total gais to all parties = Figure USD 5.0 % USD 5 % ôøøøøøøøø ôøøø ø GM øøøøøøøøö AUD.9 % FI USD 6.3 % ôøøøøøøøø øøøøøøøøö AUD 3.0 % Quatas Figure 7.0: Currecy swap betwee IBM ad BP with fiacial itermediary. øøøøøøøø ö AUD 3.0 % Valuatio of currecy swaps Like iterest rate swaps, currecy swaps ca be valued either as the differece betwee 2 bods or as a portfolio of forward cotracts

17 CMFM03 Fiacial Markets 7 Swaps & Forwards A swap ca be regarded as a coveiet way of packagig forward cotracts The plai vailla iterest rate swap i our example cosisted of 6 FRAs The fixed for fixed currecy swap i our example cosisted of a cash trasactio & 5 forward cotracts The value of the swap is the sum of the values of the forward cotracts uderlyig the swap Swaps are ormally at the moey iitially This meas that it costs othig to eter ito a swap It does ot mea that each forward cotract uderlyig a swap is at the moey iitially Credit risk A swap is worth zero to a compay iitially At a future time its value is liable to be either positive or egative The compay has credit risk exposure oly whe its value is positive Other types of swap Floatig-for-floatig iterest rate swaps, amortizig swaps, step up swaps, forward swaps, costat maturity swaps, compoudig swaps, -i-arrears swaps, accrual swaps, diff swaps, cross currecy iterest rate swaps, equity swaps, extedable swaps, puttable swaps, swaptios, commodity swaps, volatility swaps etc..

18 8 Ia Buckley Summary Most commo: IR ad FX IR swap: exchage fixed for floatig rate o otioal pricipal over period of time FX swap: exchage fixed i oe FX for fixed i other FX o pricipals i each FX Pricipal exchaged IR swap: o FX swap: yes Used to trasform paymets associated with a loa or a asset from IR fixed to floatig, or vice versa FX oe FX to aother Value IR loa as fixed ad floatig bods portfolio FRAs Fiacial istitutio exposed to credit risk. If oe couter party defaults, with +ve value, still has to hoour agreemet with other.

Introduction to Financial Derivatives

Introduction to Financial Derivatives 550.444 Itroductio to Fiacial Derivatives Determiig Prices for Forwards ad Futures Week of October 1, 01 Where we are Last week: Itroductio to Iterest Rates, Future Value, Preset Value ad FRAs (Chapter