Lecture 7: Swaps (Hull, Ch. 7) ▸ What is a simple “Interest Rate Swap”? ▸ Currency Swaps ▸ Valuation of Interest Rate Swaps Finance Gibberish: “Plain Vanilla Swap is a fixed-for-floating interest rate swap Fixed rate payer: bought a swap or “gone long” a swap” say what? Fixed-for-Variable Swap (7.1, in English) Consider the following bonds: Bond A: CI = 5% fixed Bond B: (variable) CI = 1-yr Libor, set 1-yr before it is paid $100M $100M $100M $5M $5M $5M $100M $4M ? ? |––––––|–––––––|–––––––| |––––––––|––––––|–––––––––| 0 1 2 3 0 1 2 3 Long Swap = {-Bond A, +Bond B} aka Pay Fixed, Receive Variable Lec 7 Swaps dfdf

If UTC is long this swap, then UTC has agreed ▸ to pay $5M at time 1, 2, and 3. ▸ And receive LIBOR*$100M at time 1, 2, and 3 ▸ The FV = $100M, current price of both bonds is $100M Cash Flow Analysis. Assume LIBOR will evolve as follows: Pay 1-year Receive Time fixed LIBOR Variable Net CI 0 4% 1 -$5M 5% $4M -$1M 2 -$5M 8% $5M 0 3 -$105M $108M +$3M Why would UTC be interested in this deal? Lec 7 Swaps dfdf

▸ UTC has variable rate bonds outstanding and Most likely, ▸ UTC has variable rate bonds outstanding and ▸ Expects interest rates to ↑ Long Swap will lock in a Fixed 5% rate. (But interest rates may actually ↓ ? in this case UTC would pay more not less in interest) Lec 7 Swaps dfdf

Three years ago, UTC borrowed $100M; CI variable (rate = LIBOR) Example: Three years ago, UTC borrowed $100M; CI variable (rate = LIBOR) UTC expects interest rates to ↑. To hedge this risk, UTC wants to swap into fixed rate bonds. Long Swap will lock in a Fixed 5% rate (pay Fixed, receive Variable) Strategy for UTC (swap from variable to Fixed): ➀ UTC is short a variable rate bond in the spot market @ Libor ➁ UTC will buy a variable rate bond from the FI @ Libor ➂ UTC will sell a fixed rate bond to the FI @ CI rate = 5%/yr. {Pay Variable} in the spot market, {Receive Variable from FI, Pay Fixed to FI } Lec 7 Swaps dfdf

Cash Flow Analysis. Assume LIBOR will evolve as follows: Receive 1-year Pay CI to Variable Pay Fixed Time LIBOR Bondholders from FI to FI Net CI 0 4% -$100M +$100M 0 1 5% -$4M +$4M -$5M -$5M 2 8% -$5M + $5M -$5M -$5M 3 -$108M + $108M -$105M -$105M Moral of this story: ▸ UTC has swapped from variable to fixed interest rates. ▸ IF interest rates ↑ UTC will look good ▸ But IF interest rates ↓ UTC will pay more in interest. Lec 7 Swaps dfdf

Another example: “Plain Vanilla Swap“ Companies X and Y want to invest $10M for 10-years Company Fixed Rate Variable Rate X (wants Fixed) 8%/yr LIBOR Y (wants variable) 8.8%/yr LIBOR Difference = 0.0080 = 0.0 Job of FI: create two swap contracts such that 1. FI will earn 0.2% per year ➟ Fee = 0.002*10M = $20,000 and 2. Deal equally attractive to X and Y Firm X should be able to invest $10M at a fixed CI rate = 8.3% Firm Y should be able to invest $10M at LIBOR + 30 bps How? Lec 7 Swaps dfdf

Strategy for Firm X (wants Fixed): ➀ Buy a variable rate bond in the spot market @ Libor ➁ Sell a variable rate bond to the FI @ Libor ➂ buy a fixed rate bond from the FI @ CI rate 8.3%/yr. Strategy for Firm Y (wants Variable): ➀ Buy a Fixed rate bond in the spot market @ 8.8% ➁ Sell a Fixed rate bond to the FI @ 8.8% ➂ buy a variable rate bond from the FI @ Libor + 30 bps Lec 7 Swaps dfdf

Currency Swaps (Hull, 7.8) Fixed-for-Fixed Swap ▸ Firm A (for example, BP) wants to borrow $100M in the U.S. ▸ Firm B (for example, UTC) wants to borrow £ 50M in London. Suppose the current XR is X0 = 2.0 $/ £ Company Borrow $ in US Borrow £ in the U.K. A (BP wants U.S.$ loan) r$ = 7.0% r£ = 11% B (UTC wants £ loan) r$ = 6.2% r£ = 10.6% Difference = 0.0080 = 0.0040 Total Difference = 0.0040 Given these rates, A FI may create two swap contracts that will net 0.0010/yr for the FI, and allow BP to borrow @ r$ = 6.85% and allow UTC to borrow @ r£ = 10.45% Lec 7 Swaps dfdf

Strategy for Firm A ( BP wants $Loan): ➀ Sell £ 50M Bond in spot market in the U.K. @ r£ = 11% ➁ Buy £ 50M Bond from the FI @ r£ = 11% ➂ Sell $100M Bond to FI @ r$ = 6.85% Strategy for Firm B ( UTC wants £ Loan): ➀ Sell $100M Bond in spot market @ r$ = 6.2% ➁ Buy $100M Bond from the FI @ r$ = 6.2% ➂ Sell £ 50M Bond to the FI @ r£ = 10.45% Q: How do we know that this will work? A (BP wants U.S.$ loan) B (UTC wants £ loan) CF0=+£50M-£50M+$100M=+$100M CF0=+$100M -$100M+£50M =+£50M CI1=-(£50M*0.11)+(£50M*0.11) CI1=-($100M*0.062) +($100M*0.062) -$100M*0.0685 = -$6.85M -(£50M*0.1045) = - £5.225M and so on for t=2, 3, ... , T Lec 7 Swaps dfdf

CI1=-(£50M*0.11)+$100M*0.0685 -($100M*0.062) +(£50M*0.1045)= For the FI we have: CF0=+£50M-$100M+$100M -£50M = 0 CI1=-(£50M*0.11)+$100M*0.0685 -($100M*0.062) +(£50M*0.1045)= =-£275,000 + $650,000 = ??? =-£275,000*(2 $/£) + $650,000 = + $100,000 and so on for t=2, 3, ... , T Lec 7 Swaps dfdf

|––––––––––––|––––––––––––|––––––––––––––| 0 1 2 3 Valuation of Interest Rate Swaps (7.7) (Mark to market: Interest Rate Swaps) Need ➀ PV of variable rate bond, and ➁ PV of fixed rate bond ➀ Valuation of a variable rate Bond $100M ? CI1=$4M CI2=? CI3=? |––––––––––––|––––––––––––|––––––––––––––| 0 1 2 3 Start at t=2, suppose 1-yr LIBOR rate = 10%/yr (annual compounding) ➟ CI3 =$10M What is the value at time 2 after CI2 has been paid? ➟ Value2 = (100M + 10M)/1.1 = $100M What-if 1-yr LIBOR rate = 12% per yr (annual compounding)? ➟ CI3 =$12M ➟ Value2 = (100M + 12M)/1.12 =$100M Lec 7 Swaps dfdf

|––––––––––––|––––––––––––|––––––––––––––| 0 1 2 3 $100M ? CI2=? |––––––––––––|––––––––––––|––––––––––––––| 0 1 2 3 At t=1, suppose 1-yr LIBOR rate = 8%/yr (annual compounding) ➟ CI2 =$8M What is the value at time 1 right after the CI1 has been paid? ➟ Value1 = (100M + 8M)/1.08 = $100M At t=0, suppose 1-yr LIBOR = 5 %/yr ➟CI1 =100M(0.05)=$5M Value at time 0? ➟ Value0 = (100M + 5M)/1.05 = $100M Does it matter what the actual sequence of 1-year rates is? NO Lec 7 Swaps dfdf

➁ Valuation for the fixed rate bond. Assume: LIBOR1 yr = 10%, LIBOR2 yr = 10.5%, LIBOR3 yr = 11% (all c.c.) $100M $5M $5M $5M |––––––––––––|––––––––––––––|––––––––––––––––| 0 1 2 3 PV(Bond A) = $5M(e-0.10) + $5M(e-2*0.105) + $105M(e-3*0.11) = $84.06M Mark to Market Value of Swap: Long Swap = {-Bond A, +Bond B} ➟ VSwap = -Bond ValueFixed +Bond ValueVariable = - 84.061M + 100M = $15.936M Short Swap = {+Bond A, -Bond B} ➟ VSwap = +Bond ValueFixed - Bond ValueVariable =+84.061M - 100M = -$15.936M Lec 7 Swaps dfdf

Thank You (A Favara) Lec 7 Swaps dfdf