1. 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 ? ?
|––––––|–––––––|–––––––| |––––––––|––––––|–––––––––|
Long Swap = {-Bond A, +Bond B} aka Pay Fixed, Receive Variable
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2. 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 %
1 -$5M % $4M -$1M
2 -$5M % $5M
3 -$105M $108M +$3M
Why would UTC be interested in this deal?
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3. ▸ 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)
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4. 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 Libor
➁ UTC will buy a variable rate bond from the Libor
➂ UTC will sell a fixed rate bond to the CI rate = 5%/yr.
{Pay Variable} in the spot market,
{Receive Variable from FI, Pay Fixed to FI }
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5. 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
1 5% -$4M $4M -$5M $5M
2 8% -$5M + $5M -$5M $5M
$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.
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6. 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 = = 0.0
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?
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7. Strategy for Firm X (wants Fixed):
➀ Buy a variable rate bond in the spot Libor
➁ Sell a variable rate bond to the Libor
➂ buy a fixed rate bond from the CI rate 8.3%/yr.
Strategy for Firm Y (wants Variable):
➀ Buy a Fixed rate bond in the spot 8.8%
➁ Sell a Fixed rate bond to the FI @ 8.8%
➂ buy a variable rate bond from the Libor + 30 bps
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8. 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 = =
Total Difference =
Given these rates,
A FI may create two swap contracts that will net /yr for the FI,
and allow BP to r$ = 6.85%
and allow UTC to r£ = 10.45%
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9. Strategy for Firm A ( BP wants $Loan):
➀ Sell £ 50M Bond in spot market in the 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* = -$6.85M (£50M*0.1045) = - £5.225M
and so on for t=2, 3, ... , T
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10. 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* ($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
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11. |––––––––––––|––––––––––––|––––––––––––––| 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=?
|––––––––––––|––––––––––––|––––––––––––––|
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
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12. |––––––––––––|––––––––––––|––––––––––––––| 0 1 2 3
$100M
? CI2=?
|––––––––––––|––––––––––––|––––––––––––––|
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
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13. 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
|––––––––––––|––––––––––––––|––––––––––––––––|
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 = M + 100M = $15.936M
Short Swap = {+Bond A, -Bond B}
➟ VSwap = +Bond ValueFixed - Bond ValueVariable = M - 100M = -$15.936M
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14. Thank You
(A Favara)
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