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Swaps
Financial Derivatives
Finance 206/717
Philipp ILLEDITSCH
Fall 2012
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Introduction
What are Swaps?
Commodity Swaps
Interest Rate Swaps
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Definition
A swap is an agreement between two parties to exchange cash flowsat future dates according to certain rules
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Swaps
Forward or futures are used to lock in price at one future date
e.g. price of oil one year from today
What if you want to lock in prices at different dates?
Option 1:
buy “strip” of futures contracts
Option 2:
enter into swap agreement
Swap is just sequence of forward/futures contracts, combined withborrowing and lending
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Commodity Swaps Example
Utility Inc is going to buy
100,000 barrels of oil 1 year from today
100,000 barrels of oil 2 years from today
Current forward/futures prices:
F 0,1 = $100/barrel
F 0,2 = $105/barrel
We will need interest rates: 1-year cont. comp. risk-free rate is 6%
2-year cont. comp. risk-free rate is 6.5%
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Commodity Swaps Example cont.
Swap agreement:
sign agreement with counter-party to buy 100,000 barrels in
both year 1 and year 2,
for price $X /barrel
What is swap price X ?
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Pricing Swap
What are cash flows from swap (in $100, 000)?
Year 0 Year 1 Year 2
long swap 0 S 1 − X S 2 − X
How can we synthesize swap?
Year 0 Year 1 Year 2Long 1-year forwards S 1 − F 0,1Long 2-year forwards S 2 − F 0,2
Borrow at 1-year risk-free rate − F
0,
1− X e
−r 0,1 F
0,
1− X
Invest at 2-year risk-free rate −
F 0,2 − X
e −
2r 0,2 F 0,2 − X
−
F 0,1 − X
e −r 0,1 S 1 − X S 2 − X
−
F 0,2 − X
e −2r 0,2
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Swap Price
No-arbitrage implies
− (F 0,1 − X ) e −r 0,1
− (F 0,2 − X ) e −2r 0,2 = 0
i.e.,X
e −r 0,1 + e
−2r 0,2
= F 0,1e −r 0,1 + F 0,2e
−2r 0,2
Substituting in for forward prices and interest rates
X = 100e −.06 + 105e −2×.065
e −.06 + e −2×.065 = $102.4125
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In General
Suppose forward prices of oil are F 0,t
Then price per barrel of swap covering periods t 1, t 2, . . . , t n is
X =
n
i =1 e −r 0,t i
·t i F 0,t i n
i =1 e −r 0,t i
·t i
i.e. it is price at which PV of swap paymentsn
i =1
Xe −r 0,t i
·t i
is equal to PV of payments under strip of forward agreementsn
i =1
e −r 0,t i
·t i F 0,t i
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Swap Embeds Loan
To extent to which forward prices predict spot prices, Utility Incexpects to pay above spot price at date 1, and below spot priceat date 2
That is, there is loan hidden in swap price
Of course, spot prices might turn out to be different
But Utility Inc can use swap to construct pure loan
How?
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Value of Swap
Value of swap is zero initially
Value may change because
futures prices may change
interest rates may change
swap payments are made
What is value of swap at any time t ?
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Interest Rate Swaps
We will look at “plain vanilla” interest rates swaps
These swaps are an agreement for the exchange of floating rate
interest payments and fixed rate interest rate payments
As of Dec 2011, the Bank of International Settlements estimatesa worldwide notional amount outstanding of 403 trillion USD,and a market value of 18 trillion USD
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Example
Suppose BlueChip borrows $1M in a floating rate loan in October
2010, which they expect to repay in 3 years e.g. they take 1-year loan for $1M , which they plan to roll-over
or they take 3-year floating rate loan
In either case, we assume that BlueChip pays interest of r LIBOR + 0.25%
(note: not continuously compounded)
To hedge interest rate exposure, BlueChip can use swap
agrees to pay amount equal to $1M × (r fix − r LIBOR ) tocounter-party
say LB (large bank)
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Example cont.
BlueChip’s resulting net payment is
$1M × (r LIBOR + 0.25%) + $1M × (r fix − r LIBOR )
= $1M × (r fix + 0.25%)
They have converted 3-year floating-rate loan into a 3-year fixedrate loan
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Example cont.
12-month LIBOR rates turn out to be:
Oct 2010 6%Oct 2011 5%Oct 2012 8%
i.e. if you take 12-month $1,000 loan in Oct 2010 at LIBOR,you must repay (1 + 6%) × $1, 000 in Oct 2011.
If BlueChip’s swap agreement specifies annual payments,
and uses fixed rate of 7.2%,
then loan interest payments and swap payments made byBlueChip to LB are:
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Net Payments
Oct Interest payments Swap payments/receipts Total2010
2011
2012
2013
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Net Payments
Oct Interest payments Swap payments/receipts Total2010 0 0 0
2011 $1M × (6% + 0.25%) $1M × (7.2% − 6%)
= $62,
500 = $12,
000 $74,
5002012 $1M × (5% + 0.25%) $1M × (7.2% − 5%)= $52, 500 = $22, 000 $74, 500
2013 $1M × (8% + 0.25%) $1M × (7.2% − 8%)= $82, 500 = −$8, 000 $74, 500
Floating rate used here is the one 12-months prior to paymentdate
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Fixed-Rate and Floating Rate Payers
BlueChip is fixed rate payer
LB is floating rate payer
Why would somebody prefer fixed over floating payments?
Why would somebody prefer floating over fixed payments?
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Swap Pricing—Back to Example
BlueChip entered into swap agreement with LB in October 2010
Swap agreement required BlueChip to pay $1M × (r fix − r LIBOR ) forthree years
Let r LIBOR t ,t +1 denote LIBOR rate from date t to date t + 1
What is cash flow of Blue Chips’s swap position?
2010 2011 2012 2013
swap 0 $1M ×
r LIBOR10,11
− r fix
$1M ×
r LIBOR11,12
− r fix
$1M ×
r LIBOR12,13
− r fix
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P S
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Pricing Swap
We could do this just as before with commodities
e.g. LB’s cash flow in Oct 2012 is $1M ×r fix − r LIBOR 11,12
where r LIBOR 11,12 is 12-month LIBOR rate from Oct 2011 to Oct
2012
LB could hedge floating rate, r LIBOR
by locking in forward rate; e.g. using FRA or Eurodollar futures
We could then take the present value of certain cash flows andfind the value of r fix such that the net present value is zero ...
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E i W D i P i
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Easier Way to Determine Price
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Easier Way to Determine Price
Cash flows of LB’s swap position are
2010 2011 2012 2013
LB’s swap 0 $1M ×
r fix − r LIBOR10,11
$1M ×
r fix − r LIBOR11,12
$1M ×
r fix − r LIBOR12,13
buy a fixed 0 $1M × r fix $1M × r fix $1M × r fix
rate bond −B fix2010 +$1M
sell a floating 0 −$1M × r LIBOR10,11 −$1M × r LIBOR11,12
−$1M × r LIBOR12,13
rate bond +B float2010 −$1M
Let B fix2010 denote price of fixed rate coupon bond in Oct 2010 that matures in
Oct 2013
Let B float2010 denote price of floating rate coupon bond in Oct 2010 that
matures in Oct 2013
No arbitrage implies that B float2010 = B
fix2010
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E i W t D t i P i
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Easier Way to Determine Price
Net cash flows would be same if:
BlueChip pays LB $1M × r fix in Oct 2011, 2012, 2013
and in Oct 2013 pays LB principal $1M
LB pays BlueChip $1M × r LIBOR in Oct 2011, 2012, 2013
and in Oct 2013 pays BlueChip principal $1M
That is:
BlueChip sells LB bond that has face value of $1M , and pays
coupon rate of r fix
LB sells BlueChip bond that has face value of $1M , and payscoupon rate equal to LIBOR rate
Choose r fix such that B fix2010 = B
float2010
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Wh t i Bfloat?
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What is B float2010?
What is price of bond that pays floating rate (LIBOR) as coupon?
Bond that is paying floating rate is just worth par value
Back to example
Price at maturity is equal to face value; i.e. B float2013 = $1M
What is B float2012 ?
B float2012 = $1M × r LIBOR12,13 + $1M
1 + r LIBOR12,13
= $1M
What is B float2011 ?
B float2011 =
$1M × r LIBOR11,12 + B float2012
1 +r
LIBOR11,12
= $1M
What is B float2010 ?
B float2010 =
$1M × r LIBOR10,11 + B float2011
1 + r LIBOR10,11
= $1M
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Wh t i ?
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What is r fix?
r fix is coupon rate such that B fix2010 = B
float2010 = $1M
Suppose yield curve in 2010 is: r 0,1 = 6%
r 0,2 = 6.5%
r 0,3 = 7%
We have
$1Mr fixe −0.06 + $1Mr fixe
−0.065×2 + ($1Mr fix + $1M ) e −0.07×3 = $1M
Solving for r fix:
r fix = 1 − e −0.07×3
e −0.06 + e −0.065×2 + e −0.07×3 = 7.2%
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In General
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In General
r fix is par value coupon rate
i.e. rate such that
T
t =1
e −t ×r 0,t $1M × r fix + e
−T ×r 0,T $1M = $1M
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Market Value of Swap