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Interest Rate Forwardsand Swaps
Interest Rate Forwardsand Swaps
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Outline
PART ONE
! Chapter 1: interest rate forward contracts and their
pricing and mechanics
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Outline
PART TWO
! Chapter 2: basic and customized swaps and their pricing and mechanics
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Outline
PART THREE
! Chapter 3: Cross-currency swaps
! Chapter 4: Swap revaluation and trading strategies
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Outline
PART FOUR
! Chapter 5: Quiz
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Chapter 1: Forward Rate Agreements
! Use of interest rate forwards
! Determination of forward rate
! Mechanics of FRA
! Arbitraging incorrect FRA pricing
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! Assume your bank on Jan 1, 2009 posts zero-coupon rates from 3 months to 5 years equal to the $ Libor rates
! 3-mo Libor is 3%; 6-mo Libor is 3.5%
! Depositor wants to know rate you would offer on 3-month deposit starting in 3 months, i.e. on April 1, 2009
! Usually denoted as 3x6 forward rate agreement, or FRA
Quoting a Forward Rate
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Calculating FRA rate
! Done via no-arbitrage exercise
! First you borrow for 3 months and invest for 6 months
No-arbitrage pricing exercise
0 3 MO 6 MO
3.50%
3.00% ?
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!Investment grows to (1+3.5%x181/360) = 1.0176 after 6 months
!Borrowing grows to (1 + 3% x 90/360) = 1.0075 after 3 months
!To break even you would need your borrowing in the next 3 months (91 days) to grow
by 1.0176/1.0075 = 1.0100
!This implies a borrowing rate for the 3x6 period of 3.9648%, calculated as (1.0100 - 1)
x (360/91)
!Can think of FRA rate of 3.9648% as the rate that makes an investor who invests for 6
months achieve same return as one who invests for 3 months and then again for 3 more
months at this forward rate
Calculating FRA rate
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General FRA Equation
)(][][
3601
3601
3601 }{
FSS
LL
DaysDaysR
DaysR
F "#
$
"$
%"
)()()(360
1360
1360
1FsSLL DaysFDaysRDaysR "
$""
$%"
$
or
WhereRL = spot Libor for the long periodRS = spot Libor for the short periodF = forward LiborDAYSL = number of days in the long periodDAYSS = number of days in the short periodDAYSF = number of days covered by the forward period
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! Customer does not actually need to place funds with bank quoting the FRA.
Customer places funds anywhere she likes, and collects or makes payment under
the FRA which brings her yield back to the FRA quoted rate all-in
! Example: if Libor resets 1% below FRA rate, customer collects annualized 1% from
bank, and vice-versa if Libor resets 1% above FRA rate
FRA Mechanics (1)
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FRA Settlement Diagram
Customer Bank
3.9648%
Libor
Libor
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! Libor resets at 3.75%, for period in question
! Client deposit is for $100 (so this is FRA notional amount)
! Interest earned on deposit is $100 x (3.75% x 91/360) = $0.9479
! Under the FRA, bank pays the depositor $100 x (3.9648% - 3.75%) x 91/360 = $0.0543
! Sum of these two amounts gives customer $1.0022, which is exactly equivalent to an annualized 3.9648% for a 91 day period
FRA Settlement Example
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Amount of net settlement will be known as soon as Libor setting takes place, at the beginning of the period in question.
Question is whether to wait until end of this period to pay it, or pay some discounted amount upfront.
FRA convention is to settle net payment as soon as it is possible to calculate it; this reduces counterparty and operational risk in the system; net payment due is discounted at current Libor, known as the settlement rate, i.e.
$0.0543/(1 + 3.75% x 91/360) = $0.0538
We can therefore write the formula for a FRA's net settlement amount as follows:
Notional x (Contract rate - Settlement Rate) x (Days/360)
(1 + Settlement Rate x Days/360)
FRA Mechanics (2)
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3-month Libor FRA Table
starts in 21 months, ends in 24 months21x24
starts in 9 months, ends in 12 months9x12
starts in 6 months, ends in 9 months6x9
starts in 3 months, ends in 6 months3x6
Period CoveredFRA
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6-month Libor FRA Table
starts in 54 months, ends in 60 months54x60
!!
!!
starts in 18 months, ends in 24 months18x24
starts in 12 months, ends in 18 months12x18
starts in 6 months, ends in 12 months6x12
Period CoveredFRA
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! We know from before that 6-mo. Libor is 3.50%; 12-mo. Libor is 4.00%
! As before, we can set
(1+ 4% x 365/360) = (1 + 3.5% x 181/360) x (1 + F x 184/360);
! This leads to 6x12 FRA rate = 4.41%
! You should confirm this on Worksheet FRA
Pricing 6x12 FRA
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! Assume dealer is quoting 3x6 FRA at 4.50%
! You borrow at 3.50% for 6 months, invest at 3% for 3 months and lock in 4.5% investment rate for next 3 months; at maturity your profit per dollar is $0.00136, calculated as
(1 + 3% x 90/360) x (1 + 4.50% x 91/360) – (1 + 3.5% x 181/360)
! This may seem small, but remember you can do this on large notional amounts. On $100MM, you would earn $136,309
! Conversely if dealer quotes FRA at 3.6%, you “invert” the trade: lend for 6 months at 3.5%, borrow for 3 months at 3%, and lock in the 3.6% rate at which you borrow again in 3 months’ time for 3 more months; at maturity your profit per dollar is $0.00093. You can check this calculation.
FRA Arbitrage
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FRA Arbitrage with Bid-Offer
! Bank is quoting the following rates:
4.02%4.00%3x6 FRA
3.6%3.4%6-months
3.1%2.9%3-months
OfferBid
!Bank is quoting a FRA rate that appears higher than the correct one, while quoting 3-
and 6-month interest rates that straddle the rates we had seen before
!Again, you try to arbitrage the unusually high FRA rate by implementing the first
version of the arbitrage we showed previously: you borrow for 6 months, lend for 3
months, and lock in the FRA rate to lend again for 3 months in 3 months’ time
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! Given the bid offer spread, the outcome of this trade per dollar is
(1 + 2.9% x 90/360) x (1 + 4% x 91/360) – (1 + 3.6% x 181/360),
which this time comes to a loss of $0.00067, or $66,558 if the notional is $100MM.
! If conversely you tried to lend for 6 months, borrow for 3 months, and lock in the FRA rate to borrow again for 3 months in 3 months’ time, the outcome per dollar would be
(1 + 3.4% x 181/360) – (1 + 3.1% x 90/360) x (1 + 4.02% x 91/360),
which again generates a loss, this time of $0.00090 per dollar, or $89,597 on a notional of $100MM
! Apparent error in the FRA rate was not large enough to enable you to overcome the cost disadvantages of paying the bid-offer spread on 3 separate instruments
FRA Arbitrage with Bid-Offer
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! No arbitrage is ever completely risk-free
! Main residual risks of arbitrage we illustrated are:
! Risk of default by the entity or entities to whom you lend the money
! Risk of default by the FRA counterparty when the FRA is in your favor at
settlement
! Risk that the settlement payment under the FRA, since it is discounted and
paid upfront, cannot be reinvested for the forward period at a rate at least as
high as the one used for the discounting
Final Remarks on Arbitrage
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Interest Rate Forwardsand Swaps
(Part II)
Interest Rate Forwardsand Swaps
(Part II)
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Chapter 2: Interest Rate Swaps
IRS construction and pricing
Basic interest swap pricing and use for hedging interest rate risk
Use and pricing of customized swaps for more complex situations
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Assume on Jan 1, 2009 a borrower has borrowed $100MM for 3 years at 3-mo. Libor flat.
With Libor at 3%, first interest payment will be $0.75MM.
Borrower has “exposure” to Libor for all 11 subsequent interest periods. A rising Libor will increase interest expense and reduce reported income.
Interest Rate Exposure (1)
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! Assume FRA rates you are quoting are as shown in this table:
Interest Rate Exposure (2)
6.75%92Oct 1,201133x3612
6.50%92July 1,201130x3311
6.25%91Apr 1, 201127x3010
6.00%90Jan 1,201124x279
5.75%92Oct 1,2010 21x248
5.50%92July 1,201018x217
5.25%91Apr 1, 2010 15x186
5.00%90Jan 1, 201012x155
4.75%92Oct 1,20099x124
4.50%92July 1, 20096x93
4.00%*91April 1, 20093x62
3.00%90Jan 1,2009Spot1
Rate for clientNumber of Days in period
Period Start DateContractDesignation
PeriodNumber
*/ This 4% rate for the 3x6 FRA is our old 3.9684%, adjusted to reflect the bank’s bid-offer. Similarly, all other rates include the bank’s profit margin
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! Borrower could hedge each reset using a series of FRAs, but would have unequal
and rising interest payments in each quarter.
! What the client needs is a smoother pattern of cash outflows – this is where the
interest rate swap comes in, as described in Worksheet Simplified IRS Pricing
! Swap fixed leg is single rate that makes all cash flows borrower pays have a PV
equal to PV of cash flows he would have paid under series of FRAs
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! Equation explains why swap rate is often described as the time-weighted
average of the relevant Libors
"
"
#
#
$
#12
1
12
1
i
i
i
ii
DF
DFLibor
FSo
Approximate Swap Formula
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Customized Swaps
! We will customize swaps with the following features:
! A profit margin for the dealer
! A notional that changes over time
! A forward start date
! Different settlement frequencies between the floating and fixed leg, and/or also
different daycounts
! A feature that fixes the Libor reset under the swap at the end rather than the
beginning of the period in question (“Libor-in-arrears”). Each of these except
the first is examined via its own worksheet
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Amortizing Swap (1)
! Now assume loan amortizes in last year in four equal installments
! Swap needs to amortize in the same way, otherwise borrower will end up over-hedged in year 3
! Careful with Amortizations: first amortization here occurs at end of period 9, so notional at beginning of period 9 is still $100 MM. Similarly notional at beginning of period 12 is $25 MM, not zero.
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IRS Replication (1)
! You have been asked to quote on a 5-year swap in which you will pay 6-mo
Libor and receive fixed semiannually
! You issue a 5-year bond and invest the proceeds in a 5-year floating rate note
that pays 6-month Libor flat
! Note that net initial proceeds are zero, and repayment of your investment’s
principal in 5 years’ time would repay the bond you have issued, so again there
will be no net cash flow for the principal
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IRS Replication (2)
Dealer
6-month Libor
Fixed
6-month Libor
Couponson fixed rate bond
!To break even under the swap, you would set the fixed leg equal to the fixed coupon
you are paying on your bond
!In fact any IRS can be viewed as two bond positions, one long and one short: if you
are receiving fixed under the swap and paying floating , it is as if you have issued a
Libor-based liability and invested the proceeds in a fixed rate bond; and vice-versa
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Basis Swap (1)
! Now assume your bank wants to borrow £100MM for 5 years but has no access to the £ funding market. The £/$ spot rate is 1.50
! Your bank issues a $150MM 5-year FRN which pays 6-month $ Libor and converts the proceeds into £100MM in the spot FX market
! Assume also another bank, in the UK, wants to borrow $150MM for 5 years but has no access to the $ funding market. This bank issues a £100MM 5-year FRN which pays 6-month £ Libor and converts the proceeds into USD 150MM
! Now the two banks enter into the swap described in this diagram:
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Your Bank UK Bank
£ Libor on £100 MM semi-annually
$ Libor on $150 MM semi-annually
£ 100 MM bond
at £ Libor
£100 MM at maturity
$150 MM at maturity
$150 MM bond
at $ Libor
Basis Swap (2)
This kind of swap is called a Libor basis swap, and plays a critical role in funding
operations of large banks, and also in pricing cross-currency swaps for customers
Note very carefully that unlike an IRS, a basis swap includes an exchange of
principal at maturity, without which neither party would have been properly hedged
in our example
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Libor Basis Swap Screen
8410-year
637-year
425-year
313-year
312-year
OfferBidTerm
GBP/USD 6 month Libor Basis Swap
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Interest Rate Forwardsand Swaps
(Part III)
Interest Rate Forwardsand Swaps
(Part III)
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Outline
PART THREE
! Chapter 3: Cross-currency swaps
! Chapter 4: Swap revaluation and trading strategies
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Chapter 3: Cross-currency swaps
! Pricing a basic cross-currency swap
! Alternative pricing methods
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Cross-Currency Swap
! Suppose you have issued a $150MM 3-year 5.5% bond but really needed £ to fund a UK expansion
! You convert the $ proceeds of your issuance into £, and seek to hedge your future $ coupon and principal obligations into £ by buying $ and selling £ under a series of derivative contracts
! Outright FX forwards would create liability in £ with uneven cash outflows since forward rates for successive dates will not be equal
! You would like your bank to make all payments required in $ under the bond, in return for you making to the bank, usually on the same dates, payments in £ whose profile is identical to that of a £ bond issue
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YourCompany Bank
£ Fixed% on £100 MM semi-annually
$5.5% on $150 MM semi-annually
£100 MM at maturity
$150 MM at maturity
$150 MM bond
at $5.5%
CCS Diagram
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Basis Point Conversion
! You cannot simply add or subtract equal numbers of basis points to each leg after you have finished pricing the swap
! Suppose in other words that after you priced the previous swap, it turned out that the $ bond could not be issued at 5.50%, but rather at 5.75%
! Client would logically ask you now to reset the $ leg of the swap to this level, and would understand that the £ leg also would need to be revised upward
! An identical adjustment of 25 bps to the £ leg would be incorrect, because PVs must be equal, and these PVs are obtained by discounting off two different curves
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Constructing/hedging CCS
► Most typically this is done using 3 swaps, two IRS and one Libor basis swap, as shown in this diagram:
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CCS Variations (3)
! Also frequencies and daycount for the two legs do not have to match – so we could have the $ leg based on quarterly under a 30/360 day convention, while the £ leg could be based on 6-month Libor and under an act/365 day basis
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Chapter 4: Swap Revaluation & Trading Strategies
! IRS factor sensitivities
! Effect of parallel shift and pivot on swap value
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Swap Revaluation
Fixed rate receiver has a gain when rates decline, since fixed payments are discounted at lower discount rates; it follows that receiving fixed under a swap is one way of taking a view on declining interest rates
Can reach same result by remembering that fixed rate receiver can be viewed as owning a fixed rate bond which he is funding at Libor; so he would have gain on his asset when rates decline, while liability’s value remains essentially the same
Price gain for the Receiver of a 1 bp drop in rates should be roughly equal to the price gain of owning a 3-year, $100MM quarterly-pay bond whose coupon is 5.34%
More generally DV01 of a swap is very nearly the same as DV01 of a fixed rate bond whose principal, coupon, maturity and daycount/frequency match those of the fixed leg of the swap
Swap also has convexity – like a bond – which means changes in its value are not linear relative to changes in rates
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Interest Rate Forwardsand Swaps
(Part IV)
Interest Rate Forwardsand Swaps
(Part IV)
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Question 1
On January 1, 2009, a borrower whose interest cost is Libor + 75 bps under a $100MM loan signs up for the 12x18 FRA. The rate under the FRA is 5%. 6-month Libor resets at the beginning of January 2010 at 6%. What is the exact all-in amount paid in that period by the borrower, taking into account both the loan and the FRA, and assuming the FRA settles at the end of that period?
How much money could you make if you tried to arbitrage the following rates a dealer quotes you on Jan 1, 2009? Assume a $100MM notional for your arbitrage.
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Question 4
You have entered into a 10-year interest rate swap in which you receive 6% fixed semi-annually and pay Libor semi-annually on a notional of $10MM. Right after you enter the swap, the whole Libor curve shift upwards by a parallel 100bps. Which of the following is most likely to be your approximate P&L? Do not use Excel to estimate your answer.
a) A gain of $700,000
b) A loss of $700,000
c) A gain of $300,000
d) A loss of $300,000
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Question 4 Solution
! Since you are receiving the fixed payments and rates have risen, you should expect
a loss since the PV of the cash flows you are receiving will diminish, as we
discussed before
! The duration of the swap should pretty obviously be much closer to 7 than to 3,
since the swap we modeled earlier in the module had a 3-year maturity and a
duration of around 2.8
! A duration of 7 looks about right, meaning the swap would lose 7% of its notional
given the 1% shift in rates
! The actual duration of this swap is in fact around 7.1 It would be a good exercise
for you to confirm this
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Question 5
Which of the following methods would not be a possible hedge for an interest rate swap you have entered into where you are paying fixed and receiving Libor?
a) Issue a fixed rate note and invest the proceeds in a floating rate note
b) Enter another interest rate swap with another counterparty under which you receive fixed and pay Libor
c) Enter into a series of FRAs in which you are paying the settlement rate and receiving the contract rate
d) Purchase a fixed rate bond and pledge it to a bank’s repo desk for funding at the Libor rate
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Question 5 Solution
! You are paying fixed under the swap so you need to receive fixed under your hedge
and pay Libor
! The only hedge that fails to do this is the first one
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Question 6
The Libor curve in USD is flat at 3%, while in GBP it is flat at 5%. On Jan 1, 2009, a borrower issues a GBP 200MM 3-year bond paying 7% quarterly, and wishes to swap it into a USD synthetic liability based on 3-month Libor. The FX spot rate is 1.75. The dealer would like to earn a profit margin of $500,000 on the swap. What is the all-in cost the dealer can quote the borrower?
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Question 7
Today is January 1, 2009. The USD Libor curve is monotonically downward sloping, with 6-month Libor spot at 6% and the 5-year forward 6-month Libor at 4%. You have just priced a spot-starting bullet interest rate swap, and are now asked to price a swap which starts on January 1, 2010 and amortizes by 25% of its original notional amount on each anniversary. The fixed leg of the swap will be increased as a result of:
a) Both the delayed-start date and the amortization schedule
b) The delayed-start date but not the amortization schedule
c) The amortization schedule but not the delayed-start date
d) Neither the delayed-start date nor the amortization schedule
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Question 7 Solution
! Since the curve is downward sloping, the highest values for Libor are the earlier
ones, and the lowest values are the last ones
! Postponing the start date of the swap removes the highest Libors from the
calculation of the swap rate, so reduces the swap rate
! Amortizing the swap notional towards the end puts lower weights on the last few
Libors, so increases the swap rate
! Therefore only the second effect increases the swap rate
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Question 8
A company has just issued $100 million of fixed rate 5-year bonds at a price of par with a coupon of 6.00% (annual) and up-front fees of .50. The company wishes to swap the debt into £. The spot rate is 1.25. Annual swap rates for 5-years are:
$ interest rate swaps: 6.12 / 6.15
£ interest rate swaps: 7.98 / 8.02
£/$ LIBOR basis swaps: £ Libor + 1 / £ Libor + 4
Assuming the company pays full bid-offer spreads, what is the all-in cost of the dollar debt swapped into fixed £? (Be sure to use the all-in cost of the $ debt by taking into account the up-front fees paid on the offering.)