BFW2751 Derivatives 1
S1, 2021
Lecture Week 4
Interest Rate Forwards and Futures
Chapter 6
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Lecture Objectives
Understand how Forward Rate Agreements (FRA), Eurodollar futures and
Bank‐Accepted Bill (BAB) futures work.
Know how to hedge short‐term interest rate risk using these instruments
Identify hedge strategies using FRA and IRFs
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Lecture Outline
• Day count conventions
• Forward Rate Agreements (FRA)
• Hedging with FRAs
• Eurodollar Futures
• Hedging with Eurodollar Futures
• Bank Accepted Bill Futures (BAB Futures)
• Hedging with BAB Futures
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Some Preliminaries
 Day Count Conventions:
• Day count defines the way in which interest accrues over time.
• Generally, we know the interest earned over some reference period, but would like
to know the interest earned over the sub-period of the reference period [ especially
interest earned for the time period elapsed during the holding period, which always
become a fraction of the whole].
• Day counts are usually expressed as X/Y.
• When calculating interest between two dates t1 and t2, X defines the way in which
the number of days between the two dates is calculated, and Y defines the way in
which the total number of days in the reference period is measured.
r1
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Y
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r2
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Day-Count Convention
 Interest-rate computations in these instruments is based on the money-market convention.
 In the US, the amount of interest payable per dollar of principal is computed as
 where
 ℓ is the given interest rate (typically the Libor rate).
 d is the actual number of days in the investment horizon.
 Great Britain and Australia use Actual/365.
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Starting Question to ponder
Your firm plans to borrow $10Million for 1 year in 6 months time and is concerned about
interest rate risk. What derivative position should you take to hedge the risk?
A.
A position that makes money if 1‐year rate goes up in 6 months’ time, OR
B.
A position that makes money if 1‐year rate goes down in 6 months’ time
Since your firm plans to borrow $10 mill not now but after 6 months for a period of 1 year,
the downside risk is that the interest rate may go up between now and in 6 months’ time.
How do you lock in the today’s interest rate? FRA?
How to hedge the increase in interest rate in the interest rate futures market? Eurodollars
Futures?
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Starting Question to ponder
Your firm plans to lend $10 Million for 1 year in 6 months time and is concerned about
interest rate risk. What derivative position should you take to hedge the risk?
A.
A position that makes money if 1‐year rate goes up in 6 months’ time, OR
B.
A position that makes money if 1‐year rate goes down in 6 months’ time
Since your firm plans to lend $10 mill not now but after 6 months for a period of 1 year, the
downside risk is that the interest rate may go down between now and in 6 months’ time.
How do you lock in the today’s interest rate? FRA?
How to hedge the increase in interest rate in the interest rate futures market? Eurodollars
Futures?
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Forward Rate Agreement (FRA)
 A forward rate agreement(FRA) is an OTC agreement that a certain rate will apply to a
certain principal during a certain future time period
 Formally, an FRA is an agreement to exchange:
 Interest calculated at a fixed rate RK
 For interest computed at a floating rate RM, the actual market rate (typically LIBOR–see next slide)
 On a specified principal amount L
 Over a specified reference period [T1, T2] in the future,
 And which is settled in discounted form at T1, or equivalently, undiscounted form at T2
T0
T1
T2
(Settlement date)
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LIBOR : London Interbank Offer Rate
 A reference interest rate, produced daily by the British Bankers Association, designed to
reflect the rate of interest at which banks are prepared to make large wholesale deposits
(i.e. lend) with other banks.
 LIBOR is quoted in major currencies for maturities up to 12 months (AUD LIBOR ceased
to be quoted in 2013)
 The rates are determined by demand and supply for that particular maturity
 E.g. If more banks want to borrow US dollars for 3 months than lend US dollars for 3
months, the 3‐month LIBOR rate will increase.
 LIBOR rates are used in loan or bond pricing. E.g. a borrower taking out a floating rate
loan may be charged an interest rate of LIBOR + say 0.5%
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FRA Convention
 For simplicity, in the materials on FRA, rates are all expressed with a compounding
frequency reflecting the length of the period to which they apply (if T2‐T1 = 0.5,the rates
are expressed with semi annual compounding. If T2‐T1 = 0.25, rates are with quarterly
compounding etc.)
 It should also be noted that
 Long position in the FRA
pays fixed, receives floating.
 Short position in the FRA
receives fixed, pays floating.
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FRA Terminology
 A FRA is described by the start and end dates of the underlying investment period (stated in months).
 For example, a "3 x 6 FRA" refers to a FRA whose underlying investment period begins in 3 months'
time and ends in 6 months' time.
 We use the general notation T1 x T2 FRA, with the understanding that T1 and T2 are measured in
months.
 It should also be reiterated that:
 Long position in the FRA  pays fixed (AR), receives floating (SR).
 Short position in the FRA  receives fixed (AR), pays floating (SR).
 k (fixed rate) is the Agreement Rate (AR) payable by buyer in long position
 ℓ (floating rate) is the Settlement Rate (SR) receivable by the buyer in long position
 If SR > AR = buyer in long position receives from seller in short position pays the difference
 If SR < AR = buyer in the long position pays to seller in short position the difference
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FRA Payoffs
 Consider the payoffs to a long position from a T1 x T2 FRA with fixed rate k .
 Suppose the actual Libor rate at time T1 (for the period [T1 , T2]) is ℓ .
 Then, the fixed payment due at T2 is =
 The floating payment received at T2 is =
 The difference between these amounts is =
 Thus, at time T1, the payoff received by the long FRA position is
(Note that P is the notional principle and d is the number of days)
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FRA Payoffs: An Example
 Consider a 4 x 7 FRA with a spot date of March 15.
 That is, the investment period begins on July 15 and ends on October 15.
 Given that if there are 92 days between July 15 and October 15.
 Suppose that
 The principal amount P in the FRA is $5,000,000.
 The fixed rate on the FRA is k = is 5% or 0.05.
 We compute the payoffs to a long FRA position.
 The payoffs to the short FRA are the negative of these payoffs.
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FRA Payoffs: An Example (Cont’d)
 Suppose the 3-month Libor on July 15 is ℓ = 5.40%.
 Then, the difference ℓ — k = +0.40%. Applying this to the principal amount, we obtain
[ SR (ℓ ) > AR (k)] the long position in FRA will gain from short position.
 Therefore the value of the difference will be as at T2 will be:

5,000,000 x [0.004 x (92 / 360)] = $5,111.11
 Discounting this quantity back to July 15th at the Libor rate ℓ , we obtain
 This is the amount the long position receives from the short position on July 15.
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FRA Payoffs: An Example (Cont’d)
 Alternatively, suppose the 3-month Libor on July 15 is ℓ = 4.70%.
 Then, the difference ℓ — k = — 0.30%. Applying this to the principal amount, we obtain
 [ SR (ℓ ) > AR (k)] the long position in FRA will pay the difference to the short position
 Therefore the value of the difference will be as at T2 will be:
• Discounting this quantity back to July 15th at the Libor rate ℓ , we obtain
• Now, the long position must pay the short position the amount $3,787.83
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Hedging with FRAs
 Suppose you can borrow at Libor and wish to hedge interest rate risk on a borrowing over a future
period [T1, T2 ].
 You can utilize a FRA for this purpose. Here is the strategy:
 Enter into a long T1 x T2 FRA today.
 If the SR > AR; Long position is set to gain at T1:
 Reinvest the proceeds received from the FRA at T1 at the then-prevailing Libor rate for maturity at T2.
 If the SR < AR; Long position is set to pay at T1
 At T1 , borrow the required amount at the then-prevailing Libor rate for repayment at T2 .
 Intuitively, you are borrowing floating and are being compensated by the FRA for the difference
between floating and fixed, so net you are paying fixed.
 The reinvestment of the FRA proceeds is required because the FRA is settled in discounted form.
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An Example on Hedging with FRA’s
It is April 5 today, and consider a corporation need to borrow $5,000,000 for 3 months from August 5
to November 5. Since the corporation is going to borrow, its downside risk is that floating interest rate
may go up between now and August 5. It enters into a long 4 x 7 FRA today and borrows the $5 mil at
LIBOR on August 5. The fixed rate (AR) in the FRA is k= 5.00%
We consider two possible outcomes for the LIBOR (ℓ ) on August 5,
(a) ℓ = 5.50% (b) ℓ = 4.50%
Show how by hedging with FRA, the corporations net cash flows are the same in either case.
The 3-months between August 5 and Nov 5 will have 92 days.
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An Example on Hedging with FRA’s
a) As the ℓ > k the corporation which is in a long position is set to receive the difference
i.e. 5,000,000 x [( 0.055 – 0.05) x 92/360] = $6,388.89
Discounting it back to August 5:
6,388,89 / [1+ (0.055) 92/360] = $6,300.34
Investing the $6,300.34 received at the Libor rate of 5.50% for 92 days will yield
6,300.34 x [1+ (0.055) x 92/360] = $6,388.89
However the corporation now must pay interest on the $5 mil borrowed for the 3 months at
Libor rate of 5.50%
5,000,000 x (0.055) x 92/360 = $70,277.63
By offsetting the cash flow from the FRA, the net outflow will be
$70,277.63 – $6,388.89 = $63,888.74
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An Example on Hedging with FRA’s
(ii) As the ℓ < k the corporation which is in a long position is set to pay the difference
i.e. 5,000,000 x (0.045 – 0.05) x 92/360 = - $6,388.75
Discounting it back to August 5 at 0.045
= - $6,388.75 / [1+ (0.045)x 92/360] = - $6,316.11
By borrowing this amount for 3 months at 4.50%
-$6,316.11 x [1+ (0.045) x 92/360] = -$6,388.74
However the corporation now must pay interest on the $5 mil borrowed for the 3 months at
Libor rate of 4.50%
5,000,000 x (0.045) x 92/360 = $57,499.88
The total cash outflow for the corporation will be:
6,388.74 + 57,499.88 = $63,888.62
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Eurodollar Futures
 It is a dollar deposited in a US. or in a foreign bank outside the US.
 The Eurodollar interest rate is the interest rate on such Eurodollar deposits.
 The most popular interest rate futures is the three-month Eurodollar futures
 The interest rates are based on the London Inter-bank offer rates (LIBOR)
 Futures contract that allows you to trade on the interest that will be paid (by someone who borrows
at the Eurodollar interest rate) on USD 1,000,000 for a future 3-month period.
 Allows a trader to speculate on future 3-month Eurodollar rates, or hedge against changes in
future 3-month Eurodollar rates.
 Usually trade for March, June, September and December delivery out to 10 years, short maturity
contracts trading for months other than these.
 Delivery is only by cash settlement. No physical delivery. Third Wednesday of the delivery month.
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Eurodollar Futures
• Quotations:

•
Quote (Q) = 100 – interest rate
•
Example: if the Q = 92, means that the 3-month interest rate is 100 – 92 = 8%
•
Interest rate is % p.a. compounded quarterly, actual / 360 day count convention.
•
Interest rates are based on LIBOR (London Interbank Offer Rate)
Eurodollar contracts are designed so that a 1 basis point (0.01) change in the futures price, results in a $25 gain or loss per
contract.
 Quoted price increases by 1 basis point
 Interest rate has dropped by 1 basis point
 long (short) positions will gain (lose) $25 per contract.
• $1,000,000 x 0.0001 x 0.25 = $25
One contract x one bp. x per quarter
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Eurodollar Futures
• Quotations:
 Eurodollar contracts are designed so that a 1 basis point (0.0001) change in the futures price,
results in a $25 gain or loss per contract.
Quoted price increases by 1 basis point
Interest rate has to drop by 1 basis point
long (short) positions will gain (lose) $25 per contract.
• $1,000,000 x 0.0001 x 0.25 = $25 / contract
One contract x one bp x per quarter
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Eurodollar Futures
 The CME defines the value of the underlying position in the IRF contract as
10,000 x [ 100 – 0.25 x (100 – Q)]
Assuming Q is 98.00
10,000 x [ 100 – 0.25(100 – Q)]
10,000 x [ 100 – 0.25(2.00)]
10,000 x 99.5 = 995,000
• If you look at this equation closely, it is nothing more that discounting the $1m deposit
by the quote Q.
• Could write the above as
$1,000,000 – [$1,000,000 x 0.25 x [1– (Q/100)]]
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Eurodollar Futures
Simple Example:
 The quoted settlement price of 99.28 on Sept 4, corresponds to a underlying value of
10,000 x [100 – 0.25 x (100 – 99.28)] = $998,200
 The final contract settlement price is 99.50 on Dec 4, corresponds to an underlying value
of
10,000 x (100 – 0.25 x (100 – 99.50) = $998,750
If long you gain $998,750 - $998,200 = $550
If short you lose $998,200 – $998,750 = ($550)
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An Example
Suppose it is
$100,000,000
between now
price for June
8% or 0.08).
now January 2020 and you anticipate a three-month borrowing need for
beginning June 2020. You wish to hedge the interest rate risk changes
and June 2020 using Eurodollar Futures. Suppose the Eurodollar futures
contract is currently 92.00 (implied Eurodollar interest rate is 100 – 92.00 =
What will be the hedged net interest payout for $100,000,000 borrowing in September
2020 if:
a) The Libor rate turns out to be 8.25 in June 2020?
b) The Libor rate turns out to be 7.75 in June 2020 ?
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An Example on outcome (a)
The cash market
Eurodollar Futures Market
Now: January 2020
Now: January 2020
Do nothing
Short 100 June 2020 Eurodollar futures contracts at index price of
92.00
Effectively at an interest rate of 8.00% p.a. for 3 – months
June 2020
June 2020
Borrow $100 million at 8.25% for 3 months
Long the 100 June 2020 Eurodollar Futures contracts at 91.75.
Effectively at 8.25% p.a. for three months.
(1,000,000 x 0.25) = 250,000
= 250,000 x ( 0.0825 – 0.0800) = 250,000 x 0.0025
Net gain per contract is $625 per contract;
100 contract = 100 x 625 = $62,500
Invest this gain for 3 – months at 8.25% p.a.
September 2020
September 2020
Interest payout for the $100 mil borrowing at 8.25% for three months
100,000,000 x (0.0825 x 0.25) = 2,062,500.00
Offset by the gains from EF =
63,789.06
Net payout
1,998,710.94
The 3 –month investment will yield
= 62.500 x (1+ 0.0825 x 0.25) = $63,789.06
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An Example on outcome (b)
The cash market
Eurodollar Futures Market
Now
Now
Do nothing
Short 100 June 2020 Eurodollar futures contracts at 92
Effectively at an interest rate of 8.00% p.a. for 3 – months
June 2020
June 2020
Borrow $100 million at 7.75%
Long the 100 June 2020 Eurodollar Futures contracts at 92.25.
Effectively at 7.75% p.a. for three months.
(1,000,000 x 0.25) = 250,000
= 250,000 x ( 0.0775 – 0.0800) = 250,000 x ( - 0.0025)
Net loss per contract is $625 per contract;
100 contract = 100 x 625 = - $62,500
Borrow this payout for 3 – months at 7.75% p.a.
September 2020
September 2020
Interest payout for the $100 borrowing at 7.75% for three months
100,000,000 x (0.0775 x 0.25) = 1,937,500.00
Add the EF loss outflow
=
63,710.93
Net payout
2,001,210.93
The 3 –month borrowing needs to be paid
62,500 x [1+ (0.0775 x 0.25)] =63,710.93
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Bank Accepted Bills (BAB) Futures
 90-day BABs are the most common
 Traded on Australian Stock Exchange (previously Sydney Futures Exchange). Is the
Australian counterpart of Eurodollar futures, ranking in top 10 interest rate futures in the
world.
 The underlying: 90-day bank accepted bill which is a commercial bill whose acceptor is a
bank, a common short term financing instrument (see next slide for details)
 Contract unit: A$1M face value 90-day BAB
 Contract months: March, June, Sep, Dec up to 20 quarter months ahead
 Quote: 100-yield, just like Eurodollar futures
 Settlement: physical delivery (unlike Eurodollar futures)
 A key difference from Eurodollar futures lies in the computation of the payoff / settlement.
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BAB Futures: payoff function
• Continuing with our notation, let k denote the BAB rate implied by the futures price F0 at the time of
entering into the contract. Let ℓ denote the BAB rate prevailing in June 2021 Ft.
• Note that BAB rate/yield changes daily.
• That is F0=100 – k*100 and Ft=100 - ℓ *100
• The payoff to a long BAB futures contract can also be written as:
 𝐹𝑡 − 𝐹0 =
1,000,000
1+ℓ 365
90
−
1,000,000
1+k 365
90
(𝑘−ℓ)(365)
90
= 1,000,000
(1+ℓ 365 )(1+k 365 )
90
90
• The payoff to a short BAB futures contract is
(ℓ−k)(365)
90
• 1,000,000
(1+ℓ 365 )(1+k 365 )
90
90
• You can see hedging using BAB futures is not perfect
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Example: Payoff to BAB futures
• On Feb 4, 2021, an investor wants to lock in the interest rate that will be earned on AUD5 million
investment for 90 days starting on March 16, 2021. The investor goes long on 5 BAB futures at
97.63. On March 16, 2021, the 90 day BAB yield is 2%. Compute gain/loss
• Therefore, k = 2.37% and ℓ = 2.00%
• 𝐹𝑡 − 𝐹0 = 5 𝑋
1,000,000
1+ℓ
90
365
−
1,000,000
90
1+k 365
H = FT – F0 = 5[ 1,000,000 / (1 + 0.02 * 90/365)] – [ 1,000,000 / (1+ 0.0237* 90/365)]
5 [1,000,000/1.0049] – [1,000,000 / 1.0058]
5 (995,092.80 – 994,190.24)
5 (902.56) = $4,512.80
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Example: Payoff to BAB futures
• 1,000,000
(𝑘−ℓ)(90/365)
90
90
(1+ℓ 365 )(1+k 365 )
= 5,000,000[ (0.0237 – 0.02) (90/365)/ (1 +0.02(90/365) (1 + 0.0237(90/365)]
= 5,000,000 [0.000912 / (1.004932) (1.005844)]
= 5,000,000 [0.000912/1.010804)
=5,000,000(0.000903)
= $4512.80
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Comparing Interest Rates Futures
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Summary
 In this lecture we have discussed the day-count conventions in estimating floating
rate values.
 The conventions of Forward Rate Agreement (FRA) and the use of FRA in hedging
against fixed over floating rate were also discussed.
 This lecture has introduced Eurodollar futures which are two of the four key moneymarket derivatives based on Libor.
 We have also examined the Australian version of the Eurodollar futures, the 90day
BAB futures.
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