FNCE4040 Derivatives Chapter 6 Interest Rate Futures University of - - PowerPoint PPT Presentation

University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives

FNCE4040 – Derivatives Chapter 6

Interest Rate Futures

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FORWARD RATE AGREEMENTS

REVIEW

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Forward Rate Agreement (FRA)

• A Forward Rate Agreement (FRA) is an OTC

agreement such that a certain interest rate will apply to either borrowing or lending a principal over a specified future period of time.

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Example

• For example a bank agrees to lend 1m USD

for 1 year starting in 1 year at an interest rate

• f 3%. The rate is quoted with an ACT/360

daycount basis.

Year 1

Year 2

today 1𝑛 𝑉𝑇𝐸 1𝑛 𝑉𝑇𝐸 $1𝑛 365 360 3.00% = 30,416.67

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FRA Mechanics / Valuation – part 1

• From the lender’s viewpoint
• A loan of 𝑂 from 𝑈

1 to 𝑈2 at an agreed rate 𝑆𝐿

• Let 𝐸 be the daycount fraction from 𝑈

1 to 𝑈2

– For an FRA 𝐸 =

𝐸𝑏𝑧𝑡 𝑈

2 −𝐸𝑏𝑧𝑡(𝑈 1)

360

𝑈

1

𝑈2

today

Interest owed:

𝑂 × 𝑆𝐿 × 𝐸

𝑂 𝑂

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𝑄𝑊 = −𝑂 ∙ 𝑓−𝑠1𝑈

1 + 𝑂 ∙ 1 + 𝑆𝐿 ∙ 𝐸 ∙ 𝑓−𝑠2𝑈 2

FRA Mechanics / Valuation – part 1

• We can value the FRA given the continuously

compounded zero rates 𝑠

1 and 𝑠2.

𝑈

1

𝑈2

today

Interest owed:

𝑂 ∙ 𝑆𝐿 ∙ 𝐸

𝑂 𝑂

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The Fair FRA Rate

• The fair FRA rate 𝑆𝐺 is the rate such that the

sum of the PV of both cash flows is zero. 𝑂𝑓−𝑠1𝑈

1 = 𝑂 1 + 𝑆𝐺𝐸 𝑓−𝑠2𝑈 2

Solve for 𝑆𝐺: 0 = −𝑓−𝑠1𝑈

1 + 1 + 𝑆𝐺𝐸 𝑓−𝑠2𝑈 2

𝑆𝐺 = 𝑓 𝑠2𝑈

2−𝑠1𝑈 1 − 1

𝐸

where 𝐸 = 𝐸𝑏𝑧𝑡 𝑈2 − 𝐸𝑏𝑧𝑡(𝑈

1)

360

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The Fair FRA Rate

• We know that the PV of a loan from 𝑈

1 to 𝑈2

at an agreed rate 𝑆𝐿 with daycount fraction 𝐸 from 𝑈

1 to 𝑈2 is

𝑄𝑊 = −𝑂𝑓−𝑠1𝑈

1 + 𝑂 1 + 𝑆𝐿𝐸 𝑓−𝑠2𝑈 2

• Combine this with the definition of the fair rate

and we have 𝑄𝑊 = −𝑂 1 + 𝑆𝐺𝐸 𝑓−𝑠2𝑈

2 + 𝑂 1 + 𝑆𝐿𝐸 𝑓−𝑠2𝑈 2

𝑄𝑊 = 𝑂 𝑆𝐿 − 𝑆𝐺 𝐸𝑓−𝑠2𝑈

2

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FRA Mechanics / Valuation – part 2

• A loan from 𝑈

1 to 𝑈2

• From the lender’s viewpoint

𝑈

1

𝑈2

Fair rate now for period 𝑈

1, 𝑈2 = 𝑆𝐿

today Interest owed:

𝑂 × 𝑆𝐿 × 𝑈2 − 𝑈

1

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Think of Mark-To-Market as the cost to offsetting your position

FRA Mechanics / Valuation – part 2

Interest owed:

𝑂 × 𝑆𝐿 × 𝑈2 − 𝑈

1

𝑓−𝑆2×𝑈

2

𝑈

1

𝑈2

today Fair rate for period 𝑈

1, 𝑈2 moves to 𝑺𝑮

Interest now prevailing:

𝑂 × 𝑺𝑮 × 𝑈2 − 𝑈

1

Take the difference and PV

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EURODOLLAR FUTURES

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Eurodollars

• A Eurodollar is a dollar deposited in a bank
• utside the United States (nothing to do with Euros)

– During the cold war the Soviet Union feared that its deposits in the USA would be seized or frozen – They moved their holdings to Moscow Narodny Bank, a Soviet-owned bank with a British Charter – The British bank then deposited that money in US banks – There was no chance of the US confiscating the money as it belonged to a British Bank

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Eurodollar Futures Contract specs

Feature Specs Settlement Cash Underlying The rate earned on a 3-month $1,000,000 Eurodollar time deposit. Same as 3-month LIBOR Quote 100 minus the 3-month rate Tick Size A one basis point move in the quote corresponds to a $25 price change Settlement On the Third Wednesday of the delivery month the final settlement prices is 100 minus the actual 3-month Eurodollar deposit rate Contract Months March, June, September and December, out 10 years

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Eurodollar Futures Contract

• As the futures quote is 100 minus the interest

rate, the investor who: –is long will profit when interest rates fall –is short will profit when interest rates rise

• The futures contract is equivalent to a payout
• f:

10,000 x [100 - 0.25 x (100-Q)]

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Eurodollar Futures Contract

• We should repeat that last slide:
• The futures quote is 100 minus the

interest rate. –The investor who is long will profit when interest rates fall –The investor who is short will profit when interest rates rise

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Example

• Suppose you buy (go long) a

contract on Nov-1

• Contract expires on Dec-21
• The prices are as shown
• How much do you gain/lose

a) on the 1st day $25 × 11 = $275 b) on the 2nd day $25 × −25 = −$625 c)

• ver the whole time until expiration?

$25 × 30 = $750

Date Quote Nov-1 97.12 Nov-2 97.23 Nov-3 96.98 ……. …… Dec-21 97.42

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Example (continued)

• If on Nov-1 you know that you will have $1m

to invest for 3 months on Dec-21 in the Eurodollar market.

• The contract locks in a rate of 100 - 97.12 =

2.88%.

• How???
• The contract expires on Dec-21 and the

Eurodollar rate available in the market is: 100 − 97.42 100 = 2.58%

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Example (continued)

• Your 3-month Eurodollar deposit will earn

interest of: $1m × 2.58% × 90 360 = $6,450

• You made a gain on your futures contract of:

$25 × 2.88% − 2.58% × 10,000 = $750

• Almost the same as a 3-month Eurodollar

deposit with interest 2.88%.

• The only difference is when you receive the

cash.

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HEDGING EXAMPLE ONE

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FRAs and Eurodollar Futures

• Consider a FRA

– $1,000,000 notional – Agreed rate: 0.30% – 90-day deposit – Starts 18-Mar – On Eurodollar Deposit rate Counterparty A Counterparty B

18-Mar $1,000,000 16-Jun $1,000,750

18-Mar counterparty A deposits $1m with counterparty B 16-June counterparty B gives back $1m plus interest to counterparty A

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Zero Rates

• Assume Today = 15-Feb
• If the zero rate to 18-Mar is 0.20% and this

contract is fair then what is the zero rate for 16-Jun?

– Fair means the PV of cash flows are equal – 31 days to 18-Mar and 121 days to 16-Jun 1,000,000 × 𝑓−0.0020× 31

365 = 1,000,750 × 𝑓−𝑠121 365

𝑠 = 𝑚𝑜 1,000,750 1,000,000 + 0.0020 × 31 365 121 365 = 27.74 𝑐𝑞𝑡

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Zero Rates

• If the zero rate for 18-Mar stays the same but

the forward Eurodollar rate changes to 0.40% does counterparty A make or lose money?

– A loses money as A locked a lower interest payment for the 18-Mar to 18-Jun period

• If the zero rate for 18-Mar changes to 0.30%

but the forward Eurodollar rate stays constant at 0.30% does counterparty A make or lose money?

– The FRA is still fair so its PV is still zero regardless of discounting.

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Hedging with a Eurodollar Future

• Consider a Eurodollar Futures contract

– $1,000,000 notional – 90-day deposit – Settles 18-Mar – Current Price = 99.70

• If you are counterparty A (the lender) would

you buy or sell the futures contract?

– You lose money when the forward Eurodollar rate goes up (you had locked in a lower rate), thus you should sell the futures contract

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Hedging with a Eurodollar Future

• Assume counterparty A has entered into the

FRA and sold one futures contract

– If the zero rate for 18-Mar changes to 0.30%, but the forward Eurodollar rate stays constant does counterparty A make or lose money?

• Neither as both have value zero.

– If the zero rate for 18-Mar stays the same, but the forward Eurodollar rate changes to 0.40% does counterparty A make or lose money?

• The ED position makes $250 immediately. The FRA

loses money but the loss is a little less due to discounting -$249.69. A small gain of $0.31

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What is the difference?

Two key differences:

• Margin vs. Collateral

– Book assumes that FRA is not collateralized

• Cash Flows do not line up

– The futures contract is cash-settled every day, so you collect cash flows daily until 18-Mar – The FRA has the interest paid on 16-Jun. The value of the FRA is affected twice by changes in interest rates:

a) From changes in interest payment to be PV’d b) Changes in PV, due to changes in interest rates

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Forward Rates and Eurodollar Futures

(continued)

• A “convexity adjustment” is often made
• Forward Rate = Futures Rate − 0.5 𝜏2T

1T2

– T

1 is the start of period covered by the

forward/futures rate – T2 is the end of period covered by the forward/futures rate (90 days later that T1)

• 𝜏 is the standard deviation of the change in

the short rate per year (often assumed to be about 1.2%)

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Convexity Adjustment when s=1.2%

Maturity of Futures (yrs) Convexity Adjustment (bps) 2 3.2 4 12.2 6 27.0 8 47.5 10 73.8

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HEDGING EXAMPLE TWO

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FRA

• Recall the FRA

– $1,000,000 notional – Agreed rate: 0.30% – 90-day deposit – Starts 18-Mar – On Eurodollar Deposit Counterparty A Counterparty B

18-Mar $1,000,000 16-Jun $1,000,750

18-Mar counterparty A deposits $1m with counterparty B 16-June counterparty B gives back $1m plus interest to counterparty A

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Disadvantages of FRA

• The problem with a FRA is that it is an OTC

contract

– Can be customized but – Higher transaction costs and – Costs more to unwind.

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Eurodollar Futures

• If Counterparty A wants the same exposure

as entering the FRA but does not want to trade OTC

• Should they go long or short the March

Eurodollar contract?

– If Counterparty A had entered into the FRA then they would make money when rates fall and lose money when rates rise. – They want to be long the Eurodollar futures contract.

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Eurodollar Futures Contract

• Assume counterparty A has gone long one

ED futures contract at 99.70.

– If the zero rate for 18-Mar changes to 0.30%, but the forward Eurodollar rate stays constant does counterparty A make or lose money?

• Neither as the forward Eurodollar rate is constant and

the futures price of 99.70 does not change.

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Eurodollar Futures Contract

• Assume counterparty A has gone long one

ED futures contract at 99.70.

– If the zero rate for 18-Mar stays the same, but the forward Eurodollar rate changes to 0.20% does counterparty A make or lose money?

• The futures price changes to 99.80. The position

makes $10 × 25 = $250. The FRA would have made less due to discounting $249.69.

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HEDGING EXAMPLE THREE

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FRA

• Change the FRA

– $10,000,000 notional – Agreed rate: 0.40% – 180-day deposit – Starts 18-Mar – On Eurodollar Deposit Counterparty A Counterparty B

18-Mar $1,000,000 14-Sep $1,000,750

18-Mar counterparty A deposits $1m with counterparty B 14-Sep counterparty B gives back $1m plus interest to counterparty A

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FRA

• Assume that the FRA has been executed and

counterparty B wants to hedge their interest rate exposure.

• The March ED contract starts on the right day

but ends in June.

• The June futures ED contract starts in June

and ends in September

• Which do we use to hedge?

– Both.

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Hedging With ED Futures

• We will ignore one or two days of interest for

illustrative purposes

• Does counterparty B want to be long or short

the futures contracts.

– Counterparty B loses money when rates fall so they want to be long the futures contracts

• How many contracts?

– 10 of each

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Hedging With ED Futures

• Does the hedge work?
• What is the relationship between the

– The cont. comp. zero rate from 18-Mar to 16-Jun, – The cont. comp. zero rate from 16-Jun to 15-Sep and – The cont. comp. zero rate from 18-Mar to 15-Sep – The longer rate is the average of the other two

• What is the approximate relationship if we

replace cont. comp with fair FRA rate?

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Hedging with ED Futures

• Assume the FRA has traded and

Counterparty B has gone long 10 March ED contracts and long 10 June ED contracts.

• Assume that the fair FRA rate changes from

0.4% to 0.6%.

• On the FRA counterparty B makes:

10𝑛 × 0.002 × 180 360 𝑓−𝑠211 365 ≈ 10𝑙

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Hedging with ED Futures

• Assume the FRA has traded and

Counterparty B has gone long 10 March ED contracts and long 10 June ED contracts.

• Assume that the fair FRA rate changes from

0.40% to 0.60%.

• On the futures contracts the total changes in

rates must average to 0.20% so assume they both move by 0.20%. Thus the losses are: 2 × 20 × 25 × 10 = 10𝑙

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Longer Dated FRA

• Eurodollar Futures contracts are liquid for 2-3

years.

• If we are borrowing/lending through a FRA for

less than this period then we can hedge with chains of ED futures.

• In fact to make trading easy the CMEGroup

quotes 2-year, 3-year and 5-year Eurodollar Bundle Futures which settles into 8, 12 and 20 consecutive ED futures contracts.

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CONTINUE EDITING

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BOND FUTURES

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Bond Futures

Feature Specs Settlement Physical settlement Underlying $100,000 face of US Treasury Bonds, bonds with remaining maturity of at least 15 years, but less than 25 years, from the first day of the delivery month. Quote Points (1 point = $1000) and 1/32 of a point Tick Size A 1/32 move in the quote corresponds to $31.25 move in value (1 Quote Point / 32 = $1000 / 32) Contract Months March, June, September and December

http://www.cmegroup.com/trading/interest-rates/us-treasury/30-year-us-treasury-bond_contract_specifications.html

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Bond Futures

• This is a contract to buy/sell $100,000 of a

US Treasury bond for a given price.

• Yesterday USH5, the March Long Bond

Futures contract was trading at 145-26 or 145+26/32 = 145.81

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Hedging Interest Rates

• This contract allows one to hedge longer term

interest rates

• If 15-25 year interest rates increase then the

price of the deliverable bonds decrease.

– The bond futures price goes down

• If 15-25 year interest rates decrease then the

price of the deliverable bonds increase

– The bond futures price goes up

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Open Interest and Volume

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Expiry

• Generally a trader of these bond futures is

not looking to take delivery only to hedge long term interest rates cheaply.

• As seen the contract is typically rolled into the

next contract.

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Trading and Settlement

• The last trading day for a particular contract is

– 7 business days prior to the last business day of the settlement month

• The last delivery date is the last business day
• f the settlement month
• March 2015 contract

– Last trading day – 20th March 2015 – Last delivery day – 31th March 2015

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Bond Futures Delivery

• The Treasury Bond futures contract allows

the party with the short position to deliver any bond that has a maturity between 15 and 25 years and is not callable within 15 years.

• There will be many possible deliverable

bonds all with different prices.

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Deliverable Bonds (Bloomberg DLV)

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Settlement

• The person who is short futures contracts

– Decides on which bond to deliver and delivers 100,000 face amount of a bond – In exchange (s)he receives Settlement_Price x Conversion_Factor + Accrued_Interest

• The March 2015 contract is trading at 145-26
• If it settled here then which bond would you

deliver and how much would you receive?

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Conversion Factor

• All of these bonds have different coupons,

maturities and prices. The futures contract normalizes the prices

– The conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% with semiannual compounding

• For the purposes of the calculation the bond’s

maturity and the times to the coupon payment dates are rounded down to the nearest 3 months.

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Accrued Interest

• We can use ExcelTM to compute the accrued

interest on the bonds

– COUPPCD(): returns the preceding coupon date – COUPNCD(): returns the next coupon date

• For the first bond in the table

– Preceding coupon date = 15-Nov-2014 – Next coupon date = 15-May-2015 – 181 Days in period – 95 Days between 15-Nov-2014 and 18-Feb-2015 – (6.25/2) x (95/181) = 1.64

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Cheapest to Deliver

• Because there are many bonds that can be

delivered during the delivery month

– The party with the short position can decide which of the available bonds is cheapest to deliver – This is an option that the party with the short position owns. – It is actually a very difficult option to value:

• For the March 2014 futures contract there are 10

closely related bonds many of which could be cheapest to deliver

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Note – The Wild Card Play

• Trading in the bond futures contract ends at

2pm Chicago Time

• Trading in treasury bonds continues until 4pm
• A trader with a short position has until 8pm to

issue a notice of intention to deliver

• If treasury prices fall after 2pm but before

4pm the trader with the short position can

– Buy the bonds at a lower price – Issue a notice to deliver

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What does this tell you?

• You are trading just to hedge interest rate risk
• The contract is too complicated to exploit all

nuances

• Get out before delivery takes place

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Valuing the Futures Contract

• If you assume for simplicity that the cheapest-

to-deliver bond and the delivery date is known, then the treasury bond futures contract is a futures contract on a asset that provides the holder with known income where I is the present value of the income.

F

0 = S0 - I

( )erT

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Other Treasury Futures

• 10, 5 and 2 US Treasury Note Futures

– Delivers a note with a maturity of at least 6.5 years and no more than 10 years.

• Bund, Bobl, Schatz – German bond futures

– These are the most widely traded bond futures – The Bund is similar to to the 10 yr note future – The Bobl is similar to the 5 yr note future – The Schatz is similar to the 2 yr note future