University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives FNCE4040 – Derivatives Chapter 6 Interest Rate Futures
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives REVIEW FORWARD RATE AGREEMENTS
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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.
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives Example • For example a bank agrees to lend 1m USD for 1 year starting in 1 year at an interest rate of 3%. The rate is quoted with an ACT/360 daycount basis. $1𝑛 365 1𝑛 𝑉𝑇𝐸 360 3.00% = 30,416.67 today Year 1 Year 2 1𝑛 𝑉𝑇𝐸
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives FRA Mechanics / Valuation – part 1 • From the lender’s viewpoint • A loan of 𝑂 from 𝑈 1 to 𝑈 2 at an agreed rate 𝑆 𝐿 • Let 𝐸 be the daycount fraction from 𝑈 1 to 𝑈 2 𝐸𝑏𝑧𝑡 𝑈 2 −𝐸𝑏𝑧𝑡(𝑈 1 ) 𝑂 – For an FRA 𝐸 = 360 Interest owed: 𝑂 × 𝑆 𝐿 × 𝐸 𝑈 today 𝑈 2 1 𝑂
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives FRA Mechanics / Valuation – part 1 • We can value the FRA given the continuously compounded zero rates 𝑠 1 and 𝑠 2 . 𝑂 1 + 𝑂 ∙ 1 + 𝑆 𝐿 ∙ 𝐸 ∙ 𝑓 −𝑠 2 𝑈 𝑄𝑊 = −𝑂 ∙ 𝑓 −𝑠 1 𝑈 2 Interest owed: 𝑂 ∙ 𝑆 𝐿 ∙ 𝐸 𝑈 today 𝑈 2 1 𝑂
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives The Fair FRA Rate • The fair FRA rate 𝑆 𝐺 is the rate such that the sum of the PV of both cash flows is zero. 1 = 𝑂 1 + 𝑆 𝐺 𝐸 𝑓 −𝑠 2 𝑈 𝑂𝑓 −𝑠 1 𝑈 2 Solve for 𝑆 𝐺 : 1 + 1 + 𝑆 𝐺 𝐸 𝑓 −𝑠 2 𝑈 0 = −𝑓 −𝑠 1 𝑈 2 1 − 1 𝑆 𝐺 = 𝑓 𝑠 2 𝑈 2 −𝑠 1 𝑈 𝐸 where 𝐸 = 𝐸𝑏𝑧𝑡 𝑈 2 − 𝐸𝑏𝑧𝑡(𝑈 1 ) 360
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives The Fair FRA Rate • We know that the PV of a loan from 𝑈 1 to 𝑈 2 at an agreed rate 𝑆 𝐿 with daycount fraction 𝐸 from 𝑈 1 to 𝑈 2 is 1 + 𝑂 1 + 𝑆 𝐿 𝐸 𝑓 −𝑠 2 𝑈 𝑄𝑊 = −𝑂𝑓 −𝑠 1 𝑈 2 • Combine this with the definition of the fair rate and we have 2 + 𝑂 1 + 𝑆 𝐿 𝐸 𝑓 −𝑠 2 𝑈 𝑄𝑊 = −𝑂 1 + 𝑆 𝐺 𝐸 𝑓 −𝑠 2 𝑈 2 𝑄𝑊 = 𝑂 𝑆 𝐿 − 𝑆 𝐺 𝐸𝑓 −𝑠 2 𝑈 2
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives FRA Mechanics / Valuation – part 2 • A loan from 𝑈 1 to 𝑈 2 • From the lender’s viewpoint Interest owed: 𝑂 × 𝑆 𝐿 × 𝑈 2 − 𝑈 1 Fair rate now for period 𝑈 1 , 𝑈 2 = 𝑆 𝐿 𝑈 today 𝑈 2 1
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives FRA Mechanics / Valuation – part 2 Think of Mark-To-Market as the cost to offsetting your position Interest owed: 𝑂 × 𝑆 𝐿 × 𝑈 2 − 𝑈 Fair rate for period 1 𝑈 1 , 𝑈 2 moves to 𝑺 𝑮 𝑓 −𝑆 2 ×𝑈 2 𝑈 today 𝑈 2 1 Interest now prevailing: Take the difference 𝑂 × 𝑺 𝑮 × 𝑈 2 − 𝑈 1 and PV
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives EURODOLLAR FUTURES
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives Eurodollars • A Eurodollar is a dollar deposited in a bank outside 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
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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 March, June, September and December, out 10 Months years
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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 of: 10,000 x [100 - 0.25 x (100-Q)]
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives Example • Suppose you buy (go long) a Date Quote contract on Nov-1 Nov-1 97.12 • Contract expires on Dec-21 Nov-2 97.23 • The prices are as shown • How much do you gain/lose Nov-3 96.98 a) on the 1 st day $25 × 11 = $275 ……. …… b) on the 2 nd day $25 × −25 = −$625 c) over the whole time until expiration? Dec-21 97.42 $25 × 30 = $750
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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%
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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.
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives HEDGING EXAMPLE ONE
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives FRAs and Eurodollar Futures • Consider a FRA 18-Mar counterparty A – $1,000,000 notional deposits $1m with counterparty B – Agreed rate: 0.30% – 90-day deposit 16-June counterparty B – Starts 18-Mar gives back $1m – On Eurodollar Deposit rate plus interest to counterparty A 18-Mar $1,000,000 Counterparty Counterparty A B 16-Jun $1,000,750
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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 𝑚𝑜 1,000,750 1,000,000 + 0.0020 × 31 365 𝑠 = = 27.74 𝑐𝑞𝑡 121 365
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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.
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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
University of Colorado at Boulder – Leeds School of Business – FNCE4040 Derivatives 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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