Understanding Interest Rate Futures Contracts

INTEREST RATE FUTURES: INTRODUCTION

LEARNING OUTCOMES T-Bill futures: deliverable, invoice price, futures price quote Eurodollar futures given LIBOR

T_BILL FUTURES The deliverable asset to the T-Bill futures contract is a $1,000,000 face value T-Bill that has 90 days to maturity. As of 2002, the T-Bill contract is cash settled.

T-BILL INSTRUMENT What is a Treasury Bill? It is a zero coupon bond of less than a year maturity, issued by the U.S. Treasury. The T-Bill that underlies the T-Bill futures contract is the 3-month Bill (90 days).

T-BILL How do you determine the price you pay for a T-Bill also called invoice price ? You apply a dollar discount to the face value of the T-Bill. How do you calculate the dollar amount of the discount? By multiplying the annual discount rate or yield by the time to expiration of the T-Bill expressed as a fraction of 360 day year.

T-BILLS How are T-Bills quoted? T-Bills are quoted not by their invoice price but through the annual discount rate or yield. See for example the Wall Street Journal.

T_BILL FUTURES The invoice price of a T-Bill is calculated as: dtm = - , 1 000 000 , 1 P dy 360 dtm = days to maturity dy = discount yield.

T-BILL FUTURES Quoted futures price: Futures price (IMM Index) = 100 DY (%) Example: Annual Discount yield 8.5% Futures Price = 100 8.50 = 91.50 The IMM Treasury Bill index (futures price) is the difference between the annual discount yield and 100.

T-BILL FUTURES T-Bill futures exist in quarterly expirations: March, June, September, December Example: The March T-Bill futures contract calls for the delivery of a 3 month T-Bill of 1,000,000 face value. As of 2002 the T-Bill futures contract is cash settled.

T-BILL FUTURES What is a basis point in 90-day T-Bills or T-Bill futures? A basis point is simply one hundredth of one percent or a 0.0001 change in the discount rate. What is the notional value that corresponds to one basis point for the T-Bill futures contract? bpv =1,000,000 0.0001 90 360=25

PRICE AT DELIVERY OR INVOICE AMOUNT Cash Bill invoice price = P =1,000,000 1-dy dtm 360 Example: dy=0.085 , dtm = 90 Price = 1,000,000(1-0.085*90/360) = $978,750.

INVOICE PRICE BASED ON T-BILL FUTURES QUOTE T-Bill Futures quote = 91.50 Since f = 100-DY(%), then DY = 100-91.50 = 8.50 and dy = DY/100 Invoice price = P =1,000,000 1-dy dtm 360

INVOICE PRICE BASED ON T-BILL FUTURES QUOTE T-Bill spot market price per $100 of cash price is equal to: dtm 360 CashPrice=100-DY Example: 100 8.5(90/360) = 97.875

HOW MUCH DO WE MAKE OR LOSE? Buy one T-Bill contract at 95.00 and sell one T-Bill contract at 97.00. Two ways of doing it: 1. Remember that 1 basis point equals $25 of notional. (97.00-95.00)*100 = 200 basis points or (200 x $25/bp) = $5,000

HOW MUCH DO WE MAKE OR LOSE? 2. Calculate the two invoice prices. Invoice price at 95.00? IP = 1,000,000x(1-.05x90/360)=$987,500 Invoice price at 97.00? IP = 1,000,000x(1-.03x90/360)=$992,250 Profit = $992,250 - $987,500 = $5,000

PRICING OF T-BILL FUTURES How do we find the no arbitrage or fair price for the T-Bill futures? What does the above actually mean?

CASH AND CARRY ARBITRAGE Futures Discount Yield 12.50% Futures contract maturing in 78 days Cash Bills 10.00% 168-day T-Bill (Deliverable on Futures) 78-day T-Bill 6%

UNDERSTANDING WHAT THE TABLE TELLS YOU Time line T-Bill for 168 days, 10% discount 168 0 78 T-Bill for 78 days 6% discount Futures on 90 day T- Bill, 12.5% discount

LET US BE SURE WE UNDERSTAND We observe today: 1. A cash T-Bill maturing in 168 days that is priced at a 10% dy. 2. A cash T-Bill maturing in 78 days that is priced at a 6% dy. 3. A futures contract that expires in 78 days and calls for the delivery of a 90-day T-Bill in 78 days. The price of this 90-day T-Bill deliverable under the futures in 78 days is given by a 12.5% dy.

CASH AND CARRY ARBITRAGE Remember, cash and carry arbitrage implies that this is what you should observe: F > S(1+cc)t t = t/T The futures price is higher than its fair price . No arbitrage implies what?

CASH AND CARRY ARBITRAGE Are you given futures prices? What are you given? Discount yields What is the cost of carry? WHAT DO WE DO TO FIGURE OUT IF THERE IS AN ARBITRAGE OPPORTUNITY?

CASH AND CARRY ARBITRAGE Cost of carry initial transactions: Buy underlying Sell futures Finance the purchase of the underlying by borrowing. Cost of carry terminal transactions: Deliver underlying under futures obligation Repay principal and interest

CASH AND CARRY ARBITRAGE Futures price quote= 87.50=(100-12.50) The implied cash invoice price for the implied futures discount yield is: Futures.Price=100-12.590 360=96.875 This is the invoice price of the T-Bill underlying the futures contract. (Futures Invoice Price)

CASH AND CARRY ARBITRAGE The cash instrument is the 168-day T-Bill because it is deliverable on the Futures. Its Discount yield is 10%. Using this to calculate the spot price Spot.price =100-10168 360=95.333

CASH AND CARRY ARBITRAGE In order to buy the previous cash bill, I have to borrow (sell) a 78 day T-Bill short and the discount yield for it is 6%. For this bill to have a sale value of 95.333 it has to have a face value of: 95.333= X 1-0.0678 95.333 1-0.0678 X = =96.589 360 360

CASH AND CARRY ARBITRAGE What is the face value of the short Bill? S(1+cc)t The formula above is the no arbitrage futures invoice price, because it is equal to the Cash price plus the cost of carry

CASH AND CARRY ARBITRAGE Under cash and carry arbitrage we should see that: F > S(1+cc)t Does the relationship hold? 96.875>96.589 A cash and carry arbitrage is confirmed! You will make $0.286 per 100 dollars of face value or $2,860 per million of face value.

CASH AND CARRY ARBITRAGE 2. Use the cost of carry relationship under no arbitrage, I.e compare the implied cost of carry (implied repo rate) with the actual cost of carry! Under no-arbitrage one plus the cost of carry for the carry period must be equal to one plus the implied repo rate for the carry period: (1+cc)t=F S

CASH AND CARRY ARBITRAGE Futures Invoice Price F S=Bf(90) Bc(168)=968,750 953,333=1.01617 Invoice price for cash bill The implied cost of carry or repo rate for 78-days is then 1.617%. (1.01617-1)*100

CASH AND CARRY ARBITRAGE The actual cost of carry is the cost to finance the 168-day T-Bill purchase with the issuance of the 78- day T-Bill with a discount yield of 6%, I.e: Since the face value of a T-Bill is 100, the 6% discount implies an original price of 98.70. 100 * (1-0.06*78/360) = 98.70

CASH AND CARRY ARBITRAGE Then, the cost of carry rate is given by: 100 / 98.70 = 1.01317 or 1.317% Then: 1.617%>1.317%

MORE SIMPLY It is much simpler to do the following: Total Return on Carry trade Total Financing cost Principal + interest on short bill Futures Invoice Price Bfo(90) Bo(168) 1 Bo(78) Cash bill deliverable against futures Short bill, financing of cash purchase

CASH AND CARRY ARBITRAGE Rule: If the actual cost of carry is lower than the implied repo rate, exploit cash and carry arbitrage: Borrow, buy cash bill, sell futures; hold bill and deliver against futures. If the actual cost of carry is higher than the implied repo rate, exploit reverse cash and carry arbitrage : Buy futures, sell cash bill short and invest proceeds, take delivery of bill and cover short bill position. (Reverse arbitrage might be more costly)