1

Interest Rate Futures

Chapter 6

2

Day Count Conventions in the U.S. (Pages 102-103)

Treasury Bonds:Corporate Bonds:Money Market

Instruments:

Actual/Actual (in period)

30/360

Actual/360

3

BOND FUTURES

We will study:Long-term bonds:

U.S GovernmentT-Bonds.

Short-term bonds:

Euro-dollar time deposit rates.

4

AN APPLICATION OF DURATION 4. THE PRICE SENSITIVITY HEDGE RATIO

The Hedge Value: V = S + nF.

Objective: Minimize the position’s value sensitivity to interest rate

changes.

S = The bond’s spot value.

F = The futures price.

n = The number of futures used in the hedge.

5

The objective is to take a position in n futures such that a small change in the interest rate, r, will have no effect on the hedge value.

Notationally:

o.dr

dV :n*

:is objective The

nFSV

6

.dr

dy

dy

dFn

dr

dy

dy

dS

dr

dV

:rulechain the Using

.dr

dFn

dr

dS

dr

dV nFSV

F

F

S

S

7

Solve for n that sets dV/dr = 0.

.

drdy

dydF

drdy

dydS

n*

,0dr

dV

F

F

S

S

Next, we use the following substitutions for

.dy

dF and

dy

dS

FS

8

S

S

SS

SS

y1

SD

dy

dS

dy

y + 1

S

dS - = D

F

F

FF

FF

y1

FD-

dy

dF

dy

y + 1

F

dF - = D

From the definition of DURATION:

Upon substitution in n:

9

Usually, the yields sensitivities to the interest rate, r, are assumed to be the same for the spot yield and for the futures yield . Thus:

drdydr

dy

y1FD

y1SD

n*F

S

F

F

S

S

dr

dy

dr

dy FS

10

The price sensitivity hedge ratio.

)y(1FD

)y(1SDn*

SF

FS

11

The price sensitivity hedge ratio,

Example:

S = $50,000,000 F=$80,000

DS=8; yS=7.98%; DF=8.3; yF= 7.62%

600)(1.0798)80,000(8.3

)(8)(1.076250,000,000n*

12

The price sensitivity hedge ratio with continuous rates is:

F

S

FD

SDn*

In our example:

6028.3

8

80,000

50,000,000n*

13

INTEREST RATE FUTURES

The two most traded interest rate futures:

TREASURY BONDS (CBT)

USD100,000; pts. 32nds of 100%

EURODOLLARS (CME)

Eurodollars1,000,000; pts. Of 100%

14

LONG-TERM INTEREST RATE FUTURES

The U.S. T-BOND FUTURES

trades on the CBOT

The underlying assets are Treasury Bonds with long-term maturity. It is among the most successful futures contracts of all existing contracts. On any given day, there are 40 to 50 different T-bonds traded in the cash market. Most of these are deliverable against a T-bond futures position (contract specifications below), which makes this market extremely liquid.

15

SPECIFICATIONS OF U.S. TREASURY BOND FUTURES CONTRACTS

EXCHANGE CBOT

TICKET SYMBOL US

CONTRACT SIZE $100,000 FACE VALUE

CONTRACT MONTHS MAR. JUN. SEP. DEC.

PRICE QUOTATION POINTS AND 1/32 OF A POINT. PRICES ARE BASED ON 6% COUPON RATE WITH 20 YEARS TO MATURITY

TICK SIZE 1/32 OF A POINT, = $31.25

DELIVERABLE GRADES U.S. T-BONDS THAT ARE NOT CALLABLE FOR AT LEAST 15 YEARS AND HAVE A MATURITY OF AT LEAST 15 YEARS FROM THE FIRST BUSINESS DAY OF THE DELIVERY MONTH.

LAST TRADING DAY 7TH BUSINESS DAY PRECEDING THE LAST BUSINESS DAY OF THE DELIVERY MONTH.

DELIVERY METHOD FEDERAL RESERVE BOOK-ENTRY WIRE-TRANSFER SYSTEM.

PRICE LIMITS NONE

16

Delivery

The SHORT position prerogatives:

1.On which day to deliver?

The timing option

17

Delivery

2.Which T-bond to deliver?

The bond option

Many deliverable T-bonds are available

With (call) Maturity > 15years

The short will deliver the most profitable (deliverable) T-bond

available.

This bond is called:

The Cheapest –To-Delivery bond.

18

The Cheapest –To-Delivery bond

SHORT:

On date t: Opened a short position for Ft,T.

On date T: Buys T-bond for its Cash Price

and deliver it for an adjusted futures

price. The original futures price, Ft,T was

Calculated Based on CR = 6% and time to

Maturity 20 yrs, without accrued Interest.

19

The Cheapest –To-Delivery bond

The delivered T-bond CR is most likely

different than 6% and its Maturity may be

any number greater than 15.

Finally, because of the Daily Marking-to

Market process, the original Ft,T does not

apply any more. Instead, the

last Settlement price applies.

20

The Cheapest –To-Delivery bond

The SHORT’s payment is calculated as follows:

[Last settlement price][Conversion Factor]

+ Accrued Interest.

The SHORT’s cost of delivery is:

The Quoted T-bond price

+ Accrued Interest.

21

The Cheapest –To-Delivery bond

The SHORT’s profit is:

[Last settlement price][Conversion Factor]

- [The Quoted T-bond price]

22

The Cheapest –To-Delivery bond

Definition:

Of all the deliverable T-bonds in the market

The Cheapest to Delivery yeilds the

Min{BT(Quoted) – FT(last settle)[CF]}

CF = The delivered T-bond’s Conversion Factor.

23

The Conversion Factor puts the quoted price of the delivered T-bond on the same footing with the original

futures price. It is the NPV of the difference between the two scaled by the value of one contract; $100,000 as

follows:

CF = NPV(differnece)/(100,000)

000,100.06/2)(1

100,000.06/2)(1

000(CR/2)100,

CF2M

2M

1tt

24

2M2M

2M

2M

1tt

2M

2M

1tt

1.03

1

1.03

11

2(.03)

CRCF

1.03

1

1.03

1

2

CRCF

100,000.06/2)(1

100,000.06/2)(1

000(CR/2)100,

CF

25

CBOT T-BOND CONVERSION FACTORS

YRS = Years to maturity or 1st callability.M = Number of remaining monthsCR = Coupon of the delivered T-bond.CF = Conversion factor. Round off M to: M*= 0,3,6, OR 9.

26

2YRS2YRS

0 (1.03)].03

(1.03)1[

2

CRCF

Case 2: M* = 3

4

CR)(1.03)

2

CR(CFCF .5

03

Case 1: M* = 0

27

Case 3: M* = 6

1)+(2YRS1)+(2YRS

6 (1.03)].03

(1.03)1[

2

CRCF

Case 4: M* = 9

4

CR)(1.03)

2

CR(CFCF .5

69

28

EXAMPLE:

THE CF ON DEC 1999 FOR DELIVERING THE 11 3/4s WITH MATURITY: NOV 15,

2015.ON 12.1.99 YRS = 15 until 2014.M = 11 14 DAYS are ignored.YRS = 15 M is rounded off to M*

= 9 .

First compute:1)+2(15)

1)+2(15)

6 (1.03)].03

(1.03)1[

2

.1175CF (

(

CF6 = 1.575012319.

29

CF6 = 1.575012319.

Next compute:

factor. conversion sbond' theis

61.58041884 9

CF4

.1175.5)(1.03)2

.117519(1.57501239CF

30

HEDGING WITH T-BOND FUTURES

31

A SHORT T-BOND HEDGE A bond portfolio manager decides to sell $10M FV of 11 7/8 T-bonds on March 28. Currently, FEB 26, the bond sells for S=$101/$100FV.

DATE CASH FUTURESFEB. 25 10M FV T-BONDS SELL 160 JUN T-bond

CR = 11 7/8 FUTURES.S = 10,100,000 F=70-16Ds = 7.83 Df = 7.20Ys = 11.74% Yf = 14.92%

1604)500)(1.117(7.20)(70,

.1492)100,000)(1(7.83)(10,-=*n

32

DATE CASH FUTURESMAR. 28 S = 95.6875/$100FV LONG 160 JUN T-bond

$9,568,750 FUTURESOpportunity loss <$531,250> F = 61 - 23

Futures gain: [(70-16)-(61-23)]160=(8-25)160=($8,781.25)160=$1,405,000.

Total selling price: 9,568,750 + 1,405,000 = $10,973,750

33

LONG HEDGE WITH T - BOND FUTURESDATE CASH FUTURESMAR. 29 LONG 110 SEP T-BOND Fs. F = 78-21

BY REGRESSION: n*= 110.

JUL. 15 S=107 19/32 SHORT110 SEP T-BOND Fs

$10,759,375BUY BONDS. F = 86-6

Gain from futures:110[(86-6) – (78-21)]=110[7-17]=110[$7,531.25] = $828,437.5

THUS, THE EFFECTIVE PURCHASE PRICE OF THE T- BONDS IS:$10,7593,750 - $ 828,437.5 = $9,930,937.5.

34

HEDGING A CORPORATE BOND ISSUE

FEB. 24. DECISION: ISSUE $50M CORPORATE BONDS AT PAR VALUE ON MAR. 24.EXPECTATIONS: CR = 13.76%

M = 20yrs D = 7.22

DATE CASH FUTURES2.24 DS = 7.83 SHORT 674 FUTURES.

yS = 13.6% F(JUN) = 68-11S=$50M. DF=7.83; yF = 13.6%

674.- 1376)343.75)(1.(7.83)(68,

.1360)000,000)(1(7.22)(50,- = *n

35

DATE SPOT FUTURES3.24 ISSUE BONDS LONG 674 JUN T-BOND Fs CR=13.26% F(JUN) = 55-25 S=$90.74638/$100FV

V(BOND ISSUE) = $45,373,190

Gain from futures: 674[(68-11)-(55-25)]=674[12-18]=674[$12,562.5]=$8,467,125.

TOTAL VALUE=$53,840,315.

36

SHORT-TERM INTEREST RATE FUTURES

EURODOLLAR FUTURES

The U.S. T-BILLS FUTURES

The underlying assets For the Eurodollars futures are 3-months Eurodollars time deposit. It is the

most successful futures contracts of all existing contracts. Clearly, the

underlying asset for the U.S. Gov. T-bills are T-bills.

37

CONTRACT SPECIFICATIONS FOR:

90-DAY T-BILL 3-Month EURODOLLAR FUTURES

SPECIFICATIONS 13-WEEKUS T-BILL 3-MONTH EURODOLLAR TIME DEPOSIT

SIZE USD1,000,000 Eurodollars1,000,000

CONTRACT GRADE new or dated T-bills CASH SETTLEMENT

with 13 weeks to maturity

YIELDS DISCOUNT ADD-ON

HOURS ( Chicago time) 7:20 AM-2:00PM 7:20 AM - 2:00PM

DELIVERY MONTHS MAR-JUN-SEP-DEC MAR-JUN-SEP-DEC

TICKER SYMBOL TB EB

MIN. FLUCTUATION .01(1 basis pt) .01(1 basis pt)

IN PRICE USD25/pt USD25/pt

LAST TRADING DAY The day before the 2nd London business day

first delivery day before 3rd Wednesday

DELIVERY DATE 1st day of spot month Last day of trading

on which 13-week

T-bill is issued and a 1-year

T-bill has 13 weeks to maturity

38

EURODOLLAR FUTURES

These are futures on the interest earned on Eurodollar three-month

time deposits.

The rate used is

LIBOR - London Inter-Bank Offer Rate.

These time deposits are non transferable, thus, there is no

delivery! Instead, the contracts are CASH SETTLED.

39

EURODOLLAR FUTURES PRICE

The IMM (CME) quotes the IMM index. Let the quote be denoted by Q then, the Futures price is given by:

F = 1,000,000[1 – (1 – Q/100)(.25)].

On the delivery date – the third Wednesday of the delivery month – the quote for the CASH SETTLEMENT is given by the 90-day LIBOR: Q/100 = 1 – L/100

F = 1,000,000[1 - .25L/100]

40

EURODOLLAR FUTURES PRICE:

The IMM index = 95.53 for the JUN 2001.

F = 1,000,000[1 – (1 – 95.53/100)(.25)]

=$988,825

On the delivery date the quote for the CASH SETTLEMENT is given by:

The 90-day LIBOR: 8%

F = 1,000,000[1 - .25(8)/100]=$980,000

41

r0,k = REPO RATE FOR 27 DAYS 9.45% r0,T = ED TD RATE FOR 117 DAYS 9.40% rF = ED TD FUTURES RATE: F0,k = 90.65 r F = 9.35% _|_________________|_____________________|_________TIME 0 k T

rF = 9.385% > 9.35%

1 + F

r = (1.094

11790)

(1.0945

2790)

= 1.12388671.0274595

= 1.09385

Arbitrage with Eurodollar Futures

42

Arbitrage with Eurodollar Futures continuedDATE SPOT FUTURES

MAY 23 Deposit $1,000,000 Short 1 ED futures.

in a 117 days ED time L = 9.35%

deposit to earn 9.40%

over the 117 days.

Jun 19 Borrow $1,000,000 Cash settle (Long)

for the current L. at the current L.

This is equivalent to borrow the money for 9.35%SEP 17 Receive Repay the loan

1,000,000[1+.094(117/360)] 1,000,000[1+.0935(90/360)]

=1,030,550 = 1,023,375

Arbitrage profit = $7,175

43

How to calculate the profit from a ED futures?

Assume that initially, Qt = 90.54. Thus,

Ft = 1,000,000[1 – (1 – 90.54/100)(.25)].

At a later date, k, the index dropped by exactly 100th of a point; that is, Qk = 90.53.

Fk = 1,000,000[1 – (1- 90.53/100)(.25)].

It is easily verified that the difference between the two futures prices is exactly:

$25. Thus, we have just seen that

Every 100th of the quote Q is $25.

44

Hedging with Eurodollars futures.

Eurodollar futures became the most successful contract in the world. Its enormous success is attributed to its ability to fill in the need for hedging that still remained open even with a successful market for T-bond and T-bill futures. The main attribute of the 90-day Eurodollars futures is that, unlike the T-bills futures, it is risky. This risk makes it a better hedging tool than the risk-free T-bill futures.

45

The examples below demonstrate how to hedge with ED futures using a STRIP, or a STACK. In most of the loans involved in these hedging strategies, the interest today determines the payment by the end of the period. Only interest payments are paid during the loan term and the last payment include the interest and the principal payment.

46

A STRIP HEDGE WITH EURODOLLARS FUTURESOn November 1, 2000, a firm agrees to borrow $10M for 12 months, beginning December 19,

2000 at LIBOR + 100bps.DATE CASH FUTURES Q11.1.00 LIBOR 8.44% Short 10 DEC 91.41

Short 10 MAR 91.61 Short 10 JUN 91.53 Short 10 SEP 91.39

12.19.00 LIBOR 9.54% Long 10 DEC 90.46

3.13.01 LIBOR 9.75% Long 10 MAR 90.25

6.19.01 LIBOR 9.44% Long 10 JUN 90.56

9.18.01 LIBOR 8.88% Long 10 SEP 91.12

47

PERIOD: 1 2 3 4 RATEa: 10.54% 10.75% 10.44% 9.88%

INTERESTb: $263,500 $268,750 $261,000 $247,000

FUTURESc: $23,750 $34,000 $24,250 $6,750

NETd: $239,750 $234,750 $236,750 $240,250

EFFECTIVE RATEe: 9.59% 9.39% 9.47% 9.61%UNHEDGED AVERAGE RATE: 10.40%HEDGED AVERAGE RATE: 9.52%

a. LIBOR + 100 BPSb. ($10M)(RATE)(3/12)c. (PRICE CHANGE)(25)(100)(10)d. b - ce. (NET/10M)(12/3)(100%)

48

A STACK HEDGE WITH EURODOLLAR FUTURES:

DATA ON NOVEMBER 11, 2000

VOLUME OPEN INTERESTDEC 00 46,903 185,609

MAR 01 29,236 127,714

JUN 01 5,788 77,777

SEP 01 2,672 30,152

DECISION: STACK MAR FUTURES FOR JUN AND SEP AND ROLL OVER AS SOON AS OPEN INTEREST REACHES 100,000.

49

THE STACK HEDGEDATE CASH FUTURES F. POSITION11.1.00 8.44% S 10 DEC 91.41S10DEC

S 30 MAR 91.61S30MAR12.19.00 9.54% L 10 DEC 90.46S30MAR

1.12.01 9.47% L 20 MAR 90.47S10MAR

S 20 JUN 90.42 S20JUN

3.13.01 9.75% L 10 MAR 90.25 S20JUN

3.22.01 9.95% L 10 JUN 89.78S10JUNS 10 SEP 89.82

S10SEP

6.19.01 9.44% L 10 JUN 90.56S10SEP

9.18.01 8.88% L 10 SEP 91.12 NONE

50

PERIOD: 1 2 3 4 RATE(%)a: 10.54 10.75 10.44 9.88

INTERESTb: 263,500 268,750 261,000 247,000

FUTURES($)c: 23,750 91,000 12,500 <32,500>

NET($) d: 239,750 177,750 248,500 279,500

EFFECTIVE RATE (%)e: 9.59 7.11 9.94 11.18UNHEDGED AVERAGE RATE 10.40%HEDGED AVERAGE RATE 9.455%

a. LIBOR + 100 BPSb. ($10M)(RATE)(3/12)c. (PRICE CHANGE)(25)(100)(10)d. b - ce. (NET/10M)(12/3)(100%).