1

BOND FUTURES

We will study:

1.The Cash Market for bonds

2. Short-term bonds:

U.S Government T-Bills

Eurodollar time deposit rates.

3.U.S Government T-Bonds.

2

BONDS - THE CASH MARKET

DEFINITION A BOND IS A PROMISE TO PAY CERTAIN AMOUNTS ON PRESPECIFIED FUTURE DATES.

BOND PARAMETERS

P = THE BOND CASH PRICE

FV= THE FACE VALUE or THE PAR VALUE OF THE BOND

M = THE MATURITY DATE OF THE BOND, or THE END OF THE LAST PERIOD OF THE BOND’S LIFE.

t= THE TIME INDEX; t = 1,2,……, M.

Ct= THE CASH FLOW FROM THE BOND AT TIME PERIOD t. USUALLY, THE CASH FLOW IS ASSUMED TO OCCUR AT THE PERIOD’S END.

NORMALLY:

Ct= C for t = 1, …., M – 1 and CM= C + FV.

C IS CALLED THE COUPON OF THE BOND.

CR = THE COUPON RATE.

C = (CR)(FV) = THE $ AMOUNT OF THE COUPON.

3

EXAMPLE:

A 30 YEAR TREASURY BOND WITH FACE VALUE OF $1,000, PAYS ANNUAL COUPONS AT AN 8%.

THUS:

M=30;

FV = $1,000;

CR = 8%;

C = (.08)($1,000) = $80 for t= 1, 2, …29

and

C = $1,080 for t = 30.

An investor who buys this bond will receive $80 at the end of every year for the next 29 years and $1,080 at the end of the 30-th year.

4

EXAMPLE: THE SAME T-BOND PAYS COUPONS SEMMI ANNUALLY.

IN THIS CASE, WE HAVE NEW PARAMETERS:

m= THE NUMBER OF PAYMENT PERIODS EVERY YEAR.

N = THE NUMBER OF COUPON PAYMENTS.

Thus: M=30; m = 2; N = 60; FV = $1,000; CR = 8%; C = (.08/2)($1,000) = $40 for t= 1, 2, …59 and C = $1,040 for t = 60.

An investor who buys this bond will receive $40 every six months for the next 29.5 years, plus $1,040 at the end of the 30-th year.

DEFINITION: A PURE DISCOUNT BOND PAYS THE FV AT ITS MATURITY BUT PAYS NO COUPONS (C = 0) IN ANY OF THE INTERIM PERIODS.

5

P R I C I N G B O N D S : A B O N D W I T H M A T U R I T Y O F M Y E A R S W I T H C O U P O N S P A I D

A N N U A L L Y I S P R I C E D B Y :

.r)(1

MFV

r)(1M1

rC P

:solution form closed A

,r)(1

MFVM

1t r)(1t

C t P

t.allfor tr r is convention usual The

...;)3r)(12r)(11r(1

3C

)2r)(11r(12C

1r1

C 1 P

:CFs theof NPV theis P.M

1t r)(1t

tC P

r = T H E B O N D ’ S Y I E L D T O M A T U R I T Y . ( Y T M )

6

Mr)(1FVP

F O R A B O N D W I T H S E M I A N N U A L C O U P O N P A Y M E N T S :

2M)2r(1

FV2M

1t t)2r(1

2C

Ρ

A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:

7

E X A M P L E S :

M = 3 0 ; F V = $ 1 , 0 0 0 ; C R = 8 % P A I D S E M I A N N U A L L Y . = > C = ( . 0 8 ) 1 , 0 0 0 / 2 = $ 4 0 .

r = 1 0 %

70.810$05.1

000,105.11

05.

40

05.1

040,1

)05.1(

4060

6060

60

1

t

t

T H E S A M E B O N D W I T H A N N U A L C O U P O N P A Y M E N T S W O U L D B E P R I C E D :

46.811$1.1

000,11.11

1.1

8030

30

T H E B O N D I S S O L D A T A D I S C O U N T B E C A U S E T H E Y T M I S G R E A T E R T H A N T H E C R

r = 5 %

$1,463.631.025

1,0001.0251

.025

40P

6060

T H E S A M E B O N D A S A P U R E D I S C O U N T B O N D T B O N D W O U L D B E P R I C E D :

1 0 0 0

1 13 13 0

,

.$ 5 7 .

I F I T W E R E A C O N S U L W I T H A N N U A L C O U P O N S :

$800.1

80P

8

Q U O T E S A R E G I V E N B Y C O N V E N T I O N , U S I N G A D I S C O U N T Y I E L D , d , W H E R E :

D I S C O U N T = F A C E V A L U E - M A R K E T P R I C E :

FVPFV

t360d

FV$DISCOUNT

t360d

B O N D E Q U I V A L E N T Y I E L D ( B E Y ) :

dt360365d

1

360dt1

360365di

PPFV

t365i

9

E X A M P L E :

t = 9 0 d a y s

F V = $ 1 , 0 0 0 , 0 0 0

d = 1 1 %

(11.468%). .11468360

(.11)901

360

365(.11)i

.11468972,500

972,5001,000,000

90

365iBEY

11%.1190

360

1,000,000

972,5001,000,000d

$27,500.DISCOUNT1,000,000

)($DISCOUNT

90

360.11

1

10

D U R A T I O N D E F I N I T I O N :

.

Pr1

tC

D

M

1tt

t

O B S E R V E T H A T :

,t W

P

r1CtD

M

1tt

M

1t

tt

W h e r e :

: Durationof tioninterpreta

following the toleadsresult This

.P

r1/CW and 1W

tt

t

M

1tt

11

DURATION

IS THE WIEGHTED AVERAGE OF COUPON PAYMENTS’ TIME

PERIODS, t,

WEIGHTED BY THE PROPORTION THAT THE DISCOUNTED CASH

FLOW, PAID AT EACH PERIOD, IS OF THE CURRENT BOND PRICE.

12

DURATION INTERPRETED AS A MEASURE OF THE BOND PRICE

SENSITIVITY

.DP

r)(1tC

r1r)d(1

PdP

: yieldsg,Rearrangin

.r)(1

tC

r)P(1

r)(1

P

r)(1

r)d(1

dP

. r)(1

tC

r)(1

1

r)d(1

dP

dr

dP

:1ror r respect to withderivative Its

.r1

CP:price bond The

tt

tt

tt

M

1tt

t

13

TY. ELASTICI PRICE BONDTHED

YIELDTHE OF %Δ THE

PRICE BONDTHE OF %Δ THED

.

r1r)d(1

PdP

- D

: thatstatesresult final The

The negative sign merely indicates that D changes in opposite direction to the change in the yield, r.

Next we present a closed form formula to calculate duration of a bond:

14

09.11

6100

)1(.11.1)1(.

306

100)1(.30)1(.11.11.1

D230

230

E X A M P L E :r = 1 0 % = . 1F V = $ 1 0 0C = $ 6 = > C R = 6 %M = 3 0 = 1m = 1P = $ 6 2 . 2 9

N = N U M B E R O F P A Y M E N T S

m = C O U P O N P A Y M E N T S P E R Y E A R

Θ = F R A C T I O N O F Y E A R T O T H E N E X T C O U P O N P A Y M E N T

Θ 1 2 … . y e a r s

CFV2r1

N

mr1r

m1NΘ

CFV2r

mNr1

N

mr1)r(1

D

15

M 0 2 4 6 8 1 0 1 2 1 4 1 6

5 5 4 . 7 6 4 . 5 7 4 . 4 1 4 . 2 8 4 . 1 7 4 . 0 7 3 . 9 9 3 . 9 2

1 0 1 0 8 . 7 3 7 . 9 5 7 . 4 2 7 . 0 4 6 . 7 6 6 . 5 4 6 . 3 6 6 . 2 1

1 5 1 5 1 1 . 6 1 1 0 . 1 2 9 . 2 8 8 . 7 4 8 . 3 7 8 . 0 9 7 . 8 8 7 . 7 1

2 0 2 0 1 3 . 3 3 1 1 . 2 0 1 0 . 3 2 9 . 7 5 9 . 3 6 9 . 0 9 8 . 8 9 8 . 7 4

2 5 2 5 1 4 . 0 3 1 1 . 8 1 1 1 . 8 6 1 0 . 3 2 9 . 9 8 9 . 7 5 9 . 5 8 9 . 4 5

3 0 3 0 1 4 . 0 3 1 1 . 9 2 1 1 . 0 9 1 0 . 6 5 1 0 . 3 7 1 0 . 1 8 1 0 . 0 4 9 . 9 4

3 5 3 5 1 3 . 6 4 1 1 . 8 4 1 1 . 1 7 1 0 . 8 2 1 0 . 6 1 1 0 . 4 6 1 0 . 3 6 1 0 . 2 8

4 0 4 0 1 3 . 1 3 1 1 . 7 0 1 1 . 1 8 1 0 . 9 2 1 0 . 7 6 1 0 . 6 5 1 0 . 5 7 1 0 . 5 1

5 0 5 0 1 2 . 1 9 1 1 . 4 0 1 1 . 4 0 1 0 . 9 9 1 0 . 9 1 1 0 . 8 5 1 0 . 8 1 1 0 . 7 8

1 0 0 1 0 0 1 1 . 0 2 1 1 . 0 1 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0

C O M P U T I N G D U R A T I O N

r = 1 0 %

Coupon Rate

16

N O T I C E T H A T T H E C H A N G E I N T H E B O N D ’ S P R I C E M A Y B E E S T I M A T E D U S I N G D U R A T I O N , B Y :

r1

ΔrD)(P)(ΔΡ

E X A M P L E :

M = 3 0 ; C R = 6 % ; D = 1 1 . 0 9 a n d P = $ 6 2 . 2 9 .

r = 1 0 % r = 1 1 %

P P

r r = . 0 8

P = - ( 1 1 . 0 9 ) ( 6 2 . 2 9 ) ( - . 0 2 )1 . 1 P

1 1 0 9 6 2 2 90 11 1 6 2 8 1 0 1

1 0 %

1 2 5 6 1 8 5

. ..

. . $ 5 6 .

. $ 7 4 .

17

DURATION OF A BOND PORTFOLIO

V = The total bond portfolio value

Pi = The value of the i-th bond

Ni = The number of bonds of the i-th bond in the portfolio

Vi = Pi Ni = The total value of the i-th bond in the portfolio

V = ΣPiNi The total

portfolio value.

We now prove that: DP = ΣwiDi

.

18

:yields

,P

r1by deviding and gMultiplyin

.r)d(1

dPN

r)d(1

PNd

r)d(1

dV

: toleadsmaturity -to- yieldthe

respect to withV, value,portfolio

bond theof derivative the takingThus,

.V

r)(1

r)d(1

dV

r)(1r)d(1

VdV

D

:elasticity price bond theas dinterprete be

may durationsaw that already We

i

ii

ii

P

19

:or ,V

VD

V

r1DV

r1

1D

:have we,V

r1

r)d(1

dVD

intoresult this

ngSubstituti .DVr)(1

1

r)d(1

dV

:rewrite weThus, .VPN

whileduration, i sbond' of negative

thearebracket square thein termsThe

]P

r)(1

r)d(1

dP[PN

r)(1

1

:sign summation thein termsgRearrangin

.r)(1

P

P

r)(1

r)d(1

dPN

r)d(1

dV

ip

iiP

P

ii

iii

i

iii

i

i

ii

iD

20

iiP

ii

i

iiii

iiP

Dw D

duration protfolio the,conclusion in Thus,

.1w ;V

V w

: where

,DwDV

VDV

V

1 D

is the weighted average of the durations of the bonds in the portfolio. The weights are the proportions the bond value is of the entire portfolio value.

21

Example: Consider a portfolio of two T-bonds:BOND FV

in $M

N YTM

COUPON

T-BOND

$100

15yrs

6% 5%

T-BOND

$200

30yrs

6% 15%

BOND PRICE W DT-BOND $90.2

0.1673 10.4673

T-BOND

$449.1 539.3

0.8327 1.0000

12.4674

D= (.1673)(10.4673) +(.8327)(12.4674)

D = 12.1392

22

IMMUNIZING BANK PORTFOLIOOF

ASSETS AND LIABILITIES

TIME 0 ASSETSLIABIABILITIES

$100,000,000 $100,000,000 (LOANS) (DEPOSITS)

D = 5 D = 1r = 10% r = 10%

TIME 1 r => 12%

.82$1,818,1810100,000,001.1

(.02)1ΔV

.09$9,090,9090100,000,001.1

(.02)5ΔV

L

A

BUT IF DA = DL THEY REACT TO RATES CHANGES IN EQUAL AMOUNTS. THE BANK PORTFOLIO IS IMMUNIZED , i.e., IT’S VALUE WILL NOT CHANGE FOR A “small” INTEREST RATE CHANGE, IF THE PORTFOLIO’S DURATION IS ZERO or:

DP = DA - DL = 0.

23

A 5-YEAR PLANNING PERIOD CASE OF IMMUNIZATION IN

THE CASH MARKET

BOND C FV M r D PA $100 $1,000 5 yrs 10% 4.17 $1,000B $100 $1,000 10 yrs10% 6.76 $1,000

4.17WA + 6.76WB = 5

WA + WB = 1

WA = .677953668.

WB = .322046332.

VP = $200M implies:

Hold $135,590,733.6 in bond A, And $64,409,266.4 in bond B.

Next, assume that r rose to 12%. The portfolio in which bonds A and B are held in equal proportions will change to:

24

[1 - 4.17 (.02/1.1)] 100M = $92,418,181.2

[1 - 6.76 (.02/1.1)] 100M = $87,709,090.91 TOTAL = $180,127,272.7

INVEST THIS AMOUNT FOR 5 YEARS AT 12% y CONTINUOUSLY COMPOUNDED YIELDS: $328,213,290. ANNUAL RETURN OF:

%.9.91000,000,200

290,213,3285

7.272,803,181

782.492,56

1.1

)02(.76.614.266,409,64

7.490,310,1251.1

)02(.17.416.733,590,135

AFTER 5 YEARS AT 12%: $331,267,162. ANNUAL RETURN OF:

%.101000,000,200

162,267,3315

The weighted average portfolio changes to:

25

Before turning to the futures markets we elaborate on the

common practice of: Repurchase Agreements

An integral part of trading T-bills and T-bill futures is the market for repurchase agreements, which are used in much of the arbitrage trading in T-bills. In a repurchase agreement -- also called an RP or repo -- one party sells a security (in this case, T-bills) to another party at one price and commits to repurchase the security at another price at a future date. The buyer of the T-bills in a repo is said to enter into a reverse repurchase agreement., or reverse repo. The buyer’s transactions are just the opposite of the seller’s. The figure below demonstrates the transactions in a repo.

26

Party A Party B

Date t - Close the Repo

T-Bill

P1= P0(1+r0,t )

Transactions in a Repurchase Agreement

Date 0 - Open the Repo:

Party A Party BT- Bill

PO

Example: T-bill FV = $1M.

P0 = $980,000. The repo rate = 6%. The repo time: t = 4 days.

P1= P0 [(repo rate)(n/360) + 1]

= 980,000[(.06)(4/360) + 1]

= 980,653.33

27

A repurchase agreement effectively allows the seller to borrow from the buyer using the security as collateral. The seller receives funds today that must be paid back in the future and relinquishes the security for the duration of the agreement. The interest on the borrowing is the difference between the initial sale price and the subsequent price for repurchasing the security. The borrowing rate in a repurchase agreement is called the repo rate. The buyer of a reverse repurchase agreement receives a lending rate called the reverse repo rate. The repo market is a competitive dealer market with quotations available for both borrowing and lending. As with all borrowing and lending rates, there is a spread between repo and reverse repo rates.

28

The amount one can borrow with a repo is less than the market value of the security by a margin called a haircut. The size of the haircut depends on the maturity and liquidity of the security. For repos on T-bills, the haircut is very small, often only one-eighth of a point. It can be as high as 5% for repurchase agreements on longer-term securities such as Treasury bonds and other government agency issues.Most repos are held only overnight, so those who wish to borrow for longer periods must roll their positions over every day. However, there are some longer-term repurchase agreements, called term repos, that come in standardized maturities of one, two, and three weeks and one, two, three, and six months.Some other customized agreements also are traded.

29

INTEREST RATE FUTURES

The three most traded interest rate futures are:

TREASURY BILLS (CME)

$1mil; pts. Of 100%

EURODOLLARS (CME)

$1mil; pts. Of 100%

TREASURY BONDS (CBT)

$100,000; pts. 32nds of 100%

30

CONTRACT SPECIFICATIONS FOR:

90-DAY T-BILL; 3-Month EURODOLLAR FUTURES

SPECIFICATIONS 13-WEEK 3-Month EURODOLLAR

US T-BILL TIME DEPOSIT

SIZE $1,000,000 $1,000,000

CONTRACT GRADE new or dated T-billsCASH 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 ($25/pt) ($25/pt)

LAST TRADING DAY The day before the2nd 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

31

Transactions in a Cash-and-Carry Arbitrage.

RepoMarket

Arbitrageur

T-BillDealer

FuturesMarket

PO (MONEY)

T-Bill

T-Bill

ShortPosition F0,t

P O (

MONEY)

Transactions in a Cash-and-Carry Arbitrage.

RepoMarket

Arbitrageur

FuturesMarket

P0(1+r0t)

T-Bill

Deliver

T-Bill

Receive F 0,t

Date t

Date 0

F 0,t > P0(1+r0,t)

32

Transactions in a Reverse Cash-and-Carry Arbitrage.

RepoMarket

Arbitrageur

T-BillDealer

FuturesMarket

PO (MONEY)

T-Bill

P0

LongPosition F0,t

T-Bill

Transactions in a Reverse Cash-and-Carry Arbitrage

RepoMarket

Arbitrageur

FuturesMarket

P0(1+r0,t)

T-Bill

TakeDelivery

T-Bill

Pay F 0,t

Date t

F 0,t < P0(1+r0,t)

Date 0

33

EXAMPLE: A 91- DAY T-BILL ARBITRAGE An arbitrageur observes that a 91-day T-bill yields 9.20 percent, a 182-day bill yields 9.80 percent, and a futures contract requiring delivery of a 91-day T-bill three months hence is priced so as to yield 10.2 percent. What action would he take ? TIME LINE <-------- r = 9.20% --------> <--------- 10.20% --------> -|----------------------------------------|---------------------------------------|---> 0 91 days 182 days ------------------------------- r = 9.80% ------------------------------------->

(1.098)2 = 1.092(1 + r1,2) r1,2= .1040 or 10.40% The no-arbitrage rate: 10.4% is greater than the market rate: 10.2%.

CASH - AND - CARRY

34

T I M E C A S H F U T U R E S 0 a ) B O R R O W $ 9 5 4 , 3 3 0 . 5 6 c ) S E L L A T - B I L L F U T U R E S

F O R 9 1 D A Y S

b ) B U Y A 1 8 2 - D A Y T - B I L L

9 1 D A Y S

R E P A Y T H E L O A N : D E L I V E R T H E 9 5 4 , 3 3 0 . 5 6 ( 1 . 0 9 2 ) . 2 5 9 1 - D A Y T - B I L L S F O R = $ 9 7 5 , 5 6 1 . 1 4 $ 9 7 6 , 0 1 1 . 7 5

A R B I T R A G E P R O F I T : $ 4 5 0 . 6 1 P E R C O N T R A C T

P = 1 , 0 0 0 , 0 0 0

( 1 . 0 9 8 . 5) = $ 9 5 4 , 3 3 0 . 5 6 F =

1 , 0 0 0 , 0 0 0

( 1 . 1 0 2 . 2 5) = $ 9 7 6 , 0 1 1 . 7 50 , t

35

THE SAME CASH-AND-CARRY STRATEGY .

TIME CASH FUTURES 0 a) ENTER A REPO AGREEMENT c) SHORT A

BY BORROWING $954,330.56 T-BILL FUTURES FOR 91 DAYS. F0,t = $976,011.75 b) BUY THE 182-DAY T-BILL FOR $954,330.56 AND GIVE IT TO THE OTHER PARTY OF THE REPO AGREEMENT. THUS, EFFECTIVELY, THE 91-DAY T-BILL IS SOLD TO THE OTHER PARTY FOR $954,330.56

91 DAYS RECEIVE THE T-BILL WHICH DELIVER THE

IS NOW A 91-DAY BILL. 91-DAY T-BILL FOR $976,011.75 REPAY THE REPO DEALER $975,561.14

PROFIT $450.61 / CONTRACT.

36

Transactions in a Cash-and-Carry Arbitrage

RepoMarket

Arbitrageur

T-BillDealer

FuturesMarket

PO =$954,330.56

182-day T-Bill

P0= $954,330.56

Short Position FOt = $976,011.75

182-day T-B

ill

Transactions in a Cash-and-Carry Arbitrage

RepoMarket

Arbitrageur

FuturesMarket

P0(1+r0,t) = $975,561.14

91 day T-Bill

Deliver91-day T-Bill

F 0,t = $976,011.75

Date t

Date 0

Profit = $450,61

37

I F T H E I M P L I E D F U T U R E S R A T E W A S 1 1 . 2 % T H E N : T H E O R E T I C A L = 1 0 . 4 % < 1 1 . 2 % = A C T U A L

R E V E R S E C A S H - A N D - C A R R Y T I M E C A S H F U T U R E S 0 a ) E N T E R A R E V E R S E R E P O c ) L O N G A 9 1 -D A Y

F O R $ 9 5 4 , 3 3 0 . 5 6 . T - B I L L F U T U R E S b ) S E L L 1 8 2 - D A Y S H O R T F 0 , t =

$ 9 7 3 , 8 0 9 . 0 4

9 1 D A Y S C L O S I N G T H E R E P O Y O U T A K E D E L I V E R Y

R E C E I V E $ 9 7 5 , 5 6 1 . 1 3 A N D O F 9 1 - D A Y

C L O S I N G T H E S H O R T P O S I T I O N T - B I L L F O R

Y O U D E L I V E R T H E 2 1 D A Y $ 9 7 3 , 8 0 9 . 0 4 T - B I L L S P R O F I T : $ 9 7 5 , 5 6 1 . 1 3 - $ 9 7 3 , 8 0 9 . 0 4 = $ 1 , 7 5 2 . 0 9

1 , 0 0 0 , 0 0 0

( 1 . 1 1 2 ) . 2 5

38

Transactions in a Reverse Cash-and-Carry Arbitrage

RepoMarket

Arbitrageur

T-BillDealer

FuturesMarket

PO = $954,330.56

182-day T-Bill

PO = $954,330.56

LongPosition FO,t = $973,809.04

182-day T-B

ill

Transactions in a Reverse Cash-and-Carry Arbitrage

RepoMarket

Arbitrageur

FuturesMarket

P1 = $975,561.13

91 day T-Bill

TakeDelivery91-day T-Bill

F 0,t = $973,809.04

Date t

Date 0

PROFIT = $1,752.09

39

ARBITRAGE ?

-------- 6% -------------- -----------12.5% ------------ Time|-------------------------------|----------------------------------|------------------>1.5 77 days 3.22 90 days 6.20

--------------------------------- 10% ----------------------- 167 days

(1 + .1

167360) = (1 + .06

77360) (1 +

1,2r

90360)

1,2r =

167901.1

1.779006

- 1 = .1354 ; 13.54%

THEORETICAL = 13.54% > 12.5 = ACTUAL

CASH-AND-CARRY.

40

LET THE YIELD ON THE SHORT-TERM BILL BE 8%:

(1.1) (1.08) (1 r

r 11.74%

167

360

77

3601,2

90

360

1,2

)

THEORETICAL RATE = 11.74%

IS LESS THAN

12.50% = ACTUAL MARKET RATE

REVERSE CASH-AND-CARRY

41

PRICE SENSITIVITY HEDGE RATIO:

AN APPLICATION OF DURATION.

The Hedge Value: V = S + NF.

Criterion: Minimize the position’s value sensitivity to interest rate changes.

.dr

dy

dy

dFN

dr

dy

dy

dS

dr

dV

:maturity to

yieldsfutures andspot thein changes

theof termsin expression thisrewrite

rule, chain the Using.dr

dF

dr

dS

dr

dV

F

F

S

S

42

To minimize dV/dr with respect to N, set:

.

drdy

dydF

drdy

dydS

N

,0dr

dV

F

F

S

S

Next, we use the following substitutions for

.dy

dF and

dy

dS

FS

43

SS

S S

SD = - dS

S 1 + y

dy dy1 + y

dS

S

.F

y + 1

dy

dF

dy

y + 1

F

dF - = D F

FF

FF

Recall that:

.y1

FD

dy

dS

and y1

SD

dy

dS

FF

S

SS

S

44

N = - SD

(1 + y ) (1 + y )

FD

dy

dy

dr

S

S

F

F

S

F

dr

N = - S D (1 + y )

F D (1 + y )S F

F S

Normally, the ratios of the yields sensitivities to the interest rate, r, are assumed to be zero. Thus:

This is the

price sensitivity hedge ratio.

45

A LONG HEDGE T-BILLS

CASH FUTURES

FEB. 15 DO NOTHING BUY 1 JUNE FUTURES: T-BILL

d = 8.20 => P = 100-8.2(91/360) IMMI = 91.32 => d = 8.68 =>

P = 97.927222 F = 10,000[100-8.68(90/360)]

F = $978,300

Sy = (

36591100

97.927222 - 1) = .0876

Fy = (

36591100

97.83 - 1) = .0981

MAY 17 d = 7.69 => P = 100 - 7.69(91/360) SELL 1 JUNE T-BILL FUTURES

P = 98.0561 IMMI = 92.54

BUY PER 1M $980,561 F = 10,000[1007.46(90/360)] =

OPPORTUNITY LOST <$1,289.17> $981,350

ACTUAL PRICE $977,511 PROFIT $3,050

PER 1M P = $979,272

16)300)(1.087(.25)(978,

1)272)(1.098(.25)(979,NF

46

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.

47

CALCULATING THE PROFIT ON A EURODOLLAR FUTURES TIME SPOT FUTURES MAY 23,01 $1M ED TD AT LONG 1 ED JUNE

9.35% FUTURES. F = 90.65 IMPLIED RATE: 9.35%

1I = .0935

41M = $23,375

MAY 30,01 $1M, ED,TD AT SHORT 1 ED JUNE 10.35% FUTURES. F = 89.65

IMPLIED RATE: 10.35%

2I = .1035

41M = $25,875

1M(89.65 - 90.65)(.01)90

360 = -$2,500

DIFFERENCE: 25,875 - 23,375 = $2,500

48

HOW TO LOCK IN A BORROWING RATE

TIME CASH FUTURES

MAY 23,01 90-DAY LIBOR = 9.35% F0,t = 90.65 DO NOTHING SHORT JUNE ED

FUTURES k=JUNE 19,01 BORROW $1M LONG

FOR 90 DAYS Fk,t = 100-LIBOR AT 90-DAY LIBOR

t

(i) LIBOR = 7% Fk,t = 100 - 7 = 93 (90.65 - 93)100(25) = - $5,875

NET = $23.375

I = (.07)$1M

4 = $17,500

(ii) LIBOR = 10.5% Fk,t = 100-10.5 = 89.5

(90.65-89.5)100(25)= $ 2,875

NET = 23,375

ANNUAL INTEREST RATE =

OR 9.35%

I =(.105)$1M

4 = $26,250

23,375

1,000,000

360

90.0935

49

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 ? rF = 9.35%

_|_________________|_____________________|_________TIME 0 k t

t360(1+

0,tr ) = (1+

0,kr

k360) (1+

Fr

t-k360)

1 + F

r =

tt-k(1+

0,tr )

(1+0,k

r

kt-k)

rF = 9.385% > 9.35%

1 + F

r = (1.094

11790)

(1.0945

2790)

= 1.12388671.0274595

= 1.09385

Arbitrage with Eurodollar Futures

50

$1M((1+.0935

90360)

(1+.094

117360)

) = $993,171.08

CASH FUTURES

5.23 ESTABLISH A: SHORT AN ED TD FUTURES

F0,t = 90.65

TIME DEPOSIT FOR t .

6.19 BORROW $1M AT 9.35%

9.17 RECEIVE SETTLE IN CASH:

$993,117.08(1+.094

117360) $1M(1.0935

90360)

= $1,022,597.43 = $1,022,597.43

[(1,000,000-993,171.08)/993,171.08](360/27) = 9.17%

Quasi Arbitrage or, how to borrow capital using Eurodollar

Futures.

51

HEDGE COMMERCIAL PAPER ISSUE: A SHORT HEDGE WITH EURODOLLAR FUTURES

CASH MARKET FUTURES MARKET

APR 6 d = 10.17 IMMI = 88.23 => d = 11.77F = 10,000[100-11.77(90/360)]= $970,575DF = 90/360P = 94.915 = (100 - 10.17

180

360)

SY = (

1,000,000949,150

365180) - 1

FY = (

1,000,000970,575

36591) - 1 = .1283

= .1116

1 + Y S = 1.1116

DS = 180/365 1+YF = 1.1283

DO NOTHING

20 = 1611.1

31.128

970,575

9,491,500

365/90

365/180 = N F

SELL 20 SEPT. EURODOLLAR FUTURES - BECAUSE I (ISSUE) SELL IN THE CASH MARKET.

52

JUL 20 ISSUE $10M COMMERCIAL PAPER BUY 20 SPT EURODOLLARFUTURES

MATURING IN 180 DAYS IMMI = 87.47

d = 11.34 F=10,000[100-12.53(90/360)]P = 94.33

INCOME: $9,433,000 F = $968,675

12.51% = 1 - )9,433,000

10M( 180365

: HEDGE)COST(NO

COST(WITH HEDGE): 20(970,575) + [9,433,000 - 20(968,675)] =$ 9,471,000

A C T UA L C O S T : (1 0 M

9 ,4 7 1 ,0 0 0) - 1 = 1 1 .6 5 %

3 6 5

1 8 0

PROFIT: 20[970,575-968,675] = $38,000

53

On November 1, 2000, a firm agrees to borrow $10M for 12

months, beginning December 19, 2000 at

LIBOR + 100bps.

A STRIP HEDGE WITH EURODOLLARS FUTURES

DATE CASH FUTURES F11.1.00 LIBOR 8.44% Short 10 DEC91.41

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

12.19.00 LIBOR 9.54% Long 10 DEC90.46

3.13.01 LIBOR 9.75% Long 10 MAR90.25

6.19.01 LIBOR 9.44% Long 10 JUN90.56

9.18.01 LIBOR 8.88% Long 10 SEP91.12

54

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

EFFECTIVERATEe: 9.59% 9.39% 9.47% 9.61%

MEAN RATEUNHEDGED 10.40%HEDGED 9.52%

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

55

A STACK HEDGE WITH EURODOLLAR FUTURES:

DATA ON NOVEMBER 11, 2000

VOLUME OPEN INTEREST

DEC 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.

ROLL OVER AS SOON AS OPEN INTEREST

REACHES 100,000

56

THE STACK HEDGE

DATE CASH FUTURES F. POSITION

11.1.00 8.44% S 10 DEC 91.41S10DEC

S 30 MAR 91.61S30MAR

12.19.00 9.54% L 10 DEC 90.46S30MAR

1.12.01 9.47% L 20 MAR 90.47S10MAR

S 20 JUN 90.42 S20JUN

2.22.01 9.95% L 10 JUN 89.78S10MAR

S 10 SEP 89.82 S10JUNS10SEP

3.13.01 9.75% L 10 MAR 90.25 S10JUNS10SEP

6.19.01 9.44% L 10 JUN 90.56 S10SEP

9.18.01 8.88% L 10 SEP 91.12 NONE

57

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

INTEREST($)b: 263,500 268,750 261,000 247,000

FUTURES($)c: 23,750 34,000 28,500 28.500

(3,500) 16,000 (32,500)

NET($) d: 239,750 234,750 236,000 235,000

EFFECTIVERATE (%) e: 9.59 9.39 9.44

9.40

MEAN RATEUNHEDGED 10.40%HEDGED 9.46%a. LIBOR + 100 BPSb. ($10M)(RATE)(3/12)c. (PRICE CHANGE)(25)(100)(10)d. b - ce. (NET/10M)(12/3)(100%).This completes the example.

Next, we show an example of aSTRIP hedge with T-bill futures:

58

A FIRM BORROWS $10M AT A FLOATING RATE LIBOR +1%

TIME SPOT FUTURES

SEP. 15 RECEIVE $10M SHORT 10 T-BILL FUTURESLIBOR = 8% DEC F = 91.75PAY 9% MAR F = 91.60I1 = $225,000 JUN F = 91.45

ACTUAL $225,000 9.00% R

DEC. 15 LIBOR = 9.15% LONG 10 DEC. FUTURES:PAY 10.15% F = 90.85I2 = $253,750 GAIN:

10[91.75 - 90.85]10,000 (.25) = $22,500

ACTUAL: $231,250 9.25% R

MAR. 15 LIBOR = 9.50 LONG 10 MAR FUTURESPAY 10.50% F = 90.50I3 = $262,500 GAIN:

10[91.60 - 90.50]10,000 (.25) = $27,500

ACTUAL: $235,000 9.40% R

JUNE 15 LIBOR = 10.05% LONG 10 JUN. FUTURES:PAY 11.05% F = 89.95I4 = $276,250 GAIN:

10[91.45 - 89.95]10,000 (.25) = $37,500

ACTUAL: $238,750 9.55% R

SEP. 15 PAY $10M

.4

360.

M10

ACTUAL = R.

360

90M)$10

100

100 + LIBOR( = I

59

THE BANK

QUOTEDTIME RATE SPOT FUTURES

SEP. 159.00% LIBOR = 8% SHORT 10 ED FUTURESEXTENDED BORROWS $10M DEC: 91.7510M LOAN AT 8% MAR: 91.60

JUN: 91.45NET: 1% (.01/4)10M = $25,000 ACTUAL = $25,000 ? 1%

DEC. 15 9.25% LIBOR = 9.15% LONG 10 ED FUTURESBORROWS $10M AT 90.85AT 9.15%

NET: .1% GAIN $22,500

(.001/4)10M = $2,500 ACTUAL = $25,000 ? 1%

MAR. 15 9.40% LIBOR = 9.50% LONG 10 ED FUTURESBORROWS $10M AT 90.50AT 9.50%

NET: -.1% GAIN $27,500

(-.001/4)10M = $-2,500 ACTUAL = $25,000 ? 1%

JUNE 15 9.55% LIBOR = 10.05% LONG 10 ED FUTURESBORROWS $10M AT 89.95AT 10.05%

NET: -.5% GAIN $37,500

(-.005/4)10M = $-12,500 ACTUAL = $25,000 ? 1%

SEP. 15 + $10M ACTUAL = $25,000 ? 1%

FUTURES GAIN = 10(F(SEP) - F(t)) 10,000(. 25)

60

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 give day, there are 40 to 50 different T-bonds traded in the cash market and most of these are deliverable against a T-bond futures position, which makes this market extremely liquid.

61

SPECIFICATIONS OF U.S. TREASURY BOND FUTURES CONTRACTS

EXCHANGE CBOT

DATE OF INTRODUCTION AUGUST 22, 1997

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(8% <2000)

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.

62

CONVERSION FACTORS

T-BOND FUTURES PRICES ARE BASED ON A 20 –YEAR BOND THAT PAYS 6% (8%, PRIOR TO 2000) COUPON RATE, SEMIANNUALLY.

DELIVERABLE AGAINST A SHORT POSITION IS ANY T-BOND WITH MATURITY, OR FIRST TIME TO CALLABILITY OF 15 YEARS. THUS, THE SHORT MAY CHOSE FROM A VARIETY OF BONDS AND DELIVER THE BOND THAT WILL MINIMIZE (MAXIMIZE) THE SHORT COST (REVENUE). THIUS BONMD IS LABELED “THE CHEAPEST TO DELIVERY.”

THE PROBLEM IS THAT THE ORIGINAL FUTURES PRICE CONTRACTED FOR MUST BE ADJUSTED TO CONFORM WITH THE BOND THAT IS EVENTUALLY DELIVERED. THE ADJUSMENT PUT THE BOND DELIVERED ON PAR WITH THE ORIGINAL FUTURES PRICE. THIS IS DONE WITH CONVERSION FACTORS. THE CONVERSION FACTOR OF A T-BOND REPRESENTS THE NET PRESENT VALUE OF THE COUPON PAYMENTS OF THE BOND ACTUALLY DELIVERED IN EXCESS(IN SHORT), RELATIVE TO A 20YEAR BOND WITH 6% (8%) RATE, R.

$100,000R/2)(1

$100,0000R/2)(1

$100,0002

CR

CF2Mt

63

CBOT T-BOND CONVERSION FACTORS based on 8% futures pricesYRS = YEARS TO MATURITY OR FIRST

CALLABILITYM = NUMBER OF MONTHSCR = COUPON RATECF = Conversion factor ROUND MONTHS TO: M*= 0,3,6, OR 9.

Case 1: M* = 0 2YRS

2YRS

0 (1.04)].04

(1.04)1[

2

CRCF

Case 2: M* = 3

4

CR)(1.04)

2

CR(CFCF .5

03

64

Case 3: M* = 6

1)+(2YRS1)+(2YRS

6 (1.04)].04

(1.04)1[

2

CRCF

Case 4: M* = 9

4

CR)(1.04)

2

CR(CFCF .5

69

65

EXAMPLE:

THE CONVERSION FACTOR FOR DELIVERY OF THE 11 3/4s FOR NOVEMBER 15, 2015 , ON THE

DECEMBER 1999 T-BOND FUTURES CONTRACT:

ON 12.1.99 YRS = 15 until 2014.M = 11 14 DAYS are ignored.THUS, YRS = 15 M is rounded off to M* = 9 .FIRST COMPUTE :

1)+2(15)1)+2(15)

6 (1.04)].04

(1.04)1[

2

.1175CF (

(

CF6 = 1.3298. NOW COMPUTE:

factor. conversion sbond' theis 1.3322 9

CF4

.1175.5)(1.04)2

.1175(1.32989CF

66

A bond portfolio manager decides to sell $10M FV of 11 7/8 M-19 yrs T-bonds on March 28. Currently, FEB 26, the bond sells for S=$101/$100FV.

A SHORT T-BOND HEDGE

TIME CASH FUTURES

FEB. 25 10M FV T-BONDS SELL 160 JUNCR = 11 7/8 M-19 T-BOND Fs.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

MAR. 28 S = 95.6875/$100FV LONG 16 JUN$9,568,750 T-BOND Fs

Opportunity loss <$531,250> F = 61 - 23Futures gain: [(70-16)-(61-23)]160

=(8-25)160=($8781,25)160=$1,405,000.

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

67

LONG HEDGE WITH T - BOND FUTURES

DATE CASH FUTURES

MAR. 29 LONG 110 SEP T-BOND Fs.

F = 78-21BY REGRESSION N = 110.

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

$10,759,375 Fs. 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

68

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

5.24 ISSUE BONDS LONG 674 JUN T-BOND CR=13.26% Fs. F(JUN) = 55-25 S=$907.4638/$100FV

V(BOND ISSUE) = $45,373,190Gain from futures: 674[(68-11)-(55-25)]

=674[12-18] =674[$12,562.5]=$8,467,125.

TOTAL VALUE=$53,840,315.

69

PROJECT PLANNING:

ELIMINATION OF FOREIGN INVESTMENT RISK EXPOSURE BY

FORWARD TRADING OR BY THE USE OF U.S. TREASURY SECURITIES.

An American firm expects an income of FC40,268,394 from a project in the foreign country. Its next project will begin only 90 days later in the same country. Thus, the above sum will be invested in the foreign country for 90 days. The interest rate in 90 days is not known and the firm would like to eliminate the risk exposure associated with the investment of the above sum for 90 days. Given current spot and forward exchange rates, as well as current and forward interest rates, the firm will choose between investing forward or using the U.S. T-bill futures.

70

PROJECT PLANNING:

Exchange rate Spot Interest rate

Spot: .4729 U.S. F.C.

90-day .4765 3-month: 14%9.15%

180-day .4782 6-month: 16$ 10.22%

3-month forward rates

U.S. = 18.035%. F.C. = 11.3%

I. Invest 40,268,394 FC forward at the foreign country will result in a fixed and known total sum of:

40,268,394e(.113)(.25) = 41,422,197FC

in 180 days.

Instead, the firm could entertain the following strategy:

71

Date Spot market Futures market

t=0 Do nothing. 1.Short 40,268,394 90-day forward at $.4765/FC.

2. Long 20 T-bill futures on the CBT for 959,394.49

3. Short $20M 180-day forward at $.4808/FC.

t=90

Receive Pay $40,268,394(.4765) =

40,268,394FC. = $19,187,889.80 and take delivery of $20M face value T-bills.

t = 180 T-bills Take delivery of 41,597,338 mature, Collect to close your short $20M forward position paying

$20M.

THE SECOND STRATEGY IS BETTER.