Download r = the bond`s yield to maturity.

Interest Rate
Markets
Chapter 5
1
Types of Interest Rates
• Treasury rates
• LIBOR rates
• Repurchase rates
2
QUOTES ARE GIVEN BY CONVENTION,
USING A DISCOUNT YIELD, d, WHERE:
DISCOUNT = FACE VALUE - MARKET PRICE:
d  360 DISCOUNT
t
FV
 FV  P 
360
d

t 
FV


BOND EQUIVALENT YIELD(BEY):


i  365  FV  P 
t 
P


1


365
dt
id
1 
360 
360 
365d

360  dt
3
EXAMPLE:
t = 90 days
FV = $1,000,000
d = 11%
360 (DISCOUNT)
.11 
 DISCOUNT  $27,500.
90 1,000,000
1,000,000  972,500 360
d
 .1111% 
1,000,000
90
BEY  i 
365 1,000,000  972,500 
 .11468


90 
972,500

365  (.11)90 
i  (.11)
1

360 
360 
1
 .11468 (11.468%).
4
The market for 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.
5
Transactions in a Repurchase Agreement
Date 0 - Open the Repo:
Party A
T- Bill
PO
Party B
Date t - Close the Repo
T-Bill
Party A
Party B
Pt= P0(1+r0,t )
6
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
7
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
8
between repo and reverse repo rates.
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 Tbills, 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
9
customized agreements also are traded.
Zero Rates
A zero rate (or spot rate), for
maturity T is the rate of interest
earned on an investment that
provides a payoff only at time T.
10
Example
(Table 5.1, page 95)
Maturity
(years)
0.5
Zero Rate
(% cont comp)
5.0
1.0
5.8
1.5
6.4
2.0
6.8
11
Bond Pricing
• To calculate the cash price of a bond
we discount each cash flow at the
appropriate zero rate
• In our example, the theoretical price
of a two-year bond (FV = $100)
providing a 6% coupon semiannually
is:
3e
0.050.5
 103e
 3e
0.0581.0
0.0682.0
 3e
0.0641.5
 98.39
12
Bond Yield
• The bond yield is the discount rate
that makes the present value of the
cash flows on the bond equal to the
market price of the bond
• Suppose that the market price of the
bond in our example equals its
theoretical price of 98.39
• The bond yield is given by solving
3e
 y0.5
 3e
 y1.0
 3e
 y1.5
 103e
 y2.0
 98.39
to get y=0.0676 or 6.76%.
13
Par Yield
• The par yield for a certain maturity is
the coupon rate that causes the bond
price to equal its face value.
• In our example we solve
c 0.050.5 c 0.0581.0 c 0.0641.5
e
 e
 e
2
2
2
c  0.0682.0

 100  e
 100
2

to get c = 6.87 (with s.a. compoundin g)
14
Par Yield continued
In general if m is the number of
coupon payments per year, d is the
present value of $1 received at
maturity and A is the present value of
an annuity of $1 on each coupon date
(100  100d)m
c
A
15
Sample Data for Determining
the Zero Curve (Table 5.2, page 97)
Bond
Time to
Annual
Bond
Principal
Maturity
Coupon
Price
(dollars)
(years)
(dollars)
(dollars)
100
0.25
0
97.5
100
0.50
0
94.9
100
1.00
0
90.0
100
1.50
8
96.0
100
2.00
12
101.6
16
The Bootstrapping the Zero Curve
• An amount 2.5 can be earned on 97.5
during 3 months.
• The 3-month rate is 4 times 2.5/97.5
or 10.256% with quarterly
compounding
• This is 10.127% with continuous
compounding
• Similarly the 6 month and 1 year rates
are 10.469% and 10.536% with
continuous compounding
17
The Bootstrap Method continued
• To calculate the 1.5 year rate we solve
4e
0.10469( 0.5)
 4e
0.10536(1.0)
 104e
 R (1.5)
 96
to get R = 0.10681 or 10.681%
• Similarly the two-year rate is 10.808%
18
Zero Curve Calculated from
the Data (Figure 5.1, page 98)
12
Zero
Rate (%)
11
10.681
10.469
10
10.808
10.536
10.127
Maturity (yrs)
9
0
0.5
1
1.5
2
2.5
19
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;
Ct=
THE CASH FLOW FROM THE BOND AT TIME
PERIOD t. USUALLY, THE CASH FLOW IS
ASSUMED TO OCCUR AT THE PERIOD’S END.
t = 1,2,……, M.
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.
20
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 every year for the next 30
years.
21
EXAMPLE:
THE SAME T-BOND PAYS
COUPONS SEMI 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 30 years.
DEFINITION: A PURE DISCOUNT BOND PAYS
THE FV AT ITS MATURITY BUT PAYS NO
COUPONS (C = 0) IN ANY OF THE INTERIM
PERIODS.
22
PRICING BONDS:
A BOND WITH MATURITY OF
M YEARS WITH COUPONS PAID ANNUALLY IS
PRICED BY:
M Ct
P 
. P is the NPV of the CFs :
t
t1 (1 r)
C
C
C
3
1 
2
P

 ...;
1  r1 (1  r1)(1  r2 ) (1  r1)(1  r2 )(1  r3)
The usual convention is r  rt for all t.
M Ct
FV ,
P 

t1 (1 r)t (1 r)M
A closed form solution :

M 
C
FV
P  1  (1 r)
.

M
r

(1 r)
23
r = the bond’s yield to maturity.
That is, if the investor buys the bonds at the
market price and holds it to its maturity, r is the
annual rate of return on this investment.
24
FOR A BOND WITH SEMIANNUAL
COUPON PAYMENTS:
C
2M
FV
2
Ρ 
r t
r 2M
(1  )
t1 (1  )
2
2
A PURE DISCOUNT BOND DOES NOT PAY
CUPONS UNTIL ITS MATURITY; C = 0:
P
FV
(1  r) M
25
EXAMPLES:
M = 30;
FV = $1,000;
CR = 8% PAID
SEMIANNUALLY. => C = (.08) 1,000/2=$40.
r = 10%
60

t 1


40
1,040
40
1,000
60


1

1
.
05

 $810.70
t
60
60
(1  .05)
1.05
.05
1.05
THE SAME BOND WITH ANNUAL COUPON PAYMENTS WOULD BE
PRICED:


80
1,000
 30

1  1.1

 $811.46
1.1
1.130
THE BOND IS SOLD AT A DISCOUNT BECAUSE THE YTM IS
GREATER THAN THE CR
r = 5%
P


40
1,000
1  1.02560 
 $1,463.63
 60
.025
1.025
THE SAME BOND AS A PURE DISCOUNT BOND T BOND WOULD BE
PRICED:
1,000
 
 $57.31
30
11
.
IF IT WERE A CONSUL WITH ANNUAL COUPONS:
P  80  $800
.1
26
Result:
The bond sells at par if P = FV.
If CR = r
the bond sells at par.
If CR > r
the bond sells at a premium.
If CR < r
the bond sells at a discount.
27
DURATION DEFINITION:
M
D

OBSERVE THAT:
t 1
tC t
t
1  r  .
P
 C t 1  r  t
D   t 
P
t 1 
M




M
Wt,
t 1
t
Where:
Ct / 1  r 
Wt  1 and Wt 
.

P
t 1
This result leads to the following
M
interpretation of Duration:
t
28
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.
29
Duration in continuous time
• Duration of a bond that provides cash flow c i at time t i is
 ci e
ti 

i 1
 B
n
 yti



where B is its price and y is its yield (continuously
compounded)
• This leads to
B
 Dy
B
30
DURATION INTERPRETED AS A MEASURE OF THE BOND
PRICE SENSITIVITY
M
Ct
The bond price : P  
.
t
t 1 1  r 
Its derivative with respect to r or r  1 :
dP
dP
1


dr
d(1  r)
(1  r)
tC t
 (1  r) t .
dP
(1  r)
(1  r)

d(1  r)
P
P(1  r)

tC t

  (1  r) t



.

Rearrangin g, yields :
tC t
dP

t
(1

r)
P

  D.
d(1  r)
P
1 r
31
The final result states that :
dP
P
D.
d(1  r)
1 r
THE %Δ OF THE BOND PRICE
D
THE %Δ OF THE YIELD
D  THE BOND PRICE ELASTICI TY.
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
32
duration of a bond:
COMPUTING DURATION
r = 10%
Coupon Rate
M
0
2
4
6
8
10
12
14
16
5
5
4.76
4.57
4.41
4.28
4.17
4.07
3.99
3.92
10
10
8.73
7.95
7.42
7.04
6.76
6.54
6.36
6.21
15
15
11.61
10.12
9.28
8.74
8.37
8.09
7.88
7.71
20
20
13.33
11.20
10.32
9.75
9.36
9.09
8.89
8.74
25
25
14.03
11.81
11.86
10.32
9.98
9.75
9.58
9.45
30
30
14.03
11.92
11.09
10.65
10.37
10.18
10.04
9.94
35
35
13.64
11.84
11.17
10.82
10.61
10.46
10.36
10.28
40
40
13.13
11.70
11.18
10.92
10.76
10.65
10.57
10.51
50
50
12.19
11.40
11.40
10.99
10.91
10.85
10.81
10.78
100
100
11.02
11.01
11.00
11.00
11.00
11.00
11.00
11.00
33
N = NUMBER OF PAYMENTS
m = COUPON PAYMENTS PER YEAR
Θ = FRACTION OF YEAR TO THE NEXT
COUPON PAYMENT
(1 
D

r) 1 


Θ 1
2
….years
r  N  1  r N  r 2 FV Θ  N  1

m 
m
C 
m 


r 1 


r  N  1  r 2 FV

m 
C

EXAMPLE:
r = 10% = .1
FV = $ 100
C = $6 => CR = 6%
M = 30
=1
m=1
P = $62.29


100
1.1 1.1  1  (.1)30  (.1)
30
6
D
 11.09
30
2 100
(.1) 1.1  1  (.1)
34
6
30

2

APPLICATIONS OF DURATION:
1.
Δr
ΔΡ  ( D)(P)
1 r
EXA MPLE:
M=30; CR=6%;D=11.09 and P=$62.29.
r = 10%

r = 11%
P  1109
. 62.29
r  10%

.01
11
.
 6.28  P  $56.01
1
r = .08
P = -(11.09)(62.29)
(-.02)
 12.56  P  $74.85
1.1
1
35
Duration Continued
• When the yield y is expressed with
compounding m times per year
BDy
B  
1 y m
• The expression
D
1 y m
is referred to as the “modified duration”
36
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 .
37
We already saw that duration may
be interprete d as the bond price elasticity :
dV
dV (1  r)
V
DP  

.
d(1  r)
d(1  r) V
(1  r)
Thus, taking the derivative of the bond
portfolio value, V, with respect to
the yield - to - maturity leads to :
d  N i Pi
dV
dPi

  Ni
.
d(1  r)
d(1  r)
d(1  r)
1 r
Multiplyin g and deviding by
,
Pi
yields :
38

dV
dPi  (1  r) Pi
  N i
.

d(1  r) 
d(1  r)  Pi (1  r)
Rearrangin g terms in the summation sign :
1
dPi (1  r)

N i Pi [
]

(1  r)
d(1  r) Pi
The terms in the square bracket are the
negative of bond' s i duration, while
N i Pi  Vi . Thus, we rewrite :
dV
1

Vi D i .

d(1  r) (1  r)
39
Substituti ng this result into
dV  1  r 
DP  

, we have :
d(1  r)  V 
 1
 1  r 
DP  
Vi D i  


1  r
 V 
Vi Di

Dp 
, or :
V
40
1
Vi
D P   Vi Di   Di   w i Di ,
V
V
where :
Vi
wi  ;
V
w
i
 1.
Thus, in conclusion , the protfolio duration
D P   w i Di
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
41
value.
Example: a portfolio of two T-bonds:
BOND
N
YTM
COUPON
T-BOND
FV in
$M
$100
15yrs
6%
5%
T-BOND
$200
30yrs
6%
15%
BOND
PRICE
$90.2
$449.1
539.3
T-BOND
T-BOND
D
W
0.1673
0.8327
1.0000
D
10.4673
12.4674
= (.1673)(10.4673) +(.8327)(12.4674)
= 12.1392
42
Example: a two bond portfolio:
BOND
FV
N
YTM
CUPON
T-BOND
$500M
15yrs
6%
5%
T-BOND
$200M
30yrs
6%
15%
BOND
PRICE
$451,5M
$449,1M
$900,6M
1
2
T-BOND1
T-BOND
2
D
W
0,5013
0,4987
1,0000
D
10,4673
12,4674
= (0,5013)(10,4673) +(0,4987)(12,4674)
= 11,45.
43
APPLICATION OF DURATION
2.
IMMUNIZING BANK PORTFOLIO OF ASSETS AND LIABILITIES
TIME 0
ASSETS
$100,000,000
(LOANS)
D=5
r = 10%
TIME 1
r => 12%
LIABIABILITIES
$100,000,000
(DEPOSITS)
D=1
r = 10%
(.02)
ΔVA  5
100,000,000  $9,090,909.09
1.1
(.02)
ΔVL  1
100,000,000  $1,818,181.82
1.1
44
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.
45
APPLICATIONS OF DURATION.
3.
EXAMPLE:
A 5-YEAR PLANNING PERIOD CASE OF
IMMUNIZATION IN THE CASH MARKET
BOND
A
B
C
FV
M
$100 $1,000 5 yrs
$100 $1,000 10 yrs
r
10%
10%
D
P
4.17 $1,000
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
46
to:
[1 - 4.17 (.02/1.1)] 100M
= $92,418,181.2
[1 - 6.76 (.02/1.1)] 100M
TOTAL
= $87,709,090.91
= $180,127,272.7
INVEST THIS AMOUNT FOR 5 YEARS AT 12%,
CONTINUOUSLY COMPOUNDED YIELDS:
$328,213,290. ANNUAL RETURN OF:
5
328,213,290
 1  9.9%.
200,000,000
47
The weighted average portfolio changes to:
(.02) 

135,590,733.61  4.17
 125,310,490.7

1.1 

(.02) 
56,492.782

64,409,266.41  6.76

 181,803,272.7
1
.
1


AFTER 5 YEARS AT 12%: $331,267,162. ANNUAL RETURN OF:
5
331,267,162
 1  10%.
200,000,000
48
BOND
FUTURES
We will study:
Short-term bonds:
U.S Government T-Bills
and
Euro-dollar time deposit
rates.
U.S Government T-Bonds.
49
AN APPLICATION OF DURATION
4.
THE PRICE SENSITIVITY HEDGE RATIO
The Hedge Value:
V = S + nF.
S = The bond’s spot value.
F = The futures price.
n = The number of futures used in the hedge.
Minimize the position’s value sensitivity to
interest rate changes.
First, we write the change in the hedge value when r changes:
Criterion:
50
dV dS
dF

n .
dr dr
dr
Using the chain rule :
dV dS dyS
dF dy F

n
.
dr
dyS dr
dy F dr
51
Solve for n that sets dV/dr = 0.
dV
 0,
dr
dS
dy S
n
dF
dy F
dy S
dr
.
dy F
dr
Next, we use the following substitutions for
dS
dF
and
.
dyS
dy F
52
From the definition of DURATION:
dS 1 + yS

DS = S dyS
dS SDS

dyS 1  yS
dF 1 + yF
dF FDF


DF = F dyF
dy F 1  y F
Upon substitution in n:
53
SDS
1  yS
n
FD F
1  yF
dyS
dr
dy F
dr
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:
dyS
dy F

dr
dr
54
The price sensitivity hedge ratio.
SDS (1  y F )
n
FD F (1  yS )
55
The price sensitivity hedge ratio with
continuous rates is:
SDS
n
FD F
56
INTEREST RATE FUTURES
The three most traded interest rate futures:
TREASURY BILLS (CME)
USD1,000,000; pts. Of 100%
EURODOLLARS (CME)
Eurodollars1,000,000; pts. Of 100%
TREASURY BONDS (CBT)
USD100,000; pts. 32nds of 100%
57
CONTRACT SPECIFICATIONS FOR:
90-DAY T-BILL; 3-Month EURODOLLAR FUTURES
SPECIFICATIONS
13-WEEK US 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
1st day of spot month
Last day of trading
DELIVERY DATE
on which 13-week
T-bill is issued and a 1-year
T-bill has 13 weeks to maturity
58
Arbitrage profits in the short-term futures
market involve
Activities in the futures market and the
spot repo market.
The following figures explain the cash and
carry and the reverse cash and carry
strategies:
59
Transactions in a Cash-and-Carry Arbitrage.
Repo
Market
PO
Arbitrageur
T-Bill
Short
Position F0,T
T-Bill
T-Bill
Dealer
Futures
Market
Date 0
Repo
Market
P0(1+r0T)
Arbitrageur
T-Bill
F 0,T > P0(1+r0,T)
Receive F
Deliver
0,T
T-Bill
Futures
Market
Date T
60
Transactions in a Reverse Cash-and-Carry Arbitrage.
Repo
Market
PO
Arbitrageur
T-Bill
Long
Position F0,T
P0
T-Bill
Dealer
Futures
Market
Date 0
Repo
Market
P0(1+r0,T)
Arbitrageur
T-Bill
Take
Delivery
F 0,T < P0(1+r0,T)
Date T
Pay F
0,T
T-Bill
Futures
Market
61
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 percen t. 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
62
THE CASH-AND-CARRY STRATEGY .
TIME
0
91 DAYS
CASH
a) ENTER A REPO AGREEMENT
BY BORROWING $954,330.56
FOR 91 DAYS.
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
RECEIVE THE T-BILL WHICH
IS NOW A 91-DAY BILL.
FUTURES
c) SHORT A
T-BILL FUTURES
F0,t = $976,011.75
DELIVER THE
91-DAY T-BILL
FOR
$976,011.75
REPAY THE REPO DEALER
$975,561.14
PROFIT $450.61 / CONTRACT.
63
Repo
Market
PO =$954,330.56
Arbitrageur
182-day T-Bill
Short FO,T =
$976,011.75
P0= $954,330.56
T-Bill
Dealer
Futures
Market
Date 0
Repo
Market
P0(1+r0,t) = $975,561.14
Arbitrageur
91 day T-Bill
Profit = $450,61
Deliver
91-day
T-Bill
F 0,T = $976,011.75
Futures
Market
Date T
64
IF THE IMPLIED FUTURES RATE WAS 11.2% THEN:
THEORETICAL = 10.4% < 11.2% = ACTUAL
REVERSE CASH-AND-CARRY
TIME
CASH
0
a) ENTER A REVERSE REPO
FOR $954,330.56.
b) SELL 182-DAY SHORT
FUTURES
c) LONG A 91-DAY
T-BILL FUTURES
F0,t = $973,809.04
1,000,000
(1.112).25
91 DAYS
CLOSING THE REPO YOU
RECEIVE $975,561.13 AND
CLOSING THE SHORT POSITION
YOU DELIVER THE 21 DAY
T-BILLS
PROFIT:
$975,561.13 - $973,809.04
= $1,752.09
TAKE DELIVERY
OF 91-DAY
T-BILL FOR
$973,809.04
65
PO = $954,330.56
Repo
Market
Arbitrageur
182-day T-Bill
Long
FO,T =
$973,809.04
PO = $954,330.56
Futures
Market
T-Bill
Dealer
Repo
Market
Date 0
P1 = $975,561.13
Arbitrageur
91 day T-Bill
PROFIT = $1,752.09
Date T
Take
Delivery
91-day
T-Bill
Futures
Market
F 0,T =
$973,809.04
66
We now present several example of
hedging with 90-day T-bills futures.
In order to determine whether to hedge
LONG or SHORT, one must remember
that the bonds price is reciprocal to the
interest rate and thus, hedging a falling
interest rate means hedging an increase
in the bond’s price and vice-versa.
67
A LONG HEDGE WITH T-BILLS
FEB. 15
CASH
FUTURES
DO NOTHING
LONG 1 JUN T-Bill FUTURES
d = 8.20 => P = 100-8.2(91/360) IMMI = 91.32 => d = 8.68 =>
P = 97.927222
PER 1M P = $979,272
365
91
100
yS  [
]  1  0.0876
97.927222
F = 1,000,000[1-8.68/100(.25)]
F = $978,300
100
yF  [
]
97.83
365
91
 1  0.0919
(.25)(979, 272)(1.0919)
n
1
(.25)(978, 300)(1.0876)
68
MAY 17
d = 7.69 => P = 100 - 7.69(91/360)
SHORT 1 JUNE T-BILL FUTURES
P = 98.0561
IMMI = 92.54
BUY PER 1M $980,561
F = 1,000,000[1 - 7.46/100(.25)] =
OPPORTUNITY LOST <$1,289.17>
$981,350
ACTUAL PRICE $977,511
PROFIT $3,050
69
HEDGE COMMERCIAL PAPER ISSUE: A SHORT HEDGE WITH T-bills
FUTURES
CASH MARKET
FUTURES MARKET
APR 6
d = 10.17
180
P = 94.915 = (100 - 10.17 )
360
365
180
1,000,000
yS  [
]
949,150
DS = 180/365
IMMI = 88.23 => d = 11.77
F = 10,000[100-11.77(90/360)]
= $970,575
DF = 90/365
1,000,000
]
 1  .1116 yF  [
970,575
365
91
 1  .1283
180 / 365 9,491,500 1.1283
n=


= 20
90 / 365 970,575 1.1116
Do Nothing.
Short 20 T-bills futures
70
JUL 20
ISSUE $10M COMMERCIAL PAPER
MATURING IN 180 DAYS
d = 11.34
BUY 20 SEP T-bills
FUTURES
IMMI = 87.47
F=10,000[10012.53(90/360)]
P = 94.33
INCOME: $9,433,000
F = $968,675
PROFIT: 20[970,575-968,675] = $38,000
COST(NO HEDGE) : (
10M
9,433,000
365
)180 - 1 = 12.51%
COST(WITH HEDGE): 20(970,575) + [9,433,000 - 20(968,675)]
=$ 9,471,000
10M
)
ACTUAL COST: (
9,471,000
365
180
- 1 = 11.65%
71
A FIRM BORROWS $10M AT A FLOATING RATE LIBOR +1%
TIME
SPOT
FUTURES
SEP. 15
RECEIVE $10M
LIBOR = 8%
PAY 9%
I1 = $225,000
SHORT 10 T-BILL FUTURES
DEC F = 91.75
MAR F = 91.60
JUN F = 91.45
DEC. 15
LIBOR = 9.15%
PAY 10.15%
I1 = $225,000
LONG 10 DEC. FUTURES:
F = 90.85
GAIN:
10[91.75 - 90.85]100 (25)
= $22,500
ACTUAL: $202,500
8.10% R
72
A FIRM BORROWS $10M AT A FLOATING RATE LIBOR +1%
TIME
MAR. 15
SPOT
LIBOR = 9.50
PAY 10.50%
I2 = $253,750
ACTUAL: $226,250
JUNE 15
LIBOR = 10.05%
PAY 11.05%
I3 = $262,500
ACTUAL: $225,000
SEP. 15
FUTURES
LONG 10 MAR FUTURES
F = 90.50
GAIN:
10[91.60 - 90.50]100 (25)
= $27,500
9.05% R
LONG 10 JUN. FUTURES:
F = 89.95
GAIN:
10[91.45 - 89.95]10,000 (.25)
= $37,500
9.00% R
I4 = $276,250
PAY $10M
ACTUAL: $276,250
11.05% R
Unhedged average rate: 10.175%
Hedged average rate:
9.30%.
73
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.
74
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]
75
Arbitrage with Eurodollar Futures
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
1+ r
F
=
(1.094 )
117
90
(1.0945)
1.1238867
27 = 1.0274595 = 1.09385
90
rF = 9.385% > 9.35%
76
Arbitrage with Eurodollar Futures continued
DATE
SPOT
FUTURES
MAY 23
Deposit $1,000,000
Short 1 ED futures.
in a 117 ED time deposit
L = 9.35%
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
77
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.
78
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.
79
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
determine 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.
80
A STRIP HEDGE WITH EURODOLLARS FUTURES
On November 1, 2000, a firm agrees to borrow $10M
for 12 months, beginning December 19, 2000 at
LIBOR + 100bps.
DATE
CASH
FUTURES
Q
11.1.00 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
81
PERIOD:
1
2
RATEa:
10.54%
10.75%
INTERESTb:
$263,500
FUTURESc:
NETd:
3
10.44%
9.88%
$268,750
$261,000
$247,000
$23,750
$34,000
$24,250
$6,750
$239,750
$234,750
$236,750
$240,250
EFFECTIVE RATEe: 9.59%
9.39%
9.47%
UNHEDGED AVERAGE RATE:
10.40%
HEDGED AVERAGE RATE:
9.52%
a.
b.
c.
d.
e.
4
9.61%
LIBOR + 100 BPS
($10M)(RATE)(3/12)
(PRICE CHANGE)(25)(100)(10)
b-c
(NET/10M)(12/3)(100%)
82
A STACK HEDGE WITH EURODOLLAR FUTURES:
DATA ON NOVEMBER 11, 2000
DEC 00
VOLUME
46,903
MAR 01
29,236
127,714
JUN 01
5,788
77,777
SEP 01
2,672
30,152
DECISION:
OPEN INTEREST
185,609
STACK MAR FUTURES FOR JUN AND SEP AND
ROLL OVER AS SOON AS OPEN INTEREST
REACHES 100,000.
83
12.19.00
THE STACK HEDGE
CASH
FUTURES
8.44%
S 10 DEC 91.41
S 30 MAR 91.61
9.54%
L 10 DEC 90.46
1.12.01
9.47%
L 20 MAR 90.47 S10MAR
S 20 JUN 90.42 S20JUN
2.22.01
9.95%
3.13.01
9.75%
6.19.01
9.44%
L 10 JUN 89.78 S10MAR
S 10 SEP 89.82 S10JUN
S10SEP
L 10 MAR 90.25 S10JUN
S10SEP
L 10 JUN 90.56 S10SEP
9.18.01
8.88%
L 10 SEP 91.12 NONE
DATE
11.1.00
F. POSITION
S10DEC
S30MAR
S30MAR
84
PERIOD:
1
RATE(%) a:
10.54
INTERESTb:
FUTURES($)c:
NET($) d:
2
3
4
10.75
10.44
9.88
263,500
268,750
261,000
247,000
23,750
34,000
239,750
234,750
9.59
9.39
UNHEDGED AVERAGE RATE
HEDGED AVERAGE RATE
EFFECTIVE RATE (%) e:
a.
b.
c.
d.
e.
28,500
<3,500>
28.500
16,000
<32,500>
236,000
235,000
9.44
9.40
10.40%
9.46%
LIBOR + 100 BPS
($10M)(RATE)(3/12)
(PRICE CHANGE)(25)(100)(10)
b-c
(NET/10M)(12/3)(100%).
85