Interest Rate Markets Chapter 5

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Interest Rate MarketsChapter 5
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Types of Interest Rates

• Treasury rates
• LIBOR rates
• Repurchase rates

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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 buyers transactions are just the
opposite of the sellers. The figure below
demonstrates the transactions in a repo.
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Transactions in a Repurchase Agreement Date 0 -
Open the Repo
T- Bill
Party A
Party B
PO
Date t - Close the Repo
T-Bill
Party A
Party B
Pt P0(1r0,t )
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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
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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.
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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.
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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.

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Example (Table 5.1, page 95)

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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

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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
• to get y0.0676 or 6.76.

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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

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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

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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
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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
  

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The Bootstrap Method continued

• To calculate the 1.5 year rate we solve
• to get R 0.10681 or 10.681
• Similarly the two-year rate is 10.808

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Zero Curve Calculated from the Data (Figure 5.1,
page 98)

Zero Rate ()
10.808
10.681
10.469
10.536
10.127
Maturity (yrs)
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r the bonds 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.
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A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL
ITS MATURITY C 0
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Result The bond sells at par if P FV. If CR
r the bond sells at par. If CR gt r the bond
sells at a premium. If CR lt r the bond sells at
a discount.
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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.
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Duration in continuous time

• Duration of a bond that provides cash flow c i at
  time t i is
• where B is its price and y is its yield
  (continuously compounded)
• This leads to

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DURATION INTERPRETED AS A MEASURE OF THE BOND
PRICE SENSITIVITY
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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
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Coupon Rate
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Duration Continued

• When the yield y is expressed with compounding m
  times per year
• The expression
• is referred to as the modified duration

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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 SPiNi
The total portfolio value. We now prove that DP
SwiDi .
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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.
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Example a portfolio of two T-bonds
D (.1673)(10.4673) (.8327)(12.4674)
12.1392
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Example a two bond portfolio
D (0,5013)(10,4673) (0,4987)(12,4674)
11,45.
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APPLICATION OF DURATION 2. IMMUNIZING BANK
PORTFOLIO OF ASSETS AND LIABILITIES TIME 0
ASSETS LIABIABILITIES 100,000,000
100,000,000 (LOANS)
(DEPOSITS) D 5 D 1 r 10 r
10 TIME 1 r gt 12
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BUT IF DA DL THEY REACT TO RATES CHANGES
IN EQUAL AMOUNTS. THE BANK PORTFOLIO IS
IMMUNIZED , i.e., ITS VALUE WILL NOT CHANGE
FOR A small INTEREST RATE CHANGE, IF THE
PORTFOLIOS DURATION IS ZERO or DP DA -
DL 0.
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APPLICATIONS OF DURATION. 3. EXAMPLE A 5-YEAR
PLANNING PERIOD CASE OF IMMUNIZATION IN
THE CASH MARKET
BOND C FV M r D P A
100 1,000 5 yrs 10 4.17 1,000 B
100 1,000 10 yrs 10 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
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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, CONTINUOUSLY COMPOUNDED YIELDS
328,213,290. ANNUAL RETURN OF
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The weighted average portfolio changes to
AFTER 5 YEARS AT 12 331,267,162. ANNUAL
RETURN OF
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S The bonds spot value. F The futures
price. n The number of futures used in the
hedge.
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Solve for n that sets dV/dr 0.
Next, we use the following substitutions for
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From the definition of DURATION
Upon substitution in n
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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
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The price sensitivity hedge ratio.
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The price sensitivity hedge ratio with continuous
rates is
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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
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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 Eurod
ollars1,000,000 CONTRACT GRADE new or dated
T-bills CASH SETTLEMENT with 13 weeks to
maturity YIELDS DISCOUNT ADD-ON HOURS (
Chicago time) 720 AM-200PM 720 AM -
200PM 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
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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
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Transactions in a Cash-and-Carry Arbitrage.
PO
Repo Market
Arbitrageur
T-Bill
Short Position F0,T
PO (MONEY)
T-Bill
T-Bill Dealer
Futures Market
Date 0
P0(1r0T)
Repo Market
Arbitrageur
T-Bill
Receive F 0,T
Deliver T-Bill
F 0,T gt P0(1r0,T)
Futures Market
Date T
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Transactions in a Reverse Cash-and-Carry
Arbitrage.
PO
Repo Market
Arbitrageur
T-Bill
Long Position F0,T
T-Bill
P0
T-Bill Dealer
Futures Market
Date 0
P0(1r0,T)
Repo Market
Arbitrageur
T-Bill
Take Delivery T-Bill
Pay F 0,T
F 0,T lt P0(1r0,T)
Futures Market
Date T
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PO 954,330.56
Repo Market
Arbitrageur
182-day T-Bill
Short FO,T 976,011.75
182-day T-Bill
P0 954,330.56
T-Bill Dealer
Futures Market
Date 0
P0(1r0,t) 975,561.14
Repo Market
Arbitrageur
91 day T-Bill
Deliver 91-day T-Bill
F 0,T 976,011.75
Profit 450,61
Futures Market
Date T
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PO 954,330.56
Repo Market
Arbitrageur
182-day T-Bill
Long FO,T 973,809.04
182-day T-Bill
PO 954,330.56
Futures Market
T-Bill Dealer
Date 0
P1 975,561.13
Repo Market
Arbitrageur
91 day T-Bill
Take Delivery 91-day T-Bill
F 0,T 973,809.04
PROFIT 1,752.09
Futures Market
Date T
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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
bonds price and vice-versa.
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DS 180/365 Do Nothing.
Short 20 T-bills futures
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PROFIT 20970,575-968,675 38,000
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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.
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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
1,000,0001 (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
1,000,0001 - .25L/100
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Arbitrage with Eurodollar Futures
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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
1,000,0001.094(117/360) 1,000,0001.0935(90/3
60) 1,030,550 1,023,375 Arbitrage
profit 7,175
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How to calculate the profit from a ED
futures? Assume that initially, Qt 90.54. Thus,
Ft 1,000,0001 (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
1,000,0001 (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.
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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.
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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.
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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
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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 BPS b. (10M)(RATE)(3/12) c.
(PRICE CHANGE)(25)(100)(10) d. b -
c e. (NET/10M)(12/3)(100)
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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 AND ROLL OVER AS SOON
AS OPEN INTEREST REACHES 100,000.
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THE STACK HEDGE DATE CASH FUTURES F.
POSITION 11.1.00 8.44 S 10 DEC
91.41 S10DEC S 30 MAR 91.61 S30MAR 12.19.00
9.54 L 10 DEC 90.46 S30MAR 1.12.01 9.47 L
20 MAR 90.47 S10MAR S 20 JUN
90.42 S20JUN 2.22.01 9.95 L 10 JUN
89.78 S10MAR S 10 SEP 89.82 S10JUN S1
0SEP 3.13.01 9.75 L 10 MAR
90.25 S10JUN S10SEP 6.19.01 9.44 L 10
JUN 90.56 S10SEP 9.18.01 8.88 L 10 SEP
91.12 NONE
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• 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 34,000
  28,500 28.500
• lt3,500gt 16,000
• lt32,500gt
• NET() d 239,750 234,750 236,000
  235,000
• EFFECTIVE RATE () e 9.59 9.39
  9.44 9.40
• UNHEDGED AVERAGE RATE 10.40
• HEDGED AVERAGE RATE 9.46
• a. LIBOR 100 BPS
• b. (10M)(RATE)(3/12)
• c. (PRICE CHANGE)(25)(100)(10)
• d. b - c
• (NET/10M)(12/3)(100).