Chapter 5 NEW Slides Interest Rate Markets2

7/30/2019 Chapter 5 NEW Slides Interest Rate Markets2

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Derivative Securities

Chapter 5

Interest Rate Markets


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FIN 480 ( Instructor- SfR) Chapter 5 2

Types of Rates

Treasury rates

LIBOR

Repo rates


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Treasury rates

Rates on Treasury securities

Most recently auctioned issues of a given maturity calledOn the run issues exist for following maturities: 1m, 3m, 6m, 12 m, 2 yr, 5yr, 10yr

Those with maturities 3 yrs, 7 yrs, 15 yrs, 20 yrs and30 yrs have been discontinued

Par values are used to figure out the underlying yields


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LIBOR- London Interbank offered rate

Determined based on the quotations of 16 major banks; Rate atwhich banks and FIs transact in the London Interbank market

The lender bank invests cash in a CD issued by borrower bank.

Maturity of the CD short tem i.e. overnight to 1 year

Credit rating of the borrower is AA ( LIBOR Is not completely a riskfree rate)

Currencies supported: Pound, Euro, US $, CAD, AUD, Yen, SWFrancs Called Euro currency market; outside the control of a single

government

Borrowing USD in LIBOR market would be a Eurodollar loan


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LIBOR vs. LIBID

LIBOR: London Interbank offered rate

rate at which a bank makes deposits

LIBID: London Interbank bid rate

rate at which a bank accepts deposits

LIBOR>LIBID Offer rate>bid rate


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Repurchase agreement (Repo) rates

Borrower deposits securities with custodian andborrows money from a Lender

At maturity the buyer buys back the securities at a

pre-agreed price (includes a premium)

The implied rate is the repo rate

Essentially a Forward contract

Maturity: Overnight repo and term repo


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Zero Rates

A zero rate (or spot rate), formaturity Tis the rate of interestearned on an investment thatprovides a payoff only at time T


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Spot rates Example (Table 4.2)

Maturity(years)

Zero Rate(% cont. comp.)

0.5 5.01.0 5.8

1.5 6.4

2.0 6.8

0

1

2

3

4

5

6

7

8

0.5 1 1.5 2

spot rates


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Bond Pricing

To calculate the cash price of a bond wediscount each cash flow at the appropriatezero rate

In our example, the theoretical price of atwo-year bond providing a 6% couponsemiannually is

3 3 3

103 98 39

0 05 0 5 0 058 1 0 0 064 1 5

0 068 2 0

e e e

e

. . . . . .

. ..


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Bond Yield

The bond yield is the discount rate that makes thepresent value of the cash flows on the bond equalto the market price of the bond

Suppose that the market price of the bond in ourexample equals its theoretical price of 98.39

The bond yield is given by solving

to gety = 0.0676 or 6.76%.3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y . . . . .


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Par Yield

The par yield for a certain maturity is thecoupon rate that causes the bond price toequal its face value.

In our example we solve

g)compoundins.a.(with876getto

1002

100

222

0.2068.0

5.1064.00.1058.05.005.0

.c=

ec

ec

ec

ec


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Bootstrapping to get spot curveSample Data (Table 4.3)

Bond Time to Annual BondPrincipal 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 Bootstrap Method

An amount 2.5 can be earned on 97.5during 3 months.The 3-month rate is 4 times 2.5/97.5 or 10.256%

with quarterly compounding

This is 10.127% with CC

Similarly the 6 month and 1 year rates are

10.469% and 10.536% with CC


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The Bootstrap Method continued

To calculate the 1.5 year rate wesolve

to getR = 0.10681 or 10.681%

Similarly the two-year rate is10.808%

96104445.10.110536.05.010469.0

R

eee


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Zero Curve Calculated from theData (Figure 4.1)

9

10

11

12

0 0.5 1 1.5 2 2.5

Zero

Rate (%)

Maturity (yrs)

10.127

10.469 10.53610.681 10.808


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Forward Rates:The forward rate is thefuture zero rate implied by todays term

structure of interest rates

Suppose that the zero rates for time periods T1and T2are R1 and R2 with both rates CC and letRcbe the CC forward rate for the periodbetween times T1 and T2 is

12

1122

122222

:forsolve

)(

121122

TT

TRTR

R

TTRTRTR

eee

F

F

TTRTRTR F


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Calculation of Forward RatesTable 4.5

Zero Rate for Forward Rate

an n -year Investment forn th Year

Year (n ) (% per annum) (% per annum)

1 3.0

2 4.0 5.0

3 4.6 5.8

4 5.0 6.2

5 5.3 6.5


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Theories of the Term Structure

Expectations Theory: forward rates equal expectedfuture zero rates

Market Segmentation: short, medium and longrates determined independently of each other

Liquidity Preference Theory: forward rates higherthan expected future zero rates because ofexpected risk premium for longer term rates


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Upward vs Downward SlopingYield Curve

For an upward sloping yield curve:

Fwd Rate > Zero Rate > Par Yield

For a downward sloping yield curve

Par Yield > Zero Rate > Fwd Rate


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Forward Rate Agreement

A forward rate agreement (FRA) is anagreement that a certain rate willapply to a certain principal during acertain future time period

T1 T2

Receive Rk: FRA rate

FRA contract done at time tTo receive Rk: FRA rate for aloan amount $Lat time T1 for a maturity ofT2-T1


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Forward Rate Agreementcontinued

An FRA is equivalent to anagreement where interest at apredetermined rate,RK is exchanged

at time T1 for interest at the time T1market rate

An FRA can be valued by assuming that the market rate at time T1 = forward

interest rate today


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Valuation FRA

Assume that forward rates are realized

Calculate the terminal payoffs

Discount using current spot rate

T1 T2


Receive Rk: FRA ratePay RF: LIBOR rate for T1T2 period

2212rateLIBOR-rateFRA$L

TReTT


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Valuation FRA: example

Assume that forward rates are realized

Calculate the terminal payoffs

Discount using current spot rate

T1: 3 yrs T2: 3 yrs 3 mts


Receive 4: FRA ratePay 3: LIBOR rate for T1T2 periodL= $100 mi

2212

33%-%4$100miTR

e


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Duration of a bond that provides cash flow ci at time ti is

whereB is its price andy is its yield (continuouslycompounded)

This leads to

B

ect

iyt

in

i

i

1

yDB

B

Duration


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Duration Continued

When the yield y is expressed withcompounding m times per year

The expression

is referred to as the modified duration

my

yBDB

1

D

y m1


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Duration Matching

This involves hedging against interestrate risk by matching the durations ofassets and liabilities

It provides protection against smallparallel shifts in the zero curve


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Duration-Based Hedge Ratio

FC

P

DF

PD

FC

Contract Price for Interest Rate Futures

DF Duration of Asset Underlying Futures atMaturity

P Value of portfolio being HedgedDP Duration of Portfolio at Hedge Maturity


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Example (page 144-145)

Three month hedge is required for a $10 millionportfolio. Duration of the portfolio in 3 months willbe 6.8 years.

3-month T-bond futures price is 93-02 so thatcontract price is $93,062.50

Duration of cheapest to deliver bond in 3 months is9.2 years

Number of contracts for a 3-month hedge is

42.792.950.062,93

8.6000,000,10

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Chapter 5 NEW Slides Interest Rate Markets2