Ch04HullOFOD6thEd

7/30/2019 Ch04HullOFOD6thEd

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Options, Futures, and Other Derivatives 6th Edition, Copyright John C. Hull 2005 4.1

Interest Rates

Chapter 4


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Types of Rates

Treasury rates

LIBOR rates

Repo rates


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Measuring Interest Rates

The compounding frequency usedfor an interest rate is the unit of

measurement The difference between quarterly

and annual compounding isanalogous to the differencebetween miles and kilometers


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Continuous Compounding(Page 79)

In the limit as we compound more and morefrequently we obtain continuously compoundedinterest rates

$100 grows to $100eRTwhen invested at acontinuously compounded rateRfor time T

$100 received at time Tdiscounts to $100e-RTat

time zero when the continuously compoundeddiscount rate isR


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Conversion Formulas(Page 79)

Define

Rc : continuously compounded rate

Rm: same rate with compounding m timesper year

R mR

m

R m e

c m

mR mc

ln

/

1

1


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Zero Rates

A zero rate (or spot rate), for maturity Tisthe rate of interest earned on an

investment that provides a payoff only attime T


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Example (Table 4.2, page 81)

Maturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8


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Bond Pricing

To calculate the cash price of a bond wediscount each cash flow at the appropriate zerorate

In our example, the theoretical price of a two-year bond providing a 6% coupon semiannuallyis

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 thatmakes the present value of the cash flows onthe bond equal to the market price of thebond

Suppose that the market price of the bond inour example equals its theoretical price of98.39

The bond yield (continuously compounded) is

given by solving

to gety=0.0676 or 6.76%.

3 3 3 103 98390 5 1 0 15 2 0

e 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.(withgetto 876

1002

100

222

0.2068.0

5.1064.00.1058.05.005.0

.c=

ec

ec

ec

ec


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Options, Futures, and Other Derivatives 6th

Edition, Copyright John C. Hull 2005 4.11

Par Yield continued

In general ifmis the number of couponpayments per year,Pis the present valueof $1 received at maturity andA is thepresent value of an annuity of $1 on eachcoupon date

cP m

A

( )100 100


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Sample Data (Table 4.3, page 82)

Bond Time to Annual Bond Cash

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 Bootstrap Method

An amount 2.5 can be earned on 97.5 during 3months.

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 are10.469% and 10.536% with continuous

compounding


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The Bootstrap Method continued

To calculate the 1.5 year rate we solve

to getR = 0.10681 or 10.681%

Similarly the two-year rate is 10.808%

9610444 5.10.110536.05.010469.0 Reee


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Zero Curve Calculated from the

Data (Figure 4.1, page 84)

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 the future zero rateimplied by todays term structure of interest

rates


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Calculation of Forward RatesTable 4.5, page 85

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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Formula for Forward Rates

Suppose that the zero rates for timeperiods T1and T2areR1 andR2 with bothrates continuously compounded.

The forward rate for the period betweentimes T1 and T2 is

R T R T

T T

2 2 1 1

2 1


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Instantaneous Forward Rate

The instantaneous forward rate for amaturity Tis the forward rate that appliesfor a very short time period starting at T. Itis

whereR is the T-year rate

R TR

T


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Upward vs Downward Sloping

Yield 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 will apply toa certain principal during a certain futuretime period


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Forward Rate Agreement

continued

An FRA is equivalent to an agreementwhere interest at a predetermined rate,RKis exchanged for interest at the marketrate

An FRA can be valued by assuming thatthe forward interest rate is certain to berealized


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Valuation Formulas (equations 4.9 and 4.10page 88)

Value of FRA where a fixed rateRKwill bereceived on a principalL between times T1 andT2 is

Value of FRA where a fixed rate is paid is

RFis the forward rate for the period andR2 is the

zero rate for maturity T2 What compounding frequencies are used in

these formulas forRK,RM, andR2?

22))(( 12TR

FKeTTRRL

22))(( 12TR

KFeTTRRL


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Duration of a bond that provides cash flow c iat 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 (page 89)


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Duration Continued

When the yieldy 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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Convexity

The convexity of a bond is defined as

2

1

2

2

2

)(2

1

thatso

1

yCyDB

B

B

etc

y

B

BC

n

i

yt

ii

i


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Theories of the Term StructurePage 93

Expectations Theory: forward rates equalexpected future zero rates

Market Segmentation: short, medium andlong rates determined independently ofeach other

Liquidity Preference Theory: forwardrates higher than expected future zerorates

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