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7/27/2019 Ch04HullOFOD8thEdition

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Chapter 4

Interest Rates

Options, Futures, and Other Derivatives 8th Edition,Copyright John C. Hull 2011 1


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Types of Rates

Treasury ratesLIBOR rates

Repo rates



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Treasury Rates

Rates on instruments issued by agovernment in its own currency



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LIBOR and LIBIDLIBOR is the rate of interest at which a bankis prepared to deposit money with anotherbank. (The second bank must typically have

a AA rating)LIBOR is compiled once a day by the BritishBankers Association on all major currencies

for maturities up to 12 monthsLIBID is the rate which a AA bank is preparedto pay on deposits from anther bank



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Repo Rates

Repurchase agreement is an agreementwhere a financial institution that ownssecurities agrees to sell them today forXand

buy them bank in the future for a slightlyhigher price, Y

The financial institution obtains a loan.The rate of interest is calculated from thedifference betweenXand Yand is known as

the repo rateOptions, Futures, and Other Derivatives 8th Edition,

Copyright John C. Hull 2011 5


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The Risk-Free RateThe short-term risk-free rate traditionally

used by derivatives practitioners is LIBORThe Treasury rate is considered to beartificially low for a number of reasons (See

Business Snapshot 4.1)

As will be explained in later chapters:

Eurodollar futures and swaps are used to extendthe LIBOR yield curve beyond one year

The overnight indexed swap rate is increasingly

being used instead of LIBOR as the risk-free rateOptions, Futures, and Other Derivatives 8th Edition,



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Measuring Interest Rates

The compounding frequency used foran interest rate is the unit ofmeasurement

The difference between quarterly andannual compounding is analogous tothe difference between miles and

kilometers



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Impact of Compounding

When we compound m times per year at rateR anamountA grows toA(1+R/m)m in one year


Compounding frequency Value of $100 in one year at 10%

Annual (m=1) 110.00

Semiannual (m=2) 110.25

Quarterly (m=4) 110.38

Monthly (m=12) 110.47

Weekly (m=52) 110.51

Daily (m=365) 110.52


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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 rateR for 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)

DefineRc : continuously compounded rate

Rm: same rate with compoundingm

times peryear


( )

R mR

m

R m e

cm

mR mc

= +

=

ln

/

1

1


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Examples

10% with semiannual compounding isequivalent to 2ln(1.05)=9.758% withcontinuous compounding

8% with continuous compounding isequivalent to 4(e0.08/4 -1)=8.08% with quarterlycompounding

Rates used in option pricing are nearlyalways expressed with continuouscompounding



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Zero Rates

A zero rate (or spot rate), for maturity Tis therate of interest earned on an investment thatprovides a payoff only at time 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 appropriatezero rate

In our example, the theoretical price of a two-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 YieldThe bond yield is the discount rate that makes

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

Suppose that the market price of the bond in

our 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 98 390 5 1 0 1 5 2 0e e e ey y y y + + + =. . . . .


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Par YieldThe par yield for a certain maturity is the

coupon rate that causes the bond price toequal its face value.

In our example we solve


g)compoundinsemiannual(with876getto1002100

222

020680

51064001058050050

.c=

e

c

ec

ec

ec

=

++

++

..

......


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Par Yield continuedIn general if m is the number of coupon

payments per year, dis the present value of$1 received at maturity andA is the presentvalue of an annuity of $1 on each coupon

date

(in our example, m = 2, d = 0.87284, andA =3.70027)


A

mdc

)( 100100 =


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Data to Determine Zero Curve(Table 4.3, page 82)


Bond Principal Time toMaturity (yrs)

Coupon peryear ($)*

Bond price ($)

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

* Half the stated coupon is paid each year


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The Bootstrap Method

An amount 2.5 can be earned on 97.5 during3 months.

Because 100=97.5e0.101270.25 the 3-month

rate is 10.127% with continuouscompounding

Similarly the 6 month and 1 year rates are

10.469% and 10.536% with continuouscompounding



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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 theData (Figure 4.1, page 84)


9

10

11

12

0 0.5 1 1.5 2 2.5

ZeroRate (%)

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 interestrates


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Formula for Forward Rates

Suppose that the zero rates for time periods T1 and T2areR1 andR2 with both rates continuouslycompounded.

The forward rate for the period between times T1 andT2 is

This formula is only approximately true when ratesare not expressed with continuous compounding


R T R T

T T

2 2 1 1

2 1


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Application of the Formula


Year (n) Zero rate for n-year

investment(% per annum)

Forward rate for nth

year(% per annum)

1 3.02 4.0 5.0

3 4.6 5.8

4 5.0 6.2

5 5.5 6.5


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Instantaneous Forward Rate

The instantaneous forward rate for a maturityTis the forward rate that applies for a veryshort time period starting at T. It is

whereR is the T-year rate


R T R

T+


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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 an OTCagreement that a certain rate will apply to acertain principal during a certain future time

period


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Forward Rate Agreement: KeyResults

An FRA is equivalent to an agreement where interestat a predetermined rate,RK is exchanged for interest atthe market rate

An FRA can be valued by assuming that the forwardLIBOR interest rate,RF , is certain to be realized

This means that the value of an FRA is the present

value of the difference between the interest that wouldbe paid at interest at rateRFand the interest that wouldbe paid at rateRK



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Valuation Formulas

If the period to which an FRA applies lastsfrom T1 to T2, we assume that RFandRKareexpressed with a compounding frequency

corresponding to the length of the periodbetween T1 and T2With an interest rate ofRK, the interest cash

flow isRK (T2T1) at time T2With an interest rate ofRF, the interest cashflow isR

F

(T2T

1) at time T

2Options, Futures, and Other Derivatives 8th Edition,



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Valuation Formulas continued

When the rateRKwill be received on a principal ofL

the value of the FRA is the present value of

received at time T2When the rate R

Kwill be received on a principal of L

the value of the FRA is the present value of

received at time T2


))(( 12 TTRR FK

))((12

TTRRKF


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Example

An FRA entered into some time ago ensuresthat a company will receive 4% (s.a.) on $100million for six months starting in 1 year

Forward LIBOR for the period is 5% (s.a.)

The 1.5 year rate is 4.5% with continuous

compoundingThe value of the FRA (in $ millions) is


467050050040100 510450 ..)..( .. = e


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Example continued

If the six-month interest rate in one year turnsout to be 5.5% (s.a.) there will be a payoff (in$ millions) of

in 1.5 years

The transaction might be settled at the one-year point for an equivalent payoff of


750500550040100 ..)..( =

7300

02751

750.

.

.=


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Duration (page 89-90)

Duration of a bond that provides cash flow ciat time ti is

whereB is its price andy is its yield

(continuously compounded)


=

=

B

ectD

iyt

in

i

i

1


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Key Duration Relationship

Duration is important because it leads to thefollowing key relationship between thechange in the yield on the bond and the

change in its price


yDB

B=


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Key Duration Relationship continued

When the yieldy is expressed withcompounding m times per year

The expression

is referred to as the modified duration


my

yBD

B +

= 1

D

y m1+


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Bond Portfolios

The duration for a bond portfolio is the weightedaverage duration of the bonds in the portfolio withweights proportional to prices

The key duration relationship for a bond portfoliodescribes the effect of small parallel shifts in the yieldcurve

What exposures remain if duration of a portfolio ofassets equals the duration of a portfolio of liabilities?



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ConvexityThe convexity, C, of a bond is defined as

This leads to a more accurate relationship

When used for bond portfolios it allows larger shifts inthe yield curve to be considered, but the shifts still

have to be parallelOptions, Futures, and Other Derivatives 8th Edition,


B

etc

y

B

BC

n

i

yt

iii

=

=

=

1

2

2

21

( )22

1yCyD

B

B+=


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Theories of the Term StructurePage 91-92

Expectations Theory: forward rates equalexpected future zero rates

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

Liquidity Preference Theory: forward rateshigher than expected future zero rates



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Liquidity Preference Theory

Suppose that the outlook for rates is flat andyou have been offered the following choices

Which would you choose as a depositor?Which for your mortgage?


Maturity Deposit rate Mortgage rate

1 year 3% 6%

5 year 3% 6%


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Liquidity Preference Theory cont

To match the maturities of borrowers andlenders a bank has to increase long ratesabove expected future short rates

In our example the bank might offer

Options, Futures, and Other Derivatives 8th Edition,C i ht J h C H ll 2011

Maturity Deposit rate Mortgage rate

1 year 3% 6%

5 year 4% 7%

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