bm fi6051 wk9 lecturer notes

8/14/2019 BM FI6051 WK9 Lecturer Notes

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Financial DerivativesFI6051

Finbarr MurphyDept. Accounting & FinanceUniversity of LimerickAutumn 2009

Week 9 Fixed Income


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On a point of interest, the cover slide shows a USGovernment Bond, 4% Coupon Issused in1933. Unusually, there are unused coupons

attached

Bonds are still issued by governments,municipalities, semi-state bodies and corporate

institutions. Actual certificates are now rarelyissued and the coupons are not attached to thebond. This is all done electronically but theprincipals are the same

Bonds


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Treasury Rates The interest rates applicable to the borrowings of a

government denominated in its own currency

For example, US Treasury rates apply to the borrowingsof the US government denominated in US dollars

Such debt instruments include T-bills (money markets),and T-Notes and T-bonds (capital markets)

Given that negligible default risk applies to

governmental debt, Treasury rates tend to be verylow Treasury rates are often used as a proxy for risk-free

interest rates

Types of Interest Rates


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Source: Reuters 15/09/05


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Source: Reuters 15/09/05


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LIBOR Rates Large international banks transfer funds between each

other by means of 1-, 3-, 6-, and 12-month deposits

The deposits can be denominated in any of the worldsmajor currencies

Each international bank quotes bid and offer rates forsuch interbank transfers of funds

The bid(offer) is the rate at which an international bank

is willing to accept (advance) deposits The bid rate is referred to as the London Interbank Bid

Rate or LIBID



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The offer rate is referred to as the London InterbankOffer Rate or LIBOR

LIBOR rates tend to be slightly higher thancorresponding Treasury rates

The reason for this is the LIBOR rates, unlike Treasuryrates, are not considered to be entirely risk-free

LIBOR rates however do tend to be verylowdue to thelowdefault risk involved in the interbank deposits

Therefore, LIBOR rates are often used as a proxy forrisk-free interest rates



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Repo Rates A repo is an agreement involving the sale of securities

by one party to another with a promise to repurchase ata specified price and on a specified date in the future

The underlying securities to repos are primarily Treasuryand government agent instruments

The repo allows short-term returns on excess funds,where the securities form a source ofcollateral

The difference between the sale and repurchase pricesrepresents the interest earned on the repo

The level of interest on the repo is referred to as therepo rate



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Zero rates or zero-coupon rates refer to theinterest rates applying to investments thatcontinue for some specified term

The n-yearzero rate is the interest rate thatapplies to an n-yearinvestment

All interest and principal is realized at the expiryof the investment, i.e. no intermediate payments

For instance, consider a 5% zero rate on a 5-year

investment initiated at $100

Zero Rates


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The terminal value of the investment is $128.40,i.e.

In the markets many of the interest ratesobserved are notpure zero rates Many instruments for example offer coupon payments

which are paid prior to expiry

It is however possible to determine zero ratesfrom the prices of such coupon-bearing

instruments

Zero Rates

( ) 40.128100 505.0 =e


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Bonds are long-term debt obligations issued bycorporations and governments Funds raised are generally used to support large-scale

and long-term expansion and development

Bonds are financial instruments designed to: Repay the original investment principal at a pre-

specified maturity date

Make periodic coupon interest payments over the life ofthe investment period

The theoretical price of a bond involves summing

the present value of all resulting cash flows

Bond Pricing


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Given that the cash flows occur at different pointsin time, appropriate zero-rates are used for thediscounting

To illustrate, consider the following Treasury zerorates

Bond Pricing

Maturity (Years) Zero Rate (%)

0.5 5.0

1 5.8

1.5 6.4

2 6.8


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Bond Pricing

Treasury Zero Curve

0

1

2

3

4

5

6

7

8

0.5 1 1.5 2

Years

Yield(%)


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Consider a 2-year Treasury bond with a facevalue of $100 and a coupon rate of 6% paidsemi-annually

The coupon payment on the bond is $3, which isdetermined as follows

where

the face value of the bond

the coupon rate on the bond

the (per year) payment frequency of the coupon

Bond Pricing

( )32

06.0100

==

m

rP cf

fPcr

m


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The following table details all the cash flows onthe bond, along with the present value of each

Note that the appropriate discount rates used forthe PV calculations above are the zero rates givenpreviously

Bond Pricing

Payment Date(Years)

Cash Flow Present Value of Cash Flow

0.5 3 3e-0.05(0.5)

1 3 3e-0.058(1)

1.5 3 3e-0.064(1.5)

2 103 103e-0.068(2)


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Therefore the price of the bond underconsideration is $98.39, i.e.

Bond Pricing

( ) ( ) ( ) ( ) 39.98103333 2068.05.1064.01058.05.005.0

=+++

eeee


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The yieldor yield-to-maturityon a coupon-bearing bond is the rate that equates all cashflows to its market value

Let ydenote the yield on a bond, and take thebond considered previously

The yield yon the bond may be determined bysolving the following equation

Bond Yield

( ) ( ) ( ) ( ) 100103333 25.115.0 =+++ yyyy eeee


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The solution to the above equation is non-trivialand requires a numerical search routine such asNewton-Raphson

The solution gives a value for the bond yield of6.76%, i.e. y = 6.76%

Bond Yield


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Treasury zero rates can be calculated from theprices of traded debt instruments

One common method of determining the interestrates is that ofbootstrapping

Consider 5 separate bonds, 3 of which are zero-

coupon and 2 of which are coupon-bearing

Details of the bonds are given in the next table

Determining Treasury Zero Rates


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The zero rates for the 3 zero-coupon bonds can

be calculated easily


Face Value Maturity(Years)

Annual Coupon

(Semi-Annual Payment)

Bond Price

100 0.25 0 97.50

100 0.5 0 94.90

100 1 0 90.00

100 1.5 8 96.00

100 2 12 101.60


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For this note that the zero rate on a zero-couponbond is given by the following formula

where

the face value of the bond

the current market price of the bond

the term-to-maturity of the bond

Note that the above formula gives zero ratesusing (1/T)-period compounding

That is, discrete compounding rather than continuouscompounding


TP

PP

o

f 10

fP0P

T


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In order to express these zero rates usingcontinuous compounding the following formula isused

where

the rate of interest with continous compounding

the rate of interest with discrete compounding

the compounding frequency ofRm per annum

The above formulas will be illustrated with thefirst zero-coupon bond


+= mR

mRm

c 1ln

cRmR

m


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The term-to-maturity of the zero-coupon bond isT = 0.25

So the zero rate associated with the bond is forquarterly compounding since

Therefore, the 3-month zero rate with quarterlycompounding is


41==

Tm

%256.1045.97

5.971004 =

=R


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The conversion ofR4 to the corresponding zero

rate with continuous compounding is calculatedas follows

Note now that the term-to-maturity of the secondzero-coupon bond is T = 0.5

So the zero rate associated with the bond is forsemi-annual compounding since


%127.1010127.04

10256.01ln4 == +=cR

2

1

== Tm


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Therefore, the 6-month zero rate with semi-annual compounding is

The conversion ofR2 to the corresponding zero

rate with continuous compounding is calculatedas follows


%469.1010469.02

10748.01ln2 ==

+=cR

%748.102

9.94

9.941002 =

=R


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In the same way, it can be shown that for thethird zero-coupon bond that Rc= 10.536%

Consider now the first coupon-bearing bondpresented in the bond data previously

The term-to-maturity of this bond is one and ahalf years, i.e. T = 1.5

The next table details all the cash flows resultingfrom this bond



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From the work done so far the 6-month and 1-year zero rates have already been calculated, i.e.


Payment Date(Years)

Cash Flow

0.5 4

1 4

1.5 104

%536.10

%469.10

1,

5.0,

=

=

c

c

R

R


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So the 1.5-year zero rate can be determined bythe solving the following pricing relation

Solving for Rc,1.5 proceeds as follows

( )( ) ( )

( ) ( )

( )%681.1010681.05.1

85196.0ln

85196.0ln5.1

85196.0104

4496

5.1,

5.1,

110536.05.010469.05.15.1,

===

==

=

c

c

R

R

R

eee

c


( ) ( ) ( ) 96104445.1110536.05.010469.0 5.1,

=++

cReee


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Consider now the second coupon-bearing bondpresented in the bond data previously

The term-to-maturity of this bond is two years,i.e. T = 2

The table below details all the cash flows from

this bond


Payment Date (Years) Cash Flow

0.5 6

1 6

1.5 6

2 106


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From the work done so far it is known that

So the 2-year zero rate can be determined by thesolving the following pricing relation


%681.10

%536.10

%469.10

5.1,

1,

5.0,

=

=

=

c

c

c

R

R

R

( ) ( )

( ) ( ) 6.1011066

66

25.110681.0

110536.05.010469.0

2, =++

+

cRee

ee


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Solving for Rc,2 is straightforward and proceeds as

follows

The next table summarizes the zero ratescalculated under the bootstrap method


( )

( ) ( )

( )%808.1010808.0

2

8056.0ln

8056.0ln2

8056.0

5.1,

2,

22,

===

==

c

c

R

R

R

e c


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The following diagram is a graph of the zero ratecurve given the rates tabulated above


Maturity (Years) Zero Rate (%)

0.25 10.127

0.5 10.469

1 10.5361.5 10.681

2 10.808


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9

10

11

12

0 0.5 1 1.5 2 2.5

Maturity (yrs)

10.127

10.469 10.53

6

10.68

1

10.808


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Forward Interest Rate

A Forward Interest Rate is an interest ratewhich is specified now for a loan that will occur

at a specified future date As with current interest rates, forward interest

rates include a term structure which shows thedifferent forward rates offered to loans of

different maturities.

Forward Rates


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Forward rates are those rates implied by currentzero rates for periods of time in the future

Consider two zero rates Rx and Ry, with maturitiesTx and Ty respectively (Ty > Tx)

Let RF

denote the forward rate for the period of

time between Tx and Ty

RFcan be calculated from the two zero rates

using the following general formula

Forward Rates


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Forward Rates

xy

xxyy

FTT

TRTRR

=

We are assuming continuously compounded rates

We can quickly derive this from first principles Assume the 3month EURIBOR Rate is 4.1%

And the 6month EURIBOR Rate is 4.3%

We can say that:

Now, derive the equation above!

( ) )5.0)(043.0()25.05.0()041.0)(25.0( 100100 eee FR =


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To illustrate further, consider the following zerorate data

Forward Rates

Maturity (Years) Zero Rate (%)1 10

2 10.5

3 10.8

4 11

5 11.1

We are assuming continuously compounded rates


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Treasury Zero Curve

4

4.5

5

5.5

6

6.5

7

0.5 1 1.5 2Years

Yield(%)

Tx

Forward Rates

Ty

Rx

Ry


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Let denote the forward rate for the periodbetween year 1 and year 2

According to the general formula

Similarly let denote the forward rate for the

period between year 2 and year 3

Forward Rates

2,1FR

( ) ( )

( ) ( )

%1111.0

110.02105.0

12

12 122,1

==

=

=

RRR

F

3,2FR


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The general forward rate formula gives

In the same way it is possible to calculate the 1-year forward rates for the 4th and 5th years underconsideration

The next table presents all the forward rates

Forward Rates

( ) ( )

( ) ( )%4.11114.0

2105.03108.0

23

23 233,2

==

=

=

RRR

F


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By rewriting the general forward rate formula it ispossible to establish important relationshipsbetween zero and forward rates

Forward Rates

Maturity (Years) Zero Rate (%) Forward Rates

(for n-th year)

1 10

2 10.5 11

3 10.8 11.4

4 11 11.6

5 11.1 11.5


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Forward Rates

Forward Curve

8

8.5

9

9.5

10

10.5

11

11.5

12

1 2 3 4 5Years

Yield(%

)

Zero-Rate Forward-Rate


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The general forward rate formula can berewritten as follows

If the zero curve is upward sloping, i.e. Ry>Rx,

then from the relation above RF>Ry

If the zero curve is downward sloping, i.e. Ry


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Taking limits as Ty approaches Txleads to the

following relationship

In the above equation R is the zero rate for amaturity ofT

And RF is referred to as the instantaneous forward

rate at time T That is, the forward rate that applies to an infinitesimal

time period beginning at time T

Forward Rates

T

RTRRF

+=


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A Forward Rate Agreement (FRA) is a bilateral orover the counter (OTC) interest rate contract inwhich two counterparties agree to exchange the

difference between an agreed interest rate andan as yet unknown reference rate of specifiedmaturity that will prevail at an agreed date in thefuture.

Payments are calculated against a pre-agreednotional principal

The reference rate is typically LIBOR or EURIBOR

Forward Rate Agreements (FRAs)


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Consider a FRA that is agreed between twoparties with an interest rate ofRK applying

between times T1 and T2 (T2 > T1)

The interest rate RK applies to some principal L



47/54

Let R1 and R2 denote the zero rates applying to

the maturities T1 and T2 respectively

The next table illustrates the cash flows resultingfrom the FRA

The value of the agreement at time 0, V(0), can

be found by taking the present value of thesecash flows


Date Cash Flow

T1 -L

T2 + L{exp[RK(T2-T1)]}


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Forward Rate Agreements (FRAs)Treasury Zero Curve

4

4.5

5

5.5

6

6.5

7

0.5 1 1.5 2Years

Y

ield(

T2

R1

R2

T1

FRA Buyer Lends (Pays) L at T1 What is L worth today? I.e. at T(0)? = Le-R 1T1

FRA Buyer Receives L at T2 plus interest between

T1 & T2 What is this worth today? I.e. at T(0)? = e-R 2T2(Le-R k(T 2-T 1)))

where

12

1122

TT

TRTRRK

=


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Therefore, V(0) is as follows

From this it can be noted that V(0) = 0 when


( ) 221211)0(TRTTRTR

eLeLeV K +=

( )

12

1122

221211

TT

TRTRR

TRTTRTR

K

K

=

=


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The equation for RK above corresponds to the

general forward rate equation from the lastsection

So the initial value of a FRA is zero when theagreed rate RK is set equal to the corresponding

forward rate RF



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Forward Rate Agreements are usually settled atT1 (rather than T2)

A FRA is agreed on a notional amount of 100MM The agreed Forward Rate (RK) is 4.5% between

18months and 2years

Let RM equal the actual six month spot rate in

18months time



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At T1 (in 18 months), the parties to the FRA agree

to settle the trade as RM is known at that point

According to the agreement, the lender receives

100MM(e(R K-R M)(T 2-T 1)-1) at T2 As the agreement is settled at T1, the lender

receives 100MM(e(R K-R M)(T 2-T 1)-1).e(- RM)(T 2-T 1)

Note that the lender can lose money

Use examples to confirm these cash flows



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Hull, J.C, Options, Futures & Other Derivatives,2005, 6th Ed. Chapter 4

Further reading


54/54

Hull, J.C, Options, Futures & Other Derivatives,2005, 6th Ed. Chapter 4

Further reading

bm fi6051 wk9 lecturer notes

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