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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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Types of Interest Rates
Source: Reuters 15/09/05
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Types of Interest Rates
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
Types of Interest Rates
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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
Types of 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
Types of Interest Rates
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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
Determining Treasury Zero Rates
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
Determining Treasury Zero Rates
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
Determining Treasury Zero Rates
+= 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
Determining Treasury Zero Rates
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
Determining Treasury Zero Rates
%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
Determining Treasury Zero Rates
%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
Determining Treasury Zero Rates
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From the work done so far the 6-month and 1-year zero rates have already been calculated, i.e.
Determining Treasury Zero Rates
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
Determining Treasury Zero Rates
( ) ( ) ( ) 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
Determining Treasury Zero Rates
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
Determining Treasury Zero Rates
%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
Determining Treasury Zero Rates
( )
( ) ( )
( )%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
Determining Treasury Zero Rates
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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Determining Treasury Zero Rates
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
Forward Rate Agreements (FRAs)
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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
Forward Rate Agreements (FRAs)
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
Forward Rate Agreements (FRAs)
( ) 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
Forward Rate Agreements (FRAs)
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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
Forward Rate Agreements (FRAs)
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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
Forward Rate Agreements (FRAs)
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Hull, J.C, Options, Futures & Other Derivatives,2005, 6th Ed. Chapter 4
Further reading
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Hull, J.C, Options, Futures & Other Derivatives,2005, 6th Ed. Chapter 4
Further reading