Bm Fi6051 Wk9 Lecturer Notes
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Financial Derivatives FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2009
Week 9 – Fixed Income
Bonds
On a point of interest, the cover slide shows a US Government Bond, 4¼% Coupon Issused in 1933. Unusually, there are unused coupons attached
Bonds are still issued by governments, municipalities, semi-state bodies and corporate institutions. Actual certificates are now rarely issued and the coupons are not attached to the bond. This is all done electronically but the principals are the same
Types of Interest Rates
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 borrowings of 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 very low Treasury rates are often used as a proxy for risk-free interest rates
Types of Interest Rates
Source: Reuters 15/09/05
Types of Interest Rates
Source: Reuters 15/09/05
Types of Interest Rates
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 world’s major currencies Each international bank quotes bid and offer rates for such 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
The offer rate is referred to as the London Interbank Offer Rate or LIBOR LIBOR rates tend to be slightly higher than corresponding Treasury rates The reason for this is the LIBOR rates, unlike Treasury rates, are not considered to be entirely risk-free LIBOR rates however do tend to be very low due to the low default risk involved in the interbank deposits Therefore, LIBOR rates are often used as a proxy for risk-free interest rates
Types of Interest Rates
Repo Rates
A repo is an agreement involving the sale of securities by one party to another with a promise to repurchase at a specified price and on a specified date in the future The underlying securities to repos are primarily Treasury and government agent instruments The repo allows short-term returns on excess funds, where the securities form a source of collateral The difference between the sale and repurchase prices represents the interest earned on the repo The level of interest on the repo is referred to as the repo rate
Zero Rates
Zero rates or zero-coupon rates refer to the interest rates applying to investments that continue for some specified term
The n-year zero rate is the interest rate that applies to an n-year investment
All interest and principal is realized at the expiry of the investment, i.e. no intermediate payments
For instance, consider a 5% zero rate on a 5-year investment initiated at $100
Zero Rates
The terminal value of the investment is $128.40, i.e. 0.05 ( 5 )
100e
In the markets many of the interest rates observed are not pure zero rates
= 128.40
Many instruments for example offer coupon payments which are paid prior to expiry
It is however possible to determine zero rates from the prices of such coupon-bearing instruments
Bond Pricing
Bonds are long-term debt obligations issued by corporations and governments
Bonds are financial instruments designed to:
Funds raised are generally used to support large-scale and long-term expansion and development
Repay the original investment principal at a prespecified maturity date Make periodic coupon interest payments over the life of the investment period
The theoretical price of a bond involves summing the present value of all resulting cash flows
Bond Pricing
Given that the cash flows occur at different points in time, appropriate zero-rates are used for the discounting
To illustrate, consider the following Treasury zero rates Maturity (Years)
Zero Rate (%)
0.5
5.0
1
5.8
1.5
6.4
2
6.8
Bond Pricing Treasury Zero Curve
8 7
Yield (%)
6 5 4 3 2 1 0 0.5
1
1.5 Years
2
Bond Pricing
Consider a 2-year Treasury bond with a face value of $100 and a coupon rate of 6% paid semi-annually
The coupon payment on the bond is $3, which is determined as follows
Pf × rc where
m
=
100( 0.06 ) =3 2
Pf ≡ the face value of the bond rc ≡ the coupon rate on the bond m ≡ the (per year) payment frequency of the coupon
Bond Pricing
The following table details all the cash flows on the bond, along with the present value of each 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)
Note that the appropriate discount rates used for the PV calculations above are the zero rates given previously
Bond Pricing Therefore the price of the bond under consideration is $98.39, i.e. −0.05 ( 0.5 ) −0.058 ( 1) −0.064 ( 1.5 ) −0.068 ( 2 )
3e
+ 3e
+ 3e
+ 103e
= 98.39
Bond Yield
The yield or yield-to-maturity on a couponbearing bond is the rate that equates all cash flows to its market value
Let y denote the yield on a bond, and take the bond considered previously
The yield y on the bond may be determined by solving the following equation − y ( 0.5 ) − y ( 1) − y ( 1.5 ) − y( 2)
3e
+ 3e
+ 3e
+ 103e
= 100
Bond Yield
The solution to the above equation is non-trivial and requires a numerical search routine such as Newton-Raphson
The solution gives a value for the bond yield of 6.76%, i.e. y = 6.76%
Determining Treasury Zero Rates
Treasury zero rates can be calculated from the prices of traded debt instruments
One common method of determining the interest rates is that of bootstrapping
Consider 5 separate bonds, 3 of which are zerocoupon and 2 of which are coupon-bearing
Details of the bonds are given in the next table
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
The zero rates for the 3 zero-coupon bonds can be calculated easily
Determining Treasury Zero Rates
For this note that the zero rate on a zero-coupon bond is given by the following formula Pf − P0
where
Po
×
1 T
Pf ≡ the face value of the bond P0 ≡ the current market price of the bond T ≡ the term-to-maturity of the bond
Note that the above formula gives zero rates using (1/T)-period compounding
That is, discrete compounding rather than continuous compounding
Determining Treasury Zero Rates
In order to express these zero rates using continuous compounding the following formula is used where
Rm Rc = m ln1 + m
Rc ≡ the rate of interest with continous compounding Rm ≡ the rate of interest with discrete compounding m ≡ the compounding frequency of Rm per annum
The above formulas will be illustrated with the first zero-coupon bond
Determining Treasury Zero Rates
The term-to-maturity of the zero-coupon bond is T = 0.25
So the zero rate associated with the bond is for quarterly compounding since
1 m= =4 T
Therefore, the 3-month zero rate with quarterly compounding is 100 − 97.5 R4 = × 4 = 10.256% 97.5
Determining Treasury Zero Rates
The conversion of R4 to the corresponding zero rate with continuous compounding is calculated as follows 0.10256 Rc = 4 ln1 + = 0.10127 = 10.127% 4
Note now that the term-to-maturity of the second zero-coupon bond is T = 0.5
So the zero rate associated with the bond is for semi-annual compounding since m = 1 = 2
T
Determining Treasury Zero Rates
Therefore, the 6-month zero rate with semiannual compounding is 100 − 94.9 R2 = × 2 = 10.748% 94.9
The conversion of R2 to the corresponding zero rate with continuous compounding is calculated as follows 0.10748 Rc = 2 ln1 + = 0.10469 = 10.469% 2
Determining Treasury Zero Rates
In the same way, it can be shown that for the third zero-coupon bond that Rc = 10.536%
Consider now the first coupon-bearing bond presented in the bond data previously
The term-to-maturity of this bond is one and a half years, i.e. T = 1.5
The next table details all the cash flows resulting from this bond
Determining Treasury Zero Rates
Payment Date (Years)
Cash Flow
0.5
4
1
4
1.5
104
From the work done so far the 6-month and 1year zero rates have already been calculated, i.e.
Rc ,0.5 = 10.469% Rc ,1 = 10.536%
Determining Treasury Zero Rates
So the 1.5-year zero rate can be determined by the solving the following pricing relation − Rc ,1.5 ( 1.5 ) −0.10469 ( 0.5 ) −0.10536 ( 1)
4e
+ 4e
+ 104e
= 96
Solving for Rc,1.5 proceeds as follows
96 − 4e −0.10469( 0.5 ) − 4e −0.10536( 1) e = = 0.85196 104 ⇒ − Rc ,1.5 (1.5) = ln ( 0.85196) − Rc ,1.5 ( 1.5 )
⇒ Rc ,1.5
ln( 0.85196) =− = 0.10681 = 10.681% 1. 5
Determining Treasury Zero Rates
Consider now the second coupon-bearing bond presented 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
Determining Treasury Zero Rates
From the work done so far it is known that Rc , 0.5 = 10.469% Rc ,1 = 10.536% Rc ,1.5 = 10.681%
So the 2-year zero rate can be determined by the solving the following pricing relation
6e −0.10469( 0.5 ) + 6e −0.10536( 1) + 6e
− 0.10681( 1.5 )
+ 106e
− Rc , 2 ( 2 )
= 101.6
Determining Treasury Zero Rates
Solving for Rc,2 is straightforward and proceeds as follows
e
− Rc , 2 ( 2 )
= 0.8056
⇒ − Rc , 2 ( 2 ) = ln ( 0.8056 ) ⇒ Rc ,1.5
ln ( 0.8056 ) =− = 0.10808 = 10.808% 2
The next table summarizes the zero rates calculated under the bootstrap method
Determining Treasury Zero Rates
Maturity (Years)
Zero Rate (%)
0.25
10.127
0.5
10.469
1
10.536
1.5
10.681
2
10.808
The following diagram is a graph of the zero rate curve given the rates tabulated above
Determining Treasury Zero Rates 12
11
10.469
10
10.53 6
10.127
10.808
10.68 1
9 0
0.5
1
1.5
2
2.5
Maturity (yrs)
Forward Rates
Forward Interest Rate A Forward Interest Rate is an interest rate which 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 the different forward rates offered to loans of different maturities.
Forward Rates
Forward rates are those rates implied by current zero rates for periods of time in the future
Consider two zero rates Rx and Ry, with maturities Tx and Ty respectively (Ty > Tx)
Let RF denote the forward rate for the period of time between Tx and Ty
RF can be calculated from the two zero rates using the following general formula
Forward Rates RF =
T y − Tx
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:
(100e
R y T y − R x Tx
( 0.25 )( 0.041)
)e
RF ( 0.5 − 0.25 )
= 100e
( 0.043)( 0.5 )
Now, derive the equation above! We are assuming continuously compounded rates
Forward Rates
To illustrate further, consider the following zero rate data Maturity (Years)
Zero Rate (%)
1
10
2
10.5
3
10.8
4
11
5
11.1
We are assuming continuously compounded rates
Forward Rates 7
Treasury Zero Curve
Ry
6.5
Yield (%)
6 5.5Rx 5 4.5 Tx
4 0.5
1
Years
1.5
2
Ty
Forward Rates
Let RF1, 2 denote the forward rate for the period between year 1 and year 2
According to the general formula R F1, 2
R2 ( 2 ) − R1 (1) = 2 −1 = 0.105( 2) − 0.10(1) = 0.11 = 11%
Similarly let R F2 , 3 denote the forward rate for the period between year 2 and year 3
Forward Rates
The general forward rate formula gives RF2 , 3
R3 ( 3) − R2 ( 2 ) = 3− 2 = 0.108( 3) − 0.105( 2 ) = 0.114 = 11.4%
In the same way it is possible to calculate the 1year forward rates for the 4th and 5th years under consideration
The next table presents all the forward rates
Forward Rates Maturity (Years)
Zero Rate (%)
1
10
2
10.5
11
3
10.8
11.4
4
11
11.6
5
11.1
11.5
Forward Rates (for n-th year)
By rewriting the general forward rate formula it is possible to establish important relationships between zero and forward rates
Forward Rates Forward Curve
12 11.5
Yield (%)
11 10.5 10 9.5 Zero-Rate
Forward-Rate
9 8.5 8 1
2
Years
3
4
5
Forward Rates
The general forward rate formula can be rewritten as follows Tx RF = R y + ( R y − Rx ) T y − Tx
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
Forward Rates
Taking limits as Ty approaches Tx leads to the following relationship RF = R + T
∂R ∂T
In the above equation R is the zero rate for a maturity of T
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 Rate Agreements (FRAs)
A Forward Rate Agreement (FRA) is a bilateral or ‘over the counter’ (OTC) interest rate contract in which two counterparties agree to exchange the difference between an agreed interest rate and an as yet unknown reference rate of specified maturity that will prevail at an agreed date in the future.
Payments are calculated against a pre-agreed notional principal The reference rate is typically LIBOR or EURIBOR
Forward Rate Agreements (FRAs)
Consider a FRA that is agreed between two parties with an interest rate of RK applying between times T1 and T2 (T2 > T1)
The interest rate RK applies to some principal L
Forward Rate Agreements (FRAs)
Let R1 and R2 denote the zero rates applying to the maturities T1 and T2 respectively
The next table illustrates the cash flows resulting from the FRA
Date
Cash Flow
T1
-L
T2
+ L{exp[RK(T2-T1)]}
The value of the agreement at time 0, V(0), can be found by taking the present value of these cash flows
Forward Rate Agreements (FRAs) 7
TreasuryZero Curve R2
Yield (%)
6.5 6 5.5
R1
5 4.5 4 0.5
Years
1 T1
1.5
2 T2
Lends (Pays) T1 interest between FRA Buyer Receives L at LT2atplus where R2TI.e. R1TT(0)? What is L worth today? T 2 − at 1 1 & T2 R = K = Le-R isT this worth What today? T2 −I.e. T1 at T(0)? = e-R T (Le-R (T -T )) ) 1 1
2 2
k
2
1
Forward Rate Agreements (FRAs)
Therefore, V(0) is as follows
V (0) = − Le
− R1T1
+ Le
RK ( T2 −T1 )
e
− R2T2
From this it can be noted that V(0) = 0 when
− R1T1 = RK ( T2 − T1 ) − R2T2 R2T2 − R1T1 ⇒ RK = T2 − T1
Forward Rate Agreements (FRAs)
The equation for RK above corresponds to the general forward rate equation from the last section
So the initial value of a FRA is zero when the agreed rate RK is set equal to the corresponding forward rate RF
Forward Rate Agreements (FRAs)
Forward Rate Agreements are usually settled at T1 (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)
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 -R )(T -T )-1) at T2 As the agreement is settled at T1, the lender receives 100MM(e(R -R )(T -T )-1).e(- R )(T -T ) K
M
2
1
K
M
2
1
M
2
1
Note that the lender can lose money Use examples to confirm these cash flows
Further reading
Hull, J.C, “Options, Futures & Other Derivatives”, 2005, 6th Ed.
Chapter 4
Further reading
Hull, J.C, “Options, Futures & Other Derivatives”, 2005, 6th Ed.
Chapter 4
Week 9 – Fixed Income
Bonds
On a point of interest, the cover slide shows a US Government Bond, 4¼% Coupon Issused in 1933. Unusually, there are unused coupons attached
Bonds are still issued by governments, municipalities, semi-state bodies and corporate institutions. Actual certificates are now rarely issued and the coupons are not attached to the bond. This is all done electronically but the principals are the same
Types of Interest Rates
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 borrowings of 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 very low Treasury rates are often used as a proxy for risk-free interest rates
Types of Interest Rates
Source: Reuters 15/09/05
Types of Interest Rates
Source: Reuters 15/09/05
Types of Interest Rates
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 world’s major currencies Each international bank quotes bid and offer rates for such 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
The offer rate is referred to as the London Interbank Offer Rate or LIBOR LIBOR rates tend to be slightly higher than corresponding Treasury rates The reason for this is the LIBOR rates, unlike Treasury rates, are not considered to be entirely risk-free LIBOR rates however do tend to be very low due to the low default risk involved in the interbank deposits Therefore, LIBOR rates are often used as a proxy for risk-free interest rates
Types of Interest Rates
Repo Rates
A repo is an agreement involving the sale of securities by one party to another with a promise to repurchase at a specified price and on a specified date in the future The underlying securities to repos are primarily Treasury and government agent instruments The repo allows short-term returns on excess funds, where the securities form a source of collateral The difference between the sale and repurchase prices represents the interest earned on the repo The level of interest on the repo is referred to as the repo rate
Zero Rates
Zero rates or zero-coupon rates refer to the interest rates applying to investments that continue for some specified term
The n-year zero rate is the interest rate that applies to an n-year investment
All interest and principal is realized at the expiry of the investment, i.e. no intermediate payments
For instance, consider a 5% zero rate on a 5-year investment initiated at $100
Zero Rates
The terminal value of the investment is $128.40, i.e. 0.05 ( 5 )
100e
In the markets many of the interest rates observed are not pure zero rates
= 128.40
Many instruments for example offer coupon payments which are paid prior to expiry
It is however possible to determine zero rates from the prices of such coupon-bearing instruments
Bond Pricing
Bonds are long-term debt obligations issued by corporations and governments
Bonds are financial instruments designed to:
Funds raised are generally used to support large-scale and long-term expansion and development
Repay the original investment principal at a prespecified maturity date Make periodic coupon interest payments over the life of the investment period
The theoretical price of a bond involves summing the present value of all resulting cash flows
Bond Pricing
Given that the cash flows occur at different points in time, appropriate zero-rates are used for the discounting
To illustrate, consider the following Treasury zero rates Maturity (Years)
Zero Rate (%)
0.5
5.0
1
5.8
1.5
6.4
2
6.8
Bond Pricing Treasury Zero Curve
8 7
Yield (%)
6 5 4 3 2 1 0 0.5
1
1.5 Years
2
Bond Pricing
Consider a 2-year Treasury bond with a face value of $100 and a coupon rate of 6% paid semi-annually
The coupon payment on the bond is $3, which is determined as follows
Pf × rc where
m
=
100( 0.06 ) =3 2
Pf ≡ the face value of the bond rc ≡ the coupon rate on the bond m ≡ the (per year) payment frequency of the coupon
Bond Pricing
The following table details all the cash flows on the bond, along with the present value of each 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)
Note that the appropriate discount rates used for the PV calculations above are the zero rates given previously
Bond Pricing Therefore the price of the bond under consideration is $98.39, i.e. −0.05 ( 0.5 ) −0.058 ( 1) −0.064 ( 1.5 ) −0.068 ( 2 )
3e
+ 3e
+ 3e
+ 103e
= 98.39
Bond Yield
The yield or yield-to-maturity on a couponbearing bond is the rate that equates all cash flows to its market value
Let y denote the yield on a bond, and take the bond considered previously
The yield y on the bond may be determined by solving the following equation − y ( 0.5 ) − y ( 1) − y ( 1.5 ) − y( 2)
3e
+ 3e
+ 3e
+ 103e
= 100
Bond Yield
The solution to the above equation is non-trivial and requires a numerical search routine such as Newton-Raphson
The solution gives a value for the bond yield of 6.76%, i.e. y = 6.76%
Determining Treasury Zero Rates
Treasury zero rates can be calculated from the prices of traded debt instruments
One common method of determining the interest rates is that of bootstrapping
Consider 5 separate bonds, 3 of which are zerocoupon and 2 of which are coupon-bearing
Details of the bonds are given in the next table
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
The zero rates for the 3 zero-coupon bonds can be calculated easily
Determining Treasury Zero Rates
For this note that the zero rate on a zero-coupon bond is given by the following formula Pf − P0
where
Po
×
1 T
Pf ≡ the face value of the bond P0 ≡ the current market price of the bond T ≡ the term-to-maturity of the bond
Note that the above formula gives zero rates using (1/T)-period compounding
That is, discrete compounding rather than continuous compounding
Determining Treasury Zero Rates
In order to express these zero rates using continuous compounding the following formula is used where
Rm Rc = m ln1 + m
Rc ≡ the rate of interest with continous compounding Rm ≡ the rate of interest with discrete compounding m ≡ the compounding frequency of Rm per annum
The above formulas will be illustrated with the first zero-coupon bond
Determining Treasury Zero Rates
The term-to-maturity of the zero-coupon bond is T = 0.25
So the zero rate associated with the bond is for quarterly compounding since
1 m= =4 T
Therefore, the 3-month zero rate with quarterly compounding is 100 − 97.5 R4 = × 4 = 10.256% 97.5
Determining Treasury Zero Rates
The conversion of R4 to the corresponding zero rate with continuous compounding is calculated as follows 0.10256 Rc = 4 ln1 + = 0.10127 = 10.127% 4
Note now that the term-to-maturity of the second zero-coupon bond is T = 0.5
So the zero rate associated with the bond is for semi-annual compounding since m = 1 = 2
T
Determining Treasury Zero Rates
Therefore, the 6-month zero rate with semiannual compounding is 100 − 94.9 R2 = × 2 = 10.748% 94.9
The conversion of R2 to the corresponding zero rate with continuous compounding is calculated as follows 0.10748 Rc = 2 ln1 + = 0.10469 = 10.469% 2
Determining Treasury Zero Rates
In the same way, it can be shown that for the third zero-coupon bond that Rc = 10.536%
Consider now the first coupon-bearing bond presented in the bond data previously
The term-to-maturity of this bond is one and a half years, i.e. T = 1.5
The next table details all the cash flows resulting from this bond
Determining Treasury Zero Rates
Payment Date (Years)
Cash Flow
0.5
4
1
4
1.5
104
From the work done so far the 6-month and 1year zero rates have already been calculated, i.e.
Rc ,0.5 = 10.469% Rc ,1 = 10.536%
Determining Treasury Zero Rates
So the 1.5-year zero rate can be determined by the solving the following pricing relation − Rc ,1.5 ( 1.5 ) −0.10469 ( 0.5 ) −0.10536 ( 1)
4e
+ 4e
+ 104e
= 96
Solving for Rc,1.5 proceeds as follows
96 − 4e −0.10469( 0.5 ) − 4e −0.10536( 1) e = = 0.85196 104 ⇒ − Rc ,1.5 (1.5) = ln ( 0.85196) − Rc ,1.5 ( 1.5 )
⇒ Rc ,1.5
ln( 0.85196) =− = 0.10681 = 10.681% 1. 5
Determining Treasury Zero Rates
Consider now the second coupon-bearing bond presented 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
Determining Treasury Zero Rates
From the work done so far it is known that Rc , 0.5 = 10.469% Rc ,1 = 10.536% Rc ,1.5 = 10.681%
So the 2-year zero rate can be determined by the solving the following pricing relation
6e −0.10469( 0.5 ) + 6e −0.10536( 1) + 6e
− 0.10681( 1.5 )
+ 106e
− Rc , 2 ( 2 )
= 101.6
Determining Treasury Zero Rates
Solving for Rc,2 is straightforward and proceeds as follows
e
− Rc , 2 ( 2 )
= 0.8056
⇒ − Rc , 2 ( 2 ) = ln ( 0.8056 ) ⇒ Rc ,1.5
ln ( 0.8056 ) =− = 0.10808 = 10.808% 2
The next table summarizes the zero rates calculated under the bootstrap method
Determining Treasury Zero Rates
Maturity (Years)
Zero Rate (%)
0.25
10.127
0.5
10.469
1
10.536
1.5
10.681
2
10.808
The following diagram is a graph of the zero rate curve given the rates tabulated above
Determining Treasury Zero Rates 12
11
10.469
10
10.53 6
10.127
10.808
10.68 1
9 0
0.5
1
1.5
2
2.5
Maturity (yrs)
Forward Rates
Forward Interest Rate A Forward Interest Rate is an interest rate which 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 the different forward rates offered to loans of different maturities.
Forward Rates
Forward rates are those rates implied by current zero rates for periods of time in the future
Consider two zero rates Rx and Ry, with maturities Tx and Ty respectively (Ty > Tx)
Let RF denote the forward rate for the period of time between Tx and Ty
RF can be calculated from the two zero rates using the following general formula
Forward Rates RF =
T y − Tx
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:
(100e
R y T y − R x Tx
( 0.25 )( 0.041)
)e
RF ( 0.5 − 0.25 )
= 100e
( 0.043)( 0.5 )
Now, derive the equation above! We are assuming continuously compounded rates
Forward Rates
To illustrate further, consider the following zero rate data Maturity (Years)
Zero Rate (%)
1
10
2
10.5
3
10.8
4
11
5
11.1
We are assuming continuously compounded rates
Forward Rates 7
Treasury Zero Curve
Ry
6.5
Yield (%)
6 5.5Rx 5 4.5 Tx
4 0.5
1
Years
1.5
2
Ty
Forward Rates
Let RF1, 2 denote the forward rate for the period between year 1 and year 2
According to the general formula R F1, 2
R2 ( 2 ) − R1 (1) = 2 −1 = 0.105( 2) − 0.10(1) = 0.11 = 11%
Similarly let R F2 , 3 denote the forward rate for the period between year 2 and year 3
Forward Rates
The general forward rate formula gives RF2 , 3
R3 ( 3) − R2 ( 2 ) = 3− 2 = 0.108( 3) − 0.105( 2 ) = 0.114 = 11.4%
In the same way it is possible to calculate the 1year forward rates for the 4th and 5th years under consideration
The next table presents all the forward rates
Forward Rates Maturity (Years)
Zero Rate (%)
1
10
2
10.5
11
3
10.8
11.4
4
11
11.6
5
11.1
11.5
Forward Rates (for n-th year)
By rewriting the general forward rate formula it is possible to establish important relationships between zero and forward rates
Forward Rates Forward Curve
12 11.5
Yield (%)
11 10.5 10 9.5 Zero-Rate
Forward-Rate
9 8.5 8 1
2
Years
3
4
5
Forward Rates
The general forward rate formula can be rewritten as follows Tx RF = R y + ( R y − Rx ) T y − Tx
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
Forward Rates
Taking limits as Ty approaches Tx leads to the following relationship RF = R + T
∂R ∂T
In the above equation R is the zero rate for a maturity of T
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 Rate Agreements (FRAs)
A Forward Rate Agreement (FRA) is a bilateral or ‘over the counter’ (OTC) interest rate contract in which two counterparties agree to exchange the difference between an agreed interest rate and an as yet unknown reference rate of specified maturity that will prevail at an agreed date in the future.
Payments are calculated against a pre-agreed notional principal The reference rate is typically LIBOR or EURIBOR
Forward Rate Agreements (FRAs)
Consider a FRA that is agreed between two parties with an interest rate of RK applying between times T1 and T2 (T2 > T1)
The interest rate RK applies to some principal L
Forward Rate Agreements (FRAs)
Let R1 and R2 denote the zero rates applying to the maturities T1 and T2 respectively
The next table illustrates the cash flows resulting from the FRA
Date
Cash Flow
T1
-L
T2
+ L{exp[RK(T2-T1)]}
The value of the agreement at time 0, V(0), can be found by taking the present value of these cash flows
Forward Rate Agreements (FRAs) 7
TreasuryZero Curve R2
Yield (%)
6.5 6 5.5
R1
5 4.5 4 0.5
Years
1 T1
1.5
2 T2
Lends (Pays) T1 interest between FRA Buyer Receives L at LT2atplus where R2TI.e. R1TT(0)? What is L worth today? T 2 − at 1 1 & T2 R = K = Le-R isT this worth What today? T2 −I.e. T1 at T(0)? = e-R T (Le-R (T -T )) ) 1 1
2 2
k
2
1
Forward Rate Agreements (FRAs)
Therefore, V(0) is as follows
V (0) = − Le
− R1T1
+ Le
RK ( T2 −T1 )
e
− R2T2
From this it can be noted that V(0) = 0 when
− R1T1 = RK ( T2 − T1 ) − R2T2 R2T2 − R1T1 ⇒ RK = T2 − T1
Forward Rate Agreements (FRAs)
The equation for RK above corresponds to the general forward rate equation from the last section
So the initial value of a FRA is zero when the agreed rate RK is set equal to the corresponding forward rate RF
Forward Rate Agreements (FRAs)
Forward Rate Agreements are usually settled at T1 (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)
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 -R )(T -T )-1) at T2 As the agreement is settled at T1, the lender receives 100MM(e(R -R )(T -T )-1).e(- R )(T -T ) K
M
2
1
K
M
2
1
M
2
1
Note that the lender can lose money Use examples to confirm these cash flows
Further reading
Hull, J.C, “Options, Futures & Other Derivatives”, 2005, 6th Ed.
Chapter 4
Further reading
Hull, J.C, “Options, Futures & Other Derivatives”, 2005, 6th Ed.
Chapter 4
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