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92 ©2013 Pearson Education, Inc. Publishing as Prentice Hall
Chapter 7
Interest Rate Forwards and Futures
Question 7.1
Using the bond valuation formulas (7.1), (7.3), (7.6) we obtain
the following yields and prices:
Maturity Zero-Coupon Bond Yield
Zero-Coupon Bond Price
One-Year Implied Forward Rate
Par Coupon
Cont. Comp. Zero Yield
1 0.04000 0.96154 0.04000 0.04000 0.03922
2 0.04500 0.91573 0.05003 0.04489 0.04402
3 0.04500 0.87630 0.04500 0.04492 0.04402
4 0.05000 0.82270 0.06515 0.04958 0.04879
5 0.05200 0.77611 0.06003 0.05144 0.05069
Question 7.2
The coupon bond pays a coupon of $60 each year plus the
principal of $1,000 after five years. We have cash flows of [60,
60, 60, 60, 1,060]. To obtain the price of the coupon bond, we
multiply each cash flow by the zero-coupon bond price of that year.
This yields a bond price of $1,037.25280.
Question 7.3
Maturity Zero-Coupon Bond Yield
Zero-Coupon Bond Price
One-Year Implied Forward Rate
Par Coupon
Cont. Comp. Zero Yield
1 0.03000 0.97087 0.03000 0.03000 0.02956
2 0.03500 0.93351 0.04002 0.03491 0.03440
3 0.04000 0.88900 0.05007 0.03974 0.03922
4 0.04500 0.83856 0.06014 0.04445 0.04402
5 0.05000 0.78353 0.07024 0.04903 0.04879
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Question 7.4
Maturity Zero-Coupon Bond Yield
Zero-Coupon Bond Price
One-Year Implied Forward Rate
Par Coupon
Cont. Comp. Zero Yield
1 0.05000 0.95238 0.05000 0.05000 0.04879
2 0.04200 0.92101 0.03406 0.04216 0.04114
3 0.04000 0.88900 0.03601 0.04018 0.03922
4 0.03600 0.86808 0.02409 0.03634 0.03537
5 0.02900 0.86681 0.00147 0.02962 0.02859
Question 7.5
Maturity Zero-Coupon Bond Yield
Zero-Coupon Bond Price
One-Year Implied Forward Rate
Par Coupon
Cont. Comp. Zero Yield
1 0.07000 0.93458 0.07000 0.07000 0.06766
2 0.06000 0.88999 0.05009 0.06029 0.05827
3 0.05000 0.86384 0.03028 0.05065 0.04879
4 0.04500 0.83855 0.03016 0.04578 0.04402
5 0.04000 0.82193 0.02022 0.04095 0.03922
Question 7.6
In order to be able to solve this problem, it is best to take
equation (7.6) of the main text and solve progressively for all
zero-coupon bond prices, starting with Year 1. This yields the
series of zero-coupon bond prices from which we can proceed as
usual to determine the yields.
Maturity Zero-Coupon Bond Yield
Zero-Coupon Bond Price
One-Year Implied Forward Rate
Par Coupon
Cont. Comp. Zero Yield
1 0.03000 0.97087 0.03000 0.03000 0.02956
2 0.03500 0.93352 0.04002 0.03491 0.03440
3 0.04000 0.88899 0.05009 0.03974 0.03922
4 0.04700 0.83217 0.06828 0.04629 0.04593
5 0.05300 0.77245 0.07732 0.05174 0.05164
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Question 7.7
a) We are looking for r0 (1, 3). We will use equation (7.3) of
the main text, and the known one-year and three-year zero-coupon
bond prices. We have to solve the following equation:
( ) 3 101 1,3r−
+⎡ ⎤⎣ ⎦ = ( )( )0,10,3
PP
⇔ r0 (1, 3) = ( )( )0,1 0.9433961 1 0.075040,3 0.816298
PP
− = − =
b) Let’s calculate the zero-coupon bond price from Year 1 to 2
and from Year 1 to 3, they are:
P0 (1, 2) = ( )( )0,2 0.881659 0.934560,1 0.943396
PP
= =
P0 (1, 3) = ( )( )0,3 0.816298 0.865280,1 0.943396
PP
= =
Now, we have the relevant implied forward zero-coupon prices and
can find the coupon of the par two-year coupon bond issued at time
1 according to formula (7.6).
c = ( )( ) ( )
0
0 0
1 1,3 0.13472 0.0748511,2 1,3 0.93456 0.86528
PP P
−= =
+ +
Question 7.8
a) We have to take into account the interest we (or our
counterparty) can earn on the FRA settlement if we settle the loan
on initiation day and not on the actual repayment day. Therefore,
we tail the FRA settlement by the prevailing market interest rate
of 5 percent. The dollar settlement is:
( )1 annually FRA
annually
r rr−
+× notional principal = ( )0.05 0.06
1 0.05−
+ × $500,000.00 = −$4,761.905
b) If the FRA is settled on the date the loan is repaid (or
settled in arrears), the settlement amount is determined by:
( )annually FRAr r− × notional principal = (0.05 − 0.06) ×
$500,000.00 = −$5,000 We have to pay at the settlement because the
interest rate we could borrow at is 5 percent,
but we have agreed via the FRA to a borrowing rate of 6 percent.
Interest rates moved in an unfavorable direction.
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Question 7.9
a) We have to take into account the interest we (or our
counterparty) can earn on the FRA settlement if we settle the loan
on initiation day and not on the actual repayment day. Therefore,
we tail the FRA settlement by the prevailing market interest rate
of 7.5 percent. The dollar settlement is:
( )1 annually FRA
annually
r rr−
+× notional principal = ( )0.075 0.06
1 0.075−
+ × $500,000.00 = −$6,976.744
b) If the FRA is settled on the date the loan is repaid (or
settled in arrears), the settlement amount is determined by:
( )annually FRAr r− × notional principal = (0.05 − 0.06) ×
$500,000.00 = −$5,000 We receive money at the settlement because
our hedge pays off. The market interest rate has gone up, making
borrowing more expensive. We are compensated for this loss through
the insurance that the short position in the FRA provides.
Question 7.10
We can find the implied forward rates using the following
formula:
( ) ( )( )0
0
0,1 ,
0,P t
r t t sP t s
+ = =⎡ ⎤⎣ ⎦ +
This yields the following rates on the synthetic FRAs:
r0 (90, 180) = 0.990090.97943
− 1 = 0.010884
r0 (90, 270) = 0.990090.96525
− 1 = 0.025734
r0 (90, 360) = 0.990090.95238
− 1 = 0.039596
Question 7.11
We can find the implied forward rate using the following
formula:
( ) ( )( )0
0
0,1 ,
0,P t
r t t sP t s
+ + =⎡ ⎤⎣ ⎦ +
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With the numbers of the exercise, this yields:
r0 (180, 360) = 0.979430.95238
− 1 = 0.028403
The following table follows the textbook in looking at forward
agreements from a borrower’s perspective (i.e., a borrower goes
long an FRA to hedge his position, and a lender is thus short the
FRA).
Transaction t = 0 t = 180 t = 360
Enter short FRA −10M +10M × 1.028403
= 10.28403M
Buy 9.7943M Zero Coupons maturing at −9.7943M +10M
time t = 180
Sell (1 + 0.028403) ∗ 10M ∗ 0.95238 Zero +10M × 1.028403
−10.28403M
Coupons maturing at time t = 360 ×0.95238 = 9.7943M
TOTAL 0 0 0
By entering in the above mentioned positions, we are perfectly
hedged against the risk of the FRA. Please note that we are making
use of the fact that interest rates are perfectly predictable.
Question 7.12
We can find the implied forward rate using the following
formula:
( ) ( )( )0
0
0,1 ,
0,P t
r t t sP t s
+ = =⎡ ⎤⎣ ⎦ +
With the numbers of the exercise, this yields:
r0 (270, 360) = 0.965250.95238
− 1 = 0.0135135
The following table follows the textbook in looking at forward
agreements from a borrower’s perspective (i.e., a borrower goes
long on an FRA to hedge his position and a lender is thus short the
FRA). Since we are the counterparty for a lender, we are in fact
the borrower and thus long the forward rate agreement.
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Transaction today t = 0 t = 270 t = 360
Enter long FRA 10M −10M × 1.013514
= −10.13514M
Sell 9.6525M Zero Coupons 9.6525M −10M
maturing at time t = 180
Buy (1 + 0.013514) ∗ 10M ∗ 0.95238 −10M × 1.013514
+10.13514M
Zero Coupons maturing at time t = 360 ×0.95238 = −9.6525M
TOTAL 0 0 0
By entering in the above mentioned positions, we are perfectly
hedged against the risk of the FRA. Please note that we are making
use of the fact that interest rates are perfectly predictable.
Question 7.13
First, let us calculate the value of the three-year par coupon
bond after we have held it for two years. After two years, the bond
still entitles us to receive one coupon and the repayment of the
principal. We have to discount those payments, which we receive at
t = 3, with the implied forward rate from Year 2 to 3. This
determines the value of the three-year par coupon bond after two
years. We have:
B2 = 106.954851.0800705
= 99.0258
Furthermore, after one year, we received a coupon of 6.95485
dollars, which we reinvested at the prevailing interest rate, the
implied forward rate from Year 1 to Year 2, 1.0700237, and we
receive a coupon of 6.95485 at the end of Year 2. The value of the
coupons at the end of Year 2 is:
Sum of coupon values = 6.95485 × 1.0700237 + 6.95485 =
14.3967045
Therefore, our two-year gross return is:
2 − year return = 14.3967045 99.0258100
+ − 1 = 1.134225
Finally, we annualize this return by taking its square root.
This yields an annualized gross return of 1.065, which was to be
shown.
Question 7.14
We would like to guarantee the return of 6.5 percent. We receive
payments 6.95485 after Year 1 and Year 2, and a payment of
106.95485 after Year 3. If interest rates are uncertain, we face
an
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interest rate risk for the investment of the first coupon from
Year 1 to Year 2 and for the discounting of the final payment from
Year 3 to Year 1.
Suppose we enter into a forward rate agreement to lend 6.95484
from Year 1 to Year 2 at the current forward rate from Year 1 to
Year 2, and we enter into a forward rate agreement to borrow
106.95485 tailed by the prevailing forward rate for Year 2 to Year
3, at the prevailing forward rate. This leads to the following
cash-flow table:
Transaction today t = 0 t = 1 t = 2 t = 3
Buy three-year par bond −100
Receive first coupon 0 6.95485
Enter short FRA 0 −6.95485 6.95485 × 1.0700237
Receive second coupon 0 6.95485
Enter long FRA for tailed position 0 106.95485/1.0800705
−106.95485
Receive final coupon and principal 0 = 99.025804 106.95485
TOTAL −100 0 113.4225 0
We see that we can secure the same gross return as in the
previous question, 113.4225 100÷ = 1.065. By entering appropriate
FRAs, we secured the desired return of 6.5 percent. Please note
that we made use of the fact that we knew that we wanted to undo
the position at t = 2.
Question 7.15
a) Let us follow the suggestion of the problem and buy the
two-year zero-coupon bond. We will create a synthetic lending
opportunity at the zero-coupon implied forward rate of 7.00238
percent, and we will finance it by borrowing at 6.8 percent, thus
creating an arbitrage opportunity. In particular, we will have:
Transaction Today t = 0 t = 1 t = 2
Buy 1.0700237 two-year zero-coupon bonds −0.881659 *1.0700237 0
1.0700237
= −0.943396
Sell one one-year zero coupon bond +0.943396 −1
Borrow one from Year 1 to Year 2 @ 6.8% +1 −1.06800
TOTAL 0 0 0.0020237
We see that we have created something out of nothing, without
any risk involved. We have found an arbitrage opportunity.
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b) Let us follow the suggestion of the problem and sell the
two-year zero-coupon bond. We will create a synthetic borrowing
opportunity at the zero-coupon implied forward rate of 7.00238
percent, and we will lend at 7.2 percent, thus creating an
arbitrage opportunity. In particular, we will have:
Transaction Today t = 0 t = 1 t = 2
Sell 1.0700237 two-year zero-coupon bonds 0.881659 *1.0700237 0
−1.0700237
= 0.943396
Buy one one-year zero coupon bond −0.943396 +1
Lend one from Year 1 to Year 2 @ 7.2% −1 +1.07200
TOTAL 0 0 0.0019763
We see that we have created something out of nothing, without
any risk involved. We have indeed found an arbitrage
opportunity.
Question 7.16
a) The implied LIBOR of the September Eurodollar futures of 96.4
is: 100 96.4400− = 0.9%
b) As we want to borrow money, we want to buy protection against
high interest rates, which means low Eurodollar future prices. We
will short the Eurodollar contract.
c) One Eurodollar contract is based on a $1 million three-month
deposit. As we want to hedge a loan of $50M, we will enter into 50
short contracts.
d) A true three-month LIBOR of 1 percent means an annualized
position (annualized by market conventions) of 1% ∗ 4 = 4%.
Therefore, our 50 short contracts will pay:
[96.4 − (100 − 4) × 100 × $25 × 50] = $50,000
The increase in the interest rate has made our loan more
expensive. The futures position that we entered to hedge the
interest rate exposure compensates for this increase. In
particular, we pay $50,000,000 × 0.01 − payoff futures = $500,000 −
$50,000 = $450,000, which corresponds to the 0.9 percent we sought
to lock in.
Question 7.17
aa) The interest rate is higher than the rate of the forward
rate agreement, therefore the lender must pay the borrower. If the
FRA is settled on day 60, the payment made by the lender to the
borrower is:
150
150
( )1
FRAr rr−+
× notional principal = (0.028 0.025)
1 0.028−
+ × $100,000,000.00 = $291,828.79
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ab) If the FRA is settled on the date that the loan is repaid
(or settled in arrears), the settlement amount is determined
by:
(r150 − rFRA) × notional principal = (0.028 − 0.025) ×
$100,000,000.00 = $300,000.00
The lender pays the borrower because we are in the state of the
world in which the lender does not need protection: Interest rates
have risen, and thus he makes a payment to bring back the interest
he earns to 2.5 percent.
ba) The interest rate is lower than the rate of the forward rate
agreement; therefore, the lender will receive payment from the
borrower. If the FRA is settled on day 60, the payment made by the
borrower to the lender is:
150
150
( )1
FRAr rr−+
× notional principal = (0.022 0.025)1 0.022
−+
× $100,000,000.00 = −$293,542.07
bb) If the FRA is settled on the date the loan is repaid (or
settled in arrears), the settlement amount is determined by:
(r150 − rFRA) × notional principal = (0.022 − 0.025) ×
$100,000,000.00 = −$300,000.00
The lender is paid by the borrower because we are in the state
of the world in which the lender’s protection pays off: Interest
rates have gone down, and thus she is compensated for the loss in
investment proceeds. The payment of the borrower brings back the
interest she earns to 2.5 percent.
Question 7.18
a) We face the classic problem of asset mismatch. We are
interested in locking in an interest rate for a 150-day investment,
60 days from now. However, while the Eurodollar futures matures 60
days from now, it secures a lending rate for 90 days. We face the
problem that the 90-day and 150-day interest rates may not be
perfectly correlated. (For example, the term structure could, over
the next 60 days, move from upward sloping to downward
sloping).
b) As we want to lend money, we want to buy protection against
low interest rates, which means high Eurodollar future prices. We
will therefore long the Eurodollar contract.
c) The implied LIBOR of the September Eurodollar futures of 94
is: 100 94400− = 1.5 percent.
Under the assumption that the three-month LIBOR rate and the
150-day interest rate are based on the same annualized interest
rate of 6 percent, we are able to lock in an interest rate
of: 1.5% × 15090
= 2.5 percent. Please note that this is a rather strong
assumption.
d) One Eurodollar futures contract is based on a $1 million
three-month deposit. As we want to hedge an investment of $100M, we
will enter into 100 long contracts. Again, we are making the strong
assumption that the annualized three-month LIBOR rate and the
annualized 150 day rate are identical and perfectly correlated.
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Question 7.19
We will calculate the necessary different bond prices according
to the formula:
B(y) = ( )1 1
ni
ii
Cy= +
∑
Ci are the coupon payments, and for i = n, it includes the
payment of the principal.
This yields the following bond prices, and the price difference
with respect to the base case of a yield of 7 percent:
Seven-year, 6 percent coupon bond:
Yield Price Price change relative to 7% yield price
6.5% 97.2577 2.647
6.75% 95.9226 1.3119
7% 94.6107 0
7.25% 93.3217 −1.289
7.5% 92.0551 −2.5556
10-year, 8 percent coupon bond:
Yield Price Price change relative to 7.5% yield price
7% 107.0236 3.5916
7.25% 105.2073 1.7753
7.5% 103.4320 0
7.75% 101.6966 −1.7354
8% 100.0000 −3.432
We are now able to solve for the true hedge ratio by equating
the price changes corresponding to the yield changes. We solve:
price change bond1 = −N × price change bond2
This gives for:
an increase of 25 basis points: N = −0.74277
a decrease of 25 basis points: N = −0.73897
an increase of 50 basis points: N = −0.7446
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a decrease of 50 basis points: N = −0.737
We see that the hedge ratios for increases and decreases of the
same number of basis points differ. Note as well that the
difference between the hedge ratios increases with an increase in
the basis points. This is a consequence of the convexity of the
bonds. It is caused by the changes in duration as the interest rate
changes.
Question 7.20
We will use the Excel functions Duration and Mduration to
calculate the required durations. They are of the form:
MDURATION(Start Date; Terminal Date; Coupon; Yield;
Frequency)
DURATION(Start Date; Terminal Date; Coupon; Yield;
Frequency),
where frequency determines the number of coupon payments per
year. In order to use the function, we have to give Excel a start
date and terminal date, but we can just pick two dates that are
exactly the requested number of years apart. Plugging in the values
of the exercises yields:
a) Macaulay Duration = 4.59324084
Modified Duration = 4.3983078
b) Macaulay Duration = 5.99377496
Modified Duration = 5.73566981
c) We need to find the yield to maturity of this bond first. We
can do so by using the YIELD function of Excel. Plugging in the
relevant values, we get: Yield = 0.07146759. Now we can again use
the Mduration and Duration formulas. This yields:
Macaulay Duration = 7.6955970
Modified Duration = 7.1822957
Question 7.21
b) Let us start with part (b) because we already know the Excel
function for the Macaulay duration, DURATION().
Using the equation with DURATION(01/01/1980; 01/01/2000; 0.06;
0.20; 2), and DURATION (01/01/1970; 01/01/2000; 0.06; 0.20; 2)
yields for the 20-year bond:
Macaulay Duration = 6.09533079
and for the 30-year bond:
Macaulay Duration = 5.66839682
a) Now we can back out the price value of a basis point by
multiplying the Macaulay duration by −B(y)/(1 + y). In order to do
so, we must find the prices of the two bonds. As the yield to
maturity is given, we simply have:
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B(y) = ( )1 1
ni
ii
Cy= +
∑
This yields for the two bonds:
B(6, 20y) = 31.546645
B(6, 30y) = 30.229899
Therefore, we have the following PVPBs:
PVBP(20 years) = −6.09533079 × 31.5466451.06
= −181.4031
PVBP(30 years) = −5.66839682 × 30.2298991.06
= −161.6557
c) As we can see from the above example, this statement is not
always true. The above example is an extreme one in which the yield
to maturity is extremely high. Since the Macaulay duration is a
weighted average of the time until the bond payments occur, with
the weights being the percentage of the bond price accounted for by
each payment, we see that with a very high yield, the last payments
get significantly more discounted than the previous ones. For a
coupon bond, the last payment is the largest one, being interest
plus principal. Therefore, with a high yield, the largest payment
of a long term bond gets a higher discount than a bond with the
same characteristics but a shorter maturity. Overall, this can make
the duration of a long-term bond smaller than that of a short-term
one.
Question 7.22
We will exploit equation (7.13) of the main text to find the
optimal hedge ratio:
N = ( ) ( )( ) ( )
( )( )
1 1 1 1
2 2 2 2
/ 1 6.631864 106.44 / 1.05004 672.255918/ 1 7.098302 112.29 /
1.05252 757.2951883
D B y yD B y y
+ ×− = = − =
+ ×−0.887707
Therefore, we have to short 0.887707 units of the nine-year bond
for every eight-year bond to obtain a duration-matched
portfolio.
Question 7.23
We can verify the conversion factor for the six-year semiannual
bond by calculating in Excel:
PRICE(1/1/94, 1/1/00, 0.04, 0.06, 100, 2) = 90.045996
For the eight-year bond, we calculate a conversion factor
of:
PRICE(1/1/92, 1/1/00, 0.055, 0.06, 100, 2) = 96.8597245
Note that those Excel bond values are for $100 par, so the
conversion factors are obtained by dividing the above results by
100.
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Now, we are in a position to calculate the difference between
the invoice price and the market price, using the futures price of
117.02.
six-year bond: 117.02 × 0.90046 − 102.48 = 2.8961
eight-year bond: 117.02 × 0.96860 − 113.564 = −0.2188
We see that it is more advantageous for the short position to
deliver the six-year bond, as the owner of the short position can
make a small profit by delivering the six-year bond. He would lose
money if he delivered the eight-year bond.
Question 7.24
a) Compute the convexity of a three-year bond paying annual
coupons of 4.5 percent and selling at par. For a par bond, the
yield to maturity is equal to the coupon, or 4.5 percent in our
case. We can calculate the convexity based on the formula:
Convexity ( )( )
( )( )2 22 21
1 111 1
ni ni
y
i i n nC m MB m my m y m+ +=
⎡ ⎤+ += +⎢ ⎥
+ +⎢ ⎥⎣ ⎦∑
( )( )
( )( )1 2 2 22 2
1 1 1 2 2 11 4.5 1 4.5 1100 1 11 0.045 1 0.045+ +
⎡ + += +⎢
+ +⎢⎣
( )( )3 22
3 3 1 104.51 1 0.045 1 +
⎤++ ⎥
+ ⎥⎦
= 10.3680
b) Compute the convexity of a three-year 4.5 percent coupon bond
that makes semiannual coupon payments and that currently sells at
par.
Now, m = 2. We have n = m ∗ T = 2 ∗ 3 = 6. The convexity is:
Convexity ( )( )
( )( )2 22 21
1 111 1
ni ni
y
i i n nC m MB m my m y m+ +=
⎡ ⎤+ += +⎢ ⎥
+ +⎢ ⎥⎣ ⎦∑
( )( )
( )( )1 2 2 22 2
1 1 1 2 2 11 2.25 1 2.25 1100 2 21 0.0225 1 0.0225+ +
⎡ + += +⎢
+ +⎢⎣
( )( )
( )( )3 2 4 22 2
3 3 1 4 4 12.25 1 2.25 12 21 0.0225 1 1 0.0225 1+ ++ +
+ ++ +
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( )( )
( )( )5 2 6 22 2
5 5 1 6 6 12.25 1 102.252 21 0.0225 1 1 0.0225 1+ +
⎤+ ++ + ⎥
+ + ⎥⎦
= 9.3302
c) Is the convexity different in the two cases? Why?
Yes, the convexity for the semiannual bond is smaller. We spread
the bond payments out over more periods, which makes the bond’s
duration slightly less susceptible to interest rate changes.
Question 7.25
Suppose a 10-year zero-coupon bond with a face value of $100
trades at $69.20205.
a) What is the yield to maturity and modified duration of the
zero-coupon bond?
The yield of the zero-coupon bond is equal to
(1 + y)10 = ( )01000,10P
y = 1 10100 1
69.20205⎛ ⎞ −⎜ ⎟⎝ ⎠
= 0.03750
Calculate its modified duration.
The modified duration of the ten-year zero-coupon bond is:
modified duration ( ) ( ) ( )11 1
1 1 1n
i ni
i C m n MB y y m m my m y m=
⎡ ⎤= × +⎢ ⎥
+ + +⎢ ⎥⎣ ⎦∑
=( )10
1 1 10 10069.20205 1 0.0375 1 1 0.0375
⎡ ⎤× ⎢ ⎥
+ +⎢ ⎥⎣ ⎦
= 101.0375
=9.63855
b) Calculate the approximate bond price change for a 50 basis
point increase in the yield, based on the modified duration you
calculated in part (a). Also calculate the exact new bond price
based on the new yield to maturity.
We can use the formula in the main text:
B (y + ε) = B (y) − [DMod × B (y) ε]
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106 Part Two/Forwards, Futures, and Swaps
©2013 Pearson Education, Inc. Publishing as Prentice Hall
= 69.20205 − 9.63855 × 69.20205 × 0.0050
B (0.0425) = 65.86701
New bond price:
B (0.0425) = 10100
1.0424 = 66.01703
c) Calculate the convexity of the 10-year zero-coupon bond.
Convexity ( )
( )( )
( )( )2 22 21
1 111 1
ni ni
i i n nC m MB y m my m y m+ +=
⎡ ⎤+ += +⎢ ⎥
+ +⎢ ⎥⎣ ⎦∑
( )( )10 22
10 10 11 100069.20205 1 1 0.0375 +
⎡ ⎤+= +⎢ ⎥
+⎢ ⎥⎣ ⎦
= 102.19190
d) Now use the formula (equation 7.15) that takes into account
both duration and convexity to approximate the new bond price.
Compare your result to that in part (b).
B (y + ε) = B (y) − [DMod × B (y) ε] + 0.5 × Convexity × B (y) ×
ε2 = 69.20205 − 9.63855 × 69.20205 × 0.0050 + 0.5 × 102.19190 ×
69.20205
×0.00502
B (0.0515) = 65.86701 + 0.088299
= 65.95541
The approximation is much better now compared to the result in
part (b), but it is still somewhat off the true price. The long
time to maturity and the considerable change in basis points for
this bond is responsible for the deviation.