Chapter 16
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Jan 9, 2024
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1.
Award: 10.00
points
Problems? Adjust credit for all students.
A 9-year bond paying coupons annually has a yield of 10% and a duration of 7.194 years. Assume the market yield changes by 50 basis points.
Required:
What is the percentage change in the bond’s price?
Note: Assume modified duration and a positive increase in yield change. Do not round intermediate calculations. Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.
The percentage change in the bond’s price is
(3.27)
%
Explanation:
Assume the 7.194 is the Macaulay duration (not Modified Duration). The percentage change in the bond’s price is:
− (D ÷ (1 + y
)) × Δ
y
= −(7.194 ÷ 1.10) × 0.005 = −0.0327 = −3.27%, or a 3.27% decline
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
2.
Award: 10.00
points
Problems? Adjust credit for all students.
Required:
a. Find the duration of a 6% coupon bond making annual
coupon payments if it has three years until maturity and has a yield to maturity of 6%.
Note: The face value of the bond is $1,000. Do not round intermediate calculations. Round your answer to 2 decimal places.
b. What is the duration if the yield to maturity is 10%?
Note: The face value of the bond is $1,000. Do not round intermediate calculations. Round your answer to 2 decimal places.
a. 6% YTM
2.83
years
b. 10% YTM
2.82
years
Explanation:
a. YTM = 6%
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount
rate = 6%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$ 60.00
$ 56.60
0.0566
0.0566
2
60.00
53.40
0.0534
0.1068
3
1,060.00
890.00
0.8900
2.6700
Column Sums:
$ 1,000.00
1.0000
2.8334
Duration = 2.83 years
b. YTM = 10%
(1)
Time until
Payment (Years)
(2)
Cash Flow
(3)
PV of CF
(Discount rate =
10%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$ 60.00
$ 54.55
0.0606
0.0606
2
60.00
49.59
0.0551
0.1101
3
1,060.00
796.39
0.8844
2.6531
Column Sums:
$ 900.53
1.0000
2.8238
Duration = 2.82 years, which is less than the duration at the YTM of 6%.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
3.
Award: 10.00
points
Problems? Adjust credit for all students.
Find the duration of a 6% coupon bond making semiannually coupon payments if it has three years until maturity and has a yield to maturity of 6%. What is the duration if the yield to maturity is 10%?
Note: The face value of the bond is $1,000. Do not round intermediate calculations. Round your answers to 2 decimal places.
Duration
6% YTM
2.79
years
10% YTM
2.78
years
Explanation:
For a semiannual 6% coupon bond selling at par, we use the following parameters: coupon = 3% per half-year period, y
= 3%, T
= 6 semiannual periods.
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount rate = 3%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 30.00
$ 29.13
0.02913
0.0291
2
30.00
28.28
0.02828
0.0565
3
30.00
27.45
0.02745
0.0824
4
30.00
26.65
0.02665
0.1066
5
30.00
25.88
0.02588
0.1294
6
1,030.00
862.61
0.86261
5.1757
Column Sums:
$ 1,000.00
1.00000
5.5797
D
= 5.5797 half-year periods = 2.79 years
If the bond’s yield is 10%, use a semiannual yield of 5% and semiannual coupon of 3%:
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount rate =
5%)
(4)
Weight
(5)
Column (1) × Column
(4)
1
$ 30.00
$ 28.57
0.03180
0.0318
2
30.00
27.21
0.03029
0.0606
3
30.00
25.92
0.02884
0.0865
4
30.00
24.68
0.02747
0.1099
5
30.00
23.51
0.02616
0.1308
6
1,030.00
768.60
0.85544
5.1326
Column Sums:
$ 898.49
1.00000
5.5522
D
= 5.5522 half-year periods = 2.78 years
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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4.
Award: 10.00
points
Problems? Adjust credit for all students.
You predict that interest rates are about to fall. Which bond will give you the highest capital gain?
Zero coupon, long maturity
Explanation:
Investors tend to purchase longer term bonds when they expect yields to fall so that they can capture significant capital gains, and the lack of a coupon payment ensures the capital gain will be even greater.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
5.
Award: 10.00
points
Problems? Adjust credit for all students.
In each of the following pairs of bonds, select the bond that has the highest duration or effective duration:
Required
a. Bond A
is a 6% coupon bond, with a 20-year time to maturity selling at par value. Bond B
is a 6% coupon bond, with a 20-year time to maturity selling below par value.
b. Bond A
is a 20-year noncallable coupon bond with a coupon rate of 6%, selling at par. Bond B
is a 20-year callable bond with a coupon rate of 7%, also selling at par.
a. Highest duration
Bond A
b. Highest duration
Bond A
Explanation:
a. Bond B has a higher yield to maturity than bond A since its coupon payments and maturity are equal to those of A, while its price is lower. (Perhaps the yield is higher because of differences in credit risk.) Therefore, the
duration of Bond B must be shorter.
b. Bond A has a lower yield and a lower coupon, both of which cause Bond A to have a longer duration than Bond B. Moreover, A cannot be called, so that its maturity is at least as long as that of B, which generally increases
duration.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
6.
Award: 10.00
points
Problems? Adjust credit for all students.
An insurance company must make payments to a customer of $10 million in one year and $4 million in five years. The yield curve is flat at 10%.
Required:
a. If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
b. What must be the face value and market value of that zero-coupon bond?
Note: Do not round intermediate calculations. Enter your answers in millions rounded to 2 decimal places.
$
$
a. Maturity of zero coupon bond
1.86
years
b. Face value
13.82
million
b. Market value
11.57
million
Explanation:
a. (1) Time until
Payment (Years)
(2) Cash Flow
(3) PV of CF (Discount Rate
= 10%)
(4) Weight
(5) Column (1) ×
Column (4)
1
$ 10
million
$ 9.09
million
0.7854
0.7854
5
4
million
2.48
million
0.2146
1.0729
Column Sums
$ 11.57
million
1.0000
1.8583
D
= 1.86 years = required maturity of zero-coupon bond.
b. The market value of the zero must be $11.57 million, the same as the market value of the obligations. Therefore, the face value must be:
$11.57 million × (1.10)
1.8583
= $13.82 million
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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7.
Award: 10.00
points
Problems? Adjust credit for all students.
Long-term Treasury bonds currently are selling at yields to maturity of nearly 6%. You expect interest rates to fall. The rest of the market thinks that they will remain unchanged over the coming year.
If you are correct, choose the bond that will provide the higher holding-period return over the next year in each of the following:
Required:
a. i. A Baa-rated bond with coupon rate 6% and time to maturity 20 years.
ii. An Aaa-rated bond with coupon rate of 6% and time to maturity 20 years.
b. i. An A-rated bond with coupon rate 3% and maturity 20 years, callable at 105.
ii. An A-rated bond with coupon rate 6% and maturity 20 years, callable at 105.
c. i. A 4% coupon noncallable T-bond with maturity 20 years and YTM = 6%.
ii. A 7% coupon noncallable T-bond with maturity 20 years and YTM = 6%.
a. Select the bond that will provide the higher holding-period return
Bond ii
b. Select the bond that will provide the higher holding-period return
Bond i
c. Select the bond that will provide the higher holding-period return
Bond i
Explanation:
In each case, choose the longer-duration bond in order to benefit from a rate decrease.
a. ii. The Aaa-rated bond has the lower yield to maturity and therefore the longer duration.
b. i. The lower-coupon bond has the longer duration and greater de facto call protection.
c. i. The lower coupon bond has the longer duration.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
8.
Award: 10.00
points
Problems? Adjust credit for all students.
Currently, the term structure is as follows: 1-year zero-coupon bonds yield 7%; 2-year zero-coupon bonds yield 8%; 3-year and longer-maturity zero-coupon bonds all yield 9%. You are choosing between 1-, 2-, and 3-year
maturity bonds all paying annual coupons of 8%.
Required:
a. What is the price of each bond today?
b. What will be the price of each bond in one year if the yield curve is flat at 9% at that time?
c. What will be the rate of return on each bond?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What is the price of each bond today?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Required C
$
$
$
Price (Today)
1 Year
1,009.35
2 Years
1,000.00
3 Years
974.69
Explanation:
a., b., c.
The table below shows the holding period returns for the three bonds:
Maturity
1 Year
2 Years
3 Years
YTM at beginning of year
7.00%
8.00%
9.00%
a. Beginning of year prices
$ 1,009.35
$ 1,000.00
$ 974.69
b. Prices at year-end (at 9% YTM)
$ 1,000.00
$ 990.83
$ 982.41
Capital gain
−$ 9.35
−$ 9.17
$ 7.72
Coupon
$ 80.00
$ 80.00
$ 80.00
1-year total $ return
$ 70.65
$ 70.83
$ 87.72
c. 1-year total rate of return
7.00%
7.08%
9.00%
Buy the three-year bond because it provides a 9% holding-period return over the next year, which is greater than the return on either of the other bonds.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
8.
Award: 10.00
points
Problems? Adjust credit for all students.
Currently, the term structure is as follows: 1-year zero-coupon bonds yield 7%; 2-year zero-coupon bonds yield 8%; 3-year and longer-maturity zero-coupon bonds all yield 9%. You are choosing between 1-, 2-, and 3-year
maturity bonds all paying annual coupons of 8%.
Required:
a. What is the price of each bond today?
b. What will be the price of each bond in one year if the yield curve is flat at 9% at that time?
c. What will be the rate of return on each bond?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What will be the price of each bond in one year if the yield curve is flat at 9% at that time?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Required C
$
$
$
Price (in One Year)
1 Year
1,000.00
2 Years
990.83
3 Years
982.41
Explanation:
a., b., c.
The table below shows the holding period returns for the three bonds:
Maturity
1 Year
2 Years
3 Years
YTM at beginning of year
7.00%
8.00%
9.00%
a. Beginning of year prices
$ 1,009.35
$ 1,000.00
$ 974.69
b. Prices at year-end (at 9% YTM)
$ 1,000.00
$ 990.83
$ 982.41
Capital gain
−$ 9.35
−$ 9.17
$ 7.72
Coupon
$ 80.00
$ 80.00
$ 80.00
1-year total $ return
$ 70.65
$ 70.83
$ 87.72
c. 1-year total rate of return
7.00%
7.08%
9.00%
Buy the three-year bond because it provides a 9% holding-period return over the next year, which is greater than the return on either of the other bonds.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
Your preview ends here
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8.
Award: 10.00
points
Problems? Adjust credit for all students.
Currently, the term structure is as follows: 1-year zero-coupon bonds yield 7%; 2-year zero-coupon bonds yield 8%; 3-year and longer-maturity zero-coupon bonds all yield 9%. You are choosing between 1-, 2-, and 3-year
maturity bonds all paying annual coupons of 8%.
Required:
a. What is the price of each bond today?
b. What will be the price of each bond in one year if the yield curve is flat at 9% at that time?
c. What will be the rate of return on each bond?
Required B
Required C
Complete this question by entering your answers in the tabs below.
What will be the rate of return on each bond?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Required C
Rate of Return
1 Year
7.00
%
2 Years
7.08
%
3 Years
9.00
%
Explanation:
a., b., c.
The table below shows the holding period returns for the three bonds:
Maturity
1 Year
2 Years
3 Years
YTM at beginning of year
7.00%
8.00%
9.00%
a. Beginning of year prices
$ 1,009.35
$ 1,000.00
$ 974.69
b. Prices at year-end (at 9% YTM)
$ 1,000.00
$ 990.83
$ 982.41
Capital gain
−$ 9.35
−$ 9.17
$ 7.72
Coupon
$ 80.00
$ 80.00
$ 80.00
1-year total $ return
$ 70.65
$ 70.83
$ 87.72
c. 1-year total rate of return
7.00%
7.08%
9.00%
Buy the three-year bond because it provides a 9% holding-period return over the next year, which is greater than the return on either of the other bonds.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
9.
Award: 10.00
points
Problems? Adjust credit for all students.
You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8%.
Required:
a. What are the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of
the bond and that of your tuition obligation?
d. What if rates fall immediately to 7%?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What are the present value and duration of your obligation?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Required C
Required D
$
Present value
17,832.65
Duration
1.48
years
Explanation:
a. PV of the obligation = $10,000 × Annuity factor (8%, 2) = $17,832.65
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 8%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 10,000.00
$ 9,259.259
0.51923
0.51923
2
10,000.00
8,573.388
0.48077
0.96154
Column Sums
$ 17,832.647
1.00000
1.48077
D
= 1.48 years
b. A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be:
$17,832.65 × 1.08
1.4808
= $19,985.21
c. If the interest rate increases to 9%, the zero-coupon bond would decrease:
$19,985.21 ÷ 1.09
1.4808
= $17,590.93
The present value of the tuition obligation would decrease to $17,591.11.
The net position decreases in value by $0.19.
d. If the interest rate decreases to 7%, the zero-coupon bond would increase in value to
$19,985.21 ÷ 1.07
1.4808
= $18,079.99
The present value of the tuition obligation would increase to $18,080.18 .
The net position decreases in value by $0.20.
The reason the net position changes at all: as the interest rate changes, so does the duration of the stream of tuition payments.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
9.
Award: 10.00
points
Problems? Adjust credit for all students.
You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8%.
Required:
a. What are the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of
the bond and that of your tuition obligation?
d. What if rates fall immediately to 7%?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What maturity zero-coupon bond would immunize your obligation?
Note: Do not round intermediate calculations. Round "Duration" and "Face value" to 2 decimal places.
Required A
Required B
Required C
Required D
$
Duration
1.48
years
Face value
19,985.21
Explanation:
a. PV of the obligation = $10,000 × Annuity factor (8%, 2) = $17,832.65
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 8%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 10,000.00
$ 9,259.259
0.51923
0.51923
2
10,000.00
8,573.388
0.48077
0.96154
Column Sums
$ 17,832.647
1.00000
1.48077
D
= 1.48 years
b. A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be:
$17,832.65 × 1.08
1.4808
= $19,985.21
c. If the interest rate increases to 9%, the zero-coupon bond would decrease:
$19,985.21 ÷ 1.09
1.4808
= $17,590.93
The present value of the tuition obligation would decrease to $17,591.11.
The net position decreases in value by $0.19.
d. If the interest rate decreases to 7%, the zero-coupon bond would increase in value to
$19,985.21 ÷ 1.07
1.4808
= $18,079.99
The present value of the tuition obligation would increase to $18,080.18 .
The net position decreases in value by $0.20.
The reason the net position changes at all: as the interest rate changes, so does the duration of the stream of tuition payments.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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9.
Award: 10.00
points
Problems? Adjust credit for all students.
You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8%.
Required:
a. What are the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of
the bond and that of your tuition obligation?
d. What if rates fall immediately to 7%?
Required B
Required D
Complete this question by entering your answers in the tabs below.
Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately
increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your
tuition obligation?
Note: Do not round intermediate calculations. Input the amount as a positive value. Round your answer to 2 decimal places.
Required A
Required B
Required C
Required D
Show less
$
Net position
decreases
in value by
0.19
Explanation:
a. PV of the obligation = $10,000 × Annuity factor (8%, 2) = $17,832.65
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 8%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 10,000.00
$ 9,259.259
0.51923
0.51923
2
10,000.00
8,573.388
0.48077
0.96154
Column Sums
$ 17,832.647
1.00000
1.48077
D
= 1.48 years
b. A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be:
$17,832.65 × 1.08
1.4808
= $19,985.21
c. If the interest rate increases to 9%, the zero-coupon bond would decrease:
$19,985.21 ÷ 1.09
1.4808
= $17,590.93
The present value of the tuition obligation would decrease to $17,591.11.
The net position decreases in value by $0.19.
d. If the interest rate decreases to 7%, the zero-coupon bond would increase in value to
$19,985.21 ÷ 1.07
1.4808
= $18,079.99
The present value of the tuition obligation would increase to $18,080.18 .
The net position decreases in value by $0.20.
The reason the net position changes at all: as the interest rate changes, so does the duration of the stream of tuition payments.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
9.
Award: 10.00
points
Problems? Adjust credit for all students.
You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8%.
Required:
a. What are the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of
the bond and that of your tuition obligation?
d. What if rates fall immediately to 7%?
Required C
Required D
Complete this question by entering your answers in the tabs below.
What if rates fall immediately to 7%?
Note: Do not round intermediate calculations. Input the amount as a positive value. Round your answer to 2 decimal places.
Required A
Required B
Required C
Required D
$
Net position
decreases
in value by
0.20
Explanation:
a. PV of the obligation = $10,000 × Annuity factor (8%, 2) = $17,832.65
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 8%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 10,000.00
$ 9,259.259
0.51923
0.51923
2
10,000.00
8,573.388
0.48077
0.96154
Column Sums
$ 17,832.647
1.00000
1.48077
D
= 1.48 years
b. A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be:
$17,832.65 × 1.08
1.4808
= $19,985.21
c. If the interest rate increases to 9%, the zero-coupon bond would decrease:
$19,985.21 ÷ 1.09
1.4808
= $17,590.93
The present value of the tuition obligation would decrease to $17,591.11.
The net position decreases in value by $0.19.
d. If the interest rate decreases to 7%, the zero-coupon bond would increase in value to
$19,985.21 ÷ 1.07
1.4808
= $18,079.99
The present value of the tuition obligation would increase to $18,080.18 .
The net position decreases in value by $0.20.
The reason the net position changes at all: as the interest rate changes, so does the duration of the stream of tuition payments.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
10.
Award: 10.00
points
Problems? Adjust credit for all students.
Pension funds pay lifetime annuities to recipients. If a firm will remain in business indefinitely, the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make
perpetual payments of $2 million per year to beneficiaries. The yield to maturity on all bonds is 16%.
Required:
a. If the duration of 5-year-maturity bonds with coupon rates of 12% (paid annually) is four years and the duration of 20-year-maturity bonds with coupon rates of 6% (paid annually) is 11 years, how much of each of these
coupon bonds (in market value) will you want to hold to both fully fund and immunize your obligation?
Note: Do not round intermediate calculations. Enter your answers in millions rounded to 1 decimal place.
b. What will be the par value of your holdings in the 20-year coupon bond?
Note: Do not round intermediate calculations. Enter your answer in millions rounded to 2 decimal places.
$
$
$
a. 5 year bond
6.7
million
a. 20 year bond
5.8
million
b. Par value
14.25
million
Explanation:
a. PV of obligation = $2 million ÷ 0.16 = $12.5 million
Duration of obligation = (1 + y) ÷ y = 1.16 ÷ 0.16 = 7.25 years
Call w
the weight on the five-year maturity bond (which has duration of four years). Then:
(
w
× 4) + (1 − w
) × 11 = 7.25 w
= 0.5357
Therefore: 0.5357 × $12.5 = $6.7 million in the 5-year bond and
0.4643 × $12.5 = $5.8 million in the 20-year bond.
b. The price of the 20-year bond is:
[$60 × Annuity factor (16%, 20)] + [$1,000 × PV factor (16%, 20)] = $407.12
Alternatively, PMT = $60; N = 20; I = 16%; FV = $1,000; solve for PV = $407.12.
Therefore, the bond sells for 0.4071 times its par value, and:
Market value = Par value × 0.4071
$5.8 million = Par value × 0.4071 Par value = $14.25 million
Another way to see this is to note that each bond with par value $1,000 sells for $407.12. If total market value is $5.8 million, then you need to buy approximately 14,250 bonds, resulting in total par value of $14.25 million.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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11.
Award: 10.00
points
Problems? Adjust credit for all students.
You are managing a portfolio of $1 million. Your target duration is 10 years, and you can invest in two bonds, a zero-coupon bond with maturity of five years and a perpetuity, each currently yielding 5%.
Required:
a. What weight of each bond will you hold to immunize your portfolio?
b. How will these weights change next year
if target duration is now nine years?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What weight of each bond will you hold to immunize your portfolio?
Note: Round your answers to 2 decimal places.
Required A
Required B
Zero-coupon bond
68.75
%
Perpetuity bond
31.25
%
Explanation:
a. The duration of the perpetuity is: (1 + y) ÷ y = 1.05 ÷ 0.05 = 21 years
Call w
the weight of the zero-coupon bond. Then:
(
w
× 5) + (1 − w
) × 21 = 10 w
= 11 ÷ 16 = 0.6875 = 68.75%
Therefore, the portfolio weights would be as follows: 11 ÷ 16 invested in the zero and 5 ÷ 16 in the perpetuity.
b. Next year, the zero-coupon bond will have a duration of 4 years and the perpetuity will still have a 21-year duration. To obtain the target duration of nine years, which is now the duration of the obligation, we again solve for w
:
(
w
× 4) + (1 − w
) × 21 = 9 w
= 12 ÷ 17 = 0.7059 = 70.59%
So, the proportion of the portfolio invested in the zero increases to 12 ÷ 17 and the proportion invested in the perpetuity falls to 5 ÷ 17.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
11.
Award: 10.00
points
Problems? Adjust credit for all students.
You are managing a portfolio of $1 million. Your target duration is 10 years, and you can invest in two bonds, a zero-coupon bond with maturity of five years and a perpetuity, each currently yielding 5%.
Required:
a. What weight of each bond will you hold to immunize your portfolio?
b. How will these weights change next year
if target duration is now nine years?
Required A
Required B
Complete this question by entering your answers in the tabs below.
How will these weights change next year
if target duration is now nine years?
Note: Round your answers to 2 decimal places.
Required A
Required B
Zero-coupon bond
70.59
%
Perpetuity bond
29.41
%
Explanation:
a. The duration of the perpetuity is: (1 + y) ÷ y = 1.05 ÷ 0.05 = 21 years
Call w
the weight of the zero-coupon bond. Then:
(
w
× 5) + (1 − w
) × 21 = 10 w
= 11 ÷ 16 = 0.6875 = 68.75%
Therefore, the portfolio weights would be as follows: 11 ÷ 16 invested in the zero and 5 ÷ 16 in the perpetuity.
b. Next year, the zero-coupon bond will have a duration of 4 years and the perpetuity will still have a 21-year duration. To obtain the target duration of nine years, which is now the duration of the obligation, we again solve for w
:
(
w
× 4) + (1 − w
) × 21 = 9 w
= 12 ÷ 17 = 0.7059 = 70.59%
So, the proportion of the portfolio invested in the zero increases to 12 ÷ 17 and the proportion invested in the perpetuity falls to 5 ÷ 17.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
12.
Award: 10.00
points
Problems? Adjust credit for all students.
My pension plan will pay me $10,000 once a year for a 10-year period. The first payment will come in exactly five years. The pension fund wants to immunize its position.
Required:
a. What is the duration of its obligation to me? The current interest rate is 10% per year.
b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond?
c. What will be the
face value of the holdings in each zero?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What is the duration of its obligation to me? The current interest rate is 10% per year.
Note: Do not round intermediate calculations. Round your answer to 4 decimal places.
Required A
Required B
Required C
Duration
4.7255
years
Explanation:
a. The duration of the annuity if it were to start in one year
would be:
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate =
10%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$
10,000
$ 9,090.909
0.14795
0.14795
2
10,000
8,264.463
0.13450
0.26900
3
10,000
7,513.148
0.12227
0.36682
4
10,000
6,830.135
0.11116
0.44463
5
10,000
6,209.213
0.10105
0.50526
6
10,000
5,644.739
0.09187
0.55119
7
10,000
5,131.581
0.08351
0.58460
8
10,000
4,665.074
0.07592
0.60738
9
10,000
4,240.976
0.06902
0.62118
10
10,000
3,855.433
0.06275
0.62745
Column Sums
$ 61,445.671
1.00000
4.72546
Because the payment stream starts in five years, instead of one year, we add four years to the duration, so the duration is 8.73 years.
b. The present value of the deferred annuity is:
(10,000 × Annuity factor (10 % ,10)) ÷ 1.10
4
= 41,968
Alternatively, CF 0 = 0; CF 1 = 0; N = 4; CF 2 = $10,000; N = 10; I = 10; Solve for NPV = $41,968.
Call w
the weight of the portfolio invested in the five-year zero. Then
(
w
× 5) + [(1 − w
) × 20] = 8.7255
w
= 0.7516
The investment in the five-year zero is equal to
0.7516 × $41,968 = $31,543
The investment in the 20-year zeros is equal to
0.2484 × $41,968 = $10,423
c. These are the present or market values of each investment. The face values are equal to the respective future values of the investments. The face value of the five-year zeros is
$31,543 × (1.10)
5
= $50,801
Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000, would be purchased. Similarly, the face value of the 20-year zeros is
$10,425 × (1.10)
20
= $70,123
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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12.
Award: 10.00
points
Problems? Adjust credit for all students.
My pension plan will pay me $10,000 once a year for a 10-year period. The first payment will come in exactly five years. The pension fund wants to immunize its position.
Required:
a. What is the duration of its obligation to me? The current interest rate is 10% per year.
b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond?
c. What will be the
face value of the holdings in each zero?
Required A
Required C
Complete this question by entering your answers in the tabs below.
If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be
placed in each bond?
Note: Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.
Required A
Required B
Required C
$
$
Investment
5-year zero-coupon bond
31,543
20-year zero-coupon bond
10,423
Explanation:
a. The duration of the annuity if it were to start in one year
would be:
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate =
10%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$
10,000
$ 9,090.909
0.14795
0.14795
2
10,000
8,264.463
0.13450
0.26900
3
10,000
7,513.148
0.12227
0.36682
4
10,000
6,830.135
0.11116
0.44463
5
10,000
6,209.213
0.10105
0.50526
6
10,000
5,644.739
0.09187
0.55119
7
10,000
5,131.581
0.08351
0.58460
8
10,000
4,665.074
0.07592
0.60738
9
10,000
4,240.976
0.06902
0.62118
10
10,000
3,855.433
0.06275
0.62745
Column Sums
$ 61,445.671
1.00000
4.72546
Because the payment stream starts in five years, instead of one year, we add four years to the duration, so the duration is 8.73 years.
b. The present value of the deferred annuity is:
(10,000 × Annuity factor (10 % ,10)) ÷ 1.10
4
= 41,968
Alternatively, CF 0 = 0; CF 1 = 0; N = 4; CF 2 = $10,000; N = 10; I = 10; Solve for NPV = $41,968.
Call w
the weight of the portfolio invested in the five-year zero. Then
(
w
× 5) + [(1 − w
) × 20] = 8.7255
w
= 0.7516
The investment in the five-year zero is equal to
0.7516 × $41,968 = $31,543
The investment in the 20-year zeros is equal to
0.2484 × $41,968 = $10,423
c. These are the present or market values of each investment. The face values are equal to the respective future values of the investments. The face value of the five-year zeros is
$31,543 × (1.10)
5
= $50,801
Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000, would be purchased. Similarly, the face value of the 20-year zeros is
$10,425 × (1.10)
20
= $70,123
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
12.
Award: 10.00
points
Problems? Adjust credit for all students.
My pension plan will pay me $10,000 once a year for a 10-year period. The first payment will come in exactly five years. The pension fund wants to immunize its position.
Required:
a. What is the duration of its obligation to me? The current interest rate is 10% per year.
b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond?
c. What will be the
face value of the holdings in each zero?
Required B
Required C
Complete this question by entering your answers in the tabs below.
What will be the face value
of the holdings in each zero?
Note: Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.
Required A
Required B
Required C
$
$
Face Value
5-year zero-coupon bond
50,801
20-year zero-coupon bond
70,123
Explanation:
a. The duration of the annuity if it were to start in one year
would be:
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate =
10%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$
10,000
$ 9,090.909
0.14795
0.14795
2
10,000
8,264.463
0.13450
0.26900
3
10,000
7,513.148
0.12227
0.36682
4
10,000
6,830.135
0.11116
0.44463
5
10,000
6,209.213
0.10105
0.50526
6
10,000
5,644.739
0.09187
0.55119
7
10,000
5,131.581
0.08351
0.58460
8
10,000
4,665.074
0.07592
0.60738
9
10,000
4,240.976
0.06902
0.62118
10
10,000
3,855.433
0.06275
0.62745
Column Sums
$ 61,445.671
1.00000
4.72546
Because the payment stream starts in five years, instead of one year, we add four years to the duration, so the duration is 8.73 years.
b. The present value of the deferred annuity is:
(10,000 × Annuity factor (10 % ,10)) ÷ 1.10
4
= 41,968
Alternatively, CF 0 = 0; CF 1 = 0; N = 4; CF 2 = $10,000; N = 10; I = 10; Solve for NPV = $41,968.
Call w
the weight of the portfolio invested in the five-year zero. Then
(
w
× 5) + [(1 − w
) × 20] = 8.7255
w
= 0.7516
The investment in the five-year zero is equal to
0.7516 × $41,968 = $31,543
The investment in the 20-year zeros is equal to
0.2484 × $41,968 = $10,423
c. These are the present or market values of each investment. The face values are equal to the respective future values of the investments. The face value of the five-year zeros is
$31,543 × (1.10)
5
= $50,801
Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000, would be purchased. Similarly, the face value of the 20-year zeros is
$10,425 × (1.10)
20
= $70,123
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req A
Req B
Complete this question by entering your answers in the tabs below.
Find the price of the bond if its yield to maturity falls to 7%.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
$
Price of the bond
1,620.45
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
Your preview ends here
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req A
Req C
Complete this question by entering your answers in the tabs below.
What price would be predicted by the duration rule?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
$
Predicted new price (duration rule)
1,605.28
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
Your preview ends here
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req B
Req D1
Complete this question by entering your answers in the tabs below.
What price would be predicted by the duration-with-convexity rule?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
Predicted new price (duration-with-convexity rule)
1,619.30
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req C
Req D2
Complete this question by entering your answers in the tabs below.
What is the percent error for each rule?
Note: Negative answers should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to
2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
Duration Rule
Duration-with-
Convexity Rule
Percentage error
(0.94)
%
(0.08)
%
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req D1
Req E1
Complete this question by entering your answers in the tabs below.
What do you conclude about the accuracy of the two rules?
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
What do you conclude about the accuracy of the two rules?
The duration-with-convexity rule provides more accurate approximations to the true change in price.
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req D2
Req E2
Complete this question by entering your answers in the tabs below.
Find the price of the bond if its yield to maturity increases to 9%.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
$
Price of the bond
1,308.21
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req E1
Req E3
Complete this question by entering your answers in the tabs below.
What price would be predicted by the duration rule?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
$
Predicted new price (duration rule)
1,295.34
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req E2
Req E4
Complete this question by entering your answers in the tabs below.
What price would be predicted by the duration-with-convexity rule?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
Predicted new price (duration-with-convexity rule)
1,308.21
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req E3
Req E5
Complete this question by entering your answers in the tabs below.
What is the percent error for each rule?
Note: Negative answers should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to
2 decimal places.
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
Duration Rule
Duration-with-
Convexity Rule
Percentage error
(0.98)
%
0.08
%
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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13.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%.
Required:
a.
Find the price of the bond if its yield to maturity falls to 7%.
b.
What price would be predicted by the duration rule?
c.
What price would be predicted by the duration-with-convexity rule?
d-1.
What is the percent error for each rule?
d-2.
What do you conclude about the accuracy of the two rules?
e-1.
Find the price of the bond if its yield to maturity increases to 9%.
e-2.
What price would be predicted by the duration rule?
e-3.
What price would be predicted by the duration-with-convexity rule?
e-4.
What is the percent error for each rule?
e-5.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req E4
Req E5
Complete this question by entering your answers in the tabs below.
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Req A
Req B
Req C
Req D1
Req D2
Req E1
Req E2
Req E3
Req E4
Req E5
Are your conclusions about the accuracy of the two rules consistent with parts (a) − (d)?
Yes
Explanation:
a. and e-1.
Using a financial calculator, we find that the actual price of the bond as a function of yield to maturity is:
Yield to maturity
Price
7%
$ 1,620.45
8%
$ 1,450.31
9%
$ 1,308.21
(N = 30; PMT = $120; FV = $1,000, I = 7%, 8%, and 9%; Solve for PV)
b.
Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $154.97 = $1,605.28
c.
Using duration-with-convexity rule, assuming yield to maturity falls to 7%:
Predicted price change = Predicted new price = $1,450.31 + $168.92 = $1,619.23.
d-1.
The actual price at a 7% yield to maturity is $1,620.45.
Therefore, for the duration rule:
% error = ($1,605.28 − $1,620.45) ÷ $1,620.45 = −0.0094 = −0.94% (approximation is too low)
And the Duration-Convexity Rule:
% error = ($1,619.23 − $1,620.45) ÷ $1,620.45 = −0.0008 = −0.08% (approximation is too low)
e-2, e-3, and e-4.
Using the duration rule, assuming yield to maturity increases to 9%
Predicted price change = Predicted new price = $1,450.31 − $154.96 = $1,295.34
The actual price at a 9% yield to maturity is $1,308.21 (from part a.).
Therefore:
% error = ($1,295.34 − $1,308.21) ÷ 1,308.21 = −0.0098 = −0.98% (approximation is too low)
Using duration-with-convexity rule, assuming yield to maturity rises to 9%
Predicted price change = Predicted new price = $1,450.31 − $141.02 = $1,309.29.
The actual price at a 9% yield to maturity is $1,308.21.
Therefore:
% error = ($1,309.29 − $1,308.21) ÷ 1,308.21 = 0.0008 = 0.08% (approximation is too high)
d-2. and e-5.
Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error
using only duration to estimate the price change. This is consistent with parts a.-d.
References
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Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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14.
Award: 10.00
points
Problems? Adjust credit for all students.
Frank Meyers, CFA, is a fixed-income portfolio manager for a large pension fund. A member of the Investment Committee, Fred Spice, is very interested in learning about the management of fixed-income portfolios. Spice has
approached Meyers with several questions.
Meyers decides to illustrate fixed-income trading strategies to Spice using a fixed-rate bond and note. Both the bond and note have semiannual coupon periods. Unless otherwise stated, all interest rate changes are parallel. The
characteristics of these securities are shown in the following table. He also considers a 9-year floating-rate bond (floater) that pays a floating rate semiannually and is currently yielding 5%.
Characteristics of Fixed-Rate Bond and Fixed-Rate Note
Fixed-Rate
Bond
Fixed-Rate Note
Price
107.18
100.00
Yield to maturity
5.00%
5.00%
Time to maturity (years)
9
4
Modified duration (years)
6.9848
3.5851
Spice asks Meyers about how a fixed-income manager would position his portfolio to capitalize on expectations of increasing interest rates. Which of the following would be the most appropriate strategy?
Select the most appropriate strategy.
Shorten his portfolio duration.
Explanation:
Shortening his portfolio duration makes the value of the portfolio less sensitive relative to interest rate changes. So if interest rates increase the value of the portfolio will decrease less.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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15.
Award: 10.00
points
Problems? Adjust credit for all students.
Frank Meyers, CFA, is a fixed-income portfolio manager for a large pension fund. A member of the Investment Committee, Fred Spice, is very interested in learning about the management of fixed-income portfolios. Spice has
approached Meyers with several questions.
Meyers decides to illustrate fixed-income trading strategies to Spice using a fixed-rate bond and note. Both the bond and note have semiannual coupon periods. Unless otherwise stated, all interest rate changes are parallel. The
characteristics of these securities are shown in the following table. He also considers a 9-year floating-rate bond (floater) that pays a floating rate semiannually and is currently yielding 5%.
Characteristics of Fixed-Rate Bond and Fixed-Rate Note
Fixed-Rate
Bond
Fixed-Rate Note
Price
107.18
100.00
Yield to maturity
5.00%
5.00%
Time to maturity (years)
9
4
Modified duration (years)
6.9848
3.5851
Spice asks Meyers to quantify price changes from changes in interest rates. To illustrate, Meyers computes the value change for the fixed-rate note in the table. Specifically, he assumes an increase in the level of interest rate of
100 basis points. Using the information in the table, what is the predicted change in the price of the fixed-rate note?
Note: Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places.
$
Predicted price change
(3.41)
Explanation:
Predicted price change:
=
-
D
1+
y
× ∆
y
×
P
0
= (−$3,581 ÷ 1.05) × 0.01 × 100 = −$3.41(decrease)
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
(
)
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16.
Award: 10.00
points
Problems? Adjust credit for all students.
Find the duration of a bond with a settlement date of May 27, 2025, and maturity date November 15, 2036. The coupon rate of the bond is 7%, and the bond pays coupons semiannually. The bond is selling at a bond-equivalent
yield to maturity of 8%. Use Spreadsheet 16.3
.
Note: Do not round intermediate calculations. Round your answers to 4 decimal places.
Macaulay duration
7.90
Modified duration
7.60
Explanation:
Using Spreadsheet 16.3 or a financial calculator:
Settlement date
5/27/2025
Maturity date
11/15/2036
Coupon rate
0.07
Yield to maturity
0.08
Coupons per year
2
The Macaulay Duration is 7.90 and the Modified duration is 7.60.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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17.
Award: 10.00
points
Problems? Adjust credit for all students.
A 30-year-maturity bond has a 7% coupon rate, paid annually. It sells today for $867.42. A 20-year-maturity bond has a 6.5% coupon rate, also paid annually. It sells today for $879.50. A bond market analyst forecasts that in five
years, 25-year-maturity bonds will sell at yields to maturity of 8% and 15-year-maturity bonds will sell at yields of 7.5%. Because the yield curve is upward-sloping, the analyst believes that coupons will be invested in short-term
securities at a rate of 6%.
Required:
a. Calculate the (annualized) expected rate of return of the 30-year bond over the 5-year period.
Note: Do not round your intermediate calculations. Round your answer to 2 decimal places.
b. What is the (annualized) expected return of the 20-year bond?
Note: Do not round your intermediate calculations. Round your answer to 2 decimal places.
a. 30-year bond
8.22
%
b. 20-year bond
7.76
%
Explanation:
a. The maturity of the 30-year bond will fall to 25 years, and its yield is forecast to be 8%. The forecasted price of the bond is $893.25
[Using a financial calculator: n
= 25; i
= 8%; FV = 1000; PMT = 70, solve for PV]
At a 6% interest rate, coupon payments will accumulate to $394.60 after five years.
[Using a financial calculator: n
= 5; i
= 6%; PV = 0; PMT = −70, solve for FV]
Total proceeds will be: $394.60 + $893.25 = $1,287.85
Therefore, the five-year return is ($1,287.85 ÷ $867.42) − 1 = 0.4847.
This is a 48.47% five-year return, or 8.22% annually (geometric mean).
b. The maturity of the 20-year bond will fall to 15 years, and its yield is forecast to be 7.5%. The forecasted price of the bond is $911.73.
[Using a financial calculator: n = 15; i
= 7.5%; FV = 1000; PMT = 65, solve for PV]
At a 6% interest rate, coupon payments will accumulate to $366.41 after five years.
[Using a financial calculator: n
= 5; i
= 6%; PV = 0; PMT = −65, solve for FV]
Total proceeds will be $366.41 + $911.73 = $1,278.14.
Therefore, the five-year return is: ($1,278.14 ÷ $879.50) − 1 = 0.4533
This is a 45.33% five-year return, or 7.76% annually (geometric mean).
The 30-year bond offers the higher expected return.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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18.
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points
Problems? Adjust credit for all students.
A 12.75-year-maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year-maturity 6% coupon bond making annual coupon
payments also selling at a yield to maturity of 8% has nearly identical duration—11.79 years—but considerably higher convexity of 231.2.
Required:
a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule?
b. Suppose the yield to maturity on both bonds decreases to 7%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each bond?
What percentage capital loss would be predicted by the duration-with-convexity rule?
Note: Input all amounts as positive values. Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Zero Coupon Bond
Coupon Bond
Actual
loss
11.09
%
10.72
%
Predicted
loss
11.06
%
10.63
%
Explanation:
a. The price of the zero-coupon bond ($1,000 face value) selling at a yield to maturity of 8% is $374.84 and the price of the coupon bond is $774.84.
At a YTM of 9%, the actual price of the zero-coupon bond is $333.28 and the actual price of the coupon bond is $691.79.
Zero-coupon bond:
Actual % loss = ($333.28 − $374.84) ÷ $374.84 = −0.1109 = 11.09% loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss = [(−11.81) × 0.01] + [0.5 × 150.3 × 0.01
2
] = −0.1106 = 11.06% loss
Coupon bond:
Actual % loss = ($691.79 − $774.84) ÷ $774.84 = −0.1072 = 10.72% loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss = [−11.79 × 0.01] + [0.5 × 231.2 × 0.01
2
] = −0.1063 = −10.63% loss
b. Now assume yield to maturity falls to 7%. The price of the zero increases to $422.04, and the price of the coupon bond increases to $875.91.
Zero coupon bond:
Actual % gain = ($422.04 − $374.84) ÷ $374.84 = 0.1259 = 12.59% gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain = [(−11.81) × (−0.01)] + [0.5 × 150.3 × 0.01
2
] = 0.1256 = 12.56% gain
Coupon bond:
Actual % gain = ($875.91 − $774.84) ÷ $774.84 = 0.1304 = 13.04% gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain = [(−11.79) × (−0.01)] + [0.5 × 231.2 × 0.01
2
] = 0.1295 = 12.95% gain
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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18.
Award: 10.00
points
Problems? Adjust credit for all students.
A 12.75-year-maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year-maturity 6% coupon bond making annual coupon
payments also selling at a yield to maturity of 8% has nearly identical duration—11.79 years—but considerably higher convexity of 231.2.
Required:
a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule?
b. Suppose the yield to maturity on both bonds decreases to 7%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Suppose the yield to maturity on both bonds decreases to 7%. What will be the actual percentage capital loss on each bond?
What percentage capital loss would be predicted by the duration-with-convexity rule?
Note: Input all amounts as positive values. Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Zero Coupon Bond
Coupon Bond
Actual
gain
12.59
%
13.04
%
Predicted
gain
12.56
%
12.95
%
Explanation:
a. The price of the zero-coupon bond ($1,000 face value) selling at a yield to maturity of 8% is $374.84 and the price of the coupon bond is $774.84.
At a YTM of 9%, the actual price of the zero-coupon bond is $333.28 and the actual price of the coupon bond is $691.79.
Zero-coupon bond:
Actual % loss = ($333.28 − $374.84) ÷ $374.84 = −0.1109 = 11.09% loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss = [(−11.81) × 0.01] + [0.5 × 150.3 × 0.01
2
] = −0.1106 = 11.06% loss
Coupon bond:
Actual % loss = ($691.79 − $774.84) ÷ $774.84 = −0.1072 = 10.72% loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss = [−11.79 × 0.01] + [0.5 × 231.2 × 0.01
2
] = −0.1063 = −10.63% loss
b. Now assume yield to maturity falls to 7%. The price of the zero increases to $422.04, and the price of the coupon bond increases to $875.91.
Zero coupon bond:
Actual % gain = ($422.04 − $374.84) ÷ $374.84 = 0.1259 = 12.59% gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain = [(−11.81) × (−0.01)] + [0.5 × 150.3 × 0.01
2
] = 0.1256 = 12.56% gain
Coupon bond:
Actual % gain = ($875.91 − $774.84) ÷ $774.84 = 0.1304 = 13.04% gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain = [(−11.79) × (−0.01)] + [0.5 × 231.2 × 0.01
2
] = 0.1295 = 12.95% gain
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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19.
Award: 10.00
points
Problems? Adjust credit for all students.
A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon payments coming once annually). The bond sells at par value.
Required:
a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years). Assume a par value of 100.
c. What price would be predicted by the modified duration rule What is the percentage error of that rule?
d. What price would be predicted by the modified duration-with-convexity rule What is the percentage error of that rule?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
Note: Round your answers to 3 decimal places.
Required A
Required B
Required C
Required D
Convexity
64.933
Duration
7.515
years
Explanation:
a. The following spreadsheet shows that the convexity of the bond is 64.933. The present value of each cash flow is obtained by discounting at 7%. (Since the bond has a 7% coupon and sells at par, its YTM is 7%.)
Convexity equals: the sum of the last column (7,434.175) divided by:
[
P
× (1 + y
)
2
] = 100 × (1.07)
2
= 114.49
Time (
t
)
Cash Flow (CF)
PV(CF)
t
2
+ t
(
t
2
+ t
) × PV(CF)
1
$ 7
$ 6.542
2
13.084
2
7
6.114
6
36.684
3
7
5.714
12
68.569
4
7
5.340
20
106.805
5
7
4.991
30
149.727
6
7
4.664
42
195.905
7
7
4.359
56
244.118
8
7
4.074
72
293.333
9
7
3.808
90
342.678
10
107
54.393
110
5,983.271
Sum:
$ 100.000
7,434.175
Convexity:
64.933
The duration of the bond is:
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 7%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 7
$ 6.542
0.06542
0.06542
2
7
6.114
0.06114
0.12228
3
7
5.714
0.05714
0.17142
4
7
5.340
0.05340
0.21361
5
7
4.991
0.04991
0.24955
6
7
4.664
0.04664
0.27986
7
7
4.359
0.04359
0.30515
8
7
4.074
0.04074
0.32593
9
7
3.808
0.03808
0.34268
10
107
54.393
0.54393
5.43934
Column sums
$100.000
1.00000
7.51523
D
= 7.52 years
b. If the yield to maturity increases to 8%, the bond price will fall to 93.29% of par value, a percentage decrease of 6.71%.
c. The Modified duration rule predicts a percentage price change of
This overstates the actual percentage decrease in price by 0.31%.
The price predicted by the duration rule is 7.02% less than face value, or 92.98% of face value.
d. The duration-with-convexity rule predicts a percentage price change of
The price predicted by the duration with convexity rule is 6.70% less than face value, or 93.30% of face value.
The percentage error is 0.01%, which is substantially less than the error using the duration rule.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
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19.
Award: 10.00
points
Problems? Adjust credit for all students.
A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon payments coming once annually). The bond sells at par value.
Required:
a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years). Assume a par value of 100.
c. What price would be predicted by the modified duration rule What is the percentage error of that rule?
d. What price would be predicted by the modified duration-with-convexity rule What is the percentage error of that rule?
Required A
Required C
Complete this question by entering your answers in the tabs below.
Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still
10 years). Assume a par value of 100.
Note: Round your answer to 2 decimal places.
Required A
Required B
Required C
Required D
$
Actual price of the bond
93.29
Explanation:
a. The following spreadsheet shows that the convexity of the bond is 64.933. The present value of each cash flow is obtained by discounting at 7%. (Since the bond has a 7% coupon and sells at par, its YTM is 7%.)
Convexity equals: the sum of the last column (7,434.175) divided by:
[
P
× (1 + y
)
2
] = 100 × (1.07)
2
= 114.49
Time (
t
)
Cash Flow (CF)
PV(CF)
t
2
+ t
(
t
2
+ t
) × PV(CF)
1
$ 7
$ 6.542
2
13.084
2
7
6.114
6
36.684
3
7
5.714
12
68.569
4
7
5.340
20
106.805
5
7
4.991
30
149.727
6
7
4.664
42
195.905
7
7
4.359
56
244.118
8
7
4.074
72
293.333
9
7
3.808
90
342.678
10
107
54.393
110
5,983.271
Sum:
$ 100.000
7,434.175
Convexity:
64.933
The duration of the bond is:
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 7%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 7
$ 6.542
0.06542
0.06542
2
7
6.114
0.06114
0.12228
3
7
5.714
0.05714
0.17142
4
7
5.340
0.05340
0.21361
5
7
4.991
0.04991
0.24955
6
7
4.664
0.04664
0.27986
7
7
4.359
0.04359
0.30515
8
7
4.074
0.04074
0.32593
9
7
3.808
0.03808
0.34268
10
107
54.393
0.54393
5.43934
Column sums
$100.000
1.00000
7.51523
D
= 7.52 years
b. If the yield to maturity increases to 8%, the bond price will fall to 93.29% of par value, a percentage decrease of 6.71%.
c. The Modified duration rule predicts a percentage price change of
This overstates the actual percentage decrease in price by 0.31%.
The price predicted by the duration rule is 7.02% less than face value, or 92.98% of face value.
d. The duration-with-convexity rule predicts a percentage price change of
The price predicted by the duration with convexity rule is 6.70% less than face value, or 93.30% of face value.
The percentage error is 0.01%, which is substantially less than the error using the duration rule.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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19.
Award: 10.00
points
Problems? Adjust credit for all students.
A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon payments coming once annually). The bond sells at par value.
Required:
a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years). Assume a par value of 100.
c. What price would be predicted by the modified duration rule What is the percentage error of that rule?
d. What price would be predicted by the modified duration-with-convexity rule What is the percentage error of that rule?
Required B
Required D
Complete this question by entering your answers in the tabs below.
What price would be predicted by the modified duration rule 1formula31.mml
What is the percentage error of that rule?
Note: Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places
Required A
Required B
Required C
Required D
Percentage price change
(7.02)
%
Percentage error
0.31
%
Explanation:
a. The following spreadsheet shows that the convexity of the bond is 64.933. The present value of each cash flow is obtained by discounting at 7%. (Since the bond has a 7% coupon and sells at par, its YTM is 7%.)
Convexity equals: the sum of the last column (7,434.175) divided by:
[
P
× (1 + y
)
2
] = 100 × (1.07)
2
= 114.49
Time (
t
)
Cash Flow (CF)
PV(CF)
t
2
+ t
(
t
2
+ t
) × PV(CF)
1
$ 7
$ 6.542
2
13.084
2
7
6.114
6
36.684
3
7
5.714
12
68.569
4
7
5.340
20
106.805
5
7
4.991
30
149.727
6
7
4.664
42
195.905
7
7
4.359
56
244.118
8
7
4.074
72
293.333
9
7
3.808
90
342.678
10
107
54.393
110
5,983.271
Sum:
$ 100.000
7,434.175
Convexity:
64.933
The duration of the bond is:
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 7%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 7
$ 6.542
0.06542
0.06542
2
7
6.114
0.06114
0.12228
3
7
5.714
0.05714
0.17142
4
7
5.340
0.05340
0.21361
5
7
4.991
0.04991
0.24955
6
7
4.664
0.04664
0.27986
7
7
4.359
0.04359
0.30515
8
7
4.074
0.04074
0.32593
9
7
3.808
0.03808
0.34268
10
107
54.393
0.54393
5.43934
Column sums
$100.000
1.00000
7.51523
D
= 7.52 years
b. If the yield to maturity increases to 8%, the bond price will fall to 93.29% of par value, a percentage decrease of 6.71%.
c. The Modified duration rule predicts a percentage price change of
This overstates the actual percentage decrease in price by 0.31%.
The price predicted by the duration rule is 7.02% less than face value, or 92.98% of face value.
d. The duration-with-convexity rule predicts a percentage price change of
The price predicted by the duration with convexity rule is 6.70% less than face value, or 93.30% of face value.
The percentage error is 0.01%, which is substantially less than the error using the duration rule.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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19.
Award: 10.00
points
Problems? Adjust credit for all students.
A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon payments coming once annually). The bond sells at par value.
Required:
a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years). Assume a par value of 100.
c. What price would be predicted by the modified duration rule What is the percentage error of that rule?
d. What price would be predicted by the modified duration-with-convexity rule What is the percentage error of that rule?
Required C
Required D
Complete this question by entering your answers in the tabs below.
What price would be predicted by the modified duration-with-convexity rule 1formula32.mml
What is the percentage error
of that rule?
Note: Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places.
Required A
Required B
Required C
Required D
Percentage price change
(6.70)
%
Percentage error
0.01
%
Explanation:
a. The following spreadsheet shows that the convexity of the bond is 64.933. The present value of each cash flow is obtained by discounting at 7%. (Since the bond has a 7% coupon and sells at par, its YTM is 7%.)
Convexity equals: the sum of the last column (7,434.175) divided by:
[
P
× (1 + y
)
2
] = 100 × (1.07)
2
= 114.49
Time (
t
)
Cash Flow (CF)
PV(CF)
t
2
+ t
(
t
2
+ t
) × PV(CF)
1
$ 7
$ 6.542
2
13.084
2
7
6.114
6
36.684
3
7
5.714
12
68.569
4
7
5.340
20
106.805
5
7
4.991
30
149.727
6
7
4.664
42
195.905
7
7
4.359
56
244.118
8
7
4.074
72
293.333
9
7
3.808
90
342.678
10
107
54.393
110
5,983.271
Sum:
$ 100.000
7,434.175
Convexity:
64.933
The duration of the bond is:
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount Rate = 7%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 7
$ 6.542
0.06542
0.06542
2
7
6.114
0.06114
0.12228
3
7
5.714
0.05714
0.17142
4
7
5.340
0.05340
0.21361
5
7
4.991
0.04991
0.24955
6
7
4.664
0.04664
0.27986
7
7
4.359
0.04359
0.30515
8
7
4.074
0.04074
0.32593
9
7
3.808
0.03808
0.34268
10
107
54.393
0.54393
5.43934
Column sums
$100.000
1.00000
7.51523
D
= 7.52 years
b. If the yield to maturity increases to 8%, the bond price will fall to 93.29% of par value, a percentage decrease of 6.71%.
c. The Modified duration rule predicts a percentage price change of
This overstates the actual percentage decrease in price by 0.31%.
The price predicted by the duration rule is 7.02% less than face value, or 92.98% of face value.
d. The duration-with-convexity rule predicts a percentage price change of
The price predicted by the duration with convexity rule is 6.70% less than face value, or 93.30% of face value.
The percentage error is 0.01%, which is substantially less than the error using the duration rule.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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1.
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2.
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3.
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4.
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The duration of a bond is a function of the bond's:
coupon rate.
yield to maturity.
time to maturity.
All of the options are correct.
None of the options are correct.
Duration is calculated by discounting the bond's cash flows at the bond's yield to maturity and, except for zero-coupon bonds, is always less than time to maturity.
References
Multiple Choice
Difficulty: 1 Basic
Ceteris paribus, the duration of a bond is positively correlated with the bond's:
time to maturity, only.
coupon rate, only.
yield to maturity, only.
All of the options are correct.
None of the options are correct.
Duration is negatively correlated with coupon rate and yield to maturity.
References
Multiple Choice
Difficulty: 2 Intermediate
Ceteris paribus, the duration of a bond is negatively correlated with the bond's:
time to maturity, only.
coupon rate, only.
yield to maturity, only.
coupon rate and yield to maturity.
None of the options are correct.
Duration is negatively correlated with coupon rate and yield to maturity.
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, the interest-rate risk of a coupon bond is higher when the bond's:
term to maturity is lower.
coupon rate is higher.
yield to maturity is lower.
current yield is higher.
None of the options are correct.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration
rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
References
Multiple Choice
Difficulty: 2 Intermediate
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Holding other factors constant, the interest-rate risk of a coupon bond is higher when the bond's:
term to maturity is higher, only.
coupon rate is higher, only.
yield to maturity is higher, only.
All of the options are correct.
None of the options are correct.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration
rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, the interest-rate risk of a coupon bond is higher when the bond's:
term to maturity is lower, only.
coupon rate is lower, only.
yield to maturity is higher, only.
term to maturity is lower and yield to maturity is higher.
None of the options are correct.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration
rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, the interest-rate risk of a coupon bond is lower when the bond's:
term to maturity is lower, only.
coupon rate is higher, only.
yield to maturity is lower, only.
term to maturity is lower and coupon rate is higher.
All of the options are correct.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration
rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, the interest-rate risk of a coupon bond is lower when the bond's:
term to maturity is lower.
coupon rate is higher.
yield to maturity is higher.
term to maturity is lower and coupon rate is higher.
All of the options are correct.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration
rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
References
Multiple Choice
Difficulty: 2 Intermediate
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Holding other factors constant, the interest-rate risk of a coupon bond is lower when the bond's:
term to maturity is higher.
coupon rate is lower.
yield to maturity is higher.
term to maturity is higher and coupon rate is lower.
All of the options are correct.
The longer the maturity, the greater the interest-rate risk. The lower the coupon rate, the greater the interest-rate risk. The lower the yield to maturity, the greater the interest-rate risk. These concepts are reflected in the duration
rules; duration is a measure of bond price sensitivity to interest rate changes (interest-rate risk).
References
Multiple Choice
Difficulty: 2 Intermediate
The "modified duration" used by practitioners is equal to the Macaulay duration:
times the change in interest rate.
times (one plus the bond's yield to maturity).
divided by (one minus the bond's yield to maturity).
divided by (one plus the bond's yield to maturity).
None of the options are correct.
D
* = D
÷ (1 + y
).
References
Multiple Choice
Difficulty: 2 Intermediate
The "modified duration" used by practitioners is equal to _________ divided by (one plus the bond's yield to maturity).
current yield
the Macaulay duration
yield to call
yield to maturity
None of the options are correct.
D
* = D
÷ (1 + y
).
References
Multiple Choice
Difficulty: 2 Intermediate
Given the time to maturity, the duration of a zero-coupon bond is higher when the discount rate is:
higher, only.
lower, only.
equal to the risk-free rate, only.
The bond's duration is independent of the discount rate.
None of the options are correct.
The duration of a zero-coupon bond is equal to the maturity of the bond.
References
Multiple Choice
Difficulty: 2 Intermediate
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The interest-rate risk of a bond is:
the risk related to the possibility of bankruptcy of the bond's issuer.
the risk that arises from the uncertainty of the bond's return caused by changes in interest rates.
the unsystematic risk caused by factors unique in the bond.
the risk related to the possibility of bankruptcy of the bond's issuer, and the risk that arises from the uncertainty of the bond's return caused by changes in interest rates.
All of the options are correct.
Changing interest rates change the bond's return, both in terms of the price of the bond and the reinvestment of coupon payments.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following two bonds is more price sensitive to changes in interest rates?
1. A par value bond, X, with a 5-year year to maturity and a 10% coupon rate.
2. A zero-coupon bond, Y, with a 5-year year to maturity and a 10% yield to maturity.
Bond X because of the higher yield to maturity
Bond X because of the longer time to maturity
Bond Y because of the longer duration
Both have the same sensitivity because both have the same yield to maturity.
None of the options are correct.
Duration is the best measure of bond price sensitivity; the longer the duration the higher the price sensitivity. Bond Y has a longer duration.
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, which one of the following bonds has the smallest price volatility?
5-year, 0% coupon bond
5-year, 12% coupon bond
5 year, 14% coupon bond
5-year, 10% coupon bond
Cannot tell from the information given
Duration (and thus price volatility) is lower when the coupon rates are higher.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following is not true?
Holding other things constant, the duration of a bond increases with time to maturity.
Given time to maturity, the duration of a zero-coupon decreases with yield to maturity.
Given time to maturity and yield to maturity, the duration of a bond is higher when the coupon rate is lower.
Duration is a better measure of price sensitivity to interest-rate changes than is time to maturity.
All of the options are correct.
The duration of a zero-coupon bond is equal to time to maturity and is independent of yield to maturity.
References
Multiple Choice
Difficulty: 2 Intermediate
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Which of the following statements are true?
1. Holding other things constant, the duration of a bond decreases with time to maturity.
2. Given time to maturity, the duration of a zero-coupon increases with yield to maturity.
3. Given time to maturity and yield to maturity, the duration of a bond is higher when the coupon rate is lower.
4. Duration is a better measure of price sensitivity to interest-rate changes than is time to maturity.
1 only
1 and 2
3 only
3 and 4
1, 2, and 4
The duration of a zero-coupon bond is equal to time to maturity and is independent of yield to maturity.
References
Multiple Choice
Difficulty: 2 Intermediate
The duration of a 5-year zero-coupon bond is:
smaller than 5.
larger than 5.
equal to 5.
equal to that of a 5-year 10% coupon bond.
None of the options are correct.
Duration of a zero-coupon bond equals the bond's maturity.
References
Multiple Choice
Difficulty: 1 Basic
The basic purpose of immunization is to:
eliminate default risk, only.
produce a zero net-interest-rate risk, only.
offset price and reinvestment risk, only.
eliminate default risk and produce a zero net-interest-rate risk, only.
produce a zero net-interest-rate risk and offset price and reinvestment risk.
When a portfolio is immunized, price risk and reinvestment risk exactly offset each other resulting in zero net-interest-rate risk.
References
Multiple Choice
Difficulty: 2 Intermediate
The duration of a par-value bond with a coupon rate of 8% (paid annually) and a remaining time to maturity of 5 years is:
Note: Do not round your intermediate calculations.
5 years.
5.4 years.
4.17 years.
4.31 years.
None of the options are correct.
Calculations are shown below.
Year
CF
PV of CF@08%
Weight × Year
1
$ 80
$80 ÷ 1.08 = $74.07
0.0741 × 1 = 0.0741
2
$ 80
$80 ÷ (1.08)
2
= $68.59
0.0686 × 2 = 0.1372
3
$ 80
$80 ÷ (1.08)
3
= $63.51
0.0635 × 3 = 0.1905
4
$ 80
$80 ÷ (1.08)
4
= $58.80
0.0588 × 4 = 0.2352
5
$ 1.080
$1,080 ÷ (1.08)
5
= $735.03
0.7352 × 5 = 3.6750
Sum
$1000.00
4.3120 years (duration)
References
Multiple Choice
Difficulty: 2 Intermediate
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The duration of a perpetuity with a yield of 8% is:
13.50 years.
12.11 years.
6.66 years.
Cannot be determined
None of the options are correct.
D
perpetuity
= (1 + y
) ÷ y
= 1.08 ÷ 0.08 = 13.5
References
Multiple Choice
Difficulty: 1 Basic
A seven-year par value bond has a coupon rate of 9% (paid annually) and a modified duration of:
Note: Do not round your intermediate calculations.
7.00 years.
5.49 years.
5.03 years.
4.87 years.
None of the options are correct.
Calculations are shown below.
Year
CF
PV of CF@9%
Weight × Year
1
$ 90
$ 82.57
0.0826 × 1 = 0.0826
2
$ 90
$ 75.75
0.0758 × 2 = 0.1516
3
$ 90
$ 69.50
0.0695 × 3 = 0.2085
4
$ 90
$ 63.76
0.0638 × 4 = 0.2552
5
$ 90
$ 58.49
0.0585 × 5 = 0.2925
6
$ 90
$ 53.66
0.0537 × 6 = 0.3222
7
$ 1,090
$ 596.26
0.5963 × 7 = 4.1741
Sum
$ 1000.00
5.4867 years (duration)
Modified duration = 5.4867 years ÷ 1.09 = 5.03 years
References
Multiple Choice
Difficulty: 3 Challenge
A $1,000 par-value bond XYZ has a modified duration of 6. Which one of the following statements regarding the bond is true
?
If the market yield increases by 1%, the bond's price will decrease by $60.
If the market yield increases by 1%, the bond's price will increase by $50.
If the market yield increases by 1%, the bond's price will decrease by $50.
If the market yield increases by 1%, the bond's price will increase by $60.
None of the options are correct.
Δ
P
÷ P
= −
D*
× Δ
y
→
Δ
P
= −
D
* × Δ
y
× P
= −
6 × 0.01 × $1,000 = −
$60
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following bonds has the longest duration?
An 8-year maturity, 0% coupon bond
An 8-year maturity, 5% coupon bond
A 10-year maturity, 5% coupon bond
A 10-year maturity, 0% coupon bond
Cannot tell from the information given
The longer the maturity and the lower the coupon, the greater the duration.
References
Multiple Choice
Difficulty: 2 Intermediate
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Which one of the following par-value 12% coupon bonds experiences a price change of $23 when the market yield changes by 50 basis points?
The bond with a duration of 6 years
The bond with a duration of 5 years
The bond with a duration of 2.7 years
The bond with a duration of 5.15 years
None of the options are correct.
Δ
P
÷ P
= −
D
* × Δ
y
→
Δ
P
= −
D
* × Δ
y
× P
= −
(5.15 ÷ 1.12) × 0.005 × $1,000 = −
$23
References
Multiple Choice
Difficulty: 3 Challenge
Which one of the following statements is true concerning the duration of a perpetuity?
The duration of a 15% yield perpetuity that pays $100 annually is longer than that of a 15% yield perpetuity that pays $200 annually.
The duration of a 15% yield perpetuity that pays $100 annually is shorter than that of a 15% yield perpetuity that pays $200 annually.
The duration of a 15% yield perpetuity that pays $100 annually is equal to that of a 15% yield perpetuity that pays $200 annually.
The duration of a perpetuity cannot be calculated.
Duration of a perpetuity = (1 + y
) ÷ y
; thus, the duration of a perpetuity is determined by the yield and is independent of the cash flow.
References
Multiple Choice
Difficulty: 1 Basic
The two components of interest-rate risk are:
price risk and default risk.
reinvestment risk and systematic risk.
call risk and price risk.
price risk and reinvestment risk.
None of the options are correct.
Default, systematic, and call risks are not part of interest-rate risk. Only price and reinvestment risks are part of interest-rate risk.
References
Multiple Choice
Difficulty: 1 Basic
The duration of a coupon bond:
does not change after the bond is issued.
can accurately predict the price change of the bond for any interest-rate change.
will decrease as the yield to maturity decreases.
All of the options are true.
None of the options are true.
Duration changes as interest rates and time to maturity change, can only predict price changes accurately for small interest rate changes, and increases as the yield to maturity decreases.
References
Multiple Choice
Difficulty: 1 Basic
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32.
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Indexing of bond portfolios is difficult because:
the number of bonds included in the major indexes is so large that it would be difficult to purchase them in the proper proportions.
many bonds are thinly traded, so it is difficult to purchase them at a fair market price.
the composition of bond indexes is constantly changing.
All of the options are true.
None of the options are correct.
All of the options are true statements about bond indexes.
References
Multiple Choice
Difficulty: 2 Intermediate
Duration measures:
weighted-average time until a bond's half-life.
weighted-average time until cash flow payment.
the time required to make excessive profit from the investment.
weighted-average time until a bond's half-life and the time required to make excessive profit from the investment.
weighted-average time until cash flow payment and the time required to make excessive profit from the investment.
Duration measures the weighted average time until cash flow payment, as one receives coupon payments throughout the life of the bond (for coupon bonds); thus, duration is less than time to maturity (except for zeros).
References
Multiple Choice
Difficulty: 2 Intermediate
Duration:
assesses the time element of bonds in terms of both coupon and term to maturity, only.
allows structuring a portfolio to avoid interest-rate risk, only.
is a direct comparison between bond issues with different levels of risk, only.
assesses the time element of bonds in terms of both coupon and term to maturity and allows structuring a portfolio to avoid interest-rate risk.
assesses the time element of bonds in terms of both coupon and term to maturity and is a direct comparison between bond issues with different levels of risk.
Duration is a weighted average of when the cash flows of a bond are received; thus both coupon and time to maturity are considered. If the duration of the portfolio equals the investor's horizon date, the investor is protected
against interest rate changes.
References
Multiple Choice
Difficulty: 2 Intermediate
Identify the bond that has the longest duration (no calculations necessary).
20-year maturity with an 8% coupon
20-year maturity with a 12% coupon
20-year maturity with a 0% coupon
10-year maturity with a 15% coupon
12-year maturity with a 12% coupon
The lower the coupon, the longer the duration. The zero-coupon bond is the ultimate low coupon bond and thus would have the longest duration.
References
Multiple Choice
Difficulty: 2 Intermediate
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33.
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34.
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35.
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36.
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When interest rates decline, the duration of a 10-year bond selling at a premium:
increases.
decreases.
remains the same.
increases at first, then declines.
decreases at first, then increases.
The relationship between interest rates and duration is an inverse one.
References
Multiple Choice
Difficulty: 2 Intermediate
An 8%, 30-year corporate bond was recently being priced to yield 10%. The Macaulay duration for the bond is 10.20 years. Given this information, the bond's modified duration would be:
8.05.
9.44.
9.27.
11.22.
None of the options are correct.
D
* = D
÷ (1 + y
) = 10.2 ÷ (1.1) = 9.27.
References
Multiple Choice
Difficulty: 1 Basic
An 8%, 15-year bond has a yield to maturity of 10% and duration of 8.05 years. If the market yield changes by 25 basis points, how much change will there be in the bond's price?
1.83%
2.01%
3.27%
6.44%
None of the options are correct.
Δ
P
÷ P
= (
−
D
× Δ
y
) ÷ (1 + y
) = (
−
8.05 × 0.0025) ÷ 1.10 = 1.83%
References
Multiple Choice
Difficulty: 2 Intermediate
One way that banks can reduce the duration of their asset portfolios is using:
fixed-rate mortgages.
adjustable-rate mortgages.
certificates of deposit.
short-term borrowing.
None of the options are correct.
One of the gap management strategies practiced by banks is the issuance of adjustable rate mortgages, which reduce the interest rate sensitivity of their asset portfolios.
References
Multiple Choice
Difficulty: 1 Basic
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37.
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38.
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39.
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40.
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The duration of a bond normally increases with an increase in:
term to maturity.
yield to maturity.
coupon rate.
All of the options are correct.
None of the options are correct.
The relationship between duration and term to maturity is a direct one; the relationship between duration and yield to maturity and to coupon rate is negative.
References
Multiple Choice
Difficulty: 2 Intermediate
Immunization is not a strictly passive strategy because:
it requires choosing an asset portfolio that matches an index.
there is likely to be a gap between the values of assets and liabilities in most portfolios.
it requires frequent rebalancing as maturities and interest rates change.
durations of assets and liabilities fall at the same rate.
None of the options are correct.
As time passes the durations of assets and liabilities fall at different rates, requiring portfolio rebalancing. Further, every change in interest rates creates changes in the durations of portfolio assets and liabilities.
References
Multiple Choice
Difficulty: 2 Intermediate
Some of the problems with immunization are:
1. duration assumes that the yield curve is flat.
2. duration assumes that if shifts in the yield curve occur, these shifts are parallel.
3. immunization is valid for one interest-rate change only.
4. durations and horizon dates change by the same amounts with the passage of time.
1 only
1 & 2
4 only
1 & 3
1, 2, & 3
Durations and horizon dates change with the passage of time, but not by the same amounts.
References
Multiple Choice
Difficulty: 2 Intermediate
If a bond portfolio manager believes:
1. in market efficiency, he or she is likely to be a passive portfolio manager.
2. that he or she can accurately predict interest-rate changes, he or she is likely to be an active portfolio manager.
3. that he or she can identify bond-market anomalies, he or she is likely to be a passive portfolio manager.
1 only
2 only
3 only
1 and 2
1, 2, and 3
If one believes that one can predict bond market anomalies, one is likely to be an active portfolio manager.
References
Multiple Choice
Difficulty: 2 Intermediate
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41.
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42.
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43.
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44.
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Cash flow matching on a multiperiod basis is referred to as:
immunization.
contingent immunization.
dedication.
duration matching.
rebalancing.
Cash flow matching on a multiperiod basis is referred to as a dedication strategy.
References
Multiple Choice
Difficulty: 1 Basic
Immunization through duration matching of assets and liabilities may be ineffective or inappropriate because:
conventional duration strategies assume a flat yield curve.
duration matching can only immunize portfolios from parallel shifts in the yield curve.
immunization only protects the nominal value of terminal liabilities and does not allow for inflation adjustment.
All of the options are correct.
None of the options are correct.
All of the options are correct statements about the limitations of immunization through duration matching.
References
Multiple Choice
Difficulty: 1 Basic
The curvature of the price yield curve for a given bond is referred to as the bond's:
modified duration.
immunization.
sensitivity.
convexity.
tangency.
Convexity measures the rate of change of the slope of the price-yield curve, expressed as a fraction of the bond's price.
References
Multiple Choice
Difficulty: 1 Basic
Consider a bond selling at par with modified duration of 10.6 years and convexity of 210. A 2% decrease in yield would cause the price to increase by 21.2% according to the duration rule. What would be the percentage price
change according to the duration-with-convexity rule?
21.2%
25.4%
17.0%
10.6%
P
÷ P
= D
* × Y
+ 0.5 × Convexity × (
y
)
2
= 10.6 × 0.02 + 0.5 × 210 × 0.02
2
= 0.254 = 25.4%
References
Multiple Choice
Difficulty: 3 Challenge
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45.
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46.
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47.
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48.
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A substitution swap is an exchange of bonds undertaken to:
change the credit risk of a portfolio.
extend the duration of a portfolio.
reduce the duration of a portfolio.
profit from apparent mispricing between two bonds.
adjust for differences in the yield spread.
A substitution swap is an example of bond price arbitrage, undertaken when the portfolio manager attempts to profit from apparent mispricing.
References
Multiple Choice
Difficulty: 2 Intermediate
A rate anticipation swap is an exchange of bonds undertaken to:
shift portfolio duration in response to an anticipated change in interest rates.
shift between corporate and government bonds when the yield spread is out of line with historical values.
profit from apparent mispricing between two bonds.
change the credit risk of the portfolio.
increase return by shifting into higher yield bonds.
A rate anticipation swap is pegged to interest-rate forecasting and involves increasing duration when rates are expected to fall and decreasing duration when rates are expected to rise.
References
Multiple Choice
Difficulty: 2 Intermediate
An analyst who selects a particular holding period and predicts the yield curve at the end of that holding period is engaging in:
a rate anticipation swap.
immunization.
horizon analysis.
an intermarket spread swap.
None of the options are correct.
Horizon analysis involves selecting a particular holding period and predicting the yield curve at the end of that holding period. The holding period return for the bond can then be predicted.
References
Multiple Choice
Difficulty: 1 Basic
Interest-rate risk is important to:
active bond portfolio managers, only.
passive bond portfolio managers, only.
both active and passive bond portfolio managers.
neither active nor passive bond portfolio managers.
obsessive bond portfolio managers.
Active managers try to identify interest-rate trends so they can move in the right direction before the changes. Passive managers try to minimize interest-rate risk by offsetting it with price changes in strategies such as
immunization.
References
Multiple Choice
Difficulty: 1 Basic
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49.
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50.
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51.
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52.
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Which of the following are true about the interest-rate sensitivity of bonds?
1. Bond prices and yields are inversely related.
2. Prices of long-term bonds tend to be more sensitive to interest-rate changes than prices of short-term bonds.
3. Interest-rate risk is positively correlated with the bond's coupon rate.
4. The sensitivity of a bond's price to a change in its yield to maturity is inversely related to the yield to maturity at which the bond is currently selling.
1 and 2
1 and 3
1, 2, and 4
2, 3, and 4
1, 2, 3, and 4
Number 3 is incorrect because interest-rate risk is inversely related to the bond's coupon rate.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following are false about the interest-rate sensitivity of bonds?
1. Bond prices and yields are inversely related.
2. Prices of long-term bonds tend to be more sensitive to interest-rate changes than prices of short-term bonds.
3. Interest-rate risk is correlated with the bond's coupon rate.
4. The sensitivity of a bond's price to a change in its yield to maturity is inversely related to the yield to maturity at which the bond is currently selling.
1
3
1, 2, and 4
2, 3, and 4
1, 2, 3, and 4
Number 3 is incorrect because interest-rate risk is inversely related to the bond's coupon rate.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following researchers have contributed significantly to bond portfolio management theory?
1. Sidney Homer
2. Harry Markowitz
3. Burton Malkiel
4. Martin Liebowitz
5. Frederick Macaulay
1 and 2
3 and 5
3, 4, and 5
1, 3, 4, and 5
1, 2, 3, 4, and 5
Harry Markowitz developed the mean-variance criterion, but not a theory of bond portfolio management.
References
Multiple Choice
Difficulty: 2 Intermediate
According to the duration concept,
only coupon payments matter.
only maturity value matters.
the coupon payments made prior to maturity make the effective maturity of the bond greater than its actual time to maturity.
the coupon payments made prior to maturity make the effective maturity of the bond less than its actual time to maturity.
coupon rates don't matter.
Duration considers that some of the cash flows are received prior to maturity, and this effectively makes the maturity less than the actual time to maturity.
References
Multiple Choice
Difficulty: 1 Basic
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53.
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54.
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55.
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56.
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Duration is important in bond portfolio management because:
1. it can be used in immunization strategies.
2. it provides a gauge of the effective average maturity of the portfolio.
3. it is related to the interest rate sensitivity of the portfolio.
4. it is a good predictor of interest-rate changes.
1 and 2
1 and 3
3 and 4
1, 2, and 3
1, 2, 3, and 4
Duration can be used to calculate the approximate effect of interest-rate changes on prices, but is not used to forecast interest rates.
References
Multiple Choice
Difficulty: 2 Intermediate
Two bonds are selling at par value, and each has 17 years to maturity. The first bond has a coupon rate of 6%, and the second bond has a coupon rate of 13%. Which of the following is true about the durations of these bonds?
The duration of the higher coupon bond will be higher.
The duration of the lower coupon bond will be higher.
The duration of the higher coupon bond will equal the duration of the lower coupon bond.
There is no consistent statement that can be made about the durations of the bonds.
The bond's durations cannot be determined without knowing the prices of the bonds.
In general, duration is negatively related to coupon rate. The greater the cash flows from coupon interest, the lower the duration will be. Since the bonds have the same time to maturity, that isn't a factor.
References
Multiple Choice
Difficulty: 3 Challenge
Two bonds are selling at par value, and each has 17 years to maturity. The first bond has a coupon rate of 6%, and the second bond has a coupon rate of 13%. Which of the following is false about the durations of these bonds?
1. The duration of the higher coupon bond will be higher.
2. The duration of the lower coupon bond will be higher.
3. The duration of the higher coupon bond will equal the modified duration of the lower coupon bond.
4. The modified duration of the bonds will be equal.
1 only
2 only
3 only
4 only
1, 3 & 4
In general, duration is negatively related to coupon rate. The greater the cash flows from coupon interest, the lower the duration will be. Since the bonds have the same time to maturity, that isn't a factor.
References
Multiple Choice
Difficulty: 3 Challenge
Which of the following two bonds is more price sensitive to changes in interest rates?
1. A par-value bond, A, with a 12 year to maturity and a 12% coupon rate.
2. A zero-coupon bond, B, with a 12 year to maturity and a 12% yield to maturity.
Bond A because of the higher yield to maturity
Bond A because of the longer time to maturity
Bond B because of the longer duration
Both have the same sensitivity because both have the same yield to maturity.
None of the options are correct.
Duration is the best measure of bond price sensitivity; the longer the duration the higher the price sensitivity.
References
Multiple Choice
Difficulty: 2 Intermediate
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58.
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59.
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60.
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Which of the following two bonds is more price sensitive to changes in interest rates?
1. A par-value bond, D, with a 2 year to maturity and an 8% coupon rate.
2. A zero-coupon bond, E, with a 2 year to maturity and an 8% yield to maturity.
Bond D because of the higher yield to maturity
Bond E because of the longer duration
Bond D because of the longer time to maturity
Both have the same sensitivity because both have the same yield to maturity.
None of the options are correct.
Duration is the best measure of bond price sensitivity; the longer the duration the higher the price sensitivity.
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, which one of the following bonds has the smallest price volatility?
7-year, 0% coupon bond
7-year, 12% coupon bond
7-year, 14% coupon bond
7-year, 10% coupon bond
Cannot tell from the information given
Duration (and thus price volatility) is lower when the coupon rates are higher.
References
Multiple Choice
Difficulty: 2 Intermediate
Holding other factors constant, which one of the following bonds has the smallest price volatility?
20-year, 0% coupon bond
20-year, 6% coupon bond
20-year, 7% coupon bond
20-year, 9% coupon bond
Cannot tell from the information given
Duration (and thus price volatility) is lower when the coupon rates are higher.
References
Multiple Choice
Difficulty: 2 Intermediate
The duration of a 15-year zero-coupon bond is:
smaller than 15.
larger than 15.
equal to 15.
equal to that of a 15-year 10% coupon bond.
None of the options are correct.
Duration of a zero-coupon bond equals the bond's maturity.
References
Multiple Choice
Difficulty: 1 Basic
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61.
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62.
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63.
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64.
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The duration of a 20-year zero-coupon bond is:
equal to 20.
larger than 20.
smaller than 20.
equal to that of a 20-year 10% coupon bond.
None of the options are correct.
Duration of a zero-coupon bond equals the bond's maturity.
References
Multiple Choice
Difficulty: 1 Basic
The duration of a perpetuity with a yield of 9.0% is:
13.50 years.
12.11 years.
6.66 years.
Cannot be determined
None of the options are correct.
D
Perpetuity
= (1 + y
) ÷ y
= 1.09 ÷ 0.09 = 12.11 years.
References
Multiple Choice
Difficulty: 1 Basic
The duration of a perpetuity with a yield of 6.5 is:
13.50 years.
12.11 years.
16.38 years.
Cannot be determined
None of the options are correct.
D
perpetuity
= (1 + y
) ÷ y
= 1.065 ÷ 0.065 = 16.38 years.
References
Multiple Choice
Difficulty: 1 Basic
Par-value-bond F has a modified duration of 9. Which one of the following statements regarding the bond is true
?
If the market yield increases by 1%, the bond's price will decrease by $90.
If the market yield increases by 1%, the bond's price will increase by $90.
If the market yield increases by 1%, the bond's price will decrease by $60.
If the market yield decreases by 1%, the bond's price will increase by $60.
Δ
P
= −
D
* × Δ
y
× P
= −
9 × 0.01 × $1,000 = −
$90
References
Multiple Choice
Difficulty: 2 Intermediate
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66.
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67.
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68.
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Par-value-bond GE has a modified duration of 11. Which one of the following statements regarding the bond is true
?
If the market yield increases by 1%, the bond's price will decrease by $55.
If the market yield increases by 1%, the bond's price will increase by $55.
If the market yield increases by 1%, the bond's price will decrease by $110.
If the market yield increases by 1%, the bond's price will increase by $110.
None of the options are correct.
Δ
P
= −
D
* × Δ
y
× P
= −
11 × 0.01 × $1,000 = −
$110
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following bonds has the longest duration?
A 15-year maturity, 0% coupon bond.
A 15-year maturity, 9% coupon bond.
A 20-year maturity, 9% coupon bond.
A 20-year maturity, 0% coupon bond.
Cannot tell from the information given
The longer the maturity and the lower the coupon, the greater the duration.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following bonds has the longest duration?
A 12-year maturity, 0% coupon bond.
A 12-year maturity, 8% coupon bond.
A 4-year maturity, 8% coupon bond.
A 4-year maturity, 0% coupon bond.
Cannot tell from the information given
The longer the maturity and the lower the coupon, the greater the duration.
References
Multiple Choice
Difficulty: 2 Intermediate
A 10%, 30-year corporate bond was recently being priced to a yield of 11%. The Macaulay duration for the bond is 11.3 years. Given this information, the bond's modified duration would be:
9.05.
10.09.
10.18.
11.22.
None of the options are correct.
D
* = D
÷ (1 + y
) = 11.3 ÷ 1.11 = 10.18
References
Multiple Choice
Difficulty: 1 Basic
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69.
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70.
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71.
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72.
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A 6%, 30-year corporate bond was recently being priced to a yield of 7%. The Macaulay duration for the bond is 9.4 years. Given this information, the bond's modified duration would be:
9.55.
9.24.
8.79.
7.78.
None of the options are correct.
D
* = D
÷ (1 + y
) = 9.4 ÷ 1.07 = 8.79
References
Multiple Choice
Difficulty: 1 Basic
A 9%, 16-year bond has a yield to maturity of 9% and duration of 7.25 years. If the market yield changes by 15 basis points, how much change will there be in the bond's price?
−
1.00%
−
2.01%
−
2.67%
−
3.44%
None of the options are correct.
Δ
P
÷ P
= −
D
* × Δ
y
= −
(7.25 ÷ 1.09) × 0.0015 = −
1.00%.
References
Multiple Choice
Difficulty: 2 Intermediate
A 7%, 14-year bond has a yield to maturity of 4.4% and duration of 8.5 years. If the market yield changes by 54 basis points, how much change will there be in the bond's price?
−
1.85%
−
2.91%
−
4.40%
−
6.44%
None of the options are correct.
Δ
P
÷ P
= −
D
* × Δ
y
= −
(8.5 ÷ 1.044) × 0.0054 = −
4.40%.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider a bond selling at par with modified duration of 12 years and convexity of 265. A 1% decrease in yield would cause the price to increase by 12%, according to the duration rule. What would be the percentage price change
according to the duration-with-convexity rule?
21.2%
25.4%
17.0%
13.3%
None of the options are correct.
P
÷ P
= −
D
* × y
+ 0.5 × Convexity × (
y
)
2
= −12 × (−0.01) + 0.5 × 265 × 0.01
2
= 0.133 = 13.3%
References
Multiple Choice
Difficulty: 3 Challenge
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73.
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74.
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75.
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Consider a bond selling at par with modified duration of 22 years and convexity of 415. A 2% decrease in yield would cause the price to increase by 44% according to the duration rule. What would be the percentage price change
according to the duration-with-convexity rule?
21.2%
25.4%
17.0%
52.3%
None of the options are correct.
P
÷ P
= −
D
* × y
+ 0.5 × Convexity × (
y
)
2
= 0.44 + 0.5 × 415 × 0.02
2
= 0.523 = 52.3%
References
Multiple Choice
Difficulty: 3 Challenge
The duration of a par-value bond with a coupon rate of 6.5% and a remaining time to maturity of 4 years is:
Note: Do not round your intermediate calculations.
3.65 years.
3.45 years.
3.85 years.
4.00 years.
None of the options are correct.
Calculations are shown below.
Year
CF
PV of CF@6.5%
Weight × Year
1
$ 65
$65 ÷ 1.065 = $61.03
0.0610 × 1 = 0.0610
2
$ 65
$65 ÷ (1.065)
2
= $57.31
0.0573 × 2 = 0.1146
3
$ 65
$65 ÷ (1.065)
3
= $53.81
0.0538 × 3 = 0.1614
4
$ 1,065
$1,065 ÷ (1.065)
4
= $827.85
0.8279 × 4 = 3.3116
Sum
$1,000.00
3.6486 years (duration)
References
Multiple Choice
Difficulty: 2 Intermediate
The duration of a par-value bond with a coupon rate of 7% and a remaining time to maturity of 3 years is:
3 years.
2.71 years.
2.81 years.
2.91 years.
None of the options are correct.
Calculations are shown below.
Year
CF
PV of CF@7%
Weight × Year
1
$ 70
$70 ÷ 1.070 = $65.42
0.0654 × 1 = 0.0654
2
$ 70
$70 ÷ (1.070)
2
= $61.14
0.0611 × 2 = 0.1222
3
$ 1,070
$1,070 ÷ (1.070)
3
= $873.44
0.8734 × 3 = 2.6202
Sum
$1,000.00
2.8078 years (duration)
References
Multiple Choice
Difficulty: 2 Intermediate
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76.
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77.
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78.
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The duration of a par-value bond with a coupon rate of 8.7% and a remaining time to maturity of 6 years is:
6.0 years.
5.1 years.
4.27 years.
3.95 years.
None of the options are correct.
Calculations are shown below.
Year
CF
PV of CF@8.7%
Weight × Year
1
$ 87
$87 ÷ 1.087 = $80.04
0.0804 × 1 = 0.0804
2
$ 87
$87 ÷ (1.087)
2
= $73.63
0.0736 × 2 = 0.1472
3
$ 87
$87 ÷ (1.087)
3
= $67.74
0.0677 × 3 = 0.2031
4
$ 87
$87 ÷ (1.087)
4
= $62.32
0.0623 × 4 = 0.2492
5
$ 87
$87 ÷ (1.087)
5
= $57.33
0.0573 × 5 = 0.2865
6
$ 1,080
$1,087 ÷ (1.087)
6
= $658.95
0.6590 × 6 = 3.9540
Sum
$1,000.00
4.9204 years (duration)
References
Multiple Choice
Difficulty: 2 Intermediate
Consider a six-year bond paying a 7% coupon, with a yield to maturity of 5.0%. What is the duration of the bond?
4.925
5.148
5.236
5.687
None of the options are correct.
Year
Cash Flow
PV of CF
% of PV
Sum of Years
1
$ 70
$ 66.67
0.0605
0.061
2
$ 70
$ 63.49
0.0576
0.115
3
$ 70
$ 60.47
0.0549
0.165
4
$ 70
$ 57.59
0.0523
0.209
5
$ 70
$ 54.85
0.0498
0.249
6
$ 1070
$ 798.45
0.7249
4.349
$ 1,101.51
1.0000
5.148
Duration
References
Multiple Choice
Difficulty: 3 Challenge
Consider a six-year bond paying a 5% coupon, with a yield to maturity of 4.5%. What is the duration of the bond?
4.925
5.152
5.339
5.787
None of the options are correct.
Year
Cash Flow
PV of CF
% of PV
Sum of Years
1
$ 50
$ 47.85
0.0466
0.047
2
$ 50
$ 45.79
0.0446
0.089
3
$ 50
$ 43.81
0.0427
0.128
4
$ 50
$ 41.93
0.0409
0.163
5
$ 50
$ 40.12
0.0391
0.196
6
$ 1050
$ 806.29
0.7860
4.716
$ 1,025.79
1.0000
5.339
Duration
References
Multiple Choice
Difficulty: 3 Challenge
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79.
Award: 10.00 points
80.
Award: 10.00 points
81.
Award: 10.00 points
Consider a four-year bond paying a 7% coupon, with a yield to maturity of 6.0%. What is the duration of the bond?
3.631
3.785
3.814
3.965
None of the options are correct.
Year
Cash Flow
PV of CF
% of PV
Sum of Years
1
$ 70
$ 66.04
0.0638
0.064
2
$ 70
$ 62.30
0.0602
0.120
3
$ 70
$ 58.77
0.0568
0.170
4
$ 1070
$ 847.54
0.8192
3.277
$ 1,034.65
1.0000
3.631
Duration
References
Multiple Choice
Difficulty: 3 Challenge
Consider a four year bond paying a 8.5% coupon, with a yield to maturity of 9.3%. What is the duration of the bond?
3.831
3.785
3.614
3.548
None of the options are correct.
Year
Cash Flow
PV of CF
% of PV
Sum of Years
1
$ 85
$ 77.77
0.0798
0.080
2
$ 85
$ 71.15
0.0730
0.146
3
$ 85
$ 65.10
0.0668
0.200
4
$ 1,085
$ 760.24
0.7803
3.121
$ 974.25
1.0000
3.548
Duration
References
Multiple Choice
Difficulty: 3 Challenge
Consider a four-year, zero-coupon bond, with a yield to maturity of 7.2%. What is the duration of the bond?
4.000
3.785
3.614
3.548
None of the options are correct.
The duration of a zero-coupon bond is always the same as the bond maturity.
References
Multiple Choice
Difficulty: 3 Challenge
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Related Questions
solution
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c. What will be the yield to maturity on two-year zeros? (Do not round intermediate calculations. Round your answers to 2 decimal
places.)
YTM
%
d. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint. Compute
the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal
places.)
Expected total rate of return
%
e. If you purchase a three-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint. Compute
the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal
arrow_forward
Assume that you conduct a relative value trade and close it out after 5 days. Your 5-day return is 0.84%. Annualize this return as a Bond Equivalent Yield (BEY). Assume that a year has 365 days. State your answer as a percent, but without the percent sign. So, for example, if your answer was 0.1053, which is 10.53%, you would answer 10.53.
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3. Assume you purchased a bond for $9,186. The bond pays $300 interest every six months. You sell the bond after 18
months for $10,000. Calculate the following:
a. Income.
b. Capital gain (or loss).
c. Total return in dollars and as a percentage of the original investment.
Review Only
Click the icon to see the Worked Solution.
a. The current income is $
(Round to the nearest dollar.)
b. The capital gain (or loss) is $
(Enter a loss as a negative number and round to the nearest dollar.)
c. The total return in dollars is $
(Round to the nearest dollar.)
The total return as a percentage of the original investment is
%. (Enter as a percentage and round to two
decimal places.)
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Suppose that you buy a two-year 7.3% bond at its face value.
a-1. What will be your total nominal return over the two years if inflation is 2.3% in the first year and 4.3% in the second? (Do not round
intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)
Nominal return
a-2. What will be your real return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)
Real return
%
%
Real return
Nominal return
b. Now suppose that the bond is a TIPS. What will be your total 2-year real and nominal returns? (Do not round intermediate calculations. Enter
your answers as a percent rounded to 2 decimal places.)
1%
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Use a different graph for each one and clearly label the axis and the shifting of curves. Explain clearly (in words and on the graph) whether the price and yield to maturity increased or decreased.
You buy a bond that pays annual interest payments of 8% of the bond’s face value of $1000.
You initially pay $1050 for the bond. You receive an annual interest payment after one year, then sell the bond for $1010. What is your total rate of return on the investment, expressed as a percentage of the purchase price?
arrow_forward
Suppose that you buy a two-year 8.9% bond at its face value.
a-1. What will be your total nominal return over the two years if inflation is 3.9% in the first year and 5.9% in the second? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)
a-2. What will be your total real return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)
b. Now suppose that the bond is a TIPS. What will be your total 2-year real and nominal returns? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)
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Please solve using Excel and show/explain the formulas.
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1. Consider the following bond that pays coupon interest semi-annually.
Coupon
Yield to maturity Maturity (Years)
Par Value
Bond
8%
6%
2
$100
a) What is the price value of a basis point? Assuming one basis point decrease.
b) Compute the Macaulay duration theoretically. What is the modified duration?
c) Compute the approximate duration by increasing yield by 20 basis points and compare
your answers with those calculated in part (b).
d) Compute the convexity measure theoretically.
e) When the yield increases by 20 basis points, what is the new bond price? What is the
estimated bond price using duration calculated in part (b)? What is the estimated bond
price using duration calculated in part (b) together with the convexity calculated in part
(d)?
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Quantitative Problem: Today, interest rates on 1-year T-bonds yield 1.7%, interest rates on 2-year T-bonds yield 2.5%, and interest rates on 3-year T-bonds yield 3.4%. a. If the pure expectations theory is correct, what is the yield on 1-year T-bonds one year from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
b. If the pure expectations theory is correct, what is the yield on 2-year T-bonds one year from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
c. If the pure expectations theory is correct, what is the yield on 1-year T-bonds two years from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
arrow_forward
You must use formulas and cell references PV of Payments when computing all values in the green cells
Problem 1: Calculate the duration of a $1,000, 6% coupon bond with three years to maturity. Assume that all market interest rates are 7%
Problem 1b: Consider the bond in Problem 1a. Calculatethe expected price change if interest ratesdrop to 6.75% using the duration approximation.Also calculate the actual price change using discounted cash flow.
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Please show full steps and correctly thanks
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Bond A makes semi-annual interest payments of $30. Bond B makes an annual payment of $50. All else equal, which one has the lower coupon rate?
Select one:
a.
B
b.
A=B
c.
A
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In this problem we are going to calculate bond prices and returns
Suppose that the yield on a 3 year note is 2.5%.
a)
Calculate the price of the 3 year note (face value = $1000) with three annual coupon payments (after year 1, after year 2, after
year 3) of $30, i.e., the coupon rate is 3.0%.
b)
Is this note selling at a discount or premium? Explain.
Suppose that after one year and after you receive one coupon payment, you decide to sell your note. Your note is now a two year note with
one coupon payment after 1 year and another after year 2. Consider the following two scenarios:
Scenario #1 - interest rates on what is now a two year note (i.e., your note) have fallen to 1.00%
Scenario #2 - interest rates on what is now a two year note (i.e., your note) have risen to 4%
c)
given scenan
d)
Calculate the price that you can sell your note for under scenario #1 and the associated rate of return when you sell your note
Calculate the price that you can sell your note for under scenario #2…
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Topic 2: Return measures2. Which of the following investments do you prefer?(a) Purchase a zero-coupon bond, which pays $1000 in ten years, for a price of $550.(b) Invest $550 for ten years in a bank savings account at a guaranteed annual interestrate of 5.5%.1
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Quantitative Problem:
Today, interest rates on 1-year T-bonds yield 1.3%, interest rates on 2-year T-bonds yield 2.3%, and interest rates on 3-year T-bonds yield
3.7%.
a. If the pure expectations theory is correct, what is the yield on 1-year T-bonds one year from now? Be sure to use a geometric average in
your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
%
1.3
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b. If the pure expectations theory is correct, what is the yield on 2-year T-bonds one year from now? Be sure to use a geometric average in
your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
2.4
%
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c. If the pure expectations theory is correct, what is the yield on 1-year T-bonds two years from now? Be sure to use a geometric average in
your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
3.8
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hi can explain to me how to solve the question step by step ? TQ
1. Assuming that the expectations theory is the correct theory of the term structure, calculate the interest rates in the term structure for maturities of one to five years, and plot the resulting yield curves for the following series of one-year interest rates over the next five years. how would yields curves change if people preferred short term bonds over long-term bonds?
a. 5%, 7%, 7%, 7%, 7%
b. 5%, 4%, 4%, 4%, 4%
2. If the yield curve suddenly becomes steeper, how would you revise your predictions of interest rates in the future?
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Consider a bond that has just paid a semi-annual coupon and has exactly 2.5 years to maturity and an annual coupon rate of 3.5%.
Price the bond with a yield 4.2%. This is an annuity calculation and the stated yield and stated coupon are double the semi-annual
yield and coupon. Use Table 6.1. (Do not round Intermediate calculations. Round the final answer to 2 decimal places. Omit $ sign
In your response.)
Price of the bond
$
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Suppose that you buy a TIPS (inflation-indexed) bond with a 1-year maturity and a coupon of 2% paid annually. Assume you buy the bond at its face value of $1,000, and the inflation rate is 10%.
a. What will be your cash flow at the end of the year? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
b. What will be your real return?
c. What will be your nominal return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)
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Please don't use Ai solution
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Suppose that you buy a 1-year maturity bond with a coupon of 7% paid annually. If you buy the bond at its face value, what real rate of return will you earn if the inflation rate is 4%? 6%? 8%? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places. Negative amount should be indicated by a minus sign.)
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- solutionarrow_forwardc. What will be the yield to maturity on two-year zeros? (Do not round intermediate calculations. Round your answers to 2 decimal places.) YTM % d. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint. Compute the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal places.) Expected total rate of return % e. If you purchase a three-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint. Compute the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimalarrow_forwardAssume that you conduct a relative value trade and close it out after 5 days. Your 5-day return is 0.84%. Annualize this return as a Bond Equivalent Yield (BEY). Assume that a year has 365 days. State your answer as a percent, but without the percent sign. So, for example, if your answer was 0.1053, which is 10.53%, you would answer 10.53.arrow_forward
- 3. Assume you purchased a bond for $9,186. The bond pays $300 interest every six months. You sell the bond after 18 months for $10,000. Calculate the following: a. Income. b. Capital gain (or loss). c. Total return in dollars and as a percentage of the original investment. Review Only Click the icon to see the Worked Solution. a. The current income is $ (Round to the nearest dollar.) b. The capital gain (or loss) is $ (Enter a loss as a negative number and round to the nearest dollar.) c. The total return in dollars is $ (Round to the nearest dollar.) The total return as a percentage of the original investment is %. (Enter as a percentage and round to two decimal places.)arrow_forwardSuppose that you buy a two-year 7.3% bond at its face value. a-1. What will be your total nominal return over the two years if inflation is 2.3% in the first year and 4.3% in the second? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) Nominal return a-2. What will be your real return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) Real return % % Real return Nominal return b. Now suppose that the bond is a TIPS. What will be your total 2-year real and nominal returns? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.) 1%arrow_forwardUse a different graph for each one and clearly label the axis and the shifting of curves. Explain clearly (in words and on the graph) whether the price and yield to maturity increased or decreased. You buy a bond that pays annual interest payments of 8% of the bond’s face value of $1000. You initially pay $1050 for the bond. You receive an annual interest payment after one year, then sell the bond for $1010. What is your total rate of return on the investment, expressed as a percentage of the purchase price?arrow_forward
- Suppose that you buy a two-year 8.9% bond at its face value. a-1. What will be your total nominal return over the two years if inflation is 3.9% in the first year and 5.9% in the second? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) a-2. What will be your total real return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) b. Now suppose that the bond is a TIPS. What will be your total 2-year real and nominal returns? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)arrow_forwardPlease solve using Excel and show/explain the formulas.arrow_forward1. Consider the following bond that pays coupon interest semi-annually. Coupon Yield to maturity Maturity (Years) Par Value Bond 8% 6% 2 $100 a) What is the price value of a basis point? Assuming one basis point decrease. b) Compute the Macaulay duration theoretically. What is the modified duration? c) Compute the approximate duration by increasing yield by 20 basis points and compare your answers with those calculated in part (b). d) Compute the convexity measure theoretically. e) When the yield increases by 20 basis points, what is the new bond price? What is the estimated bond price using duration calculated in part (b)? What is the estimated bond price using duration calculated in part (b) together with the convexity calculated in part (d)?arrow_forward
- Quantitative Problem: Today, interest rates on 1-year T-bonds yield 1.7%, interest rates on 2-year T-bonds yield 2.5%, and interest rates on 3-year T-bonds yield 3.4%. a. If the pure expectations theory is correct, what is the yield on 1-year T-bonds one year from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places. b. If the pure expectations theory is correct, what is the yield on 2-year T-bonds one year from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places. c. If the pure expectations theory is correct, what is the yield on 1-year T-bonds two years from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places.arrow_forwardYou must use formulas and cell references PV of Payments when computing all values in the green cells Problem 1: Calculate the duration of a $1,000, 6% coupon bond with three years to maturity. Assume that all market interest rates are 7% Problem 1b: Consider the bond in Problem 1a. Calculatethe expected price change if interest ratesdrop to 6.75% using the duration approximation.Also calculate the actual price change using discounted cash flow.arrow_forwardPlease show full steps and correctly thanksarrow_forward
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