CHAPTER 16: MANAGING BOND PORTFOLIOS
CHAPTER 16: MANAGING BOND PORTFOLIOS
PROBLEM SETS
1. While it is true that short-term rates are more volatile than long-term rates, the
longer duration of the longer-term bonds makes their prices and their rates of
return more volatile. The higher duration magnifies the sensitivity to interest-
rate changes.
2. Duration can be thought of as a weighted average of the maturities of the cash
flows paid to holders of the perpetuity, where the weight for each cash flow is
equal to the present value of that cash flow divided by the total present value of
all cash flows. For cash flows in the distant future, present value approaches
zero (i.e., the weight becomes very small) so that these distant cash flows have
little impact and, eventually, virtually no impact on the weighted average.
3. The percentage change in the bond’s price is:
D 7.194
− y = − 0.005 = −0.0327 = −3.27%, or a 3.27% decline
1+ y 1.10
4. a. YTM = 6%
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 6%) Weight 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 = 2.833 years
16-1
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of McGraw-Hill Education.
,CHAPTER 16: MANAGING BOND PORTFOLIOS
b. YTM = 10%
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 10%) Weight Column (4)
1 $ 60.00 $ 54.55 0.0606 0.0606
2 60.00 49.59 0.0551 0.1102
3 1,060.00 796.39 0.8844 2.6532
Column sums $900.53 1.0000 2.8240
Duration = 2.824 years, which is less than the duration at the YTM of 6%.
5. 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) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 3%) Weight Column (4)
1 $ 30.00 $ 29.13 0.02913 0.02913
2 30.00 28.28 0.02828 0.05656
3 30.00 27.45 0.02745 0.08236
4 30.00 26.65 0.02665 0.10662
5 30.00 25.88 0.02588 0.12939
6 1030.00 862.61 0.86261 5.17565
Column sums $1000.00 1.00000 5.57971
D = 5.5797 half-year periods = 2.7899 years
If the bond’s yield is 10%, use a semiannual yield of 5% and semiannual
coupon of 3%:
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 5%) Weight Column (4)
1 $ 30.00 $ 28.57 0.03180 0.03180
2 30.00 27.21 0.03029 0.06057
3 30.00 25.92 0.02884 0.08653
4 30.00 24.68 0.02747 0.10988
5 30.00 23.51 0.02616 0.13081
6 1030.00 768.60 0.85544 5.13265
Column sums $898.49 1.00000 5.55223
D = 5.5522 half-year periods = 2.7761 years
16-2
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent
of McGraw-Hill Education.
, CHAPTER 16: MANAGING BOND PORTFOLIOS
6. If the current yield spread between AAA bonds and Treasury bonds is too wide
compared to historical yield spreads and is expected to narrow, you should shift
from Treasury bonds into AAA bonds. As the spread narrows, the AAA bonds
will outperform the Treasury bonds. This is an example of an intermarket spread
swap.
7. D. Investors tend to purchase longer term bonds when they expect yields to fall
so they can capture significant capital gains, and the lack of a coupon payment
ensures the capital gain will be even greater.
8. 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.
9. a.
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Rate = Column (1)
(Years) Cash Flow 10%) Weight Column (4)
1 $10 million $ 9.09 million 0.7857 0.7857
5 4 million 2.48 million 0.2143 1.0715
Column sums $11.57 million 1.0000 1.8572
D = 1.8572 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)1.8572 = $13.81 million
10 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.
16-3
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent
of McGraw-Hill Education.
CHAPTER 16: MANAGING BOND PORTFOLIOS
PROBLEM SETS
1. While it is true that short-term rates are more volatile than long-term rates, the
longer duration of the longer-term bonds makes their prices and their rates of
return more volatile. The higher duration magnifies the sensitivity to interest-
rate changes.
2. Duration can be thought of as a weighted average of the maturities of the cash
flows paid to holders of the perpetuity, where the weight for each cash flow is
equal to the present value of that cash flow divided by the total present value of
all cash flows. For cash flows in the distant future, present value approaches
zero (i.e., the weight becomes very small) so that these distant cash flows have
little impact and, eventually, virtually no impact on the weighted average.
3. The percentage change in the bond’s price is:
D 7.194
− y = − 0.005 = −0.0327 = −3.27%, or a 3.27% decline
1+ y 1.10
4. a. YTM = 6%
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 6%) Weight 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 = 2.833 years
16-1
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent
of McGraw-Hill Education.
,CHAPTER 16: MANAGING BOND PORTFOLIOS
b. YTM = 10%
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 10%) Weight Column (4)
1 $ 60.00 $ 54.55 0.0606 0.0606
2 60.00 49.59 0.0551 0.1102
3 1,060.00 796.39 0.8844 2.6532
Column sums $900.53 1.0000 2.8240
Duration = 2.824 years, which is less than the duration at the YTM of 6%.
5. 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) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 3%) Weight Column (4)
1 $ 30.00 $ 29.13 0.02913 0.02913
2 30.00 28.28 0.02828 0.05656
3 30.00 27.45 0.02745 0.08236
4 30.00 26.65 0.02665 0.10662
5 30.00 25.88 0.02588 0.12939
6 1030.00 862.61 0.86261 5.17565
Column sums $1000.00 1.00000 5.57971
D = 5.5797 half-year periods = 2.7899 years
If the bond’s yield is 10%, use a semiannual yield of 5% and semiannual
coupon of 3%:
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Column (1)
(Years) Cash Flow Rate = 5%) Weight Column (4)
1 $ 30.00 $ 28.57 0.03180 0.03180
2 30.00 27.21 0.03029 0.06057
3 30.00 25.92 0.02884 0.08653
4 30.00 24.68 0.02747 0.10988
5 30.00 23.51 0.02616 0.13081
6 1030.00 768.60 0.85544 5.13265
Column sums $898.49 1.00000 5.55223
D = 5.5522 half-year periods = 2.7761 years
16-2
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent
of McGraw-Hill Education.
, CHAPTER 16: MANAGING BOND PORTFOLIOS
6. If the current yield spread between AAA bonds and Treasury bonds is too wide
compared to historical yield spreads and is expected to narrow, you should shift
from Treasury bonds into AAA bonds. As the spread narrows, the AAA bonds
will outperform the Treasury bonds. This is an example of an intermarket spread
swap.
7. D. Investors tend to purchase longer term bonds when they expect yields to fall
so they can capture significant capital gains, and the lack of a coupon payment
ensures the capital gain will be even greater.
8. 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.
9. a.
(1) (2) (3) (4) (5)
Time until PV of CF
Payment (Discount Rate = Column (1)
(Years) Cash Flow 10%) Weight Column (4)
1 $10 million $ 9.09 million 0.7857 0.7857
5 4 million 2.48 million 0.2143 1.0715
Column sums $11.57 million 1.0000 1.8572
D = 1.8572 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)1.8572 = $13.81 million
10 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.
16-3
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent
of McGraw-Hill Education.
