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Chapter 11 - Managing Bond Portfolios
CHAPTER 11
11-1
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
MANAGING BOND PORTFOLIOS
1. 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
2. A low coupon, long maturity bond will have the highest duration and will, therefore,
3. An intermarket spread swap should work. The trade would be to long the corporate
4. Change in Price = – (Modified Duration Change in YTM) Price
=
−Macaulay's Duration
1+ YTM
Change in YTM Price
1+ 0.0 6
5. d. None of the above.
6. The increase will be larger than the decrease in price.
7. While it is true that short-term rates are more volatile than long-term rates, the longer
8. When YTM = 6%, the duration is 2.8334.
(1) (2) (3) (4) (5)
Time until
Payment
(Years)
Payment
Payment
Discounted at
6%
Weight
Column (1)
×
Column (4)
1 60 56.60 0.0566 0.0566
(1) (2) (3) (4) (5)
Time until
Payment
(Years)
Payment
Payment
Discounted at
10%
Weight
Column (1)
×
Column (4)
11-2
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
1 60 54.55 0.0606 0.0606
When the yield to maturity increases, the duration decreases.
9. Using Equation 11.2, the percentage change in the bond price is:
P
P
0327.0
10.1
0050.0
194.7
1
y
y
10. The computation of duration is as follows:
Interest Rate (YTM) is 10%.
(1) (2) (3) (4) (5)
Time until
Payment
(Years)
Payment
(in millions
of dollars)
Payment
Discounted
At 10%
Weight
Column (1)
×
Column (4)
1
1
0.9091
0.2744
0.2744
11. The duration of the perpetuity is: (1 + y)/y = 1.10/0.10 = 11 years
Let w be the weight of the zero-coupon bond. Then we find w by solving:
12. Using Equation 11.2, the percentage change in the bond price will be:
P
P
00463.0
08.1
0010.0
0.5
y1
y
13.
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
11-3
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
b. Bond A has a lower yield and a lower coupon, both of which cause it to have a
14. Choose the longer-duration bond to benefit from a rate decrease.
a. The Aaa-rated bond has the lower yield to maturity and therefore the longer
b. The lower-coupon bond has the longer duration and more de facto call protection.
c. The lower coupon bond has the longer duration.
15.
a. The present value of the obligation is $17,832.65 and the duration is 1.4808 years,
as shown in the following table:
(1) (2) (3) (4) (5)
Time until
Payment
(Years)
Payment
Payment
Discounted
at 8%
Weight
Column (1)
×
Column (4)
1
10,000
9,259.26
0.5192
0.51923
b. To immunize the obligation, invest in a zero-coupon bond maturing in 1.4808 years.
Since the present value of the zero-coupon bond must be $17,832.65, the face value
(i.e., the future redemption value) must be:
c. If the interest rate increases to 9%, the zero-coupon bond would fall in value to:
92.590,17$
)09.1(
26.985,19$
4808.1
The present value of the tuition obligation would fall to $17,591.11, so that the net
position changes by $0.19.
If the interest rate falls to 7%, the zero-coupon bond would rise in value to:
99.079,18$
)07.1(
26.985,19$
4808.1
The present value of the tuition obligation would increase to $18,080.18, so that the
net position changes by $0.19.
The reason the net position changes at all is that, as the interest rate changes, so
does the duration of the stream of tuition payments.
92.590,17$
)09.1(
26.985,19$
4808.1
99.079,18$
)07.1(
26.985,19$
4808.1
11-4
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
16.
a. PV of obligation = $2 million/0.16 = $12.5 million
b. The price of the 20-year bond is:
[60 Annuity factor(16%, 20)] + [1000 PV factor(16%, 20)] = $407.1
17. a. Shorten his portfolio duration to decrease the sensitivity to the expected rate
increase.
18. Change in price = – (Modified duration Change in YTM) Price
19.
a. The duration of the perpetuity is: 1.05/0.05 = 21 years
Let w be the weight of the zero-coupon bond, so that we find w by solving:
b. The zero-coupon bond will then have a duration of 4 years while the perpetuity
will still have a 21-year duration. To have a portfolio with duration equal to nine
20. Macaulay Duration and Modified Duration are calculated using Excel as follows:
Inputs Formula in column B
Settlement date 5/27/2012 =DATE(2012,5,27)
11-5
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
Maturity date 11/15/2021 =DATE(2021,11,15)
Coupon rate 0.07 0.07
21. Macaulay Duration and Modified Duration are calculated using Excel as follows:
Inputs Formula in column B
Settlement date 5/27/2012 =DATE(2012,5,27)
Generally, we would expect duration to increase when the frequency of payment
decreases from two payments per year to one payment per year because more of the
bond’s payments are made further in to the future when payments are made annually.
However, in this example, duration decreases as a result of the timing of the settlement
22.
a. The duration of the perpetuity is: 1.10/0.10 = 11 years
The present value of the payments is: $1 million/0.10 = $10 million
Let w be the weight of the five-year zero-coupon bond and therefore (1 – w) is
the weight of the twenty-year zero-coupon bond. Then we find w by solving:
b. Face value of the five-year zero-coupon bond is:
$6 million (1.10)5 = $9,663,060.00
11-6
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
23. Convexity is calculated using the Excel spreadsheet below:
24.
a. Interest rate = 12%
Time until
Payment
(Years)
Payment
Payment
Discounted
at 12%
Weight
Time
×
Weight
8% coupon 1 80 71.429 0.0790 0.0790
2 80 63.776 0.0706 0.1411
Zero-coupon 1 0 0.000 0.0000 0.0000
11-7
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
b. Continue to use a yield to maturity of 12%:
Time until
Payment
(Years)
Payment
Payment
Discounted
at 12%
Weight
Time
×
Weight
12% coupon 1 120 107.143 0.1071 0.1071
25.
a.
b.
26. Using a financial calculator, we find that the price of the bond is:
For yield to maturity of 7%: $1,620.45
Using the duration rule, assuming yield to maturity falls to 7%:
11-8
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 11 - Managing Bond Portfolios
Chapter 11 - Managing Bond Portfolios
% error
%075.000075.0
45.620,1$
23.619,1$45.620,1$
Using the duration-with-convexity rule, assuming yield to maturity rises to 9%:
Predicted price change
=
[
−Duration × ∆y
1 + y +
(
0.5 × Convexity × (∆y)2
)
]
P0
=
[
(
−11.54 × + 0.01
1.08
)
+
(
0.5 × 192.4 × ( -0.01)2
)
]
$1,450.31
= –$141.02
Therefore: Predicted price = –$141.02 + $1,450.31 = $1,309.29
The actual price at a 9% yield to maturity is $1,308.21. Therefore:
11-10
© 2013 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
