Chapter 11 – Managing Bond Portfolios
CHAPTER 11
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.
a. Engage in active bond management, specifically bond swaps
4. Change in Price = – (Modified Duration Change in YTM) Price
5. d. None of the above.
7. 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 rates of return more volatile. The higher
8. When YTM = 6%, the duration is 2.8334.
Chapter 11 – Managing Bond Portfolios
(1)
(2)
(3)
(4)
(5)
(Years)
Column (4)
1
60
2
60
3
Time until
Payment
Payment
Payment
Discounted at
Weight
Column (1)
×
(1)
(2)
(3)
(4)
(5)
(Years)
Column (4)
1
60
54.55
2
60
49.59
3
Time until
Payment
Payment
Payment
Discounted at
Weight
Column (1)
×
9. Using Equation 11.2, the percentage change in the bond price is:
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
2
3
3.3133
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:
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
b. Bond A has a lower yield and a lower coupon, both of which cause it to have a
longer duration than that of Bond B. Moreover, Bond A cannot be called.
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
duration.
b. The lower-coupon bond has the longer duration and more de facto call protection.
a. The present value of the obligation is $17,832.65 and the duration is 1.4808 years,
as shown in the following table:
Computation of duration, interest rate = 8%
(1)
(2)
(3)
(4)
(5)
Time until
Payment
(Years)
Payment
Payment
Discounted
at 8%
Weight
Column (1)
×
Column (4)
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:
d. If the interest rate falls to 7%, the zero-coupon bond would rise in value to:
16. a. PV of obligation = $2 million/0.16 = $12.5 million
Therefore:
0.1875 $12.5 = $2.3 million in the 5-year bond, and
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
= – 3.5851 0.01 $100
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
years, which is now the duration of the obligation, we again solve for w:
20. Macaulay Duration and Modified Duration are calculated using Excel as follows:
Inputs
Formula in column B
Settlement date
5/27/2016
=DATE(2016,5,27)
Maturity date
11/15/2025
=DATE(2025,11,15)
Coupon rate
Yield to maturity
Coupons per year
2
Outputs
Macaulay Duration
=DURATION(B2,B3,B4,B5,B6)
Modified Duration
=MDURATION(B2,B3,B4,B5,B6)
21. Macaulay Duration and Modified Duration are calculated using Excel as follows:
Inputs
Formula in column B
Settlement date
5/27/2016
=DATE(2016,5,27)
Maturity date
11/15/2025
=DATE(2025,11,15)
Coupon rate
Yield to maturity
Coupons per year
1
Outputs
Macaulay Duration
6.8844
=DURATION(B2,B3,B4,B5,B6)
Modified Duration
=MDURATION(B2,B3,B4,B5,B6)
Chapter 11 – Managing Bond Portfolios
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
23. Convexity is calculated using the Excel spreadsheet below:
Chapter 11 – Managing Bond Portfolios
Time (t) Cash flow PV(CF) t + t^2 (t + t^2) x PV(CF)
Coupon 6 1 6 5.556 2 11.111
YTM 0.08 2 6 5.144 6 30.864
Maturity 7 3 6 4.763 12 57.156
7106 61.85 56 3463.599
8 0 0 72 0
9 0 0 90 0
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
3
768.723
0.8504
2.5513
903.927
1.0000
2.7714
Zero-coupon
1
0
0.000
0.0000
0.0000
2
0
0.000
0.0000
0.0000
3
711.780
1.0000
3.0000
711.780
1.0000
3.0000
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
2
120
95.663
0.0957
0.1913
3
797.194
0.7972
2.3916
1.0000
2.6901
Chapter 11 – Managing Bond Portfolios
25.
a.
A B C D E F G
1Time (t) Cash flow PV(CF) t + t^2 (t + t^2) x PV(CF)
2 Coupon 80 180 72.727273 2 145.4545455
b.
A B C D E F G
1Time (t) Cash flow PV(CF) t + t^2 (t + t^2) x PV(CF)
2YTM 0.1 1 0 0 2 0
26.
a. Using a financial calculator, we find that the price of the bond is:
For yield to maturity of 7%: $1,620.45
b. Using the duration rule, assuming yield to maturity falls to 7%:
Predicted price change = – Duration
0
P
y1
y
+
c. Using the duration-with-convexity rule, assuming yield to maturity falls to 7%:
Chapter 11 – Managing Bond Portfolios
d. The actual price at a 7% yield to maturity is $1,620.45. Therefore:
28.605,1$45.620,1$ ==
−
e. For yield to maturity of 9%, the price of the bond is $1,308.21
Using the duration rule, assuming yield to maturity increases to 9%:
Predicted price change = – Duration
0
P
y1
y
+
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
27. You should buy the three-year bond because it will offer a 9% holding-period return
over the next year, which is greater than the return on either of the other bonds, as
shown below:
Maturity One year Two years Three years
YTM at beginning of year 7.00% 8.00% 9.00%
Beginning of year price $1,009.35 $1,000.00 $974.69
Capital gain −$ 9.35 −$ 9.17 $7.72
Coupon $80.00 $80.00 $80.00
One year total $ return $70.65 $70.83 $87.72
28.
a. The maturity of the 30-year bond will fall to 25 years, and the yield is forecast to
be 8%. Therefore, the price forecast for the bond is:
At a 6% interest rate, the five coupon payments will accumulate to $394.60 (FV)
Therefore, total proceeds will be:
The five-year return is therefore: ($1,287.85/867.42) – 1 = 1.48469 – 1 = 48.469%
b. The maturity of the 20-year bond will fall to 15 years, and its yield is forecast to be
7.5%. Therefore, the price forecast for the bond is:
$911.73 [n = 15; i = 7.5; FV = 1000; PMT = 65]
At a 6% interest rate, the five coupon payments will accumulate to $366.41 after
Chapter 11 – Managing Bond Portfolios
29.
a. Using a financial calculator, we find that the price of the zero-coupon bond
(with $1000 face value) is:
For yield to maturity of 8%: $374.84
i. Zero Coupon Bond
Coupon Bond
ii. Zero Coupon Bond
Predicted % loss = [(–11.81) 0.01] + [0.5 150.3 (0.01)2]
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
Chapter 11 – Managing Bond Portfolios
The percentage gain predicted by the duration-with-convexity rule is:
c. The 6% coupon bond (which has higher convexity) outperforms the zero
regardless of whether rates rise or fall. This is a general property which can be
understood by first noting from the duration-with-convexity formula that the
duration effect resulting from the change in rates is the same for the two bonds
d. This situation cannot persist. No one would be willing to buy the lower
convexity bond if it always underperforms the other bond. The price of the
lower convexity bond will fall and its yield to maturity will rise. Thus, the lower
CFA 1
Answer:
C: Highest maturity, zero coupon
D: Highest maturity, next-lowest coupon
CFA 2 Answer:
a. Modified duration =
YTM1
durationMacaulay
+
Chapter 11 – Managing Bond Portfolios
b. For option-free coupon bonds, modified duration is better than maturity as a
measure of the bond’s sensitivity to changes in interest rates. Maturity considers
c. i. Modified duration increases as the coupon decreases.
CFA 3
Answer:
a. Scenario (i): Strong economic recovery with rising inflation expectations.
Interest rates and bond yields will most likely rise, and the prices of both bonds
will fall. The probability that the callable bond will be called declines, so that it
will behave more like the non-callable bond. (Notice that they have similar
b. If yield to maturity (YTM) on Bond B falls by 75 basis points:
c. For Bond A (the callable bond), bond life and therefore bond cash flows are
uncertain. If one ignores the call feature and analyzes the bond on a “to maturity”
basis, all calculations for yield and duration are distorted. Durations are too long
Chapter 11 – Managing Bond Portfolios
CFA 4
Answer:
a. The Aa bond initially has the higher yield to maturity (yield spread of 40 b.p.
versus 31 b.p.), but the Aa bond is expected to have a widening spread relative
to Treasuries. This will reduce rate of return. The Aaa spread is expected to be
stable. Calculate comparative returns as follows:
b. Other variables that one should consider:
• Potential changes in issue-specific credit quality: If the credit quality of the
bonds changes, spreads relative to Treasuries will also change.
• Changes in relative yield spreads for a given bond rating: If quality spreads
CFA 5 Answer:
∆P/P = −D* ∆y
For Strategy I:
CFA 6
Chapter 11 – Managing Bond Portfolios
Answer:
a. For an option-free bond, the effective duration and modified duration are
approximately the same. The duration of the bond described in Table 22A is
calculated as follows:
b. The total percentage price change for the bond described in Table 22A is
estimated as follows:
CFA 7
Answer:
CFA 8
Answer:
a. The two risks are price risk and reinvestment rate risk. The former refers to
bond price volatility as interest rates fluctuate, the latter to uncertainty in the
CFA 9
Answer:
The economic climate is one of impending interest rate increases. Hence, we will want
to shorten portfolio duration.
Chapter 11 – Managing Bond Portfolios
CFA 10
Answer:
a. (4) A low coupon and a long maturity
CFA 11
Answer:
a. A manager who believes that the level of interest rates will change should
CFA 12
Answer:
Chapter 11 – Managing Bond Portfolios
a. This swap would have been made if the investor anticipated a decline in long-term
interest rates and an increase in long-term bond prices. The deeper discount, lower
b. This swap was probably done by an investor who believed the 24 basis point yield
spread between the two bonds was too narrow. The investor anticipated that, if the
spread widened to a more normal level, either a capital gain would be experienced
on the Treasury note or a capital loss would be avoided on the Phone bond, or both.
c. This swap would have been made if the investor were bearish on the bond market.
The zero coupon note would be extremely vulnerable to an increase in interest rates
d. These two bonds are similar in most respects other than quality and yield. An
investor who believed the yield spread between Government and Al bonds was too
narrow would have made the swap either to take a capital gain on the Government
e. The principal differences between these two bonds are the convertible feature of the
Z mart bond, and, for the Lucky Duck debentures, the yield and coupon advantage,
and the longer maturity. The swap would have been made if the investor believed
some combination of the following:
Chapter 11 – Managing Bond Portfolios