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 zero coupon, long maturity bond will have the highest duration and will, therefore,
produce the largest price change when interest rates change.
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.
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.
Chapter 11 – Managing Bond Portfolios
(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
2
60
53.40
0.0534
0.1068
3
1060
890.00
0.8900
2.6700
Column Sum:
1000.00
1.0000
2.8334
When YTM = 10%, the duration is 2.8238
(1)
(2)
(3)
(4)
(5)
Time until
Payment
(Years)
Payment
Payment
Discounted at
10%
Weight
Column (1)
×
Column (4)
1
60
54.55
0.0606
0.0606
2
60
49.59
0.0551
0.1101
3
1060
796.39
0.8844
2.6531
Column Sum:
900.53
1.0000
2.8238
When the yield to maturity increases, the duration decreases.
9. Using Equation 11.2, the percentage change in the bond price is:
P
P
10.1
0050.0
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
2
2
1.6529
0.4989
0.9977
3
1
0.7513
0.2267
0.6803
Column Sum:
3.3133
1.0000
1.9524
Duration = 1.9524 years
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
08.1
0010.0
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
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
duration.
b. The lower-coupon bond has the longer duration and more de facto call protection.
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:
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)
1
10,000
9,259.26
0.5192
0.51923
2
10,000
8,573.39
0.4808
0.96154
Column Sum:
17,832.65
1.0000
1.48077
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 $17,832.65, the face value
Chapter 11 – Managing Bond Portfolios
c. If the interest rate increases to 9%, the zero-coupon bond would fall in value to:
26.985,19$
d. If the interest rate falls to 7%, the zero-coupon bond would rise in value to:
26.985,19$
16. a. PV of obligation = $2 million/0.16 = $12.5 million
Duration of obligation = 1.16/0.16 = 7.25 years
Call w the weight on the five-year maturity bond (with duration of 4 years). Then:
b. The price of the 20-year bond is:
[60 Annuity factor(16%, 20)] + [1000 PV factor(16%, 20)] = $407.1
Therefore, the bond sells for 0.4071 times its par value, so that:
17. a. Shorten his portfolio duration to decrease the sensitivity to the expected rate increase.
Chapter 11 – Managing Bond Portfolios
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18. Change in price = – (Modified duration Change in YTM) Price
= – 3.5851 0.01 $100
= – $3.5851
The price will decrease by $3.59.
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:
(w 5) + [(1 – w) 21] = 10 w = 11/16 = 0.6875
20. Macaulay Duration and Modified Duration are calculated using Excel as follows:
Inputs
Formula in column B
Settlement date
5/27/2018
=DATE(2018,5,27)
Maturity date
11/15/2028
=DATE(2027,11,15)
Coupon rate
0.07
0.07
Yield to maturity
0.08
0.08
Coupons per year
2
2
Outputs
Macaulay Duration
6.9659
=DURATION(B2,B3,B4,B5,B6)
Modified Duration
6.6980
=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
5/27/2018
=DATE(2018,5,27)
Maturity date
11/15/2027
=DATE(2027,11,15)
Coupon rate
0.07
0.07
Yield to maturity
0.08
0.08
Coupons per year
1
1
Outputs
Macaulay Duration
6.8844
=DURATION(B2,B3,B4,B5,B6)
Modified Duration
6.3745
=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.
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:
(w 5) + [(1 – w) 20] = 11 w = 9/15 = 0.60
23. Convexity is calculated using the Excel spreadsheet below:
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
Price $89.59 4 6 4.41 20 88.204
5 6 4.083 30 122.505
6 6 3.781 42 158.803
7106 61.85 56 3463.599
8 0 0 72 0
9 0 0 90 0
10 0 0 110 0
Sum: 89.58726 3932.242
Convexity: 37.631057
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
1080
768.723
0.8504
2.5513
Sum:
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
1000
711.780
1.0000
3.0000
Sum:
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
Chapter 11 – Managing Bond Portfolios
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3
1120
797.194
0.7972
2.3916
Sum:
1000.000
1.0000
2.6901
The weights of the earlier payments are higher when the coupon increases.
Therefore, duration falls.
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
3YTM 0.1 280 66.115702 6 396.6942149
4Maturity 3 31080 811.420 12 9737.040
5Price $950.263 Sum 950.263 10279.189
6
7 Convexity: 8.939838
=G5/(E5*(1+B2)^2)
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
3Maturity 3 2 0 0 6 0
4Price $751.315 31000 751.315 12 9015.778
5 Sum 751.315 9015.778
6
7 Convexity: 9.917355
=G5/(E5*(1+B2)^2)
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%:
y
c. Using the duration-with-convexity rule, assuming yield to maturity falls to 7%:
Chapter 11 – Managing Bond Portfolios
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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 = $168.92
Therefore: Predicted price = $168.92 + $1,450.31 = $1,619.23
d. The actual price at a 7% yield to maturity is $1,620.45. Therefore:
28.605,1$45.620,1$ ==
−
45.620,1$
Conclusion: The duration-with-convexity rule provides more accurate
approximations to the actual change in price. In this example, the percentage
error using convexity with duration is less than one-tenth the error using
duration only to estimate the price change.
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
+
97.154$31.450,1$
08.1
01.0 −=
+
21.308,1$
Using the duration-with-convexity rule, assuming yield to maturity rises to 9%:
Predicted price change =
Chapter 11 – Managing Bond Portfolios
Therefore: Predicted price = –$141.02 + $1,450.31 = $1,309.29
21.308,1$
21.308,1$29.309,1$ ==
−
Conclusion: Similar to the decrease in YTM, using the duration-with-convexity
rule for the increase in YTM predicts a more fine-tuned estimate of price than the
duration rule alone.
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
End of year price (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
One year total $ return $70.65 $70.83 $87.72
One year total rate of return 7.00% 7.08% 9.00%
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:
$893.25 [n = 25; i = 8; FV = 1,000; PMT = 70]
Therefore, total proceeds will be:
$394.60 + $893.25 = $1,287.85
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]
Chapter 11 – Managing Bond Portfolios
Therefore, total proceeds will be:
29.
a. Using a financial calculator, we find that the price of the zero-coupon bond
(with $1000 face value) is:
The price of the 6% coupon bond is:
i. Zero Coupon Bond
84.374$28.333$−=
−
Chapter 11 – Managing Bond Portfolios
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
84.374$04.422$=
−
Chapter 11 – Managing Bond Portfolios
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B: Lower yield to maturity than bond E
E: Highest coupon, shortest maturity, highest yield of all bonds
CFA 2 Answer:
a. Modified duration =
YTM1
durationMacaulay
+
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
b. If yield to maturity (YTM) on Bond B falls by 75 basis points:
Chapter 11 – Managing Bond Portfolios
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
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
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.
CFA 5 Answer:
Chapter 11 – Managing Bond Portfolios
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25-year maturity: ∆P/P = −23.81 × 0.50% = −11.9050%
Strategy I: ∆P/P = (0.5 × 3.6225%) + [0.5 × (−11.9050%)] = −4.1413%
For Strategy II:
15-year maturity: ∆P/P = −14.35 × 0.25% = −3.5875%
CFA 6
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:
Modified Duration = – ∆P/(P × ∆y)
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
rate at which coupon income can be reinvested.
b. Immunization is the process of structuring a bond portfolio in such a manner that
the value of the portfolio (including proceeds reinvested) will reach a given target
Chapter 11 – Managing Bond Portfolios
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consent of McGraw-Hill Education.
c. Duration matching is superior to maturity matching because bonds of equal
duration — not those of equal maturity — are equally sensitive to interest rate
fluctuations.
CFA 9
Answer:
The economic climate is one of impending interest rate increases. Hence, we will want
to shorten portfolio duration.
a. Choose the short maturity (2019) bond.
b. The Arizona bond likely has lower duration. Coupons are about equal, but the
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
engage in a rate anticipation swap, lengthening duration if rates are expected to
fall, and shortening duration if rates are expected to rise.
b. A change in yield spreads across sectors would call for an inter-market spread
Chapter 11 – Managing Bond Portfolios
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c. A belief that the yield spread on a particular instrument will change calls for a
substitution swap in which that security is sold if its relative yield is expected to rise
or is bought if its yield is expected to fall compared to other similar bonds.
CFA 12
Answer:
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
coupon 2⅜% bond would provide more opportunity for capital gains, greater call
protection, and greater protection against declining reinvestment rates at a cost of
only a modest drop in yield.
b. This swap was probably done by an investor who believed the 24 basis point yield
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
since the yield to maturity, determined by the discount at the time of purchase, is
locked in. This is in contrast to the floating rate note, for which interest is adjusted
periodically to reflect current returns on debt instruments. The funds received in
interest income on the floating rate notes could be used at a later time to purchase
long-term bonds at more attractive yields.
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First, that the appreciation potential of the Z mart convertible, based primarily
on the intrinsic value of Z mart common stock, was no longer as attractive as it
had been.
Second, that the yields on long-term bonds were at a cyclical high, causing bond
portfolio managers who could take A2-risk bonds to reach for high yields and long
maturities, either to lock them in or take a capital gain when rates subsequently
declined.
Third, while waiting for rates to decline, the investor will enjoy an increase in
coupon income. Basically, the investor is swapping an equity-equivalent for a long-
term corporate bond.