Chapter 23 Solutions
1.
a) Riding the yield curve. If the yield curve is expected to remain upward
sloping and unchanged, the investor can purchase a longer term security than
the investor holding period and sell it for a going at the end of the holding
period. The gain as the rule will be greater than the loss of reinvestment
income thereby providing an increase in the realized yield of the investment.
b) Swapping the buying and selling of similar types of securities in order to
increase the realized yield without increasing the risk.
c) Work out period – the amount of time it takes for the mispricing of a security
to be corrected.
d) Duration-a measure of the life or maturity of a bond which is calculated as
the present value of each cash flow relative to the total present value of the
bond multiplied by when the cash flow occurs.
e) Immunization – Regardless of the changes in interest rate, the yield of the
holding period of a bond is unchanged, i.e. the promise yield equals the
realized yield.
f)
Cross over yield – that yield where the yield to maturity is equal to the yield
to call.
2.
CFt
(1  r )t
Duration   N
CFt
t 1

t 1 (1  r )t
N
tx
Cashflow
1.
2.
3.
PV Factor CFXPVF
Weight
Duration
150
.8696
130.44
130.44
=
999.98
.1304 × 1 =
.1304
150
.7561
113.415
113.415
=
999.98
.1304 × 2 =
.2268
1150
.6575
726.125
999.98
726.125
=
999.98
.7561 × 3 =
2.2688
D = 2.6 years
3.
2.6255
Reinvestment Rate Risk – the risk that the future cash flow of the bond will have
to be invested at lower (higher) interest rates than the yield to maturity.
Interest Rate Risk – the change in the value of the bond caused by increases or
decreases in the future interest rate and its effect on realized yield.
4.
H
P
Original Investment
Two coupons
Interest on coupon @10% for 1/2 yr.
50(1.1).5 – 50 =
Principal Value at end of year @10%
Total # accrued
Total Gain
Gain per $ invested
$1,000
100
$
2.44
$ 950
100
$ 2.44
1,000
$ 1,102.44
$ 102.44
.10244
1,000
$ 1,102.44
$ 152.44
.15244
Realized coupon yield
(1 + x/2)2 = 1.10244/1
Value of Swap
9.999 ≈ 10% (1 + x/2)2 = 1.10244/1
14.7%
470 basis points
5.
Interest yield to maturity
TM at work out
H
10%
10%
P
11%
10.5%
Speed narrows from 100 basis points to 50 basis points
Workout 1 year
Reinvestment rate 10%
Original Investment
Two coupons during year
Interest on the coupon @10%
for 6 months 25(1.1)5 − 25
Principal Value at end of year
$ 680.00
50
40(1.1)5 − 50
1.22
$ 658.00
$1,000
100
Total # accrued
Total Gain
$ 736.22
$ 56.22
$1,117.44
$ 117.44
Gain per $ invested
Realized coupon yield
(1 + x/2)2 = 1.08267/1
Value of Swap
.08267
8.10%
2.44
$1,015.00
.11744
11.42%
2
(1 + x/2) = 1.11744/1
332 basis points
6.
The longer the workout period, the smaller the value of the swap.
7.
If an investor expects that interest rates will change dramatically i.e., the investor
expects rates to go up. He sells his current long term investments and puts the
funds into short term investments. If rates increase, he can then reinvest his funds
in the new higher long term yield. He is hoping that the loss of going from long
term to short term is less than the gain between current long term yields and
expected future long term yields.
8.
The maturity and duration for a zero coupon bond are equal.
The duration is always less than the maturity for a coupon bond selling at par.
The duration is less than and then it becomes greater than the maturity for a
coupon bond selling at a discount. This depends on the maturity of the bond as
shown below:
For coupon bonds selling at a premium, the duration is always less than the
maturity of the bond.
9.
The WATM only considers the timing of the cash flows and not the time value of
money, whereas the duration considers both the timing of the cash flows and the
time value of money. Hence, the WATM is always larger than the duration of a
bond.
10.
N
WATM  
t 1
tx CFt
N
 CF
t 1
t
A, B, and C
CASHFLO
CF/TCF
Tx = CF/TCF
50
 .04
1250
.04(1) = .04
50
 .04
1250
.04(2) = .08
yield to maturity of A
W
1
2
50
50
5
$957.80  
t 1
50
1000
t
(1  y ) (1  y )5
YTM  6%
yield to maturity of B
3
50
50
 .04
1250
.04(3) = .12
5
$1044.65  
t 1
4
50
50
 .04
1250
.04(4) = .16
5
1050
1050
 .84
1250
.84(5) = 4.20
TCF = 1250
1.00
WATM = 4.60
N
Duration   tx CFt (1  y )t
t 1
CF
APV
CF×PV
1
50
.943
47.15
2
50
.890
3
50
4
5
50
1000
t
(1  y ) (1  y )5
YTM  4%
yield to maturity of C
5
1000   50 (1  y )t  1000 (1  y)5
YTM  5%
t 1
wt
Duration
47.15

957.60
.049 × 1
.049
44.5
44.5

957.60
.046 × 2
.092
.840
42
42

957.60
.044 × 3
.132
50
.792
39.6
39.6

957.60
.041 × 4
.164
1.050
.747
748.35
748.35

957.60
.819 × 5
4.095
Duration = 4.532
Duration B
CF
APV
CF×PV
1
50
.962
48.1
2
50
.925
46.25
wt
Duration
48.1

1044.65
.046 × 1
.046
46.25

1044.65
.044 × 2
.088
3
50
.889
44.45
44.45

1044.65
.043 × 3
.129
4
50
.855
42.75
42.75

1044.65
.041 × 4
.164
5
1050
.822
863.10
863.10

1044.65
.826 × 5
4.13
1044.65
Duration = 4.557
Duration C
CF
APV
CF×PV
wt
Duration
1
50
.952
46.70
46.70

1, 000
.048 × 1
.048
2
50
.907
45.35
45.35

1, 000
.045 × 2
.09
3
50
.864
43.2
43.2

1, 000
.043 × 3
.129
4
50
.823
41.15
41.15

1, 000
.041 × 4
.164
5
1050
.784
823.2
823.2

1, 000
.823 × 5
4.115
1000.00
Duration = 4.546
The change in YTM does not effect WATM it is 4.6 in all three cases. The
change in YTM causes the duration to change.
A. 4.53
B. 4.55
the higher the yield to maturity the lower the duration
C. 4.54
11.
If you expect interest rates to fall, the portfolio should have a duration longer
than the length of the investment horizon.
12.
The higher the coupon rate for a given maturity bond the lower the duration.
13.
The duration of a callable bond is shorter than the duration of a noncallable
bond, sentis paribus.
14. The convexity of a bond is the second-order approximation of the bond price
changes when the yield to maturity changes. The percentage of bond price can
be calculated as the equation (23.6) when the YTM changes.
P
  D * i  0.5  Convexity  (i ) 2
P
(23.6)
Where the Convexity is the rate of change of the slope of the price-yield curve as
follows
Convexity 
1 2 P
1
 2 
P i
P  (1  i)2
n
CFt
[ (1  i)
t 1
t
(t 2  t )]
(23.7)
In Equation (23.6), the first term of on the right-hand side is the duration rule, and the
second term is the modification for convexity. Notice that for a bond with
positive convexity, the second term is positive, regardless of whether the yield
rises or falls.
The convexity is positively related to the bond price regardless of whether the yield
rises or falls. The value of duration rule with convexity is larger than the value
of duration rule without convexity. In other words, the bond price estimated by
the duration rule without convexity is always less than the bond price estimated
by the duration rule with convexity. Therefore, the bond price estimated by the
duration rule with convexity is more accurate.
15.
price  principal
years to call
Average Investment
Interest 
Cross over yield 

Cross over yield 
1100  1000
10
1100  1000
2
100 
110
 10.47%
1050
16.
CF
APV
CF×PV
wt
Duration
1
100
.9091
90.91
90.91

1000
.0909 × 1
.0909
2
100
.8264
82.64
82.64

1000
.0826 × 2
.1652
3
100
.7513
75.13
75.13

1000
.0751 × 3
.2253
4
100
.6830
68.30
68.30

1000
.041 × 4
.2732
5
1150
.6209
714.04
714.04

1000
.7140 × 5
3.57
1000
Duration = 4.32
17.
By equation (23.1), we can calculate the duration of bond as following formula
 C 
t
t


t
t 0 1 kd  

D n 
,
Ct
n
 1 k
t 0

d

t
Where Ct = the payment at time t, kd = the YTM or required rate of return of the
bondholders in the market; and n= the maturity in years.
The coupon payment is Ct  1000(0.06)  60 , yield to maturity kd is 6%, and present
bond value is sold at par $1000.
1
Time t in years
1  0.06 
1
0.943396
0.943396
1.88679
2
0.889996
1.779993
5.33998
3
0.839619
2.518858
10.07543
4
0.792094
3.168375
15.84187
5
0.747258
3.736291
22.41775
6
0.704961
4.229763
29.60834
7
0.665057
4.655400
37.24320
8
0.627412
5.019299
45.17369
t
Sum
n

1
 t  1  k
t 0
t2  t
t

1  0.06 
t
26.051374
1  0.06 
t
167.58705
 8 

1

t


 =26.051374

t
t
t  0  1  0.06  
d 



 1  8 2


1
2
(
t

t
)

(
t

t
)



 =167.58705


t
t
t 0
 1  kd   t 0
 1  0.06  
n
The duration of bond is
8

 1000(8)
1
60 t 

t 
8
t  0  1  0.06  
1  0.06  60(26.051374)  1000(5.019299)


D

 6.582381
1000
1000
The modified duration is
D* 
D
6.582381

 6.2098
(1  kd )
1.06
By using equation (23.7),
1 2 P
1
Convexity   2 
P i
P  (1  kd ) 2
P  P 
 108[

]
P
P
n
CFt
[ (1  k
t 1
d
)
t
(t 2  t )]
The exactly value of convexity is
8
CFt
1
[
(t 2  t )]

2
t
1000  (1  kd ) t 1 (1  kd )
2
8
t t
82  8
60 [
]

1000
t
(1  0.06)8
t 1 (1  0.06)

1000  (1  0.06) 2
60(167.58705)  1000(45.17369)

 49.153536
1000  (1  0.06) 2
Convexity 
The approximation value needs the information of P  is the capital loss from a
one-basis-point (0.0001) increase in interest rates and P  is the capital gain
from a one-basis-point (0.0001) decrease in interest rates.
1
1
Time t in years
1  0.0601
1
0.943307
0.943485
2
0.889829
0.890164
3
0.839382
0.839857
4
0.791795
0.792393
5
0.746906
0.747611
6
0.704562
0.70536
7
0.664618
0.665496
8
0.626939
0.627886
t
1  0.0599 
t
Bond price
8

t 0
Ct
999.3793
1000.621
1  kd 
t
Therefore, the approximation value of convexity is
P  P 

]
P
P
(1000  999.3793) (1000.621  1000)
 108 [

]  49.153541
1000
1000
Convexity  108 [
18. Based on the solution in question 17, when yield to maturity increases from 6% to
8%, the percentage change of bond price is
P
  D * i  6.2098(0.02)  0.1241959, or  12.41959%
P
The bond price estimated by the duration rule is
1000+1000(-12.41959%) = 875.804
By duration-with-convexity rule, the percentage change of bond price is
P
  D * i  0.5  Convexity  (i)2
P
 6.2098  0.02  0.5  49.1535  (0.02) 2  0.114365, or  11.4365%
Then, the bond price estimated by the duration-with-convexity rule is
1000+1000(-0.114365) = 885.635
The actual bond price when YTM is 8% is 885.0672 which can be calculated as the
table below:
1
Time t in years
1  0.08 
1
0.925926
2
0.857339
t
3
0.793832
4
0.73503
5
0.680583
6
0.63017
7
0.58349
8
0.540269
Bond price
8

t 0
Ct
885.0672
1  kd 
t
19. Based on the solution in question 17, when yield to maturity decreases from 6% to
5.5%, the percentage change of bond price is
P
  D * i  6.2098(0.005)  0.031049, or 3.1049%
P
The bond price estimated by the duration rule is
1000+1000(3.1049%) = 1031.049
By duration-with-convexity rule, the percentage change of bond price is
P
  D * i  0.5  Convexity  (i)2
P
 6.2098  (0.005)  0.5  49.1535  (0.005) 2  0.0316634, or 3.16634%
Then, the bond price estimated by the duration-with-convexity rule is
1000+1000(3.16634%) = 1031.6634
The actual bond price when YTM is 5.5% is 1031.67283 which can be calculated as
the table below:
1
Time t in years
1  0.055 
t
1
0.947867
2
0.898452
3
0.851614
4
0.807217
5
0.765134
6
0.725246
7
0.687437
8
0.651599
Bond price
8

t 0
Ct
1  kd 
t
1031.67283