Week10 Homework Assignment Solution
BKM page 480
2.
The effective annual yield on the semiannual coupon bonds is 8.16%. If the
annual coupon bonds are to sell at par they must offer the same yield, which
requires an annual coupon rate of 8.16%.
3.
The bond callable at 105 should sell at a lower price because the call provision is
more valuable to the firm. Therefore, its yield to maturity should be higher.
4.
The bond price will be lower. As time passes, the bond price, which is now above
par value, will approach par.
5.
Yield to maturity: Using a financial calculator, enter the following:
n = 3; PV = −953.10; FV = 1000; PMT = 80; COMP i
This results in: YTM = 9.88%
Realized compound yield: First, find the future value (FV) of reinvested coupons
and principal:
FV = ($80 × 1.10 × 1.12) + ($80 × 1.12) + $1,080 = $1,268.16
Then find the rate (yrealized ) that makes the FV of the purchase price equal to
$1,268.16.
$953.10 × (1 + yrealized )3 = $1,268.16 ⇒ yrealized = 9.99% or approximately
9.
10%
a.
On a financial calculator, enter the following:
n = 40; FV = 1000; PV = –950; PMT = 40
You will find that the yield to maturity on a semi-annual basis is 4.26%.
This implies a bond equivalent yield to maturity equal to: 4.26% × 2 =
8.52%
Effective annual yield to maturity = (1.0426)2 – 1 = 0.0870 = 8.70%
b.
Since the bond is selling at par, the yield to maturity on a semi-annual basis is
the same as the semi-annual coupon rate, i.e., 4%. The bond equivalent yield to
maturity is 8%.
Effective annual yield to maturity = (1.04)2 – 1 = 0.0816 = 8.16%
c.
Keeping other inputs unchanged but setting PV = –1050, we find a bond
equivalent yield to maturity of 7.52%, or 3.76% on a semi-annual basis.
Effective annual yield to maturity = (1.0376)2 – 1 = 0.0766 = 7.66%
1
10.
Since the bond payments are now made annually instead of semi-annually, the
bond equivalent yield to maturity is the same as the effective annual yield to
maturity. Using a financial calculator, enter: n = 20; FV = 1000; PV = –price,
PMT = 80.
The resulting yields for the three bonds are:
Bond Price
$950
$1,000
$1,050
Bond equivalent yield =
Effective annual yield
8.53%
8.00%
7.51%
The yields computed in this case are lower than the yields calculated with semiannual payments. All else equal, bonds with annual payments are less attractive to
investors because more time elapses before payments are received. If the bond
price is the same with annual payments, then the bond's yield to maturity is lower.
11.
Assuming semi-annual coupon payments:
Price
$400.00
$500.00
$500.00
$376.89
$456.39
$400.00
21.
Maturity
(years)
20.00
20.00
10.00
10.00
10.00
11.68
Maturity
(half-years)
40.00
40.00
20.00
20.00
20.00
23.36
Semi-annual
YTM
2.317%
1.748%
3.526%
5.000%
4.000%
4.000%
Bond equivalent
YTM
4.634%
3.496%
7.052%
10.000%
8.000%
8.000%
The stated yield to maturity, based on promised payments, equals 16.075%.
[n = 10; PV = –900; FV = 1000; PMT = 140]
Based on expected coupon payments of $70 annually, the expected yield to
maturity is 8.526%.
31 e.
(ii)
f.
(iii)
BKM page 557
1.
The percentage change in the bond’s price is:
−
2.
Duration
7.194
× Δy = −
× 0.005 = −0.0327 = −3.27% or a 3.27% decline.
1+ y
1.10
a. YTM = 6%
(1)
Time until
Payment
(2)
Cash Flow
(3)
PV of CF
(Discount
(4)
Weight
(5)
Column (1) ×
Column (4)
2
(years)
1
2
3
$60.00
$60.00
$1,060.00
Column Sums
rate = 6%)
$56.60
$53.40
$890.00
$1,000.00
0.0566
0.0534
0.8900
1.0000
0.0566
0.1068
2.6700
2.8334
(4)
(5)
Weight
Column (1) ×
Column (4)
0.0606
0.0551
0.8844
1.0000
0.0606
0.1102
2.6532
2.8240
Duration = 2.833 years
b. YTM = 10%
(1)
Time until
Payment
(years)
1
2
3
(2)
Cash Flow
$60.00
$60.00
$1,060.00
Column Sums
(3)
PV of CF
(Discount
rate = 10%)
$54.55
$49.40
$796.39
$900.53
Duration = 2.824 years, which is less than the duration at the YTM of 6%.
3.
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. Using Rule
8, we find:
D = (1.03/0.03) × [1 – (1/1.03)6] = 5.58 half-year periods = 2.79 years
If the bond’s yield is 10%, use Rule 7, setting the semiannual yield to 5%, and
semiannual coupon to 3%:
D=
1.05 (1.05) + [6 × (0.03 − 0.05)]
−
= 21 − 15.4478
0.05 0.03 × [(1.05) 6 − 1] + 0.05
= 5.5522 half-year periods = 2.7761 years
5.
a.
(1)
Time until
Payment
(years)
1
5
(2)
(3)
PV of CF
Cash Flow
(Discount rate =
10%)
$10 million
$9.09 million
$4 million
$2.48 million
Column Sums $11.57 million
(4)
(5)
Weight
Column (1) ×
Column (4)
0.7857
0.2143
1.0000
0.7857
1.0715
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
8.
c.
d.
(i)
(i)
[9/1.10 = 8.18]
3
18. a.
From Rule 6, the duration of the annuity if it were to start in 1 year would be:
1.10
10
−
= 4.7255 years
0.10 (1.10)10 − 1
Because the payment stream starts in five years, instead of one year, we add
four years to the duration, so the duration is 8.7255 years.
4
b.
The present value of the deferred annuity is:
10,000 × Annuity factor (10%,10)
= $41,968
1.10 4
Call w the weight of the portfolio invested in the 5-year zero. Then:
(w × 5) + [(1 – w) × 20] = 8.7255 ⇒ w = 0.7516
The investment in the 5-year zero is equal to:
0.7516 × $41,968 = $31,543
The investment in the 20-year zeros is equal to:
0.2484 × $41,968 = $10,425
These are the present or market values of each investment. The face
values are equal to the respective future values of the investments. The
face value of the 5-year zeros is:
$31,543 × (1.10)5 = $50,800
Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000,
would be purchased. Similarly, the face value of the 20-year zeros is:
$10,425 × (1.10)20 = $70,134
5