CHAPTER 5
ANSWERS
5-1
The value of any asset is determined by computing the present value of the future cash flows the
asset is expected to generate.
5-2
True. The value of a share of stock is the PV of all its expected future dividends. If the two
investors expect the same future dividend stream and they agree on the stock’s riskiness, then they
should reach similar conclusions as to the stock’s value regardless of their investment horizons.
5-3
A perpetual bond is similar to a no-growth stock and to a share of preferred stock in the following
ways:
(1)
All three derive their values from a series of cash inflows—that is, coupon payments from
the perpetual bond and dividends from both types of stock.
(2)
All three are assumed to have indefinite lives with no maturity value (M) for the perpetual
bond and no capital gains yield for the stocks.
5-4
If interest rates increase after a bond has been issued, the market value of the bond will decrease.
Investors can earn the higher yields on alternative investments with similar risk, and thus they will
only buy previously issued bonds if their prices have declined to the point where the yields to
maturity on the outstanding bonds are equal to the yields on similar risk alternative investments.
Everything else equal, the market prices of bonds with longer terms to maturity will change more
than the prices of bonds with shorter terms to maturity.
5-5
If interest rates decline significantly, the values of callable bonds will not rise by as much as bonds
without call provisions. It is likely that the callable bonds will be called in prior to maturity so that
the issuer can take advantage of the new, lower rates.
5-6
Yes. If a company decides to increase its dividend payout ratio, then the dividend yield component
will rise but the expected long-term capital gains yield will decline. If a greater percentage of
earnings is paid as dividends, less is available for reinvestment in the firm.
5-7
If investors demand a higher required rate of return for investing in Microsoft’s stock, then, all else
equal, the price of the stock would have to drop to provide the higher return. In our valuation
model, the value for ks in the denominator would increase, which would cause the result of the
computation for the value of the stock to decrease. If Microsoft had to pay a $10 million fine for
unfair trade practices, all else equal, the value of the stock should decrease because (a) the expected
future cash flows of Microsoft will decrease due to the unexpected cash outflow, and (b) investors
might now demand a higher rate of return for investing in Microsoft.
5-8
The general principle behind valuing a real asset is the same as for a financial asset. The principal
difference is that it probably is more difficult to estimate values for the needed inputs to the
valuation model—that is, future cash flows and the required rate of return. We discuss this concept
in detail later in the book.
88
________________________________________________________
SOLUTIONS
5-1
a.
The bonds now have an 8-year, or a 16-semiannual period, maturity, and their value is
calculated as follows:
1  (1.031 )16 
 1 
  1,000 
Vd  50 
16 
 0.03 
 (1.03) 


 50(12.61102 )  1,000 (0.62317 )  628 .05  623 .17  1,251 .22
Calculator solution: Input N = 16, I = 3, PMT = 50, and FV = 1,000, compute PV =
1,251.22.
b.
Calculator solution: Change inputs from part a to I = 6 and compute PV = 898.94
1  (1.061 )16 
 1 
  1,000 
Vd  50 
16 
 0.06 
 (1.06) 


 50(10.10590 )  1,000 (0.39365 )  505 .29  393 .65  898 .94
5-2
c.
The price of the bond will decline from $1,251.22 toward $1,000, hitting $1,000 (plus
accrued interest) at the maturity date eight years (16 six-month periods) from now.
a.
Vd =
b.
c.
d.
Vd =
Vd =
INT
=
$100
= $1,250
0.08
=
$100
= $833.33
0.12
=
$100
= $1,000 = par value
0.10
kd
INT
kd
INT
kd
Assuming annual interest payments: at 8%: Vd = $100(9.81815) + $1,000(0.21455) =
$981.81 + $214.55 = $1,196.36
Calculator solution: Input N = 20, I = 8, PMT = 100, and FV = 1000; compute PV =
1,196.36.
At 12%: Vd = $100(7.46944) + $1,000(0.10367) = $746.94 + $103.67 = $850.61
Calculator solution: Change I = 12; compute PV = $850.61
At 10%: Vd = $100(8.51356) + $1,000(0.14864) = $851.36 + $148.64 = $1,000
Calculator solution: Change I = 10; compute PV = $1,000.00
89
If the bonds are selling at a premium, the value of the 20-year bond will be less than the
value of the perpetuity, whereas the perpetuity will have a lower value if the bonds are
selling at a discount. The value of the shorter, 20-year bond fluctuates less than the longerterm, perpetual bond because the value of the perpetuity’s distant coupon payments
fluctuate significantly as the yield changes.
1
2
3
4
D0 =$2.00
^
D1
^
D2
^
D
3
^
P3 =$34.73
^
D4
D̂1 = $2(1.05)1
= $2.10
D̂ 2
= $2(1.05)2
= $2.21
D̂ 3
$2(1.05)3
= $2.32.
0
5-3
k=12%
g=5%
a.
b.
=
2.10
2.21
2.32


1
2
(1.12)
(1.12)
(1.12) 3
 2.10(0.89286 )  2.21(0.79719 )  2.32(0.71178 )  5.29
PV 
Calculator solution: Input 0, 2.10, 2.21, and 2.32 into the cash flow register, input I = 12;
compute PV = 5.29
c.
34.73
(1.12) 3
 34.73(0.71178 )  24.72
PV 
Calculator solution: Input N = 3, I = 12, and FV = 34.73; compute PV = -24.72
d.
P̂0 = $24.72 + $5.29 = $30.01 = Maximum price to pay for the stock.
e.
P̂0 =
f.
$2.10
D0 (1 + g)
= D̂1 =
= $30.00
ks - g
ks - g 0.12 - 0.05
The value of a share of stock is the PV of all its expected future dividends, so it doesn’t
matter whether your planned holding period is two years, three years, five years, or any
other length of time. For example, if you buy the stock and hold it for two years, you would
receive dividends at the end of Year 1 and Year 2, and then sell the stock at the end of Year
2. The price of the stock at the end of Year 2 should be the present value of the dividend
that will be paid in Year 3 and the price of the stock in Year 3, which represents the present
value of the dividends stockholders will receive in Year 4, Year 5, Year 6, and so on. Thus,
the value of the stock at the end of Year 2 is:
P̂2 
D̂ 3

P̂3
(1  k )
 (1  k )1
2.32
34.73


 33.08
1
(1.12)
(1.12)1
The value of the stock to an investor who plans to hold the investment for two years is:
1
90
P̂0 
D̂1

D̂ 2  P̂2
(1  k )
 (1  k ) 2
2.10
2.21  33.08


 1.88  28.13  30.01 (rounding difference)
1
(1.12)
(1.12) 2
5-4
a.
1
g = $1.1449/$1.07 - 1.0 = 7%.
Calculator solution: Input N = 1, PV = -1.07, and FV = 1.1449; compute I = 7.00
or Input N = 2, PV = -1.07, and FV = 1.2250; compute I = 7.00
5-5
b.
Dividend yield = $1.07/$21.40 = 5%.
c.
k̂ s = D̂1 /P0 + g = $1.07/$21.40 + 7% = 5% + 7% = 12%.
a.
(1)
P̂0 =
$2(1 - 0.05) $1.90
=
= $9.50
0.15 + 0.05 0.20
(2)
P̂0 =
$2
= $13.33
0.15
(3)
P̂0 =
$2(1 + 0.05) $2.10
=
= $21.00
0.15 - 0.05
0.10
(4)
P̂0 =
$2(1 - 0.10) $2.20
=
= $44.00
0.15 - 0.10
0.05
(1)
P̂0 = $2.30/(0.15 - 0.15) = $2.30/0, which is undefined.
(2)
P̂0 = $2.40/(0.15 - 0.20) = $2.30/(-0.05) = -$48, which is nonsense.
b.
These results show that the constant growth formula does not make sense if the required
rate of return is equal to or less than the expected growth rate.
5-6
c.
No. The results in part b show why.
a.
PV 
$2.50
$3.00
$4.00


1
2
(1.14)
(1.14)
(1.14) 3
 $2.50(0.87719)  $3.00(0.76947)  $4.00(0.67497)  $7.20
91
Calculator solution: Input 0, 2.50, 3.00, and 4.00 into the cash flow register, input I = 14;
compute PV = $7.20.
5-7
$4(1 + 0.04) $4.16
=
= $41.60
0.14 - 0.04
0.10
b.
P̂3 =
c.
P̂0 = PV of dividends + PV of price in Year 3
$41.60
 $7.20 
 $35.28
(1.14) 3
a.
The coupon interest is $50 = 0.05($1,000) per year, and the original yield to maturity was
5.0 percent; so both bonds would have sold for par, or for $1,000, at the time of issue. To
see this, compute the value of the IBM bond:
10
VIBM  
t =1
$50
(1.05 )
t
+
$1,000
(1.05 )10
1  1 10 


(1.05)
  $1,000  1 
 $50 
10
 0.05 
 (1.05) 


 $50(7.72173) + $1,000(0.61391) = $1,000
Calculator solution: Input N = 10, FV = 1,000, PMT = 50, and I = 5; compute PV = -1,000
b.
On January 1, 2005, the IBM bond had a remaining life of 9 years. Thus, its value is
calculated as follows:
9
V IBM = 
t =1
$50
(1.08 )t
+
$1,000
(1.08 )9
$50(6.24689) + $1,000(0.50025) = $812.59
With a calculator, enter I = 8, N = 9, FV = 1000 and PMT = 50; compute PV = -812.59
On January 1, 2005, the GM bond had a remaining life of 19 years. Thus, its value is
calculated as follows:
19
VGM = 
t =1
$50
(1.08 )
t
+
$1,000
(1.08 )19
$50(9.60360) + $1,000(0.23171) = $711.89
With a calculator, just change N to 19; compute PV = -711.89
92
c.
The capital gains yields for the bonds are:
$812.59 - $1,000
= - 18.74%
$1,000
$711.89 - $1,000
=
= - 28.81%
$1,000
Capital gains IBM =
Capital gains GM
d.
The current yield for both bonds was $50/$1,000 = 0.05 = 5.0%
e.
Total returnIBM = 5% + (-18.75%) = –13.75%
Total returnGM = 5% + (-28.81%) = –23.81%
f.
The IBM bond, which has the shorter term to maturity, lost less than the GM bond. The
price of the shorter-term bond changes less with each change in interest rates (yields), so
when the market rates increase, the prices for shorter-term bonds will decrease less than for
longer-term bonds.
g.
On January 1, 2010, the IBM bond will have a remaining life of four years. Thus, its value
will be:
4
V IBM = 
t =1
$50
(1.08 )
t
+
$1,000
(1.08 )4
$50(3.31213) + $1,000(0.73503) = $900.64
With a calculator, enter I = 8, N = 4, FV = 1000 and PMT = 50; compute PV = $900.64
On January 1, 2010, the GM bond will have a remaining life of 14 years, and its value will
be:
14
VGM = 
t =1
$50
(1.08 )
t
+
$1,000
(1.08 )14
$50(8.24424) + $1,000(0.34046) = $752.67
With a calculator, just change N to 14; compute PV = $752.67
At maturity, the values of both bonds must equal $1,000, because this is the face value of
both. So, if market rates do not change, as each bond gets closer to its maturity date, its
market value will approach its $1,000 face value. The closer the maturity date, the quicker
the market value will approach its face value (the value of the IBM bond will increase more
than the GM bond during the next six years because its maturity date is closer).
5-8
a.
D̂ t
= D0(1 + g)t
D̂ 2005 = $1.75(1.15)1 = $2.01
D̂ 2006 = $1.75(1.15)2 = $1.75(1.3225) = $2.31
D̂ 2007 = $1.75(1.15)3 = $1.75(1.5209) = $2.66
D̂ 2008 = $1.75(1.15)4 = $1.75(1.7490) = $3.06
D̂ 2009 = $1.75(1.15)5 = $1.75(2.0114) = $3.52
93
b.
Step 1
PV of
dividends

5
 (1 +D̂k )
t

t
s
t =1
$2.01
$2.31
$2.66
$3.06
$3.52




 $9.47
1
2
3
4
(1.12)
(1.12)
(1.12)
(1.12)
(1.12) 5
Step 2
P̂2009 

D̂ 2010 D̂2009 (1 + g)
=
ks - gn
ks - gn
$3.52(1.05 ) $3. 696
=
= $52.80
0.12 - 0.05
0.07
This is the price of the stock five years from now. The PV of this price, discounted back
five years, is as follows:
PV of P̂2009 
$52.80
 $52.80(0.56743)  $29.96
(1.12) 5
Step 3
The price of the stock today is as follows:
P0 = PV dividends Years 2005-2009 + PV of P2009
= $9.47 + $29.96 = $39.43
This problem could also be solved by substituting the proper values into the following
equation:
t
D0 (1 + gs )  D̂6   1 
+
P0 = 
t
5

t =1 (1 + k s )
 ks - g n   1 + ks  
5
Calculator solution: Input 0, 2.01, 2.31, 2.66, 3.06, 56.32(3.52 + 52.80) into the cash flow
register, and input I = 12; compute PV = $39.43.
c.
2005
D̂1 /P0 = $2.01/$39.43
Capital gains yield
Expected total return
= 5.10%
= 6.90*
= 12.00%
94
2009
D̂ 6 /P5 = $3.70/$52.80 = 7.00%
Capital gains yield
Expected total return
= 5.00
= 12.00%
*We know that k is 12 percent, and the dividend yield is 5.10 percent; therefore, the capital
gains yield must be 6.90 percent.
The main points to note here are as follows:
(1) The total yield is always 12 percent (except for rounding differences).
(2) The capital gains yield starts relatively high and then declines as the supernormal
growth period approaches its end. The dividend yield rises.
(3) After 12/31/09, the stock will grow at a 5 percent rate. The dividend yield will equal 7
percent, the capital gains yield will equal 5 percent, and the total return will be 12
percent.
5-9
d.
People in high income tax brackets will be more inclined to purchase ”growth” stocks to
take the capital gains and thus delay the payment of taxes until a later date. Swink’s stock
is ”mature” at the end of 2009.
e.
Because Swink’s supernormal and normal growth rates are lower, the dividends and, hence,
the present value of the stock price will be lower. The total return from the stock will still be
12 percent, but the dividend yield will be larger and the capital gains yield will be smaller
than they were with the original growth rates. This result occurs because we assume the
same last dividend but a much lower current stock price.
f.
As the required return increases, the price of the stock goes down, but both the capital gains
and dividend yields increase initially. Of course, the long-term capital gains yield still is 4
percent, so the long-term dividend yield is 10 percent.
a.
Part 1.
D̂1 = D0(1 + gs) = $1.60(1.20) = $1.92
D̂ 2 = D0(1 + gs)2 = $1.60(1.20)2 = $2.304
P̂2 =
D̂3 = $2.304(1.0 6) = 61.06
0.10 - 0.06
ks - gn
D̂1 + D̂2 + P̂2
1 + ks 1 1 + ks 2
= $1.92(0.90 909) + $2.304(0.8 264 5) + $61.06(0.8 264 5) = $54.11
P̂0 = PV(D̂1 ) + PV(D̂ 2 ) + PV(P̂2 ) =
Calculator solution: Input 0, 1.92, 63.364 = (2.304 + 61.06) into the cash flow register, and
input I = 10 compute PV = $54.11.
95
Part 2.
Expected dividend yield: D̂1 /P0 = $1.92/$54.11 = 3.55%
Capital gains yield: First, find P̂1 which equals the sum of the present values of D̂ 2 and
P̂2 , discounted for one year.
P̂1 

D̂2 +
P̂2
1
(1  k s )
(1  k s )1
$2.304 + $61.06
(1.10 )1
= $57.60
Calculator solution: Input FV = 63.364 = (2.304 + 61.06) , N =1, and I = 10; compute PV
= $57.60.
Second, find the capital gains yield:
P̂1 - P0 = $57.60 - $54.11 = 0.0645 = 6.45%
$54.11
P0
Dividend yield
Capital gains yield
5-10
= 3.55%
= 6.45
10.00% = ks.
b.
Due to the longer period of supernormal growth, the value of the stock will be higher for
each year. Although the total return will remain the same, ks = 10%, the distribution
between dividend yield and capital gains yield will differ: The dividend yield will start off
lower and the capital gains yield will start off higher for the five-year supernormal growth
condition, relative to the two-year supernormal growth state. The dividend yield will
increase and the capital gains yield will decline over the five-year period until dividend
yield = 4% and capital gains yield = 6%.
c.
Throughout the supernormal growth period, the total yield will be 10 percent, but the
dividend yield is relatively low during the early years of the supernormal growth period and
the capital gains yield is relatively high. As we near the end of the supernormal growth
period, the capital gains yield declines and the dividend yield rises. After the supernormal
growth period has ended, the capital gains yield will equal gnorm = 6%. The total yield must
equal ks = 10%, so the dividend yield must equal 10% - 6% = 4%.
d.
Some investors need cash dividends (retired people) whereas others would prefer growth.
Also, investors must pay taxes each year on the dividends received during the year whereas
taxes on capital gains can be delayed until the gain is actually realized.
a.
ks = kRF + (kM - kRF)βs = 11% + (14% - 11%)1.5 = 15.5%
b.
P0 = D̂1 / (ks - g) = $2.25/(0.155 - 0.05) = $21.43.
96
c
ks = 9% + (12% - 9%)1.5 = 15.5%; P0 = $2.25/(0.135 - 0.05) = $26.47
d.
ks = 9% + (11% - 9%)1.5 = 12.0%; P0 = $2.25/(0.12 - 0.05) = $32.14
e.
New data given: kRF = 9%; kM = 11%; g = 6%, βs = 1.3
ks = kRF + (kM - kRF)βs = 9% + (11% - 9%)1.3 = 11.6%
D0 = $2.25/1.05 = $2.14
P0 = D̂1 /(ks - g) = [$2.14(1.06)]/(0.116 - 0.06) = $2.27/0.056 = $40.54
5-11
a.
Old ks = kRF + (kM - kRF)βs = 9% + (12% - 9%)1.2 = 12.6%
New ks = 9% + (3%)0.9 = 11.7%
$2(1.07)
Old price : P̂0  D̂1 =
= $38.21
g
0.126
- 0.07
ks
New price : P̂0 
$2(1.05)
= $31.34
0.117 - 0.05
Because the new price is lower than that old price, the expansion in consumer products
should be rejected. The decrease in risk is not sufficient to offset the decline in profitability
and the reduced growth rate.
b.
P̂0 ld = $38.21
P̂ New =
$2(1.05)
k s - 0.05
Solving for ks we have the following:
$38.21 =
$2.10
ks - 0.05
$2.10 = $38.21(ks) - $1.9105
$4.0105 = $38.21(ks)
ks = 0.10496.
Solving for ßs:
10.496%
1.496%
ßs
= 9% + 3%(ßs)
= 3%(ßs)
= 0.49865
Check: ks = 9% + (3%)0.49865 = 10.4960%.
P̂0 =
$2.10
= $38.21
0.10496 - 0.05
97
Therefore, only if management’s analysis concludes that risk can be lowered to ßs =
0.49865, or approximately 0.5, should the new policy be put into effect.
5-12
a.
1  (11k ) N
Vd  PMT
k



M

 (1  k ) N

(1) 5%:
Bond L: Vd = $100(10.37966) + $1,000(0.48102) = $1,518.99
Bond S: Vd = ($100 + $1,000)(0.95238) = $1,047.62
(2) 8%:
Bond L: Vd = $100(8.55948) + $1,000(0.31524) = $1,171.19
Bond S: Vd = ($100 + $1,000)(0.92593) = $1,018.52
(3) 12%:
Bond L: Vd = $100(6.81086) + $1,000(0.18270) = $863.79
Bond S: Vd = ($100 + $1,000)(0.89286) = $982.14
Calculator solutions:
b.
5-13
(1) 5%:
Bond L: Input N = 15, I = 5, PMT = 100, and FV = 1000;
compute PV = -1,518.98
Bond S: Change N = 1; compute PV = -1,047.62
(2) 8%:
Bond L: Input N = 15, I = 8, PMT = 100, and FV = 1000;
compute PV = -1,171.19
Bond S: Change N = 1; compute PV = -1,018.52
(3) 12%:
Bond L: Input N = 15, I = 12, PMT = 100, and FV = 1000;
compute PV = -863.78.
Bond S: Change N = 1; compute PV = -982.14
Think about a bond that matures in one month. Its present value is influenced primarily by
the maturity value, which will be received in only one month. Even if interest rates double,
the price of the bond still will be close to $1,000. The value of a one-year bond would
fluctuate more than the value of a one-month bond’s value because of the difference in the
timing of receipts. However, its value would still be fairly close to $1,000 even if interest
rates doubled. A long-term bond paying semiannual coupons, on the other hand, will be
dominated by distant receipts, receipts that are multiplied by 1/(1 + kd/2)2N, and if kd
increases, these multipliers will decrease significantly. Another way to view this problem is
from an opportunity point of view. A one-month bond can be reinvested at the new rate
very quickly, and hence the opportunity to invest at this new rate is not lost; however, the
long-term bond locks in subnormal returns for a long period of time.
1  (11k ) N
Vd  PMT
k


a.

M

; k = YTM
 (1  k ) N

M = $1,000
INT = 0.095($1,000) = $95
N = 28 years in 2000
Vd = $1,165.75
98
Using Equation 5-3, the approximate yield to maturity is:
Approximat e
yield to

INT +  M N- V d 


 2( V d ) + M 


3


maturity


5-14
$1,165.75 

$95 +  $1,000 -28

 2($1,165.75) + $1,000 


3


$89.08
= 0.0802 = 8.02%
$1,110.50
b.
Calculator solution: Input N = 28, PV = -1165.75, PMT = 95, FV =1000; compute I =
8.00% = YTM
a.
M = $1,000
INT = 0.09($1,000) = $90
Approximat e
yield to
=
maturity
INT +  M N- V d 


 2( V d ) + M 


3


(1) Vd = $829
Approximat e
YTM
=
$90 +  $1,0004- $829 
 2($829)+ $1,000 


3


=
$132.75
= 0.1498 = 14.98%
$886
Calculator solution: YTM = 14.99%
(2) Vd = $1,104
Approximat e
YTM
=
$90 +  $1,0004- $1,104 
 2($1,104)+ $1,000 


3



$64
= 0.0599 = 5.99%
$1,069.33
Calculator solution: YTM = 6.00%
b.
1
1  (11k ) N 
1  (1.12
 $1,000
M
)4




Vd  PMT


$
90
k

 (1  k ) N
 0.12  (1.12) 4




 $90(3.03735 )  $1,000 (0.63552 )  $908 .88
Calculator solution: Enter I = 12, N = 4, FV = 1,000, and PMT = 90; compute PV = 908.88. Thus, if the bond is selling at a price of $829, it would be a bargain.
5-15
Vd = INT/kd; therefore, kd = INT/Vd.
INT = 0.08($1,000) = $80
a.
kd = $80/$600
= 13.3%
99
5-16
b.
kd = $80/$800
c.
kd = $80/$1,000 = 8.0%
d.
kd = $80/$1,500 = 5.3%
P̂0 =
=
5-17
a.
= 10.0%
D̂1 = D0 (1 + g)
ks - g
ks - g
$5[1 + (-0.05)] $4.75
=
= $23.75
0.15 - (-0.05)
0.20
ks = kRF + (kM - kRF)ßs
kC = 9% + (13% - 9%)0.4 = 10.6%
kD = 9% + (13% - 9%)(-0.5) = 7%.
Note that kD is below the risk-free rate. But because this stock is like an insurance policy
because it pays off when something bad happens (the market falls), the low return is not
unreasonable.
b.
In this situation, the expected rate of return is as follows:
k̂ C 
D̂1
$1.50
g
 0.04  0.06  0.04  0.10  10.0%
P0
$25
However, the required rate of return, kC, is 10.6 percent. Investors will seek to sell the stock,
dropping its price to the following:
Pc =
$1.50
= $22.73
0.106 - 0.04
At this point,
k̂ c =
$1.50
+ 4% = 10.6%
$22.73
and the stock will be in equilibrium.
5-18
D0
k
g1
g2
gnorm
= $1
= 7% + 6% = 13%
= 50%
= 25%
= 6%
100
0
ks=13%
g1=50%
1
2
g2=25%
1.50
3
gnorm=6%
…
1.875
+ 28.393 
1.327
4
1.875(1.06)
0.13  0.06
= 30.268
23.704
25.031
D̂1 =$1(1.50)
= $1.50
D̂ 2 =$1.50(1.25)
= $1.875
D̂ 3 =$1.875(1.06) = $1.9875
Nonconstant growth ends at the end of Year 2, thus P̂2 can be computed using the constant growth
dividend discount model:
P̂2 
D̂ 3
$1.9875

 $28.393
k s  g n 0.13  0.06
The current price is the present value of D̂1 , D̂ 2 , and P̂2 :
P̂0 
5-19
$1.50 $1.875  $28.393

 $1.50(0.88496 )  $30.268(0.78315 )  $25.03
(1.13)1
(1.13) 2
Bonds: Price $897.40, 20 years, PMT = $40 per 6 mos., N = 40 periods, M = $1,000
Preferred: $2/quarter forever, $95/share
Bonds: Selling at a discount, so kd > coupon. Using Equation 5-3, the approximate YTM is:
Approximat e
YTM
=
- $897.40 

$40 +  $1,00040

 2($897.40)+ $1,000 


3


=
$42.565
= 0.0456 = 4.569% per 6 months
$931.60
With a calculator, input N = 40, PV = -897.40, PMT = 40, and FV = 1000; compute I = 4.563 =
periodic rate (per six months)
EAR = (1.04563)2 – 1 = 9.33%, YTM = 9.33%.
Preferred: $2/$95 = periodic (quarterly) rate of 2.11%.
EAR = (1.0211)4 – 1 = 8.69%.
101
5-20
The bond is selling at a large premium, which means that its coupon rate is much higher than the
going market rate of interest. Using Equation 5-3, the approximate YTM is:
Approximat e
YTM
=
$1,353.54 

$70 +  $1,000 -60

 2($1,353.54) + $1,000 


3


=
$64.11
= 0.0519 = 5.19% per 6 months
$1,235.69
With a calculator, enter N = 60, PV = -1,353.54, PMT = 70, and FV = 1,000; compute I = 5.10
The actual periodic rate is 5.10 percent, so the nominal YTM is 2 x 5.10% = 10.2%. This would
be close to the going rate, and it is about what Tapely would have to pay on new bonds.
5-21
Calculate the dividend stream, and place them on a cash flow time line. Also, calculate the price of
the stock at the end of the supernormal growth period, and include it, along with the dividend to be
paid in Year 5
D0 = D̂1  D̂ 2  0
D̂ 3 = 1.00
D̂ 3 = 1.00(1.5) = 1.5
D̂ 3 = 1.00(1.5)2 = 2.25
D̂ 4 = 1.00(1.5)2(1.08) = 2.43
P̂3 
0
D̂ 4
$2.25(1.08) $2.43


 $34.7143
k s  g n 0.15  0.08 0.07
ks=15%
1
2
3
4
g1=50%
1.00
0.6575
0.8576
1.50
5
6
gnorm=8%
2.25
2.25(1.08)
+ 34.7143 
0.15  0.08
= 36.9643
18.3778
19.8929
$0
$0
$1.00
$1.50 $2.25  $34.7143




1
2
3
(1.15)
(1.15)
(1.15)
(1.15) 4
(1.15) 5
 $1.00(0.65752 )  $1.50(0.57175 )  $36.9643 (0.49718 )  $19.89
P̂0 
With a calculator, enter the cash flows as shown on the time line into the cash flow register, enter
the required rate of return as I = 15, and then find the value of the stock using the NPV calculation.
Be sure to enter CF0 = 0, or else your answer will be incorrect.
CF0 = 0; CF1-2 = 0; CF3 = 1.0; CF4 = 1.5; CF5 = 36.9643; I = 15%.
102
With these cash flows in the cash flow register, press NPV to get the value of the stock today:
NPV = $19.89.
^
5-22
a.
Year
1
2
3
4
5
6
Net Cash Flow, CF t
$18,000
12,000
15,000
10,000
10,000
10,000
^
PVIF12%,n
0.89286
0.79719
0.71178
0.63552
0.56743
0.50663
PV of CF t
$16,071.48
9,566.28
10,676.70
6,355.20
5,674.30
5,066.30
Σ = $53,410.26
If you use the cash flow, CF, register on your calculator, input 0 for CF0 and then enter the
cash flows in the above order, enter I = 12, and press NPV. The result from the calculator is
53,410.22.
^
b.
Year
1
2
3
4
5
6
Net Cash Flow, CF t
$18,000
12,000
15,000
10,000
10,000
10,000
^
PVIF15%,n
0.86957
0.75614
0.65752
0.57175
0.49718
0.43233
PV of CF t
$15,652.26
9,073.68
9,862.80
5,717.50
4,971.80
4,323.30
Σ = $49,601.34
If you use the cash flow, CF, register on your calculator, input 0 for CF0 and then enter the
cash flows in the above order, enter I = 15, and press NPV. The result from the calculator is
49,601.22.
5-23
The solution is given in the Instructor's Manual, Solutions to Integrative Problems.
5-24
Computer-Related Problem
a.
Supernormal growth rate = 12%; normal growth rate = 4%
INPUT DATA:
KEY OUTPUTS:
Supernormal growth 12.00%
Normal growth rate
4.00%
Req. rate of return 12.00%
Last dividend (D0)
$1.75
Supernormal period
5 years
103
Current price (P0)
Price at 12/31/2004
Dividend yield 2000
Dividend yield 2004
Cap. gains yield 2000
Cap. gains yield 2004
Total return both yrs
$31.50
$40.09
6.22%
8.00%
5.78%
4.00%
12.00%
MODEL-GENERATED DATA:
Expected dividends:
PV of dividends:
2000
2001
2002
2003
2004
2000
2001
2002
2003
2004
Stock price
at 12/31/2004: $40.09
Stock price
at 1/1/2000: $31.50
Yields in 2004:
Yields in 2000:
Dividend
Capital Gain
Total
b.
$1.96
2.20
2.46
2.75
3.08
1.
8.00%
4.00%
12.00%
Dividend
Capital Gain
Total
Supernormal growth
Normal growth rate
Req. rate of return
Last dividend (D0)
Supernormal period 5
KEY OUTPUTS:
12.00%
4.00%
13.00%
$1.75
years
Current price (P0)
Price at 12/31/2004
Dividend yield 2000
Dividend yield 2004
Cap. gains yield 2000
Cap. gains yield 2004
Total return both yrs
$27.86
$35.64
7.03%
9.00%
5.97%
4.00%
13.00%
k = 15 percent
INPUT DATA:
Supernormal growth
Normal growth rate
Req. rate of return
Last dividend (D0)
Supernormal period 5
3.
6.22%
5.78%
12.00%
k = 13 percent
INPUT DATA:
2.
$1.75
1.75
1.75
1.75
1.75
KEY OUTPUTS:
12.00%
4.00%
15.00%
$1.75
years
Current price (P0)
Price at 12/31/2004
Dividend yield 2000
Dividend yield 2004
Cap. gains yield 2000
Cap. gains yield 2004
Total return both yrs
$22.59
$29.16
8.68%
11.00%
6.32%
4.00%
15.00%
k = 20 percent
INPUT DATA:
Supernormal growth
Normal growth rate
Req. rate of return
Last dividend (D0)
Supernormal period 5
KEY OUTPUTS:
12.00%
4.00%
20.00%
$1.75
years
104
Current price (P0)
Price at 12/31/2004
Dividend yield 2000
Dividend yield 2004
Cap. gains yield 2000
Cap. gains yield 2004
Total return both yrs
$15.20
$20.05
12.89%
16.00%
7.11%
4.00%
20.00%