FIN303
Answers to the recommended problems
Chapter 9
9-3
Yes. If a company decides to increase its payout ratio, then the dividend yield
component will rise, but the expected long-term capital gains yield will decline.
9-4 Yes. The value of a share of stock is the PV of 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.
9-1
D0 = $1.50; g1-3 = 7%; gn = 5%; D1 through D5 = ?
D1 = D0(1 + g1) = $1.50(1.07) = $1.6050.
D2 = D0(1 + g1)(1 + g2) = $1.50(1.07)2 = $1.7174.
D3 = D0(1 + g1)(1 + g2)(1 + g3) = $1.50(1.07)3 = $1.8376.
D4 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn) = $1.50(1.07)3(1.05) = $1.9294.
D5 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn)2 = $1.50(1.07)3(1.05)2 = $2.0259.
9-2
D1 = $0.50; g = 7%; rs = 15%; P̂0 = ?
P̂0 
9-3
D1
$0.50

 $6.25.
rs  g 0.15  0.07
P0 = $20; D0 = $1.00; g = 6%; P̂1 = ?; rs = ?
P̂1 = P0(1 + g) = $20(1.06) = $21.20.
r̂s =
D1
+g
P0
$1.00(1.06)
+ 0.06
$20
$1.06
=
+ 0.06 = 11.30%. rs = 11.30%.
$20
=
9-4
a. The horizon date is the date when the growth rate becomes constant. This
occurs at the end of Year 2.
b.
0
|
1.25
rs = 10%
gs = 20%
1
|
1.50
gs = 20%
2
| gn = 5%
1.80
37.80 =
3
|
1.89
1.89
0.10  0.05
The horizon, or continuing, value is the value at the horizon date of all
dividends expected thereafter. In this problem it is calculated as follows:
$1.80 (1.05)
 $37.80.
0.10  0.05
c. The firm’s intrinsic value is calculated as the sum of the present value of all
dividends during the supernormal growth period plus the present value of the
terminal value. Using your financial calculator, enter the following inputs:
CF0 = 0, CF1 = 1.50, CF2 = 1.80 + 37.80 = 39.60, I/YR = 10, and then solve
for NPV = $34.09.
9-5
The firm’s free cash flow is expected to grow at a constant rate, hence we can
apply a constant growth formula to determine the total value of the firm.
Firm value
= FCF1/(WACC – gFCF)
= $150,000,000/(0.10 – 0.05)
= $3,000,000,000.
To find the value of an equity claim upon the company (share of stock), we must
subtract out the market value of debt and preferred stock. This firm happens to be
entirely equity funded, and this step is unnecessary. Hence, to find the value of a
share of stock, we divide equity value (or in this case, firm value) by the number
of shares outstanding.
Equity value per share = Equity value/Shares outstanding
= $3,000,000,000/50,000,000
= $60.
Each share of common stock is worth $60, according to the corporate valuation
model.
9-11
First, solve for the current price.
P̂0 = D1/(rs – g)
= $0.50/(0.12 – 0.07)
= $10.00.
If the stock is in a constant growth state, the constant dividend growth rate is also
the capital gains yield for the stock and the stock price growth rate. Hence, to
find the price of the stock four years from today:
P̂4 = P0(1 + g)4
= $10.00(1.07)4
= $13.10796 ≈ $13.11.
9-13
The problem asks you to determine the value of P̂3 , given the following facts: D1 =
$2, b = 0.9, rRF = 5.6%, RPM = 6%, and P0 = $25. Proceed as follows:
Step 1: Calculate the required rate of return:
rs = rRF + (rM – rRF)b = 5.6% + (6%)0.9 = 11%.
Step 2: Use the constant growth rate formula to calculate g:
r̂s 
D1
g
P0
$2
g
$25
g  0.03  3%.
0.11 
Step 3: Calculate P̂3 :
P̂3 = P0(1 + g)3 = $25(1.03)3 = $27.3182  $27.32.
Alternatively, you could calculate D4 and then use the constant growth rate
formula to solve for P̂3 :
D4 = D1(1 + g)3 = $2.00(1.03)3 = $2.1855.
P̂3 = $2.1855/(0.11 – 0.03) = $27.3182  $27.32.
9-15
a. Horizon value =
b.
0
|
$40 (1.07)
0.13  0.07
WACC = 13%
($ 17.70)
23.49
522.10
$527.89
 1/1.13
=
$42.80
0.06
1
|
-20
 1/(1.13)2
 1/(1.13)3
2
|
30
= $713.33 million.
3
|
40
gn = 7%
4
|
42.80
Vop3 = 713.33
753.33
Using a financial calculator, enter the following inputs: CF0 = 0; CF1 = -20;
CF2 = 30; CF3 = 753.33; I/YR = 13; and then solve for NPV = $527.89
million.
c. Total valuet=0 = $527.89 million.
Value of common equity = $527.89 – $100 = $427.89 million.
Price per share =
9-16
$427 .89
10.00
= $42.79.
The value of any asset is the present value of all future cash flows expected to be generated
from the asset. Hence, if we can find the present value of the dividends during the period
preceding long-run constant growth and subtract that total from the current stock price, the
remaining value would be the present value of the cash flows to be received during the
period of long-run constant growth.
D1 = $2.00  (1.25)1 = $2.50
$2.2321
D2 = $2.00  (1.25)2 = $3.125
$2.4913
D3 = $2.00  (1.25)3 = $3.90625
PV(D1) = $2.50/(1.12)1
=
PV(D2) = $3.125/(1.12)2
=
PV(D3) = $3.90625/(1.12)3
=
$2.7804
 PV(D1 to D3)
= $7.5038
Therefore, the PV of the remaining dividends is: $58.8800 – $7.5038 = $51.3762.
Compounding this value forward to Year 3, we find that the value of all dividends
received during constant growth is $72.18. [$51.3762(1.12)3 = $72.1799 
$72.18.] Applying the constant growth formula, we can solve for the constant
growth rate:
P̂3 = D3(1 + g)/(rs – g)
$72.18
$8.6616 – $72.18g
$4.7554
0.0625
6.25%
= $3.90625(1 + g)/(0.12 – g)
= $3.90625 + $3.90625g
= $76.08625g
=g
= g.
9-17
0
1
rs = 12%
|
|
g = 5%
D0 = 2.00
D1
2
|
D2
3
|
D3
4
|
D4
P̂3
a. D1 = $2(1.05) = $2.10; D2 = $2(1.05)2 = $2.2050; D3 = $2(1.05)3 =
$2.31525.
b. Financial calculator solution: Input 0, 2.10, 2.2050, and 2.31525 into the cash
flow register, input I/YR = 12, and solve for NPV = PV = $5.28.
c. Financial calculator solution: Input 0, 0, 0, and 34.73 into the cash flow
register, I/YR = 12, and solve for NPV = PV = $24.72.
d. $24.72 + $5.28 = $30.00 = Maximum price you should pay for the stock.
P̂0 
e.
f.
D 0 (1  g)
D1
$2.10


 $30.00.
rs  g
rs  g 0.12  0.05
No. The value of the stock is not dependent upon the holding period. The value calculated in Parts a through d is the
value for a 3-year holding period. It is equal to the value calculated in Part e. Any other holding period would produce the same
value of
P̂0 ; that is, P̂0
= $30.00.
Chapter 10
10-1
rd(1 – T) = 0.12(0.65) = 7.80%.
10-2
Pp = $47.50; Dp = $3.80; rp = ?
rp =
Dp
Pp
=
$3.80
= 8%.
$47.50
10-3
40% Debt; 60% Common equity; rd = 9%; T = 40%; WACC = 9.96%; rs = ?
10-4
WACC
= (wd)(rd)(1 – T) + (wc)(rs)
0.0996 = (0.4)(0.09)(1 – 0.4) + (0.6)rs
0.0996 = 0.0216 + 0.6rs
0.078 = 0.6rs
rs = 13%.
P0 = $30; D1 = $3.00; g = 5%; rs = ?
a. rs =
D1
$3.00
+g=
+ 0.05 = 15%.
P0
$30.00
b. F = 10%; re = ?
re =
D1
$3.00
+g =
+ 0.05
P0 (1  F)
$30(1  0.10)
=
10-8
$3.00
+ 0.05 = 16.11%.
$27.00
Debt = 40%, Common equity = 60%.
P0 = $22.50, D0 = $2.00, D1 = $2.00(1.07) = $2.14, g = 7%.
rs =
D1
$2.14
+g=
+ 7% = 16.51%.
P0
$22.50
WACC
= (0.4)(0.12)(1 – 0.4) + (0.6)(0.1651)
= 0.0288 + 0.0991 = 12.79%.
Chapter 11
11-2
The regular payback method has three main flaws: (1) Dollars received in
different years are all given the same weight. (2) Cash flows beyond the payback
year are given no consideration whatever, regardless of how large they might be. (3)
Unlike the NPV, which tells us by how much the project should increase shareholder
wealth, and the IRR, which tells us how much a project yields over the cost of capital,
the payback merely tells us when we get our investment back. The discounted
payback corrects the first flaw, but the other two flaws still remain.
11-3
The NPV is obtained by discounting future cash flows, and the discounting
process actually compounds the interest rate over time. Thus, an increase in the
discount rate has a much greater impact on a cash flow in Year 5 than on a cash
flow in Year 1.
11-4
Mutually exclusive projects are a set of projects in which only one of the projects
can be accepted. For example, the installation of a conveyor-belt system in a
warehouse and the purchase of a fleet of forklifts for the same warehouse would
be mutually exclusive projects—accepting one implies rejection of the other.
When choosing between mutually exclusive projects, managers should rank the
projects based on the NPV decision rule. The mutually exclusive project with the
highest positive NPV should be chosen. The NPV decision rule properly ranks
the projects because it assumes the appropriate reinvestment rate is the cost of
capital.
Project X should be chosen over Project Y. Since the two projects are mutually
exclusive, only one project can be accepted. The decision rule that should be used
is NPV. Since Project X has the higher NPV, it should be chosen. The cost of
capital used in the NPV analysis appropriately includes risk.
11-8
11-1
Financial calculator solution: Input CF0 = -52125, CF1-8 = 12000, I/YR = 12, and
then solve for NPV = $7,486.68.
11-2
11-4
11-5
Financial calculator solution: Input CF0 = -52125, CF1-8 = 12000, and then solve
for IRR = 16%.
Since the cash flows are a constant $12,000, calculate the payback period as:
$52,125/$12,000 = 4.3438, so the payback is about 4 years.
Project K’s discounted payback period is calculated as follows:
Annual
Discounted @12%
Period
Cash Flows
Cash Flows
Cumulative
0
($52,125)
($52,125.00)
($52,125.00)
1
12,000
10,714.29
(41,410.71)
2
12,000
9,566.33
(31,844.38)
3
12,000
8,541.36
(23,303.02)
4
12,000
7,626.22
(15,676.80)
5
12,000
6,809.12
(8,867.68)
6
12,000
6,079.57
(2,788.11)
7
12,000
5,428.19
2,640.08
8
12,000
4,846.60
7,486.68
The discounted payback period is 6 +
11-6
$2,788. 11
years, or 6.51 years.
$5,42 8.19
a. Project A: Using a financial calculator, enter the following:
CF0 = -25, CF1 = 5, CF2 = 10, CF3 = 17, I/YR = 5; NPV = $3.52.
Change I/YR = 5 to I/YR = 10; NPV = $0.58.
Change I/YR = 10 to I/YR = 15; NPV = -$1.91.
Project B: Using a financial calculator, enter the following:
CF0 = -20, CF1 = 10, CF2 = 9, CF3 = 6, I/YR = 5; NPV = $2.87.
Change I/YR = 5 to I/YR = 10; NPV = $1.04.
Change I/YR = 10 to I/YR = 15; NPV = -$0.55.
b. Using the data for Project A, enter the cash flows into a financial calculator
and solve for IRRA = 11.10%. The IRR is independent of the WACC, so it
doesn’t change when the WACC changes.
Using the data for Project B, enter the cash flows into a financial calculator
and solve for IRRB = 13.18%. Again, the IRR is independent of the WACC,
so it doesn’t change when the WACC changes.
c. At a WACC = 5%, NPVA > NPVB so choose Project A.
At a WACC = 10%, NPVB > NPVA so choose Project B.
At a WACC = 15%, both NPVs are less than zero, so neither project would be
chosen.
11-7
a. Project A:
CF0 = -6000; CF1-5 = 2000; I/YR = 14.
Solve for NPVA = $866.16. IRRA = 19.86%.
MIRR calculation:
0 14% 1
|
|
-6,000 2,000
2
|
2,000
3
|
2,000
 (1.14)2
 (1.14)3
 (1.14)4
4
|
2,000
 1.14
5
|
2,000
2,280.00
2,599.20
2,963.09
3,377.92
13,220.21
Using a financial calculator, enter N = 5; PV = -6000; PMT = 0; FV =
13220.21; and solve for MIRRA = I/YR = 17.12%.
Payback calculation:
0
1
|
|
-6,000 2,000
Cumulative CF:-6,000 -4,000
2
|
2,000
-2,000
3
|
2,000
0
4
|
2,000
2,000
5
|
2,000
4,000
Regular PaybackA = 3 years.
Discounted payback calculation:
0 14% 1
2
3
4
5
|
|
|
|
|
|
-6,000 2,000
2,000
2,000
2,000
2,000
Discounted CF:-6,000 1,754.39 1,538.94 1,349.94 1,184.16 1,038.74
Cumulative CF:-6,000 -4,245.61-2,706.67-1,356.73 -172.57 866.17
Discounted PaybackA = 4 + $172.57/$1,038.74 = 4.17 years.
Project B:
CF0 = -18000; CF1-5 = 5600; I/YR = 14.
Solve for NPVB = $1,225.25. IRRB = 16.80%.
MIRR calculation:
0 14% 1
|
|
-18,000 5,600
2
|
5,600
3
|
5,600
4
|
5,600
 1.14
 (1.14)2
 (1.14)3
 (1.14)
4
5
|
5,600
6,384.00
7,277.76
8,296.65
9,458.18
37,016.59
Using a financial calculator, enter N = 5; PV = -18000; PMT = 0; FV =
37016.59; and solve for MIRRB = I/YR = 15.51%.
Payback calculation:
0
1
2
|
|
|
-18,000 5,600
5,600
Cumulative CF:-18,000-12,400 -6,800
3
|
5,600
-1,200
Regular PaybackB = 3 + $1,200/$5,600 = 3.21 years.
4
|
5,600
4,400
5
|
5,600
10,000
Discounted payback calculation:
0 14% 1
2
3
4
5
|
|
|
|
|
|
-18,000 5,600
5,600
5,600
5,600
5,600
Discounted CF:-18,000 4,912.28 4,309.02 3,779.84 3,315.65 2,908.46
Cumulative CF:-18,000-13,087.72-8,778.70-4,998.86-1,683.211,225.25
Discounted PaybackB = 4 + $1,683.21/$2,908.46 = 4.58 years.
Summary of capital budgeting rules results:
NPV
IRR
MIRR
Payback
Discounted payback
Project A
$866.16
19.86%
17.12%
3.0 years
4.17 years
Project B
$1,225.25
16.80%
15.51%
3.21 years
4.58 years
b. If the projects are independent, both projects would be accepted since both of
their NPVs are positive.
c. If the projects are mutually exclusive then only one project can be accepted,
so the project with the highest positive NPV is chosen. Accept Project B.
d. The conflict between NPV and IRR occurs due to the difference in the size of
the projects. Project B is 3 times larger than Project A.
11-10 Project A: Using a financial calculator, enter the following data: CF0 = -400; CF13 = 55; CF4-5 = 225; I/YR = 10. Solve for NPV = $30.16.
Project B: Using a financial calculator, enter the following data: CF0 = -600; CF12 = 300; CF3-4 = 50; CF5 = 49; I/YR = 10. Solve for NPV = $22.80.
The decision rule for mutually exclusive projects is to accept the project with the
highest positive NPV. In this situation, the firm would accept Project A since
NPVA = $30.16 is greater than NPVB = $22.80.
11-11 Project S: Using a financial calculator, enter the following data: CF0 = -15000;
CF1-5 = 4500; I/YR = 14. NPVS = $448.86.
Project L: Using a financial calculator, enter the following data: CF0 = -37500;
CF1-5 = 11100; I/YR = 14. NPVL = $607.20.
The decision rule for mutually exclusive projects is to accept the project with the
highest positive NPV. In this situation, the firm would accept Project L since
NPVL = $607.20 is greater than NPVS = $448.86.
11-12 Input the appropriate cash flows into the cash flow register, and then calculate
NPV at 10% and the IRR of each of the projects:
Project S: CF0 = -1000; CF1 = 900; CF2 = 250; CF3-4 = 10; I/YR = 10. Solve for
NPVS = $39.14; IRRS = 13.49%.
Project L: CF0 = -1000; CF1 = 0; CF2 = 250; CF3 = 400; CF4 = 800; I/YR = 10.
Solve for NPVL = $53.55; IRRL = 11.74%.
Since Project L has the higher NPV, it is the better project, even though its IRR is
less than Project S’s IRR. The IRR of the better project is IRR L = 11.74%.