Chapter 7
Corporate Valuation and Stock Valuation
ANSWERS TO END-OF-CHAPTER QUESTIONS
7-1 a. A proxy is a document giving one person the authority to act for another, typically the
power to vote shares of common stock. If earnings are poor and stockholders are
dissatisfied, an outside group may solicit the proxies in an effort to overthrow
management and take control of the business, known as a proxy fight. The preemptive
right gives the current shareholders the right to purchase any new shares issued in
b. The free cash flow model defines the total value of a company as the value of operations
plus the value of nonoperating assets.
The value of operations is the present value of all the future expected free cash
flows when discounted at the weighted average cost of capital:
c. Constant growth occurs when a firm’s earnings, dividends, and free cash flows grow
at some constant long-term rate. In this situation, the constant growth model can be
used to estimate the present value of the growing cash flows or dividends. For free cash
flows, the present value is:
The horizon date is the last year in a cash flow forecast. Cash flows may grow
unevenly during the forecast period, but are assumed to grow at a constant rate for all
periods after the horizon date.
The horizon value is the value all cash flows beyond the horizon date when
discounted back to the horizon date.
When applied to dividends, the horizon value is the intrinsic stock price at the end
of the explicit forecast period. It is equal to the present value of all dividends beyond
the forecast period, discounted back to the end of the forecast period at the required
rate of return on stock. Because growth after the horizon is constant, the constant
growth model can be applied at the horizon date:
Horizon value for stock = P
̂T= DT+1
rs-gL
= DT(1+gL)
rs-gL
d. A multistage model is used when the growth rate is nonconstant for several years before
becoming constant. In this case, the constant growth model is applied at the end of the
e. Estimated value (
0
P
ˆ
) is the present value of the expected future cash flows. The market
price (P0) is the price at which an asset can be sold.
f. The required rate of return on common stock, denoted by rs, is the minimum acceptable
rate of return considering both its riskiness and the returns available on other
investments. The expected rate of return, denoted by ^
rs, is the rate of return expected
on a stockn given its current price and expected future cash flows. If the stock is in
equilibrium, the required rate of return will equal the expected rate of return. The
realized (actual) rate of return, denoted by ¯rs, is the rate of return that was actually
realized at the end of some holding period. Although expected and required rates of
return must always be positive, realized rates of return over some periods may be
negative.
h. Preferred stock is a hybrid—it is similar to bonds in some respects and to common stock
in other respects. Preferred dividends are similar to interest payments on bonds in that
they are fixed in amount and generally must be paid before common stock dividends
can be paid. If the preferred dividend is not earned, the directors can omit it without
throwing the company into bankruptcy. So, although preferred stock has a fixed
payment like bonds, a failure to make this payment will not lead to bankruptcy. Most
preferred stocks entitle their owners to regular fixed dividend payments.
7-2 True. The value of a share of stock is the PV of its expected future dividends. If the two
7-3 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—coupon payments from the
perpetual bond, and dividends from both types of stock.
7-4 The first step is to find the value of operations by discounting all expected future free cash
flows at the weighted average cost of capital. The second step is to find the total corporate
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
7-1 a. Sales_1 = $800(1 + 0.10) =$900 million
Sales_2 = $00(1 + 0.05) =$945 million
b. NOPAT_1 = 0.10($900) = $90.00 million
NOPAT_2 = 0.10($945) = $94.50 million
7-2 Value of operations = Vop = PV of expected future free cash flow
Vop =
gWACC
)g1( FCF
−
+
=
05.012.0
)05.1(000,400$
−
= $6,000,000.
7-3 The growth rate in FCF from 2021 to 2022 is g = ($750.000-$707.547)/$707.547 = 0.06.
HV2022 = VOp at 2022 =
2022 L
L
FCF (1 g )
WACC g
+
−
=
$750 (1.06)
0.11 0.06−
= $15,900 million.
7-4 HV2020 = (FCF2021 (1 + gL))/(WACC – gL)
= ($500(1 + 0.04))/(0.09 – 0.04) = $10,400 million.
7-5
V_op
$800
+ ST investments
$70
Total value
$870
-Total debt
$200
-Preferred stock
$50
Intrinsic value of equity
$620
Divided by # shares
Intrinsic stock price
7-6 D0 = $1.50; g1-3 = 5%; gn = 10%; D1 through D5 = ?
D1 = D0(1 + g1) = $1.50(1.05) = $1.5750.
D2 = D0(1 + g1)(1 + g2) = $1.50(1.05)2 = $1.6538.
7-7 D1 = $1.50; g = 6%; rs = 13%;
0
P
ˆ
= ?
0
P
ˆ
=
gr
D
s
1
−
=
06.013.0
50.1$
−
= $21.43.
7-8 P0 = $22; D0 = $1.20; g = 10%;
1
P
ˆ
= ?;
r
s= ?
1
P
ˆ
= P0(1 + g) = $22(1.10) = $24.20.
7-9 0 1 2 3
| | | |
D0 = 2.00 D1 D2 D3
2
P
ˆ
Step 1: Calculate the required rate of return on the stock:
rs = rRF + (rM – rRF)b = 7.5% + (4%)1.2 = 12.3%.
Step 3: Calculate the PV of the expected dividends:
PVDiv = $2.40/(1.123) + $2.88/(1.123)2 = $2.14 + $2.28 = $4.42.
Step 4: Calculate
2
P
ˆ
:
2
P
ˆ
= D3/(rs – g) = $3.08/(0.123 – 0.07) = $58.11.
2
P
ˆ
Alternatively, using a financial calculator, input the following:
CF0 = 0, CF1 = 2.40, and CF2 = 60.99 (2.88 + 58.11) and then enter I/YR = 12.3 to solve
for NPV = $50.50.
7-10 Dps = $5.00; Vps = $50; rps = ?
rps =
ps
ps
v
D
=
00.50$
00.5$
= 10%.
7-11 a. 1. Vop =
05.013.0
)05.01(3$
+
−
=
18.0
85.2$
= $15.83.
2. Vop = $3/0.13 = $23.08.
7-12 a. HV2 =
08.012.0
000,108$
−
= $2,700,000.
7-13 a. HV3 =
07.013.0
)07.1( 40$
−
= $713.33.
b. 0 1 2 3 4 N
| | | | | • • • |
-20 30 40
($ 17.70)
23.49
3
op
V
= 713.33
7-14
0
P
ˆ
=
gr
D
s
1
−
=
gr
)g1(D
s
0
−
+
=
)]04.0(14.0
)]04.0(1[6$
−−
−+
=
18.0
76.5$
= $32.00.
7-15 The problem asks you to determine the constant growth rate, given the following facts: P0
= $80, D1 = $4, and rs = 14%. Use the constant growth rate formula to calculate gL:
WACC = 13%
g = 7%
7-16 The problem asks you to determine the value of
3
P
ˆ
, given the following facts: D1 = $3, b
= 0.8, rRF = 5.2%, RPM = 6%, and P0 = $40. Proceed as follows:
Step 1: Calculate the required rate of return:
rs = rRF + (rM – rRF)b = 5.2% + (6%)0.8 = 10%.
Step 3: Calculate
3
P
ˆ
:
3
P
ˆ
= P0(1 + g)3 = $40(1.025)3 = $43.076 ≈ $43.08.
Alternatively, you could calculate D4 and then use the constant growth rate formula to solve
for
3
P
ˆ
:
3
P
ˆ
7-17 Vps = Dps/rps; therefore, rps = Dps/Vps.
a. rps = $3.5/$30 = 11.67%.
b. rps = $3.5/$40 = 8.75%.
7-18 D0 = $1, rS = 7% + 6% = 13%, g1 = 50%, g2 = 25%, gn = 6%.
0 1 2 3 4
| | | | |
1.50 1.875 1.9875
1.327 + 28.393 = 1.9875/(0.13 – 0.06)
7-19 Calculate the dividend stream and place them on a time line. Also, calculate the price of
the stock at the end of the nonconstant growth period, and include it, along with the
dividend to be paid at t = 5, as CF5. Then, 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.
0 1 2 3 4 5 6
| | | | | | |
0.50 0.90 1.62
0.32 19.26
0.50 20.88
9.94
$10.76 =
0
P
ˆ
5
P
ˆ
= D6/(rs – g) = 1.7334/(0.16 – 0.07) = 19.26. This is the price of the stock at the end of
Year 5.
rs = 13%
g1 = 50%
g2 = 25%
gn = 6%
07.016.0
7334.1
−
rs = 16%
g = 7%
g = 80%
7-20 a. Vps =
ps
ps
r
D
=
08.0
10$
= $125.
b. Vps =
12.0
10$
= $83.33.
7-21 a. g = $1.1449/$1.07 – 1.0 = 7%.
7-22 0 g=6% 1 2 3 4
| | | | |
D0 = 1.50 D1 D2 D3 D4
3
P
ˆ
a. D1 = $1.5(1.06) = $1.59. D2 = $1.50(1.06)2 = $1.69. D3 = $1.5(1.06)3 = $1.79.
c. $27.05(0.6930) = $18.74.
Calculator solution: Input 0, 0, 0, and 27.05 into the cash flow register, I/YR = 13, PV
= ? PV = $18.74.
d. $18.74 + $3.97 = $22.71 = Maximum price you should pay for the stock. (rounding
differences may give you $22.72.)
7-23 a. End of Year: 0 1 2 3 4 5 6
| | | | | | |
FCF0 = 1.75 FCF1 FCF2 FCF3 FCF4 FCF5 FCF6
FCFt = FCF0(1 + g)t
FCF1 = $1.75(1.15)1 = $2.0125 million.
FCF2 = $1.75(1.15)2 = $2.3114 million.
= 12%
g = 5%
g = 15%
d. Total PV of FCF1 through FCF5 = $2.0125(1 + 0.12)1 + $2.3114(1 + 0.12)2
+ $2.6615(1 + 0.12)3 + $3.0608(1 + 0.12)4
+ = $3.5199(1 + 0.12)5
= $9.48
Calculator solution: Input 0, 2.0125, 2.3114, 2.6615, 3.0608, 3.5199 into the cash flow
register, input I/YR = 12, PV = ? PV = $9.48.
e. The total value of operations today is the sum or the PV of the horizon value and the
PVs of the free cash flows from Year 1 through 5:
Vop,0 = PV FCF Years 1 through 5 + PV of HV5
= $9.48 + $29.96 = $39.44 million.
7-21 a. Graphical representation of the problem:
Nonconstant Normal
growth growth
0 1 2 3 ∞
| | | |
D0 D1 (D2 +
2
P
ˆ
) D3 D∞
PVD1
PVD2
2
P
ˆ
0
P
ˆ
= PV(D1) + PV(D2) + PV(
2
P
ˆ
)
=
2
s
2
2
s
2
s
1
)r1(
P
ˆ
)r1(
D
)r1(
D
+
+
+
+
+
=
22
$3.25 $4.225 $90.415
(1 0.12) (1 0.12) (1 0.12)
++
+ + +
= $78.3482.
b.
1
ˆ
P
=
22
s
HV D
1r
+
+
=
$90.415 $4.225
1 0.12
+
+
= $84.50.
c. Expected dividend yield: D1/P0 = $3.25/$78.3482 = 4.148%.
Capital gains yield =
0
01
P
PP
ˆ−
=
$78.3482
$78.3482
$84.50 −
= 7.852%.
Total return = Dividend yield + capital gains yield = 4.148% + 7.852% =12.00%.
Notice that this is the same as the stock’s required return, rs.
SOLUTION TO SPREADSHEET PROBLEMS
7-25 The detailed solution for the spreadsheet problem, Ch07 P25 Build a Model Solution.xlsx,
is available at the textbook’s Web site.
7-26 The detailed solution for the spreadsheet problem, Ch07 P26 Build a Model Solution.xlsx,
is available at the textbook’s Web site.
MINI CASE
Your employer, a mid-sized human resources management company, is considering
expansion into related fields, including the acquisition of Temp Force Company, an
employment agency that supplies word processor operators and computer programmers to
businesses with temporary heavy workloads. Your employer is also considering the purchase
of Biggerstaff & McDonald (B&M), a privately held company owned by two friends, each
with 5 million shares of stock. B&M currently has free cash flow of $24 million, which is
expected to grow at a constant rate of 5%. B&M’s financial statements report short-term
investments of $100 million, debt of $200 million, and preferred stock of $50 million. B&M’s
weighted average cost of capital (WACC) is 11%. Answer the following questions.
a. Describe briefly the legal rights and privileges of common stockholders.
Answer: The common stockholders are the owners of a corporation, and as such, they have
certain rights and privileges as described below.
b. What is free cash flow (FCF)? What is the weighted average cost of capital? What
is the free cash flow valuation model?
Answer: Free cash flow (FCF) is the cash flow available for distribution to all of a company’s
investors. FCF is generated by a company’s operations.