Chapter 13 – Equity Valuation
CHAPTER 13
EQUITY VALUATION
1. Theoretically, dividend discount models can be used to value the stock of rapidly
growing companies that do not currently pay dividends; in this scenario, we would be
2. It is most important to use multi-stage dividend discount models when valuing
companies with temporarily high growth rates. These companies tend to be companies
3. The intrinsic value of a share of stock is the individual investor’s assessment of the true
worth of the stock. The market capitalization rate is the market consensus for the
4. Intrinsic value = V0 = D1
1 + k + D2
(1 + k)2 + … + DH + PH
(1 + k)H
5. Intrinsic value = V0 = D0× (1 + g)
k － g :
6. Intrinsic value = V0 = D0× (1 + g)
8. Market value of the firm
= Market value of assets – Market value of debts
9. g = ROE b = 0.10 0.6 = 0.06 or 6%
P/E = 1 － b
k － g = 1 － 0.6
0.08 － 0.06 = 20
11. Given EPS = $6, ROE = 15%, plowback ratio = 0.6, and k = 10%, we first calculate the
price with the constant dividend growth model:
12. FCFF = EBIT(1 – tc) + Depreciation – Capital expenditures – Increase in NWC
13. FCFE1 = FCFF – Interest expenses(1 – tc) + Increases in net debt
Chapter 13 – Equity Valuation
14. Cost of equity = rf + E(Risk premium) = 7% + 4% = 11%
Because the dividends are expected to be constant every year, the price can be
calculated as the no-growth-value per share:
15. k = rf + β [E(rM) – rf ] = 0.05 + 1.5 (0.10 – 0.05) = 0.125 or 12.5%
16.
a. False. Higher beta means that the risk of the firm is higher and the discount rate
applied to value cash flows is higher. For any expected path of earnings and
cash flows, the present value of the cash flows, and therefore, the price of the
firm will be lower when risk is higher. Thus the ratio of price to earnings will be
lower.
a. Using the constant-growth DDM, P0 = D1
k － g :
18. ROE = 20%, b = 0.3, EPS = $2, k = 12%
a. P/E Ratio
We can calculate the P/E ratio by dividing the current price by the projected
earnings:
b. Present Value of Growth Opportunities (PVGO)
g = ROE b = 0.20 0.3 = 0.6
g = ROE b = 0.20 0.2 = 0.04 = 4%
D1 = EPS (1 – b) = $2 (1 – 0.2) = $1.6
19. ROE = 16%, b = 0.5, EPS = $2, k = 12%
a. P0 = D1
k － g =EPS × (1－b)
k － (ROE × b) = $2 × (1 － 0.5)
0.12 － 0.16 × 0.5 = $1
0.12 － 0.08 = $25
20.
a. k = rf + β [E(rM) – rf ] = 0.06 + 1.25 (0.14 – 0.06) = 0.16 or 16%
g = ROE b = 0.09 (2/3) = 0.06 or 6%
Chapter 13 – Equity Valuation
b. Leading P0/E1 = $10.60/$3.18 = 3.33
c. PVGO = P0 – E1
k = $10.60 – $3.18
0.16 = $9.28
d. Now, you revise the plowback ratio in the calculation so that b = 1/3:
g = ROE b = 0.09 1/3 = 0.03 or 3%
e. V0 increases because the firm pays out more earnings instead of reinvesting
21. FI Corporation
22. Nogro Corporation
a. D1 = E1 (1 – b) = $2 0.5 = $1
Therefore:
Chapter 13 – Equity Valuation
b. Since k = ROE, the NPV of future investment opportunities is zero:
c. Since k = ROE, the stock price would be unaffected if Nogro were to cut its
d. Again, if Nogro eliminated the dividend, this would have no impact on Nogro’s
23. Xyrong Corporation
a. k = rf + β [E(rM) – rf ] = 0.08 + 1.2 (0.15 – 0.08) = 0.164 or 16.4%
b. P1 = V1 = V0 (1 + g) = $101.82 (1 + 0.12) = $114.04
24. Before-tax cash flow from operations $2,100,000
Depreciation 210,000
Taxable income 1,890,000
The value of the firm (i.e., debt plus equity) is:
25. Use this spreadsheet for all answers (Chart output taken directly from spreadsheet.
Answers reflect each scenario change).
Inputs
Year Dividend Div growth
Investor
CF
beta 0.9
2012 0.72 0.72
mkt_prem
0.08
2013 0.81 0.81
rf 0.029
2014 0.91 0.91
2015 1.00 1.00
2016 1.12 0.1157 1.12
2017 1.24 0.1116 1.24
2018 1.37 0.1076 1.37
2019 1.52 0.1035 1.52
2020 1.67 0.0994 1.67
2021 1.83 0.0954 1.83
2022 1.99 0.0913 1.99
2023 2.17 0.0872 2.17
2024 2.35 0.0831 2.35
2025 2.53 0.0791 2.53
2026 2.72 0.0750 2.72
2027 2.93 0.0750 120.96 123.89
Price = $36.79
26. The solutions derived from Spreadsheet 13.2 are as follows:
a.
Intrinsic val Equity val Intrin/share
b.
Intrinsic val Equity val Intrin/share
c.
Intrinsic val Equity val Intrin/share
27.
Chapter 13 – Equity Valuation
a. g = ROE b = 0.20 0.5 = 0.10 or 10%
b.
Time
EPS
Dividend
BVPS
Comment
0
$1.0000
$0.5000
$5.5000
Book Value Per Share is $5.5.
1
$1.1000
$0.5500
$6.0500
g = 10%, plowback = 0.50
payout ratio = 0.60
15% (1 – 0.6) = 6% from next year.
(Because the market is unaware of the changed competitive situation, it believes
the stock price should grow at 10% per year.)
P2 = D3
k － g = $0.5881
0.15 － 0.06 = $6.5340 after the market becomes aware of the
d.
Year
Return
55.0$)11$10.12($ ==
+−
e.
Year
Return
CFA 1
Chapter 13 – Equity Valuation
Answer:
a. This director is confused. In the context of the constant growth model, it is true
that price is higher when dividends are higher holding everything else (including
CFA 2
Answer:
a. It is true that NewSoft sells at higher multiples of earnings and book value than
Capital. But this difference may be justified by NewSoft’s higher expected
b. The most important weakness of the constant-growth dividend discount model in
CFA 3
Answer:
a. The industry’s estimated P/E can be computed using the following model:
P0/E1 = payout ratio/(r − g)
Therefore:
Chapter 13 – Equity Valuation
Chapter 13 – Equity Valuation
b.
i. Forecast growth in real GDP would cause P/E ratios to be generally higher
for Country A. Higher expected growth in GDP implies higher earnings
growth and a higher P/E.
CFA 4
Answer:
a. k = rf + β [E(rM) – rf ] = 0.045 + 1.15 (0.145 − 0.045) = 0.16 or 16%
b. Year Dividends
2013 $1.72
Present value of dividends paid in years 2014 to 2016:
Year PV of Dividends
2013 $1.93/1.16 = $1.66
c. The table presented in the problem indicates that QuickBrush is selling below
intrinsic value, while we have just shown that SmileWhite is selling somewhat
above the estimated intrinsic value. Based on this analysis, QuickBrush offers
Chapter 13 – Equity Valuation
d. Strengths of two-stage DDM compared to constant growth DDM:
• The two-stage model allows for separate valuation of two distinct periods in
a company’s future. This approach can accommodate life cycle effects. It
also can avoid the difficulties posed when the initial growth rate is higher
than the discount rate.
CFA 5
Answer:
a. The value of a share of Rio National equity using the Gordon growth model and
the capital asset pricing model is $22.40, as shown below.
Calculate the required rate of return using the capital asset pricing model:
b. The sustainable growth rate of Rio National is 9.97%, calculated as follows:
g = ROE × b = ROE × Retention Rate = ROE × (1 – Payout Ratio)
CFA 6
Answer:
a. To obtain free cash flow to equity (FCFE), the two adjustments that Shaar
should make to cash flow from operations (CFO) are:
Chapter 13 – Equity Valuation
b. Note 1: Rio National had $75 million in capital expenditures during the year.
Adjustment: Negative $75 million
Note 2: A piece of equipment that was originally purchased for $10 million was
sold for $7 million at year-end, when it had a net book value of $3 million.
Equipment sales are unusual for Rio National.
Note 3: The decrease in long-term debt represents an unscheduled principal
repayment; there was no new borrowing during the year.
Note 4: On 1 January 2015, the company received cash from issuing 400,000
shares of common equity at a price of $25.00 per share.
No adjustment
Note 5: A new appraisal during the year increased the estimated market
value of land held for investment by $2 million, which was not recognized
Chapter 13 – Equity Valuation
c. Free cash flow to equity (FCFE) is calculated as follows:
FCFE = NI + NCC – FCINV – WCINV + Net borrowing
where NCC = non-cash charges
FCINV = investment in fixed capital
WCINV = investment in working capital
Million $
Explanation
NI =
$30.16
From Table 13.6
NCC =
+$67.17
FCINV =
$75.00 (capital expenditures from Note 1)
Net Borrowing =
FCFE =
$71.17 (depreciation and amortization from Table 13.6)
CFA 7
Answer:
Rio National’s equity is relatively undervalued compared to the industry on a P/E-to–
growth (PEG) basis. Rio National’s PEG ratio of 1.33 is below the industry PEG ratio
of 1.66. The lower PEG ratio is attractive because it implies that the growth rate at Rio
National is available at a relatively lower price than is the case for the industry. The
PEG ratios for Rio National and the industry are calculated below:
Rio National
Chapter 13 – Equity Valuation
CFA 8
Answer:
Using a two-stage dividend discount model, the current value of a share of Sundanci is
calculated as follows:
where:
E0 = $0.952
D0 = $0.286
E1 = E0 (1.32)1 = $0.952 1.32 = $1.2566
CFA 9
Answer:
a. Free cash flow to equity (FCFE) is defined as the cash flow remaining after
meeting all financial obligations (including debt payment) and after covering
capital expenditure and working capital needs. The FCFE is a measure of how
much the firm can afford to pay out as dividends, but in a given year may be
more or less than the amount actually paid out.
Chapter 13 – Equity Valuation
b. The FCFE model requires forecasts of FCFE for the high growth years (2017
and 2018) plus a forecast for the first year of stable growth (2019) in order to
allow for an estimate of the terminal value in 2015 based on perpetual growth.
The following table shows the process for estimating Sundanci’s current value
on a per share basis:
Free Cash Flow to Equity
Base Assumptions
Shares outstanding: 84 millions
Required return on equity (r): 14%
Actual
2016
Projected
2017
Projected
2018
Projected
2019
Growth rate (g)
27%
27%
13%
Total
Per share
Earnings after tax
$80
$0.952
$1.2090
$1.5355
$1.7351
Terminal value
Discounted value
$40.7859****
* Projected 2015 Terminal value = (Projected 2016 FCFE)/(r − g)
** Projected 2015 Total cash flows to equity
c. i. The following limitations of the dividend discount model (DDM) are
addressed by the FCFE model. The DDM uses a strict definition of cash flows
to equity, i.e. the expected dividends on the common stock. In fact, taken to its
Chapter 13 – Equity Valuation
needs have been met. Thus the FCFE model explicitly recognizes the firm’s
investment and financing policies as well as its dividend policy. In instances of
a change of corporate control, and therefore the possibility of changing dividend
policy, the FCFE model provides a better estimate of value. The DDM is biased
ii. The following limitations of the DDM are not addressed by the FCFE model.
Both two-stage valuation models allow for two distinct phases of growth, an
initial finite period where the growth rate is abnormal, followed by a stable
growth period that is expected to last indefinitely. These two-stage models share
the same limitations with respect to the growth assumptions. First, there is the
difficulty of defining the duration of the extraordinary growth period. For
example, a longer period of high growth will lead to a higher valuation, and
there is the temptation to assume an unrealistically long period of extraordinary
CFA 10
Answer:
a. The formula for calculating a price earnings ratio (P/E) for a stable growth firm
is the dividend payout ratio divided by the difference between the required rate
of return and the growth rate of dividends. If the P/E is calculated based on
trailing earnings (year 0), the payout ratio is increased by the growth rate. If the
P/E is calculated based on next year’s earnings (year 1), the numerator is the
payout ratio.
P/E on trailing earnings:
Chapter 13 – Equity Valuation
b. The P/E ratio is a decreasing function of riskiness; as risk increases the P/E ratio
decreases. Increases in the riskiness of Sundanci stock would be expected to
lower the P/E ratio.