Chapter 18 – Equity Valuation Models
CHAPTER 18: EQUITY VALUATION MODELS
PROBLEM SETS
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
required rate of return for the stock. If the intrinsic value of the stock is equal to its
4. a. k = D1/P0 + g
b. P0 = D1/(k – g) = $2/(0.16 – 0.05) = $18.18
The price falls in response to the more pessimistic dividend forecast. The
18-2
5. a. g = ROE b = 16% 0.5 = 8%
6. a. k = rf + (rM) – rf ] = 6% + 1.25(14% – 6%) = 16%
g = 2/3 9% = 6%
0.060.16
gk
0=
−
−
b. Leading P0/E1 = $10.60/$3.18 = 3.33
16.0
18.3$
k
E
0−=−=−=
The low P/E ratios and negative PVGO are due to a poor ROE (9%) that is less
than the market capitalization rate (16%).
d. Now, you revise b to 1/3, g to 1/3 9% = 3%, and D1 to:
7. Since beta = 1.0, then k = market return = 15%
Therefore:
Chapter 18 – Equity Valuation Models
18-3
05.010.0
8$
gk
D
0=
−
−
b. The dividend payout ratio is 8/12 = 2/3, so the plowback ratio is b = 1/3. The
c. Assuming ROE = k, price is equal to:
120$
10.0
12$
k
E
P1
0===
Therefore, the market is paying $40 per share ($160 – $120) for growth
opportunities.
9. a. k = D1/P0 + g
b. Since k = ROE, the NPV of future investment opportunities is zero:
010$10$
k
E
PPVGO 1
0=−=−=
c. Since k = ROE, the stock price would be unaffected by cutting the dividend and
10. a. k = rf +[E(rM ) – rf ] = 8% + 1.2(15% – 8%) = 16.4%
12.0164.0
gk
0=
−
−
b. P1 = V1 = V0(1 + g) = $101.82 1.12 = $114.04
Chapter 18 – Equity Valuation Models
11.
Time:
0
1
5
6
E t
$10.000
$12.000
$24.883
$29.860
D t
$0.000
$0.000
$0.000
$11.944
b
1.00
1.00
1.00
0.60
g
20.0%
20.0%
20.0%
9.0%
944.11$
D
Chapter 18 – Equity Valuation Models
18-5
13. The solutions derived from Spreadsheet 18.2 are as follows:
Intrinsic value:
FCFF
Intrinsic value:
FCFE
Intrinsic value
per share: FCFF
Intrinsic value
per share: FCFE
a.
81,171
68,470
36.01
37.83
b.
59,961
49,185
24.29
27.17
c.
69,813
57,913
29.73
32.00
14.
Time:
0
1
2
3
D t
$1.0000
$1.2500
$1.5625
$1.953125
g
25.0%
25.0%
25.0%
5.0%
a. The dividend to be paid at the end of year 3 is the first installment of a dividend
stream that will increase indefinitely at the constant growth rate of 5%. Therefore, we
can use the constant growth model as of the end of year 2 in order to calculate intrinsic
value by adding the present value of the first two dividends plus the present value of
the price of the stock at the end of year 2.
The expected price 2 years from now is:
The PV of expected dividends in years 1 and 2 is:
13.2$
20.1
5625.1$
20.1
25.1$
2=+
Thus the current price should be: $9.04 + $2.13 = $11.17
b. Expected dividend yield = D1/P0 = $1.25/$11.17 = 0.112 = 11.2%
c. The expected price one year from now is the PV at that time of P2 and D2:
The implied capital gain is:
The sum of the implied capital gains yield and the expected dividend yield is equal
to the market capitalization rate. This is consistent with the DDM.
Chapter 18 – Equity Valuation Models
18-6
15.
Time:
0
1
4
5
E t
$5.000
$6.000
$10.368
$12.4416
D t
$0.000
$0.000
$0.000
$12.4416
Dividends = 0 for the next four years, so b = 1.0 (100% plowback ratio).
4416.12$
D
15.1
)k1(
0==
+
b. Price should increase at a rate of 15% over the next year, so that the HPR will
equal k.
16. Before-tax cash flow from operations $2,100,000
Depreciation 210,000
After-tax cash flow from operations
(After-tax unleveraged income + depreciation) 1,438,500
New investment (20% of cash flow from operations) 420,000
The value of the firm (i.e., debt plus equity) is:
000,550,14$
05.012.0
500,018,1$
1
0=
−
=
−
=gk
C
V
Since the value of the debt is $4 million, the value of the equity is $10,550,000.
17. a. g = ROE b = 20% 0.5 = 10%
11$
10.015.0
10.150.0$
gk
)g1(D
gk
D
P0
1
0=
−
=
−
+
=
−
=
Chapter 18 – Equity Valuation Models
b.
Time
EPS
Dividend
Comment
0
$1.0000
$0.5000
1
$1.1000
$0.5500
g = 10%, plowback = 0.50
2
$1.2100
$0.7260
EPS has grown by 10% based on last
year’s earnings plowback and ROE; this
year’s earnings plowback ratio now falls
to 0.40 and payout ratio = 0.60
3
$1.2826
$0.7696
EPS grows by (0.4) (15%) = 6% and
payout ratio = 0.60
At time 2:
551.8$
06.015.0
7696.0$
gk
D
P3
2=
−
=
−
=
)15.1(
551.8$726.0$
15.1
55.0$
0=
+
c. P0 = $11 and P1 = P0(1 + g) = $12.10
(Because the market is unaware of the changed competitive situation, it believes the
stock price should grow at 10% per year.)
Year
Return
1
%0.15150.0
11$
55.0$)11$10.12($ ==
+−
2
%3.23233.0
10.12$
726.0$)10.12$551.8($ −=−=
+−
3
%0.15150.0
551.8$
7696.0$)551.8$064.9($ ==
+−
Moral: In “normal periods” when there is no special information,
CFA PROBLEMS
18-9
5. 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.
Sundanci’s FCFE for the year 2008 is computed as follows:
FCFE =
Earnings after tax + Depreciation expense − Capital expenditures − Increase in NWC
FCFE per share = FCFE/number of shares outstanding
At the given dividend payout ratio, Sundanci’s FCFE per share equals dividends
per share.
b. The FCFE model requires forecasts of FCFE for the high growth years (2009 and
2010) plus a forecast for the first year of stable growth (2011) in order to to allow
for an estimate of the terminal value in 2010 based on perpetual growth. Because
Chapter 18 – Equity Valuation Models
18–10
Free Cash Flow to Equity
Base Assumptions
Shares outstanding: 84 million
Required return on equity (r): 14%
Actual
2008
Projected
2009
Projected
2010
Projected
2011
Growth rate (g)
27%
27%
13%
Total
Per share
Earnings after tax
$80
$0.952
$1.2090
$1.5355
$1.7351
Plus: Depreciation expense
$23
$0.274
$0.3480
$0.4419
$0.4994
Less: Capital expenditures
$38
$0.452
$0.5740
$0.7290
$0.8238
Less: Increase in net working capital
$41
$0.488
$0.6198
$0.7871
$0.8894
Equals: FCFE
$24
$0.286
$0.3632
$0.4613
$0.5213
Terminal value
$52.1300*
Total cash flows to equity
$0.3632
$52.5913**
Discounted value
$0.3186***
$40.4673***
Current value per share
$40.7859****
*Projected 2010 Terminal value = (Projected 2011 FCFE)/(r − g)
**Projected 2010 Total cash flows to equity =
Projected 2010 FCFE + Projected 2010 Terminal value
c. i. 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 extreme, the DDM cannot be used to estimate
the value of a stock that pays no dividends. The FCFE model expands the definition of
cash flows to include the balance of residual cash flows after all financial obligations
and investment needs have been met. Thus the FCFE model explicitly recognizes the
Chapter 18 – Equity Valuation Models
18–11
ii. 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 growth. Second, the assumption of a sudden shift from high
growth to lower, stable growth is unrealistic. The transformation is more likely to occur
6. 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:
P/E on next year’s earnings:
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.
The P/E ratio is an increasing function of the growth rate of the firm; the higher the
expected growth, the higher the P/E ratio. Sundanci would command a higher P/E if
analysts increase the expected growth rate.
Chapter 18 – Equity Valuation Models
18–12
7. a. The sustainable growth rate is equal to:
plowback ratio × return on equity = b × ROE
where
In 2005:
b = [208 – (0.80 × 100)]/208 = 0.6154
In 2008:
b. i. The increased retention ratio increased the sustainable growth rate.
Retention ratio = [Net Income – (Dividend per share × shares outstanding)]/Net Income
ii. The decrease in leverage reduced the sustainable growth rate.
Financial leverage = (Total Assets/Beginning of year equity)
8. a. The formula for the Gordon model is:
V0 = [D0 × (1 + g)]/(r – g)
where:
Chapter 18 – Equity Valuation Models
18–13
b. Use of the Gordon growth model would be inappropriate to value Dynamic’s
common stock, for the following reasons:
9. a. The industry’s estimated P/E can be computed using the following model:
P0/E1 = payout ratio/(r − g)
However, since r and g are not explicitly given, they must be computed using the
following formulas:
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.
Chapter 18 – Equity Valuation Models
10. a. k = rf + (rM) – rf ] = 4.5% + 1.15(14.5% − 4.5%) = 16%
b.
Year
Dividend
2009
$1.72
2010
$1.72 1.12 =
$1.93
2011
$1.72 1.122 =
$2.16
2012
$1.72 1.123 =
$2.42
2013
$1.72 1.123 1.09 =
$2.63
Present value of dividends paid in 2010 – 2012:
Year
PV of Dividend
2010
$1.93/1.161 =
$1.66
2011
$2.16/1.162 =
$1.61
2012
$2.42/1.163 =
$1.55
Total =
$4.82
63.2$2013 =
D
18–15
11. a. The value of a share of Rio National equity using the Gordon growth model and
the capital asset pricing model is $22.40, as shown below.
40.22$
12.013.0
)12.01(20.0$
gk
g)(1D
Po
0=
−
+
=
−
+
=
b. The sustainable growth rate of Rio National is 9.97%, calculated as follows:
35.270$
16.30$
Equity Beginning
IncomeNet
12. a. To obtain free cash flow to equity (FCFE), the two adjustments that Shaar should
make to cash flow from operations (CFO) are:
1. Subtract investment in fixed capital: CFO does not take into account the
2. Add net borrowing: CFO does not take into account the amount of capital
supplied to the firm by lenders (e.g., bondholders). The new borrowings, net of
b. Note 1: Rio National had $75 million in capital expenditures during the year.
Adjustment: negative $75 million
Chapter 18 – Equity Valuation Models
18–16
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.
Adjustment: positive $3 million
In calculating FCFE, only cash flow investments in fixed capital should be
considered. The $7 million sale price of equipment is a cash inflow now available to
equity holders and should be added to net income. However, the gain over book
Note 3: The decrease in long-term debt represents an unscheduled principal
repayment; there was no new borrowing during the year.
Adjustment: negative $5 million
Note 4: On January 1, 2008, 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 in 2008 income.
No adjustment
Chapter 18 – Equity Valuation Models
18–17
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 18G
NCC =
+$67.17
$71.17 (depreciation and amortization from Table 18G)
– $4.00* (gain on sale from Note 2)
FCINV =
–$68.00
$75.00 (capital expenditures from Note 1)
– $7.00* (cash on sale from Note 2)
WCINV =
–$24.00
–$3.00 (increase in accounts receivable from Table 18F) +
–$20.00 (increase in inventory from Table 18F) +
–$1.00 (decrease in accounts payable from Table 18F)
Net Borrowing =
+(–$5.00)
–$5.00 (decrease in long-term debt from Table 18F)
FCFE =
$0.33
*Supplemental Note 2 in Table 18H affects both NCC and FCINV.
13. 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
Rio National
Current Price = $25.00
Normalized Earnings per Share = $1.71
Industry
Price-to-Earnings Ratio = 19.90