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Stock Valuation
One Period Valuation Model
• To value a stock, you first find the present
discounted value of the expected cash flows.
• P0 = Div1/(1 + ke) + P1/(1 + ke) where
–
–
–
–
P0 = the current price of the stock
Div = the dividend paid at the end of year 1
ke = required return on equity investments
P1 = the price at the end of period one
One Period Valuation Model
• P0 = Div1/(1 + ke) + P1/(1 + ke)
– Let ke = 0.12, Div = 0.16, and P1 = $60.
• P0 = 0.16/1.12 + $60/1.12
• P0 = $0.14285 + $53.57
• P0 = $53.71
– If the stock was selling for $53.71 or less, you
would purchase it based on this analysis.
Generalized Dividend Valuation Model
• The one period model can be extended to any
number of periods.
– P0 = D1/(1+ke)1 + D2/(1+ke)2 +…+ Dn/(1+ke)n +
Pn/(1+ke)n
• If Pn is far in the future, it will not affect P0
• Therefore, the model can be rewritten as:
–
∞
t
P0 = S
D
/(1
+
k
)
t
e
t=1
Generalized Dividend Valuation Model
• The model says that the price of a stock is determined
only by the present value of the dividends.
– If a stock does not currently pay dividends, it is assumed
that it will someday after the rapid growth phase of its life
cycle is over.
• Computing the present value of an infinite stream of
dividends can be difficult.
• Simplified models have been developed to make the
calculations easier.
Gordon Growth Model
• Assumptions:
– Dividends continue to grow at a constant rate for
an extended period of time.
– The growth rate is assumed to be less than the
required return on equity, ke.
• Gordon demonstrated that if this were not so, in the long
run the firm would grow impossibly large.
The Gordon Growth Model
Firms try to increase their dividends at a constant rate.
P0 = D0(1+g)1 + D0(1+g)2 +…..+ D0(1+g)∞
(1+ke)1
(1+ke)2
(1+ke)∞
D0 = the most recent dividend paid
g = the expected growth rate in dividends
ke = the required return on equity investments
The model can be simplified algebraically to read:
P0 = D0(1 + g)
D1
=
(ke – g)
(ke – g)
Gordon Model: Example
• Find the current price of Coca Cola stock
assuming the following:
– g = 10.95%
– D0 = $1.00
– ke = 13%.
P0 = D0(1 + g)/ke – g
P0 = $1.00(1.1095)/0.13 - 0.1095
P0 = $1.1095/0.0205 = $54.12
Gordon Model: Conclusions
• Theoretically, the best method of stock
valuation is the dividend valuation approach.
• But, if a firm is not paying dividends or has an
erratic growth rate, the simple model will not
work.
• Consequently, other methods are required.
Non-constant Growth
• Firms typically go through life cycles.
– Early in the cycle, their growth is much faster than
that of the economy as a whole.
– Later in the cycle, their growth matches the
economy’s growth.
– Finally, their growth is less than the economy’s.
• Non-constant or supernormal growth occurs
during that part of the life cycle when the firm
grows faster than the economy as a whole.
Dividend Growth Rates
Div($)
Normal growth = 8%
Supernormal growth = 30%
Normal growth = 8%
1.15
0
Zero growth = 0%
1
2
3
4
Declining growth = -8%
Years
5
Firm Valuation with Non-constant
Growth
• The value of the firm equals the present value
of its expected future dividends.
• Process:
1 Find the present value of the dividends during
the period of non-constant growth.
2 Find the price of the stock at the end of the nonconstant growth period and discount this price
back to the present.
3 Add the two components to find the value of the
stock, P0.
Firm Valuation with Non-constant
Growth: Problem
• Assume the following:
–
–
–
–
–
k = 13.4% (required rate of return
N = 3 (years of supernormal growth)
gs = 30% (rate of growth in supernormal period)
gn = 8% (rate of growth in normal period)
D0 = $1.15 (last dividend paid)
• Find the value of the stock.
Firm Valuation with Non-constant
Growth: Problem
• Step 1:
– Calculate the dividends expected at the end of each
year during the supernormal period.
•
•
•
•
Dn = Dn-1(1 + gs)
D1 = $1.15(1 + .3) = $1.495
D2 = $1.495(1 + .3) = $1.9435
D3 = $1.9435(1 + .3) = $2.5265
Firm Valuation with Non-constant
Growth: Problem
• Step 2:
– Calculate the price of the stock during the normal
growth period using the Gordon model.
• Calculate the dividend in the fourth period.
• Use the constant growth formula to find P3.
– D4 = $2.5265(1 + 0.08) = $2.7286
– P3 = $2.7286/0.134 – 0.08 = $50.53
– If the stockholder sold the stock in period 3, he would
receive $50.53. Total cash flow at time 3 equals
D3
+ P3 = $53.0576.
Firm Valuation with Non-constant
Growth: Problem
• Step 3:
– Discount the cash flows found in steps 1 and 2 and
sum the amounts to find the value of the
supernormal growth stock.
•
•
•
•
•
D1 = $1.4950/(1.134) =
D2 = $1.9435/1.2859 =
D3 = $2.5265/1.4583 =
P3 = $50.5310/1.4583 =
Value of growth stock =
$1.3183
$1.5113
$1.7325
$34.65
$39.21
Errors in Valuation
• Problems with Estimating Growth
– Growth can be estimated by computing historical growth
rates in dividends, sales, or net profits.
– But, this approach fails to consider any changes in the firm
or economy that may affect the growth rate.
• Competition, for example, will prevent high growth firms from
being able to maintain their historical growth rate.
• Nevertheless, stock prices of historically high growth firms tend to
reflect a continuation of the high growth rate.
• As a result, investors receive lower returns than they would by
investing in mature firms.
Estimating Growth: Table 1
Stock Prices for a Security with D0 = $2.00, ke = 15%,
and Constant Growth Rates as Listed
Growth(%)
1
3
5
10
11
12
13
14
Price
$14.43
17.17
21.00
44.00
55.50
74.67
113.00
228.00
Errors in Valuation
• Problems with Estimating Risk
– The dividend valuation model requires the analyst
to estimate the required return for the firms equity.
– However, a share of stock offering a $2 dividend
and a 5% growth rate changes with different
estimates of the required return.
Estimating Risk: Table 2
Stock Prices for a Security with D0 = $2.00, g = 5%,
and Required Returns as Listed
Required Return(%)
10
11
12
13
14
15
Price
$42.00
35.00
30.00
26.25
23.33
21.00
Errors in Valuation
• Problems with Forecasting Dividends
– Many factors can influence the dividend payout
ratio. They include:
• The firm’s future growth opportunities
• Management’s concern over future cash flows
• Conclusion:
– Analysts are seldom certain that the stock price
projections are accurate.
– This is why stock prices fluctuate widely on news
reports.
Price Earnings Valuation Method
• The price earnings ratio (PE) is a widely
watched measure of how much the market is
willing to pay for $1 of earnings from a firm.
• A high PE has two interpretations:
– A higher than average PE may mean that the
market expects earnings to rise in the future.
– A high PE may indicate that the market thinks the
firm’s earnings are very low risk and is therefore
willing to pay a premium for them.
Price Earnings Valuation Method
• The PE ratio can be used to estimate the value
of a firm’s stock.
• Firms in the same industry are expected to
have similar PE ratios in the long run.
• The value of a firm’s stock can be found by
multiplying the average industry PE times the
expected earnings per share.
P/E x E = P
Price Earnings Model: Example
• The average industry PE ratio for restaurants
similar to Applebee’s is 23. What is the
current price of Applebee’s if earnings per
share are projected to be $1.13?
– P0 = P/E x E
– P0 + 23 x $1.13 = $26.
Price Earnings Valuation Method
• Advantages:
– Useful for valuing privately held firms and firms
that do not pay dividends.
• Disadvantages:
– By using an industry average PE ratio, firmspecific factors that might contribute to a longterm PE ratio above or below the average are
ignored.
Non-constant Growth Model
• The non-constant growth model can be used to
estimate the value of a stock that does not pay
dividends during its early years, if it is
expected to pay dividends in the future.
Non-constant Growth Model
• Process:
– Estimate the following:
•
•
•
•
•
•
when dividend will be paid
the amount of the first dividend
the growth rate during the supernormal period
the length of the supernormal period
the long-run constant growth rate
the rate of return required by investors.
– Use the constant growth model to determine the price
of the stock after the firm reaches stable growth.
– Find all the cash flows, take the present value of each
and sum.
Setting Security Prices
• Stock prices are set by the buyer willing to pay
the highest price.
– The price is not necessarily the highest price that
the stock could get, but it is incrementally greater
than what any other buyer is willing to pay.
• The market price is set by the buyer who can
take best advantage of the asset.
Setting Security Prices
• Superior information about an asset can
increase its value by reducing its risk.
– The buyer who has the best information about
future cash flows will discount them at a lower
interest rate than a buyer who is uncertain.