Download Topic Note-3

1
MF807
Prof. Thomas Chemmanur
Topic Note-3
STOCK AND FIRM VALUATION
Common stock represents ownership of the firm: stockholders elect the board of
directors which controls the firm and are entitled to the cash flows generated by the
firm. However, stock holders are residual claimants: they get only the cash flow left
over after paying interest to debt holders and other claimants. However, at this stage,
we will assume that the firm is 100% equity financed, postponing a discussion of the
impact of other claims till we talk about the firm's capital structure choice.
1. Stock valuation with the constant dividend growth model
Consider a firm which pays dividends D1, D2, ...etc for the first H periods. Let PH
be the price of the stock at t = H. Then, from our present value class, we know that the
current market price of the stock, P0 is given by,
(1)
D1
D2
D3
DH +PH
P0 = (1+r) + (1+r)2 + (1+r)3 +...+ (1+r)H
where r is the discounting rate which corresponding to the riskiness of the stock. But
we know that PH is given by,
(2)
PH 
DH 1
D
 H  2 2  ...
(1  r ) (1  r )
2
(3)
P0 
D3
D1
D2
DH
DH 1
DH  2


 ... 


 ...
2
3
H
H 1
(1  r ) (1  r )
(1  r )
(1  r )
(1  r )
(1  r ) H  2
Using (2) in (1) and simplifying,
The problem in using (3) is that, to begin with, we don't know what the dividend
stream is! Let us make the simplifying assumption that dividends grow at a constant
rate g. Then, if D0 was the dividend in the last (current) period,
D1 = D0 (1 + g);
D2 = D1 (1+g) = D0(1+g)2;
D3 = D0(1+g)3, etc. Using this
assumption in (3),
(4)
1  g  (1  g ) (1  g ) 2 (1  g ) 3
(1  g )
P0  D0


 ... 
1 
2
3
1  r  (1  r ) (1  r )
(1  r )
(1  r ) H
H

, (1  r ) H   .........

The term in the square brackets is an infinite geometric series of the form I
talked about in the class on present values, with common ratio (1+g)/(1+r). Applying
the formula for the sum of an infinite geometric series, (4) reduces to:
(5)
P0 =
D0(1+g)
D1
=
r-g
r-g
(5) is often referred to as the constant growth formula for stock valuation. The
above formula can be used only when r > g, since otherwise the common ratio
(1+g)/(1+r) of the geometric series is greater than 1, and the stock price becomes
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infinity. It is important to remember that (5) is true only if the underlying assumption
of a constant growth rate in dividends is true (which is often not the case). The above
formula assumes that we know r and g, and want to find the price P 0. If we do not
know g, we have to make additional assumptions to find it.
3. Approximating g from ROE and the plow-back ratio
It is important to remember that return on equity (ROE) is an accounting
concept. Unfortunately, accounting numbers represent the past, while in finance, we
want to know about future cash flows to value securities. (Accounting figures also have
the problem that they can differ according to the choice of accounting method).
However, they can act as a starting place.
Earnings per share (EPS) = Net Income (obtained from the income statement)/Number
of shares
Book Value per share = Book value of equity (obtained from the balance
sheet)/Number of shares.
Return on equity (ROE) = EPS/Book Value Per Share
Dividend payout ratio = Dividend per share/EPS
Plow-back ratio = 1 - Dividend payout ratio.
Then g ≈ Plow-back ratio x ROE, assuming that the firm's earnings re-invested in the
company earn the same return as in the past. (Again, it goes without saying that this
assumption may not always be true).
3. Finding r given P0, D0 and g
From (5),
4
D1
r = P + g
0
(6)
Again assuming constant dividend growth.
4. Stock Valuation with different growth rates in dividends
Often, the assumption of a constant growth rate in dividends is not at all
realistic. The dividend may grow at a certain rate for the first few years, and then settle
down to a steady rate afterward. To illustrate how to price equity in this case, consider
the following problem.
Problem 1: A company's dividend grows at 10% rate for the first five years, and
thereafter settles down to a steady growth rate of 6%. If stockholders expect a rate of
return of 14% from investing in the equity, what is its share price? D0 = $2 per share.
5. Stock valuation from the NPV of growth opportunities
Sometimes, it may be easier to value equity as the sum of two components: (a)
The value of a share if the firm does not make any future investment, and therefore
pays out all earnings as dividends (b) The NPV of future growth opportunities.1
(a)
If the firm does not make any investment, let us assume that it will earn EPS1
next year, and continue to earn the same amount for ever. In this case, its value will be
EPS1/r, since the earnings stream will be a perpetuity.
(b) If the firm does make the investment, it will plow back a fraction of EPS1 into the
firm. In that case, it will obtain additional earnings from this new project. Denote the
NPV of this new investment per share by PVGO. Then, price per share,
1Notice that we are adding only the NPV of growth opportunities. The investment required for these is assumed
to come from retaining earnings from existing projects, which is part of (a). Therefore adding the investment for the
growth opportunities in (b) will be double-counting.
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(7)
P0= (a) + (b) =
EPS1
r + PVGO
Re-arranging (7) we get,
(8)
EPS1
PVGO

P0 = r 1 - P0 
The importance of (8) is that it gives a relationship between the earnings-to-price
ratio of a company and its market capitalization rate r (rate of return expected by
investors from the equity). When PVGO is zero (i.e., when the firm has no growth
opportunities) both are the same.
Problem 2: Consider a firm with existing assets that generate an EPS of $5. If the firm
does not invest any further, EPS is expected to remain constant at this level. However,
starting next year, the firm has the chance to invest $3 per share a year in developing
a newly discovered geothermal steam source for electricity generation. Each investment
is expected generate a perpetual 20 % return. However, the source will be fully
developed by the fifth year. What will be the stock price and earnings-price ratio
assuming investors require a 12% rate of return?
Problem 3: Consider a firm with the following data: D1 = $10 per share; dividends are
expected to grow by 5% a year; the company plows back 20% of earnings. (i) What is
the price of a share? (ii) What is next year's expected earnings? (iii) What is the return
on book equity? (iv) What is PVGO? (r = 15%).
6. Stock price as the present value of free cashflow per share:
We can also express the price per share as:
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t 
(9)
P0 

t 1
Free cash flow per share at date t
(1  r ) t
where Free cash flow = Revenue - Costs - Investment. However, free cash flow per
share is the amount that the firm is free to pay out as dividends per share. Therefore
(9) is essentially identical to (3).
7. A note on the price-earnings (P/E) ratio
The P/E ratio is available in financial newspapers. It is the reciprocal of the
EPS/Price ratio. Therefore, from (8), we can see that the P/E ratio of a company will be
higher if (a) if its r is low or (b) it has higher growth opportunities (i.e., a higher PVGO),
or a combination of the two. It is sometimes useful in stock valuation. If we can find a
similar firm with the same riskiness and growth opportunities, we can use the following
rule:
Value of the firm's equity = Earnings of the firm to be valued x P/E of the similar firm.
(The problem is, of course in finding such a very similar firm).