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Equity Valuation
Lecture Notes # 8
Contents
1 About Valuation
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2 Present-Values
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3 Choice of the Appropriate Discount Rate
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4 Future Cash Flows: the Dividend Discount Model (DDM)
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5 The Two-Stage Dividend-Growth Model
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6 Dividends and Earnings
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7 The EVA Model
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8 Alternative Valuation Techniques
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9 Applications: Boeing and Exxon
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10 Practice Problems
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About Valuation
• Sometimes we can observe a market value for the security, and we are
interested in assessing whether it is overvalued or undervalued. (E.g.:
stock analysts.)
• Sometimes there is no market value and we are trying to construct one
for bargaining or transaction purposes. (E.g.: a corporation wants to
sell a division. What should the price be?)
• Both of these situations call for equity valuation, and this is the most
common kind of valuation problem.
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Present-Values
• The stock is bought, held for some period of time (dividends are collected) and then sold. The share is valued as the present value of the
expected dividends and the proceeds from the sale: intrinsic value, V .
• Dividends are paid annually, and the time-0 dividend (D0 ) has just been
paid. The stock will be bought and held for one year. The present value
is
E(D1 + P1 )
V0 =
,
1+k
• If V0 = P0 =⇒expected rate of return coincides with the discount rate
k:
E(D1 + P1 )
.
1+k ≡
P0
• Two main issues arise in the valuation problem:
1. Choice of the appropriate discount rate
2. Estimation of future cash-flows
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Choice of the Appropriate Discount Rate
• The discount rate can be found using the CAPM. From the SML:
k = rf + β[E(rM ) − rf ].
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• Consider the following valuation problem for ConEdison at the beginning
of 1994.
The beta for ConEdison is:
βCE = .75.
The average annual rate of return on the SP 500 index, 82-94, is:
E(rM ) = .1497.
The risk-free rate (annual rate) is rf = .075. We have
k = .075 + .75 × .0747 = .1310.
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Future Cash Flows: the Dividend Discount Model (DDM)
• Assume constant growth of dividends. Dividends are expected to grow
at a constant rate g forever:
E(D2 ) = E(D1 )(1 + g), E(D3 ) = E(D1 )(1 + g)2 , . . .
The present value of all the cash-flows generated by the stock is given by
E(D1 )
E(D2 )
E(D3 )
+
+
+ ...
1+k
(1 + k)2 (1 + k)3
D0 (1 + g) D0 (1 + g)2 D0 (1 + g)3
=
+
+
+ ...
1+k
(1 + k)2
(1 + k)3
D0 (1 + g) E(D1 )
=
=
.
k−g
k−g
Note that this result is valid only when k > g. If g > k, the present
value of the dividends is infinite.
V0 =
• Note that, based on the model above, the expected rate of capital gains
on a stock equals
E(D2 ) − E(D1 ) E(D1 )(1 + g) − E(D1 )
E(V1 ) − V0
=
=
= g.
V0
E(D1 )
E(D1 )
Hence, we have
E(D1 )
E(D1 ) E(V1 ) − V0
+
=
+ g.
V0
V0
V0
The expected rate of return on a stock is the sum of expected dividend
yield and expected capital gains, where the latter coincides with the
growth rate of dividends.
k=
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• A “growth” stock is one whose expected rate of return is mainly due to
the expected growth of cash flows.
• Conversely a “value” stock is one whose expected rate of return is mainly
due to the expected dividend yield.
• In the case of ConEdison, 1993 dividends amounted to $.50, or, disregarding compounding issues, an annual dividend of
D0 = 2.
Then, we can proxy the rate of growth of dividends g with the expected
growth rate of earnings for the next five years (from analysts forecasts):
g = 5%.
In summary,
V0 = 2 × 1.05/(.1310 − .05) = 25.92.
This theoretical valuation is remarkably close to the actual price of
$26.75.
• V0 < P0 is equivalent to the asset plotting “below” the SML.
• V0 > P0 is equivalent to the asset plotting “above” the SML.
• The intrinsic value of a stock is very sensitive to changes in k and g:
dV0 1
1
=−
dk V0
k−g
and
1+k 1
dV0 1
=
.
dg V0
1+gk−g
For ConEdison we have
dV0 1
= −12.34
dk V0
and
dV0 1
= 13.29.
dg V0
This means that if k increases by 1%, the price of ConEdison drops
by 12.34%. If g increases by 1%, the price of ConEdison increases by
13.29%.
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The Two-Stage Dividend-Growth Model
• The discussion above assumes that dividends grow at a constant rate
forever. In practice, firms tend to go through different phases.
• Let g1 denote the growth rate for the first n years.
Let g2 denote the growth rate for the remaining life of the firm.
We have
1 + g1
1 + g1
V0 = D0 ×
1−
k − g1
1+k
"
!n #
+ D0 × (1 + g1 )n ×
1 + g2
1
×
.
k − g2 (1 + k)n
The intrinsic value of the firm equals the present value of a growing
annuity plus the present value of a deferred growing perpetuity.
Example 1 Consider again ConEdison stock at the beginning of 1994.
Assume g1 = .05 for the first five years, and then g2 = .06 forever. The
intrinsic value is given by

V0
6
1.05 
1.05
= 2×
1−
.131 − .05
1.131
= 8.05 + 20.59 = 28.64.
!5 

1.05
+2×
1.131
!5
×
1.06
.131 − .06
Dividends and Earnings
• While the cash flows generated by stock ownership are the cash dividends
distributed by the firm, stock valuation is sometimes performed in terms
of earnings, rather than dividends. Hence, it is important to understand
the relationship between the two.
• Assume a constant rate of growth of dividends:
1+g =
Dt
.
Dt−1
• Dividends: constant fraction (1−b) of the earnings produced by the firm,
Dt = (1 − b) × Et .
b: retention or plowback ratio
1 − b: payout ratio
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• Earnings: proportional to the physical assets (book value) of the equity
shares, S,
Et = ROE × St−1 .
• Hence, we have
St = St−1 + bEt = St−1 + bROE × St−1 = St−1 (1 + bROE).
The capital stock, and hence earnings and dividends, grows at the rate
g = bROE.
The rate of growth of dividends increases with b, the retention ratio, and
ROE.
• In turn, the rate of return on equity depends on the rate of return on
assets, ROA, the book value of equity, S, the book value of debt, Debt,
the rate of interest paid on the debt, i, and the tax rate:
!
S + Debt
S + Debt
ROE =
× ROA + 1 −
× i × (1 − t)
S
S
Debt
= ROA +
× [ROA − i × (1 − t)],
S
where 1 − (S + Debt)/S = Debt/S. Equity is equivalent to a portfolio
long 1 + Debt/S in the assets of the firm and short Debt/S in corporate
bonds. Since interest payments are tax deductible, the rate of return on
the corporate bonds is reduced by a factor equal to the corporate tax
rate.
• The share price depends on the decision to reinvest in the firm:
V0 =
E(E1 )(1 − b)
E(D1 )
=
.
k−g
k − (b × ROE)
Similarly, we can write the price-earnings ratio as
(1 + g)(1 − b)
V0
=
.
E0
k − (b × ROE)
• If ROE = k, V0 = E(E1 )/k: the current intrinsic value does not depend
on b.
• Intuition:
k is the rate at which you are “borrowing;”
ROE is the rate at which you are investing.
If ROE = k, you are “breaking even.”
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• If ROE > k, you want to increase b: V0 → +∞ as b → k/ROE < 1.
If ROE < k, you want to decrease b: V0 → 0 as b → 1.
Also, note that ROE > k means that the assets of the firm plot above
the SML. Vice versa, ROE < k means that the assets of the firm plot
below the SML.
• E(E1 )/k: value of the firm when
i) ROE = k, and/or
ii) b = 0.
• Difference between P0 and E(E1 )/k: value of reinvesting future earnings
into the firm, the present value of growth opportunities (PVGO);
PVGO ≡ P0 −
E(E1 )
.
k
• High PVGO stock: “growth” stock.
Low PVGO stock: “value” stock.
Example 2 Consider Eli Lilly at the beginning of 1992. We have
E0 = 4.50,
and
D0 = 2.00.
Assume the growth rate of earnings
g1 = .1654
for the first five years and the growth rate
g2 = .06
afterwards. Also, assume the retention ratio for the first five years to be
equal to the current retention ratio
b1 = (4.5 − 2)/4.5 = .5556.
In the long-run, the retention ratio is derived from the expected growth
rate g2 , using ROA = .18, Debt/S = 23.2, and i × (1 − t) = .0475:
b2 × ROE = g2 .
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Hence
g2
.06
=
= .2847.
ROE .18 + .2320 × (.18 − .0475)
The cost of capital for Eli Lilly is estimated at
b2 =
k = .1378.
Based on this information, we can compute the present value of the cash
flows for the first five years:

1 + .1654 
1 + .1654
2×
1−
.1378 − .1654
1 + .1378
!5 

= 10.75.
The theoretical value at time 5 is computed according to the second part
of the formula for the two-stage dividend formula
V5 = (E0 × (1 + g1 )5 × (1 − b2 )) ×
V5 = 4.5 × 1.16545 × (1 − .2847) ×
1 + g2
1
×
.
k − g2 (1 + k)5
1 + .06
1
×
= 49.44
.1378 − .06 (1 + .1378)5
Note that since we are not the given dividend growth gdiv , we cannot use
the first term D0 × (1 + gdiv )5 = E(D5 ) of the 2-stage formula directly.
Instead, we can compute the expected date-5 dividend and the growth
after date 5 from the earnings growth: E(D5 ) = E0 × (1 + g1 )n × (1 − b2 ),
where we used the earnings growth g1 until year 5 to get the expected
earnings at date 5, and the estimated payout (1 − b2 ) after year 5 for the
expected dividends after year 5.
Hence, the theoretical value at time 0 is given by
V0 = 10.75 + V5 = 10.75 + 49.44 = 60.19
which should be compared to an actual price of $79 in February 1992.
We can now calculate the present value of growth opportunities as
PVGO = 79 − (4.5 × 1.1654)/.1378 = 40.94.
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The EVA Model
• In recent years an alternative formulation of the dividend growth model
has gained popularity: the Economic Value Added or EVA model.
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• The EVA model is based on the intuition that the current value of the
firm depends on the current market value of the assets in place, S0 , plus
the discounted value of the cash flows generated by the future assets in
place in excess of the cost of equity:
V0 = S0 +
∞
X
St−1
t=1
(ROE − k)
,
(1 + k)t
where
St+1 = St + bEt+1 = St (1 + bROE)
and hence
St = S0 (1 + bROE)t
• Since
∞
X
t=1
St−1
∞
X
(ROE − k)
(ROE − k)
t (ROE − k)
=
S
(1+b×ROE)
=
S
0
0
(1 + k)t
(1 + k)t
k − (b × ROE)
t=1
the EVA formula implies that
V0 = S0 + S0
ROE(1 − b)
(ROE − k)
= S0
.
k − (b × ROE)
k − (b × ROE)
Note that S0 ROE = E(E1 ), and hence
V0 =
E(E1 )(1 − b)
,
k − (b × ROE)
which is the dividend-discount model expressed in terms of earnings.
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Alternative Valuation Techniques
• In practice, the cash-flow analysis described before is complemented by
information based on industry ratios.
• Price-earnings ratio: given the average price-earnings ratio for the industry, we can calculate the intrinsic value of the stock. We have
P0
.
V0 = E0 ×
E0
!
• Similarly, we can use the average market-to-book ratio:
P0
.
V0 = S 0 ×
S0
!
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• Also, we can use the average price-to-sales ratio:
P0
V0 = S 0 ×
.
S0
!
• Note that we can also relate cash-flow analysis to these other valuation
techniques.
For the price-earnings ratio, we have
V0
D0 1 + g
1+g
=
= (1 − b)
,
E0
E0 k − g
k−g
where D0 /E0 = 1 − b: pay-out ratio.
For the market-to-book ratio, we have
V0
E0 D0 1 + g
V0
=
= ROE ×
,
S0
S0 E0 k − g
E0
where E0 /S0 = ROE: rate of return on equity.
For the price-to-sales ratio, we have
V0
E0 D0 1 + g
V0
=
=π×
,
S0
S0 E0 k − g
E0
where E0 /S0 = π: profit margin.
These theoretical ratios can be compared to the actual ones to have
a better understanding of the relationship between the actual and the
theoretical valuation.
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Applications: Boeing and Exxon
Example 3 At the beginning of 1995 Boeing had the following ratios:
P0
= 18.15
E0
!
P0 /E0 = 27.09
vs
P0
P0 /S0 = 2.39 vs
= 2.57
S0
!
P0
= .81
P0 /S0 = 1.11 vs
S0
Hence, based on these comparisons it is hard to tell whether Boeing is undervalued or overvalued relative to the rest of the industry.
!
Example 4 Consider Exxon at the beginning of 1995. We have
1 − b = .74,
expected growth rate g = .06. We also have
k = .1113
ROE = .15
and
π = .047.
We can calculate theoretical ratios. We have
1.06
V0 /E0 = .74 ×
= 15.29
.1113 − .06
V0 /S0 = .15 × 15.29 = 2.29
V0 /S0 = .047 × 15.29 = .719
These theoretical ratios can be compared to actual ratios of 17.02, 2.44,
and .78. Hence, we may conclude that Exxon is slightly overvalued.
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Practice Problems
1. The Two-Stage Dividend Growth Model
The Patagon Company currently does not pay any dividends, and is not
expected to pay any dividend for the next six years. Seven years from
now Patagon is expected to pay the first dividend of $0.5 per share. At
that point the rate of return on assets is expected to be ROA = 20%.
The debt-equity ratio is expected to equal D/E = .3, with an after-tax
cost of debt of 4.5%. The expected payout ratio is (1 − b) = 0.6.
Patagon’s beta is 1.2, the risk free rate is 5% and the premium on the
market is 6%.
Calculate the expected value of the stock seven years from now, E(V7 ).
Also, calculate the current value of Patagon, V0 . Assume Patagon stock
is fairly priced, calculate the component of the expected rate of return
between year zero and year one which is due to the dividend yield, and
that which is due to capital gains. Redo your calculations for the expected rate of return between year seven and year eight.
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