UGBA 103 (Parlour, Spring 2015), Section 8
Valuing stocks
Raymond C. W. Leung
University of California, Berkeley
Haas School of Business, Department of Finance
Email: [email protected]
Website: faculty.haas.berkeley.edu/r_leung/ugba103
April 3, 2015
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
1 / 38
Outline
1
Quiz #2 selected solutions review
2
Core Ideas of Valuation
3
Examples
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
2 / 38
Section 1
Quiz #2 selected solutions review
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
3 / 38
Quiz #2 selected solutions review
The solutions are online right now.
I’ll just highlight a few questions that deserve some extra mention and fill in some extra details.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Question 2
Question 2
Assume that both portfolios A and B are well diversified, and that expected returns are
E[RA ] = 12% and E[RB ] = 9%. If the CAPM is true and βA = 1.2, whereas βB = 0.8, what is the
risk-free rate?
Solution. Since the CAPM holds, we have that,
E[RA ] = rf + βA (E[RM ] − rf ),
E[RB ] = rf + βB (E[RM ] − rf ).
But then rearranging the above, it implies the market risk premium must also satisfy,
E[RM − rf ] =
E[RB ] − rf
E[RA ] − rf
=
.
βA
βB
Using the last equality, it further means, substituting in the numbers,
E[RA ] − rf
E[RB ] − rf
=
βA
βB
⇐=
0.12 − rf
0.09 − rf
=
,
1.2
0.8
from which we solve to get rf = 3%.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Question 5
Question 5
An equally weighted portfolio is comprised of three assets: A, B, C, with expected returns and
standard deviations of
Asset
A
B
C
Expected return
1.6%
2.4%
2.0%
Standard deviation
1.89%
2.60%
2.24%
Further, the correlation coefficients between asset A and B is ρA,B = 0.78. The correlation
coefficient between asset A and C is ρA,C = 0.76, and the correlation coefficient between asset B
and C is ρB,C = 0.82.
What is the expected return and volatility (standard deviation) of this portfolio?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Question 5
Solution. 1 The keyword is that the portfolio is equally weighted, so wA = wB = wC = 1/3. And
so the portfolio return is the random variable,
RP = wA RA + wB RB + wC RC =
1
(RA + RB + RC ).
3
The expected return is simple,
E[RP ] = (1/3)(E[RA ] + E[RB ] + E[RC ]) = (1/3)(0.016 + 0.024 + 0.020) = 2%.
The variance is the trickiest part. Suppose for whatever reason, you only remember the box
method for the 2 asset case. Then what do you do? In this case, let X = RB + RC be another
random variable. Then the portfolio returns become RP = (1/3)(RA + X ). Now, apply the box
method, and we get,
Var(RP ) = Var ((1/3)(RA + X ))
= (1/3)2 Var(RA + X )
= (1/9) (Var(RA ) + Var(X ) + 2Cov(RA , X ))
= (1/9)(σA2 + Var(X ) + 2Cov(RA , X )).
1
I just want to illustrate an alternative solution method that requires “less” memorizing.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
7 / 38
Question 5
Solution. It remains now to compute Var(X ) and Cov(RA , X ). For Var(X ), this is another variance
of two random variables and so we can use the box method again, to obtain,
2
Var(X ) = Var(RB ) + Var(RC ) + 2Cov(RB , RC ) = σB2 + σC
+ 2ρB,C σB σC .
And furthermore, from the properties of covariance,
Cov(RA , X ) = Cov(RA , RB + RC )
= Cov(RA , RB ) + Cov(RA , RC )
= ρA,B σA σB + ρA,C σA σC .
Put everything together, and we get,
(
)
2
Var(RP ) = (1/9) σA2 + σB2 + σC
+ 2ρB,C σB σC + 2(ρA,B σA σB + ρA,C σA σC ) .
Plug in the √
given numbers, and we get Var(RP ) = 4.33%2 and so,
Std(RP ) = Var(RP ) = 2.081%.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
8 / 38
Section 2
Core Ideas of Valuation
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Basic valuation formula
Like any investment, a stock is valued by discounting its future cash flows:
P0 =
∞
∑
t=1
Dt
,
(1 + r )t
where Dt are the (expected value of) dividends paid by the stock at time t. Note here we are
discounting by some discount rate r that is constant and non-random over time.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Special case: Dividends grow at rate g forever
Suppose that D2 = D1 (1 + g), . . . , Dt = Dt−1 (1 + g) for t ≥ 2. Then this is our familiar growing
perpetuity formula. Thus, the value of the stock today P0 is,
P0 =
D1
.
r −g
This is also often called as the Gordon growth model.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Dividend Yields, Capital Gains, and Total Returns
Now, let’s rewrite the formula a little bit more. We have that, if P1 denotes the value of the stock at
time t = 1,
D1
D1 (1 + g)
D1 (1 + g)2
+
+
+ ...
1
2
(1 + r )
(1 + r )
(1 + r )3
|
{z
}
P0 =
PV today = P1 /(1 + r )
D1
P1
=
+
1+r
1+r
D1 + P1
=
.
1+r
Rearranging the above,
r =
D1
P0
|{z}
Dividend yield
+
P1 − P0
P0
| {z }
.
Capital gains / Capital loss
Rate of return of a stock
That is, the required rate of return of the stock is equal to its dividend yield plus capital gains / loss.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Practice: A mixture of both
Forecasting dividends to T = ∞ is (obviously) difficult! But why not mix? Suppose T be the
maximum number of years over which accurate forecasting of financial statements is possible.
And suppose that at this terminal time T , the value of the firm will be TVT . Then the value of the
stock today must be,
P0 =
T
∑
t=1
Raymond C. W. Leung (Berkeley-Haas)
Dt
TVT
+
.
(1 + r )t
(1 + r )T
UGBA 103 Spring 2015, §8
April 3, 2015
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Estimating terminal value
Method 1: Perpetual growth method
The method is that we assume at time T + 1 (i.e. one period after the terminal time T ), dividends
will grow at constant rate g forever thereafter. Then this implies (i.e. by growing perpetuity formula),
TVT =
DT +1
.
r −g
And so,
P0 =
T
∑
t=1
Dt
1
DT +1
+
.
(1 + r )t
(1 + r )T r − g
Comments:
This method is highly sensitive (obviously) to both your choice of the growth rate g
assumption, discount rate r assumption and the dividend forecast DT +1 . In particular, it is
especially sensitive to the growth rate g assumption! So sensitive that a few percentage point
changes could result in very wide swings in valuation. . .
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Estimating terminal value
Method 2: Exit multiples method
This method is slightly “closer to the market”. Assume that you will sell the business in year T .
Use comparable companies analysis to estimate how much you will get form a sale — estimate
the firm’s earnings or book value in year T or year T + 1, and apply a multiple based on how peers
are currently trading. Some commonly used multiples:
Market / Book
Aggregate value / EBITDA
Price / Earnings
Interpretation — let’s focus on the case of P/E, but the other multiples have similar interpretations.
Also, this may sounds “trivial”, but it aids with intuition. If we have a P/E ratio of 12x, this
should be read as: “Investors are willing to pay $12 in share price for each $1 of current
earnings”. Thus, financial ratios like the P/E captures the “valuation” of a stock.
In particular, it means that if firm A has a share price of $100 and another firm A′ that is
similar to A has a share price of $10, you cannot claim that firm A is “more expensive” than
firm A′ ! You need to “normalize”! 2
2
Indeed, Warren Buffet’s Berkshire Hathaway BRK.A (A class share) has a per unit share price of $214,800 (as of Nov 6, 2014)!
Is that enough to say that the share is expensive or cheap? As a reference, it has P/E of 16.7x (as of Nov 6, 2014).
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Decomposing the stock price
Let’s define,
Dt : Dividends per share at date t;
Et : Earnings (cash flow from operations) per share at date t;
It : New investment per share at time t.
Since the earnings of the firm are either distributed to the shareholders as dividends or
reinvested into the firm, we have that,
Earnings = Dividends + Investments
Et = Dt + It ,
This means that all the earnings of the firm are either given out as dividends or investments.
Or alternatively, we can think of dividends as,
Dividends = Earnings − Investments
Dt = Et − It .
This implies that whatever earnings Et of the firm are, less investments It , will then be
distributed out as dividends.
But substituting this accounting identity into the valuation formula, we get that,
P0 =
∞
∑
t=1
Raymond C. W. Leung (Berkeley-Haas)
∞
∑ Et − It
Dt
=
.
(1 + r )t
(1 + r )t
t=1
UGBA 103 Spring 2015, §8
April 3, 2015
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Decomposing the stock price
PVGO
Assumption: Suppose a firm without any new investment opportunity in the future (i.e. It = 0
for all t = 1, 2, . . .) would generate earnings per share equal to E = E1 for all t = 1, 2, . . ..
But suppose that a firm that makes a new investments in the future, it’s time t earnings Et
must be,
Et =
E
+
∆t
.
|{z}
|{z}
|{z}
Flat old earnings
Earnings
Earnings from
new investments
Thus, going back to the valuation formula,
Earnings
Existing
investments
z}|{
z}|{
∞
∑
Et −
It
P0 =
t
(1
+
r
)
t=1
=
=
∞
∑
(E + ∆t ) − It
(1 + r )t
t=1
∞
∑
t=1
|
∞
∑
E
∆t − It
+
t
(1 + r )
(1
+ r )t
t=1
{z
} |
{z
}
Perpetuity
PVGO
E
=
+ PVGO,
r
where PVGO represents the present value of growth opportunities.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
17 / 38
Decomposing the stock price
PVGO
We can rearrange the previous term and express that (see lecture notes),
P0
1
=
E
r
(
1−
PVGO
P0
)−1
.
The key points here are that:
A high P/E ratio can be the result of low expected future returns (i.e. low systematic risk), or of
large potential growth (growth stocks).
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
18 / 38
(OPTIONAL) Philosophical statement about stock valuation
Again, just more personal opinions of this GSI
Stock valuation is notariously difficult! The problem is that the inputs are so ridiculously complicated! If we
think about it, each of these terms are very difficult to project and estimate:
▶ Discount rate
▶ Cash flows
▶ Uncertainty and randomness between both the discount rate and the cash flows
▶ The information that the investors see and process at any given point in time
If these formulas shown are correct, then what causes the share price to move in the first place? That is, if
we have, the share price at time t,
∞
∑
Ds
Pt =
,
(1 + r )s
s=t
but if all the terms D and r on the right hand side are non-random, then what is the variance (volatility) of
this stock price? It would simply be,
Var0 (Pt ) = 0.
That is, at time 0, we expect the time t stock price to have zero variation — this says that standing at today,
you would have zero uncertainty (i.e. 100% sure) what the stock price tomorrow, next month, 10 years from
now is! But if that’s true, what moves the prices today?
Indeed, the more “modern” view of the price Pt of an asset at time t is of the form,
[∫ ∞
]
Ms
Pt = E
Cs ds Ft ,
M
t
t
where M what is called the stochastic discount factor (SDF), C are the cash flows of the asset, and Ft is
the information available to the investor at time t.
▶ This more modern view is how financial engineers and finance academics view financial markets.
This is the study of asset pricing theory and empirical asset pricing.
Indeed, a well known professor once told me personally that, despite decades of finance research (both in
academia and in industry), while we have a pretty good (but still incomplete) understanding of asset returns
(i.e. price changes), we still do not have a good understanding of asset prices (i.e. price level)!
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
19 / 38
Section 3
Examples
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April 3, 2015
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Problem 9-1
Assume Evco, Inc., has a current price of $50 and will pay a $2 dividend in one year, and its equity
cost of capital is 15%. What price must you expect it to sell for right after paying the dividend in
one year in order to justify its current price?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
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Problem 9-1
Solution
Since the stock price today is it’s expected discounted cash flows (and note here we have no
uncertainty),
Dividends tomorow + Price tomorrow
1 + Equity cost of capital
D + P1
=⇒ P0 =
1 + rE
2 + P1
=⇒ 50 =
1 + 0.15
=⇒ P1 = $55.50.
Price today =
At a current price of $50, we can expect Evco stock to sell for $55.50 immediately after the firm
pays the dividend in one year.
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UGBA 103 Spring 2015, §8
April 3, 2015
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Problem 9-2
Anle Corporation has a current price of $20, is expected to pay a dividend of $1 in one year, and
its expected price right after paying that dividend is $22.
(a) What is Anle’s expected dividend yield?
(b) What is Anle’s expected capital gain rate?
(c) What is Anle’s equity cost of capital?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
23 / 38
Problem 9-2
Solution
(a) Expected dividend yield = Expected dividends/Current price = $1/$20 = 5%
(b) Expected capital gains rate = Capital gains/Initial purchase price = ($22 − $20)/$20 = 10%
(c) Equity cost of capital = Expected dividend yield + Expected capital gains rate = 5% + 10% = 15%.
Remark
In part (c), it is important to note that the equity cost of capital is equal to the expected dividend
yield plus the expected capital gains rate. The keyword expected is crucial here!
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
24 / 38
Problem 9-3
Suppose Acap Corporation will pay a dividend of $2.80 per share at the end of this year and $3
per share next year. You expect Acap’s stock price to be $52 in two years. If Acap’s equity cost of
capital is 10%:
(a) What price would you be willing to pay for a share of Acap stock today, if you planned to hold
the stock for two years?
(b) Suppose instead you plan to hold the stock for one year. What price would you expect to be
able to sell a share of Acap stock for in one year
(c) Given your answer in part (b), what price would you be willing to pay for a share of Acap stock
today, if you planned to hold the stock for one year? How does this compare to you answer in
part (a)?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
25 / 38
Problem 9-3
Solution
Just draw a time line here.
(a) P(0) = 2.80/1.10 + (3.00 + 52.00)/1.102 = $48.00
(b) P(1) = (3.00 + 52.00)/1.10 = $50.00
(c) P(0) = (2.80 + 50.00)/1.10 = $48.00
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UGBA 103 Spring 2015, §8
April 3, 2015
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Problem 9-8
Kenneth Cole Productions (KCP), suspended its dividend at the start of 2009 and as of the middle
of 2012, has not reinstated its dividend. Suppose you do not expect KCP to resume paying
dividends until July 2014. You expect KCP’s dividend in July 2014 to be $1.00 (paid annually), and
you expect it to grow by 5% per year thereafter. If KCP’s equity cost of capital is 11%, what is the
value of a share of KCP in July 2012?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
27 / 38
Problem 9-8
Solution
We have that,
Div2014
$1.00
=
= $16.67
r −g
.11 − .05
P2013
$16.67
=
=
= $15.02
1 + 0.11
1 + 0.11
P2013 =
P2012
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
28 / 38
Problem 9-9
In 2006 and 2007, Kenneth Cole Productions (KCP) paid annual dividends of $0.72. In 2008, KCP
paid an annual dividend of $0.36, and then paid no further dividends through 2012. Suppose KCP
was acquired at the end of 2012 for $15.25 per share.
(a) What would an investor with perfect foresight of the above been willing to pay for KCP at the
start of 2006? (Note: Because an investor with perfect foresight bears no risk, use a risk-free
equity cost of capital of 5%.)
(b) Does your answer to (a) imply that the market for KCP stock was inefficient in 2006?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
29 / 38
Problem 9-9
Solution
(a) Since the investor has perfect foresight, we we use the risk free rate rf = 5% to discount. We
compute the price,
Div2006
Div2008
P2012
Div2007
+
+
+
1 + rf
(1 + rf )2
(1 + rf )3
(1 + rf )7
$0.72
$0.36
$15.25
$0.72
=
+
+
+
(1 + 0.05)1
(1 + 0.05)2
(1 + 0.05)3
(1 + 0.05)7
P2006 =
= $12.49.
(b) No, in 2006 investor expectations were likely very different — KCP might have continued to
grow. Ex-post the stock is likely to do better or worse than investors expectations. The market
would be inefficient only if the stock was overpriced relative to what would have been
reasonable expectations in 2006.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
30 / 38
Problem 9-11
Cooperton Mining just announced it will cut its dividend from $4 to $2.50 per share and use the
extra funds to expand. Prior to the announcement, Cooperton’s dividends were expected to grow
at a 3% rate, and its share price was $50. With the new expansion, Cooperton’s dividends are
expected to grow at a 5% rate. What share price would you expect after the announcement?
(Assume Cooperton’s risk is unchanged by the new expansion.) Is the expansion a positive NPV
investment?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
31 / 38
Problem 9-11
Solution
The idea here is that we should first figure out what is the appropriate discount rate rE is
being used to price this firm. Here, we see before the announcement, the firm’s dividends
structure is essentially that of a growing perpetuity. Thus, we want to solve,
Pold =
Dold
rE − gold
=⇒
50 =
4
rE − 0.03
=⇒
rE = 11%.
Note here that nothing in the question suggests that changing its dividend policy affects the
riskiness of the firm. That implies, the same discount rate rE applies to the case before the
dividend change announcement and after. Upon the change, the dividend per share changes
to $2.50 and the growth rate changes to 5%. We still use the same perpetuity formula,
Pnew =
Dnew
rE − gnew
=⇒
Pnew =
2.50
= $41.67
0.11 − 0.05
Thus in this case, the NPV must be such that,
Pnew = Pold + NPV ,
which of course is,
NPV = Pnew − Pold
=⇒
NPV = 41.67 − 50 < 0.
Hence, cutting dividends to expand is not a positive NPV investment for the shareholders of
the firm.
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
32 / 38
Problem 9-15
Halliford Corporation expects to have earnings this coming year of $3 per share. Halliford plans to
retain all of its earnings for the next two years. For the subsequent two years, the firm will retain
50% of its earnings. It will then retain 20% of its earnings from that point onward. Each year,
retained earnings will be invested in new projects with an expected return of 25% per year. Any
earnings that are not retained will be paid out as dividends. Assume Halliford’s share count
remains constant and all earnings growth comes from the investment of retained earnings. If
Halliford’s equity cost of capital is 10%, what price would you estimate for Halliford stock?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
33 / 38
Problem 9-15
Solution
See the spreadsheet for the dividend forecast:
Year
Earnings
1 EPS Growth Rate (vs. prior yr)
2 EPS
Dividends
3 Retention Ratio
4 Dividend Payout Ratio
5 Div (2 × 4)
0
1
2
3
$3.00
25%
$3.75
25%
$4.69
100% 100% 50%
0%
0%
50%
—
— $2.34
4
5
6
12.5% 12.5% 5%
$5.27 $5.93 $6.23
50%
50%
$2.64
20%
80%
$4.75
20%
80%
$4.98
From year 5 onward, dividends grow at a constant rate of 5%. Therefore, the value of the stock at
t = 4 is,
4.75
P(4) =
= $95.
0.10 − 0.05
Thus, discounting all the cash flows back to t = 0, we have that the price of the stock today at
t = 0 is,
2.34
2.64 + 95
P(0) =
+
= $68.45.
(1 + 0.10)3
(1 + 0.10)4
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
34 / 38
Problem 9-14
What is the value of a firm with initial dividend Div1 , growing for n years (i.e., until year n + 1) at
rate g1 and after that at rate g2 forever, when the equity cost of capital is r ?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
35 / 38
Problem 9-14
Solution
We compute,
P0 =
Div1
r − g1
|
(
(
) )
1 + g1 n
1
Div1 (1 + g2 )n
1−
+
1+r
(1 + r )n
r − g2
{z
} |
{z
}
n-year, constant growth annuity
(
Div1
r − g1
| {z }
=
+
|
Constant growth perpetuity
Raymond C. W. Leung (Berkeley-Haas)
1 + g1
1+r
PV of terminal value
)n (
)
Div1
Div1
−
r − g2
r − g1
{z
}
PV of difference of perpetuities in year n
UGBA 103 Spring 2015, §8
April 3, 2015
36 / 38
Problem 9-17
Maynard Steel plans to pay a dividend of $3 this year. The company has an expected earnings
growth rate of 4% per year and an equity cost of capital of 10%.
(a) Assuming Maynard’s dividend payout rate and expected growth rate remains constant, and
Maynard does not issue or repurchase shares, estimate Maynard’s share price.
(b) Suppose Maynard decides to pay a dividend of $1 this year and use the remaining $2 per
share to repurchase shares. If Maynard’s total payout rate remains constant, estimate
Maynard’s share price.
(c) If Maynard maintains the dividend and total payout rate given in part (b), at what rate are
Maynard’s dividends and earnings per share expected to grow?
Raymond C. W. Leung (Berkeley-Haas)
UGBA 103 Spring 2015, §8
April 3, 2015
37 / 38
Problem 9-17
Solution
(a) Earnings growth = EPS growth = Dividend growth = 4%. Thus,
P0 =
$3
= $50.
0.10 − 0.04
P=
$1 + $2
= $50.
0.10 − 0.04
(b) Using the total payout model,
(c) From the model,
P0 =
Raymond C. W. Leung (Berkeley-Haas)
Div1
rE − g
Div1
P0
=⇒
rE − g =
=⇒
g = rE − DivYield
=⇒
g = 0.10 −
UGBA 103 Spring 2015, §8
$1
= 8%.
$50
April 3, 2015
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