FIN 614
Common Stock Valuation
Professor Robert B.H. Hauswald
Kogod School of Business, AU
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
1
Stocks vs. Bonds
• Stock valuation should be easy
– just discount cash flows and sum them
– find appropriate discount rate
• Where is the rub?
– how do stocks and bonds differ?
• Common stock valuation models
– DCF based models: dividend discount model
– constant and non-constant growth assumptions
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Stock Valuation
© Robert B.H. Hauswald
2
Common Stock Valuation
• Valuation of stock is more difficult than of bonds:
– the cash flows are not explicit,
– the life is forever, and
– the market discount rate is not easily observed
• Types of stock: control vs. income rights
– common: receive residual profits, last in priority order
but: have a vote in corporate affairs
– preferred: fixed dividend, no vote
– which one resemble more debt?
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Stock Valuation
© Robert B.H. Hauswald
3
Stock Market Reporting
52 WEEKS
YLD
VOL
NET
HI
LO STOCK SYM DIV % PE 100s HI LO CLOSE CHG
52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75
Gap has
been as
high as
$52.75 in
the last
year.
Gap pays a
dividend of 9
cents/share
Given the
current price,
the dividend
yield is ½ %
Gap has
been as low
as $19.06 in
1/18/2011
the last year.
Given the
current price, the
PE ratio is 15
Stock Valuation
© Robert B.H. Hauswald
times
earnings
Gap ended trading
at $19.25, down
$1.75 from
yesterday’s close
6,517,200 shares
traded hands in the
last day’s trading
4
Interpretation
52 WEEKS
YLD
VOL
NET
HI
LO STOCK SYM DIV % PE 100s HI LO CLOSE CHG
52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75
• Gap Incorporated is having a tough year, trading near their 52week low. Imagine how you would feel if within the past year
you had paid $52.75 for a share of Gap and now had a share
worth $19.25! That 9-cent dividend wouldn’t go very far in
making amends.
• Yesterday, Gap had another rough day in a rough year. Gap
“opened the day down” beginning trading at $20.50, which was
down from the previous close of $21.00 = $19.25 + $1.75
• Looks like cargo pants aren’t the only things on sale at Gap.
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
5
Common Stock Cash Flows
–
–
The cash flow to holders of common stock consists of
dividends plus a future sale price
Valuation: using recursive substitution ,
•
the current price of a stock can be written as
P0 =
•
•
–
D1
D2
D3
D4
P4
+
+
+
+
2
3
4
(1 + r ) (1 + r ) (1 + r ) (1 + r ) (1 + r )4
assuming the stock is sold right after the 4th dividend is
received
for ALL stock problems, assume annual compounding
Generalization: dividend stream, discount rates
PV0 = D1/(1 + r)1 + D2/(1 + r)2 + D3/(1 + r)3 + . . . forever. .
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Stock Valuation
© Robert B.H. Hauswald
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DDM: Dividend Discount Model
• Dividend discount model: value of stock
determined by future cash flows =
dividends
– problem?
• Three special cases: dividend growth
– 0 growth model
– constant growth model
– differential growth model
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Stock Valuation
© Robert B.H. Hauswald
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Case 1: Zero Growth
• Assume that dividends will remain at the same
level forever
Div1 = Div 2 = Div 3 = L
• Since future cash flows are constant, the value of a zero
growth stock is the present value of a perpetuity
• note the timing in the general case
Div 3
Div1
Div 2
+
+
+L
(1 + r )1 (1 + r ) 2 (1 + r ) 3
Div t +1
Div
P0 =
or, more generally, Pt =
r
r
P0 =
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Stock Valuation
© Robert B.H. Hauswald
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Case 2: Constant Growth
• Assume that dividends will grow at a constant rate, g,
forever. i.e.
Div1 = Div 0 (1 + g )
Div 2 = Div1 (1 + g ) = Div 0 (1 + g ) 2
Div 3 = Div 2 (1 + g ) = Div 0 (1 + g ) 3
...
• Since future cash flows grow at a constant rate
forever, the value of a constant growth stock is the
present value of a growing perpetuity:
P0 =
1/18/2011
Div t +1
Div1
or, more generally, Pt =
r−g
r−g
Stock Valuation
© Robert B.H. Hauswald
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Growing Perpetuity
– An amount that grows at a constant rate forever is
called a growing perpetuity. In this case the expression
for the value of a stock now becomes:
P0 =
∞
∑
t =1
(1 + g ) t
D0
(1 + r ) t
– As long as g < r, the present value at the rate r of
dividends growing at the rate g is:
P0 =
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Stock Valuation
D1
r−g
© Robert B.H. Hauswald
10
Required Returns on Equity: ROE
• Decomposition of the required returns
– dividend yield
– capital gains yield
• Components of the required return:
– Rearrange P0 = D1/(r-g) to give:
r = D1/P0 + g
dividend yield = D1/Po
capital gains yield = g (price appreciation)
• ROE: rt = dividend yield + capital gains yield
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Stock Valuation
© Robert B.H. Hauswald
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Case 3: Differential Growth
• Assume that dividends will grow at different rates in
the foreseeable future and then will grow at a
constant rate thereafter
– a mix of “supernormal” growth early on and then a
constant, “normal” growth rate later.
• To value a Differential Growth Stock, we need to:
– estimate future dividends in the foreseeable future
– estimate the future stock price when the stock becomes a
Constant Growth Stock (case 2)
– compute the total present value of the estimated future
dividends and stock price at the appropriate discount rate
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Stock Valuation
© Robert B.H. Hauswald
12
Differential Growth: Two
Approaches
• A common stock just paid a dividend of $2.
– The dividend is expected to grow at 8% for 3 years,
– then it will grow at 4% in perpetuity.
• What is the stock worth?
– ROE (required return on equity): r = 12%
• To find the answer
– draw time line of growth rates and calculate dividends
– or: use both the annuity and perpetuity formulae
• Two approaches: formula vs. cash flows
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
13
A Differential Growth Example
•
•
•
•
•
•
r = 12% (required return)
g1 = g2 = g3 = 8%
D0 = $2
D1 = $2 x 1.08 = $2.16, D2 = $2.33, D3 = $2.52
g4 = gn = 4%
Constant growth rate applies to D4
– use Case 2 (constant growth) to compute P3
• D4 = $2.52 x 1.04 = $2.62
• P3 = $2.62 / (.12 - .04) = $32.75
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Stock Valuation
© Robert B.H. Hauswald
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Timeline and Cash Flows
$2(1.08)
$2(1.08) 2
$2(1.08) 3 $2(1.08)3 (1.04)
…
0
1
2
$2.16
0
1
3
$2.52 +
$2.33
2
4
$2.62
.08
3
P3 =
P0 =
1/18/2011
The constant
growth phase
beginning in year 4
can be valued as a
growing perpetuity
at time 3.
$2.62
= $32.75
.08
$2.16 $2.33 $2.52 + $32.75
+
+
= $28.89
1.12 (1.12) 2
(1.12) 3
Stock Valuation
© Robert B.H. Hauswald
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Formula
 Div N +1 


C  (1 + g1 ) T   r − g 2 
P=
1 −
+
r − g1 
(1 + r ) T  (1 + r ) N
 $2(1.08) 3 (1.04) 


3
.12 − .04
$2 × (1.08)  (1.08)  

P=
+
1 −

3
3
.12 − .08  (1.12) 
(1.12)
P = $54 × [1 − .8966] +
P = $5.58 + $23.31
1/18/2011
Stock Valuation
($32.75)
(1.12) 3
P = $28.89
© Robert B.H. Hauswald
16
Price Earnings Ratio
• Analysts frequently relate earnings per share to price.
• The price earnings ratio is a.k.a the multiple
– Calculated as current stock price divided by annual EPS
– The Wall Street Journal uses last 4 quarter’s earnings
P/E ratio =
Price per share
EPS
• Firms whose shares are “in fashion” sell at high
multiples. Growth stocks for example.
• Firms whose shares are out of favor sell at low
multiples. Value stocks for example.
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Stock Valuation
© Robert B.H. Hauswald
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Parameters Estimates
• The value of a firm depends upon its growth rate,
g, and its discount rate, r.
– Where does g come from?
– Where does r come from?
• Formula for Firm’s Growth Rate
g = Retention ratio × Return on retained earnings
• Where does r come from?
– The discount rate can be broken into two parts.
• The dividend yield
• The growth rate (in dividends)
– In practice, there is a great deal of estimation error
involved in estimating r.
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
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Growth Opportunities
• Growth opportunities are opportunities to
invest in positive NPV projects.
• The value of a firm can be conceptualized
as the sum of the value of a firm that pays
out 100-percent of its earnings as dividends
and the net present value of the growth
EPS
opportunities.
P=
+ NPVGO
r
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Stock Valuation
© Robert B.H. Hauswald
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Dividend Discount Models and
Returns on Equity (ROE)
• Important: Don’t Trust the GDGM for ROEs
– Dividends are very unstable.
• In fact, there is a fairly strong irrelevance
proposition here.
– Given its underlying projects, it should not matter whether
the firm pays out $1 or $10 in dividends.
– What it does not pay out in dividends today will make
more hey next year.
– Thus, expected rates of returns obtained from the Gordon
model are highly suspect.
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
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Summary
• Basic stock valuation models are variants of the
dividend discount models
• Two inputs for DDM
– Cash Flows = Dividend + Capital Gains
– Discount Rate
– assumes a lot of information!
• Simple growth models when world is simple
– provides benchmark
– often used as plausibility check
• NB: no risk adjustment at this point
– risk and return: next set of lectures
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Stock Valuation
© Robert B.H. Hauswald
21
Appendix: More on Stock Valuation
•
•
•
•
•
Further Example: Fudgit
Differential Growth: the formulae
Parameter estimates: r and g
Growth Opportunities
P/E Ratio and relation to DDM
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
22
Non-Constant Growth Example
• Discount individual "high" growth dividends and discount
the dividend growth model stock value at the future,
constant growth date.
• The next three dividends for Fudgit Co. are expected to be
$0.50, $1.00, and $1.50. Then the dividends are expected
to grow at a constant 5% forever. If the required return on
Fudgit is 10%, what is P0?
Total Present Value = PV(Dividends) + PV(P3)
where P3 = [D3 x (1+g)]/(r - g)
= $1.5(1.05)/(.10 - .05) = $1.575/.05
= $31.50
Total Present Value = PV(Dividends 1 to 3) + PV(P3)
= $0.454 + $0.826 + $1.127 + $23.67 = $26.07
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
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Differential Growth
• Dividends will grow at rate g1 for N years and
grow at rate g2 thereafter
Div 0 (1 + g1 ) Div 0 (1 + g1 ) 2
…
0
1
2
Div 0 (1 + g1 )
N
Div N (1 + g 2 )
= Div 0 (1 + g1 ) N (1 + g 2 )
…
…
N
1/18/2011
Stock Valuation
N+1
© Robert B.H. Hauswald
24
Divided and Conquer
We can value this as the sum of:
an N-year annuity growing at rate g1
C  (1 + g1 )T 
PA =
1 −

r − g1  (1 + r )T 
plus the discounted value of a perpetuity growing at rate
g2 that starts in year N+1
 Div N +1 


r
−
g
2 
PB = 
(1 + r ) N
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
25
Formulae vs. Cash Flows
To value a Differential Growth Stock, we can use
 Div T +1 


C  (1 + g1 )T   r − g 2 
P=
1 −
+
r − g1 
(1 + r )T  (1 + r )T
• Or we can tough it out with full cash flow
1/18/2011
Stock Valuation
© Robert B.H. Hauswald
26
Calculator: Differential Growth
• A common stock just paid a dividend of $2. The dividend is
expected to grow at 8% for 3 years, then it will grow at 4%
in perpetuity. What is the stock worth?
$28.89 = 5.58 + 23.31
• First find the PV of the supernormal dividend stream then
find the PV of the steady-state dividend stream.
N
N
3
1.12
I/Y
3.70 =
PV
– 5.58
PMT
FV
1/18/2011
$2 =
0
1.08
–1 ×100
2×1.08
1.08
Stock Valuation
3
I/Y
12
PV
– 23.31
PMT
0
2×(1.08)3 ×(1.04)
FV 32.75 =
© Robert B.H. Hauswald
.08 27