Corporate Finance
Stock Valuation
Prof. André Farber
SOLVAY BUSINESS SCHOOL
UNIVERSITÉ LIBRE DE BRUXELLES
Stock Valuation
•
Objectives for this session :
1. Introduce the dividend discount model (DDM)
2. Understand the sources of dividend growth
3. Analyse growth opportunities
4. Examine why Price-Earnings ratios vary across firms
5. Introduce free cash flow model (FCFM)
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DDM: one-year holding period
•
•
•
Review: valuing a 1-year 4% coupon bond

Face value:
€ 50

Coupon:
€2

Interest rate
5%
Bond price P0 = (50+2)/1.05 = 49.52
How much would you be ready to pay for a stock with the following
characteristics:

Expected dividend next year: € 2

Expected price next year: €50
Looks like the previous problem. But one crucial difference:
–
Next year dividend and next year price are expectations, the realized
price might be very different. Buying the stock involves some risk. The
discount rate should be higher.
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Dividend Discount Model (DDM): 1-year
horizon
• 1-year valuation formula
div1  P1
P0 
1 r
• Back to example. Assume r = 10%
2  50
P0 
 47.27
1  0.10
Expected price
r = expected return on shareholders'equity
= Risk-free interest rate + risk premium
Dividend yield = 2/47.27 = 4.23%
Rate of capital gain = (50 – 47.27)/47.27 = 5.77%
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DDM: where does the expected stock price
come from?
• Expected price at forecasting horizon depends on expected dividends
and expected prices beyond forecasting horizon
• To find P2, use 1-year valuation formula again:
• Current price can be expressed as:
• General formula:
P0 
P0 
P1 
div 2  P2
1 r
div1
div 2
P2


1  r (1  r ) 2 (1  r ) 2
div1
div 2
divT
PT


...


1  r (1  r ) 2
(1  r ) T (1  r ) T
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DDM - general formula
• With infinite forecasting horizon:
P0 
div3
divt
div1
div2



...

 ...
t
(1  r ) (1  r ) 2 (1  r ) 3
(1  r )
• Forecasting dividends up to infinity is not an easy task. So, in practice,
simplified versions of this general formula are used. One widely used
formula is the Gordon Growth Model base on the assumption that
dividends grow at a constant rate.
 DDM with constant growth g
div1
P0 
rg
 Note: g < r
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DDM with constant growth : example
Data
Next dividend: 6.00
Div.growth rate: 4%
Discount rate:
10%
Year
Dividend
DiscFac
0
Price
100.00
1
6.00
0.9091
104.00
2
6.24
0.8264
108.16
3
6.49
0.7513
112.49
4
6.75
0.6830
116.99
5
7.02
0.6209
121.67
6
7.30
0.5645
126.53
7
7.59
0.5132
131.59
8
7.90
0.4665
136.86
9
8.21
0.4241
142.33
10
8.54
0.3855
148.02
P0= 6/(.10-.04)
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Differential growth
• Suppose that r = 10%
• You have the following data:
Year
1
2
3
4 to ∞
Dividend
2
2.40
2.88
3.02
20%
20%
5%
Growth rate
• P3 = 3.02 / (0.10 – 0.05) = 60.48
P0 
2
2.40
2.88
60.48



 51.40
2
3
3
1.10 (1.10)
(1.10)
(1.10)
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A formula for g
• Dividend are paid out of earnings:
 Dividend = Earnings × Payout ratio
• Payout ratios of dividend paying companies tend to be stable.
 Growth rate of dividend g = Growth rate of earnings
• Earnings increase because companies invest.
 Net investment = Retained earnings
• Growth rate of earnings is a function of:
 Retention ratio = 1 – Payout ratio
 Return on Retained Earnings
g = (Return on Retained Earnings) × (Retention Ratio)
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Example
• Data:
 Expected earnings per share year 1: EPS1 = €10
 Payout ratio : 60%
 Required rate of return r : 10%
 Return on Retained Earnings RORE: 15%
• Valuation:
 Expected dividend per share next year: div1 = 10 × 60% = €6
 Retention Ratio = 1 – 60% = 40%
 Growth rate of dividend g = (40%) × (15%) = 6%
• Current stock price:
 P0 = €6 / (0.10 – 0.06) = €150
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Return on Retained Earnings and Debt
• Net investment = Total Asset
• For a levered firm:
 Total Asset = Stockholders’ equity + Debt
• RORE is a function of:
 Return on net investment (RONI)
 Leverage (L = D/ SE)
RORE = RONI + [RONI – i (1-TC)]×L
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Growth model: example
Dep/TotAsset
TaxRate
Year
Payout
RORE
10%
40%
0
Depreciation
Net Income
Dividend
Cfop
Cfinv
Cffin
Change in cash
Total Assets
Book Equity
1,000.00
1,000.00
1
60%
25%
2
60%
20%
3 4 to infinity
60%
100%
15%
15%
100.00
400.00
240.00
116.00
440.00
264.00
133.60
475.20
285.12
152.61
503.71
503.71
500.00
-260.00
-240.00
0.00
556.00
-292.00
-264.00
0.00
608.80
-323.68
-285.12
0.00
656.32
-152.61
-503.71
0.00
1,160.00
1,160.00
1,336.00
1,336.00
1,526.08
1,526.08
1,526.08
1,526.08
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Valuing the company
• Assume discount rate r = 15%
• Step 1: calculate terminal value
 As Earnings = Dividend from year 4 on
 V3 = 503.71/15% = 3,358
• Step 2: discount expected dividends and terminal value
V0 
240
264
285.12 3,358.08



 2,803.78
1.15 (1.15) 2 (1.15) 3
(1.15) 3
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Valuing Growth Opportunities
• Consider the data:
 Expected earnings per share next year EPS1 = €10
 Required rate of return r = 10%
Cy A
Cy B
Cy C
Payout ratio
60%
60%
100%
Return on Retained Earnings
15%
10%
-
Next year’s dividend
€6
€6
€10
g
6%
4%
0%
€150
€100
€100
Price per share P0
• Why is A more valuable than B or C?
• Why do B and C have same value in spite of different investment
policies
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NPVGO
• Cy C is a “cash cow” company
 Earnings = Dividend (Payout = 1)
 No net investment
• Cy B does not create value
 Dividend < Earnings, Payout <1, Net investment >0
 But: Return on Retained Earnings = Cost of capital
 NPV of net investment = 0
• Cy A is a growth stock
 Return on Retained Earnings > Cost of capital
 Net investment creates value (NPV>0)
 Net Present Value of Growth Opportunities (NPVGO)
 NPVGO = P0 – EPS1/r = 150 – 100 = 50
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Source of NPVG0 ?
• Additional value if the firm retains earnings in order to fund new
projects
P0 
EPS
 PV ( NPV1 )  PV ( NPV2 )  PV ( NPV3 )  ...
r
• where PV(NPVt) represent the present value at time 0 of the net
present value (calculated at time t) of a future investment at time t
• In previous example:
Year 1: EPS1 = 10 div1 = 6  Net investment = 4
EPS = 4 * 15% = 0.60 (a permanent increase)
NPV1 = -4 + 0.60/0.10 = +2 (in year 1)
PV(NPV1) = 2/1.10 = 1.82
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NPVGO: details
P0
PV g = 0
NPVGO
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
150.00
100.00
50.00
EPS1
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
Y1 to Y25
Y26 to 50
Y51 to 75
Y76 to 100
EPSt
10.00
10.60
11.24
11.91
12.62
13.38
14.19
15.04
15.94
16.89
17.91
18.98
20.12
21.33
22.61
23.97
25.40
26.93
28.54
30.26
Net Inv.
4.00
4.24
4.49
4.76
5.05
5.35
5.67
6.01
6.38
6.76
7.16
7.59
8.05
8.53
9.04
9.59
10.16
10.77
11.42
12.10
30.19
11.96
4.74
1.88
 EPS
0.60
0.64
0.67
0.71
0.76
0.80
0.85
0.90
0.96
1.01
1.07
1.14
1.21
1.28
1.36
1.44
1.52
1.62
1.71
1.82
NPV
2.00
2.12
2.25
2.38
2.52
2.68
2.84
3.01
3.19
3.38
3.58
3.80
4.02
4.27
4.52
4.79
5.08
5.39
5.71
6.05
PV(NPV)
1.82
1.75
1.69
1.63
1.57
1.51
1.46
1.40
1.35
1.30
1.26
1.21
1.17
1.12
1.08
1.04
1.01
0.97
0.93
0.90
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What Do Price-Earnings Ratios mean?
• Definition: P/E = Stock price / Earnings per share
1 NPVGO
P/E  
• Why do P/E vary across firms?
r
EPS
• As: P0 = EPS/r + NPVGO

• Three factors explain P/E ratios:
 Accounting methods:
– Accounting conventions vary across countries
 The expected return on shareholders’equity
– Risky companies should have low P/E
 Growth opportunities
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Beyond DDM: The Free Cash Flow Model
• Consider an all equity firm.
• If the company:
– Does not use external financing (not stock issue, # shares constant)
– Does not accumulate cash (no change in cash)
 Then, from the cash flow statement:
» Free cash flow = Dividend
» CF from operation – Investment = Dividend
– Company financially constrained by CF from operation
• If external financing is a possibility:
» Free cash flow = Dividend – Stock Issue
 Market value of company = PV(Free Cash Flows)
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FCFM: example
Current
situation
# shares: 100m
Euro m
Year
Net Income
Depreciation
Investment
Dividends
1
100
50
50
100
3-¥
100
50
50
100
2
100
50
50
100
Market value of company (r = 10%) V0 = 100/0.10 = €1,000m
Price per share P0 = €1,000m / 100m = €10
Project
Year
Investment
Net Income
Depreciation
1
100
2
110
50
10
3-¥
20
100
20
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Free Cash Flow Calculation
Year
Net income
Depreciation
CF from op
Replacement
Expansion
CF investment
Free cash flow
1
100
50
150
50
100
-150
0
2
150
60
210
60
100
-160
50
3-¥
200
70
270
70
0
-70
200
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Self financing – DIV = FCF, no stock issue
Free cash flow
Dividends
Stock issue
0
0
0
50
50
0
200
200
0
Market value of equity with project:
(As the number of shares is constant, discounting free cash
flows or total dividends leads to the same result)
V
0
50
1
200


 1,694
1.10 (1.10) 2 (1.10) 2 0.10
NPV = increase in the value of equity due to project
NPV = 1,694 – 1,000 = 694
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Outside financing : Dividend = Net
Income, SI = Div. – FCF
Free cash flow
Dividends
Stock issue
0
100
100
50
150
100
200
200
0
Market value of equity with project:
(Discount free cash flow, not total dividends)
V
0
50
1
200


 1,694
1.10 (1.10) 2 (1.10) 2 0.10
Same value as before!
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Why not discount total dividends?
Because part of future total dividends will be paid to new
shareholders. They should not be taken into account to value
the shares of current shareholders.
To see this, let us decompose each year the value of all shares
between old shares (those outstanding one year before) and
new shares (those just issued)
Year
Vt
Old shares
New shares
0
1,694
1,694
1
1,864
1,764
100
2
2,000
1,900
100
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Year
Total div.
Nb shares
Div./share
Price per
share
0
16.94
1
100
100
1
17.64
2
150
105.67
1.42
17.98
3
200
111.23
1.7981
The price per share is obtained by dividing the market value of old share by the number of
old shares:
Year 1:
Number of old shares = 100
P1 = 1,764 / 100 = 17.64
The number of shares to issue is obtained by dividing the total stock issue by the number of
shares:
Year 1:
Number of new shares issued = 100 / 17.74 = 5.67
Similar calculations for year 2 lead to:
Number of old shares = 105.67
Price per share P2 = 1,900 / 105.67 = 17.98
Number of new share issued = 100 / 17.98 = 5.56
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From DDM to FCFM: formulas
• Consider an all equity firm
• Value of one share: P0 = (div1 + P1)/(1+r)
• Market value of company = value of all shares
 V0 = n0P0 = (n0div1 + n0P1)/(1+r)
• n0 div1 = total dividend DIV1 paid by the company in year 1
• n0 P1 = Value of “old shares”
 New shares might be issued (or bought back) in year 1
 V1 = n1P1 = n0P1
+
(n1-n0)P1
• Statement of cash flow (no debt, cash constant):
 FCF1 = DIV1 – (n1-n0)P1
→ DIV1 + n0P1 = FCF1 + V1
• Conclusion:
 V0 = (FCF1 + V1) /(1+r)
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