1. Théorie Financière 4. Evaluation d’actions et d’entreprises
Professeur André Farber

2. Stock Valuation Objectives for this session :
Introduce the dividend discount model (DDM)
Understand the sources of dividend growth
Analyse growth opportunities
Examine why Price-Earnings ratios vary across firms
Introduce free cash flow model (FCFM)
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3. DDM: one-year holding period
Review: valuing a 1-year 4% coupon bond
Face value: € 50
Coupon: € 2
Interest rate 5%
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.
Bond price P0 = (50+2)/1.05 = 49.52
The starting point to value a stock is to use the same approach as when valuing a one-year risk-free bond.
But, for a stock, both the future dividend and the future stock price are not known. The current value is based on expectations. The expected stock price in our example might, for instance, be derived from the following data:
Stock price next year = 80 with probability ½ Stock price next year = 20 with probability ½
We will later on analyze how to obtains these expectations. So, let’s for a moment suppose that they are available.
The next question then is: what discount factor to use?
If we were to use the risk-free interest rate, the price of the stock in our example would be equal to the price of the one-year risk-free bond. But this would mean that both the bond and the stock have the same expected return.
Risk averse investor will require a higher expected return on the stock. The discount rate should
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4. Dividend Discount Model (DDM): 1-year horizon
1-year valuation formula
Back to example. Assume r = 10%
Expected price
r = expected return on shareholders'equity
= Risk-free interest rate + risk premium
Dividend yield = 2/47.27 = 4.23%
The 1-year valuation formula is closely related to the calculation of the rate of return that investor expect from this share over the next year.
Expected return = r =[div1 + (P1-P0)]/P0
This expected return is the sum of:
* the expected dividend yield div1/P0
* the expected capital gain (P1 - P0) / P0
Some historical perspective: US S&P index
Average real return 8.81% 8.30% 9.62%
Dividend yield 4.70% 5.34% 3.70%
Rate of capital gain 4.11% 2.96% 5.92%
Source: Fama, Eugene F., and Kenneth R. French, The Equity Premium, Journal of Finance, 57, 2 (April 2002) pp
Rate of capital gain = (50 – 47.27)/47.27 = 5.77%
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5. 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:
The current price can be decomposed into:
* the present value of dividends up to the investment horizon
* the present value of the terminal value at the investment horizon
As the horizon recedes, the present value of dividends increases and the present value of the terminal value decreases.
As an illustration, suppose the dividend per share is assume to remain constant at 10 per share. If the return required by investor is 10%, the current stock price (using the formula for a perpetuity) is:
P0 = 10 / 0.10 = 100
If the horizon is 10 years, the decomposition of the current stock price is:
* Present value of dividends = * Present value of terminal value =
The following table illustrates this decomposition for different horizon
Horizon PV(Dividends) PV(PT)
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6. DDM - general formula With infinite forecasting horizon:
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
Note: g < r
The Gordon Growth Model can also be expressed as:
r = div1/P0 + g
This formula is sometimes used to calculate the cost of equity capital (the rate of return required by shareholders) of regulated companies
This alternative formulation says the market capitalization rate equals the dividend yield plus the expected growth rate of dividend.
The growth rate of dividend is also the expected capital gain (as a fraction of current price).
To see this, note that:
P1 = div2/(r-g) = div1(1+g)/(r-g) = P0 (1+g)
Historical perspective on g (from Fama French 2002 Table 1)
Real growth rate 2.08% 2.74% 1.05% of dividends
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7. DDM with constant growth : example
Data
Next dividend: Div.growth rate: 4% Discount rate: %
Year
Dividend
DiscFac
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/( )
This example illustrates the Gordon growth model.
Note that Price P1 = 6.24/( ) = 100 (1.04) = 104
The market capitalization r rate is 10%. You can check that it is the sum of:
* the expect dividend yield 6.00/100 = 6%
* the expect capital gain ( )/100 = 4%
The current price can be decomposed into:
* present value of dividends 1 to 10 = 42.94
* present value of terminal value = 57.06
Note that dividends to be receive in more than 10 years account for more than 50% of the current price. Claiming that stock prices are unduly affected by short term forecasts of dividends seems exaggerated.
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8. Differential growth Suppose that r = 10% You have the following data:
Year 1 2 3
4 to ∞
Dividend
2.40
2.88
3.02
Growth rate
20% 5%
We first calculate the terminal value. We set the horizon at year 3 and we calculate the stock price P3.
Remember that the formula for the present value of a growing perpetuity gives the value one year before the first cash flow. So, the calculation of P3 is based on the dividend of year 4.
We then apply the general formula to obtain the current price.
Note that the same result would have been obtained by setting the horizon at the end of year 2. The dividend for year 2 is 2.88 and it starts growing at a rate of 5% per annum from year 3 on. So, the stock price at the end of year 2 is:
P2 = 2.88 / (0.10 – 0.05) = 57.60
You can check that this price is indeed the present value of the next dividend plus the next price:
P2 = ( ) / (1.10) = 57.60
The current price is:
P0 = 2/(1.10) /(1.10)² /(1.10)² = 51.40
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9. g = (Return on Retained Earnings) × (Retention Ratio)
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
For the calculation of g, I use the Return on Retained Earnings (RORE). This ratio is defined as:
RORE = Earnings/Net Investment
where Earnings is the permanent increase in earnings generated by the net investment.
Attentive readers of Brealey and Myers migth notice that these esteemed professors use ROE for calculating the dividend growth rate (see p.69). I tend to think that is is a source of confusion. ROE is defined as as the ratio between total earnings and book equity. This ratio is an average figure whereas for our purpose we need to consider an incremental (or marginal ratio).
This is illustrated in the following example for an all equity firm.
Year Net Investment Earnings RORE % % %
At the end of year 3:
Book equity = 300 Earnings = 60 ROE = 20% but the most recent investment yields only 10%!
g = (Return on Retained Earnings) × (Retention Ratio)
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10. 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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11. Review: dividend discount model (DDM)
g = (Return on Retained Earnings) (Retention Ratio)
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12. 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
Notations:
TA = total asset
SE = Stockholders’ equity
D = Financial debt
L = Leverage ratio L = D/ SE (a constant by assumption)
i = interest rate of financial debt
TC = corporate tax rate
RORE = return on retained earnings
RONI = return on net investment
TA = SE + D = SE(1+L)
SE = Earnings × Retention ratio
Earnings = (EBIT - i D)(1-TC)
RORE = Earnings/ SE
RONI = EBIT(1-TC)/ TA
Combining these expression leads to: RORE = RONI + [RONI-i(1-TC)]×L
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13. Growth model: example
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14. Valuing the company Assume discount rate r = 15%
Step 1: calculate terminal value
As Earnings = Dividend from year 4 on
V3 = /15% = 3,358
Step 2: discount expected dividends and terminal value
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15. Valuing Growth Opportunities
Consider the data:
Expected earnings per share next year EPS1 = €10
Required rate of return r = 10%
Why is A more valuable than B or C?
Why do B and C have same value in spite of different investment policies
Cy A
Cy B
Cy C
Payout ratio
60%
100%
Return on Retained Earnings
15%
10% -
Next year’s dividend €6
€10 g 6% 4% 0%
Price per share P0
€150
€100
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16. 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
Remember the Gordon Dividend Growth Model:
P0 = div1/(r-g) = (EPS1× Payout)/(r-g)
g = RORE (1 – Payout)
If a company does not have a competitive advantage, RORE = r
Remember, in a competitive equilibrium, profit is zero.
If RORE = r :
r-g = r × Payout
P0 = (EPS1 × Payout)/(r × Payout) = EPS1 / r
The market value of a company with a payout ratio less than one is equal to the market value of a company that would pay all its earnings as dividend.
This is why companies B and C in our example have the same price.
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17. Source of NPVG0 ?
Additional value if the firm retains earnings in order to fund new projects
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 = /0.10 = +2 (in year 1)
PV(NPV1) = 2/1.10 = 1.82
The net present value of each investment is:
NPVt = - NetInvestmentt + (RORE ×NetInvestmentt) / r
= EPSt ×Retention ratio × (-1 + RORE/r)
= EPSt ×Retention ratio × (RORE – r) /r
Note that NPVt = 0 if RORE = r
The growth rate of NPVt is g (net investment is proportional to earnings)
Using the constant growth formula
NPVGO = NPV1/(r – g)
= [EPS1×Retention ratio × (RORE – r) ] / [r ×(r – g)]
In our example, for company A:
NPVGO = [10 × 0.40 × (0.15 – 0.10)] / [ 0.10 × (0.10 – 0.06)] = 50
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18. NPVGO: details
In the Gordon growth model, the growth rate of dividend is a constant. This is a very strong assumption. If the payout ratio is stable, assuming a constant growth rate is the same as assuming that the return on reinvested capital is a constant. In other word, the Gordon growth model is based on the assumption that the company has some sort of competitive advantage forever. The net present value of net investment is positive year after year up to infinity!
Economics teaches us that competitive advantages are hard to maintain. Because of competition, returns on new investments tend to revert to the cost of capital.
The impact on NPVGO of limiting the duration of the competitive advantage (defined as the number of year with RORE > r) is dramatic. The calculations in the slide shows that NPVGO would drop to 30 if the period of competitive advantage is limited to 25 years.
Exhibit: NPVGO as a function of number of years of competitive advantage
# years NPVGO ∞ 50.00
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19. What Do Price-Earnings Ratios mean?
Definition: P/E = Stock price / Earnings per share
Why do P/E vary across firms?
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
The intuitive interpretation of the P/E is that provides the price to pay per unit of earnings. As a consequence, shares of companies with high P/E are sometimes considered as “expensive”.
This is not necessarily correct.To see this, consider the following example.
Companies A and B have the same risk. Investors require the same expected return r = 10% They have following characteristics:
EPS Payout g Price
A % 0% 100
B 10 60% 6% 150
Their respective P/E are:
A 100/10 = 10
B 150/10 = 15
The P/E is higher for B than for A because, when buying B, not only do you acquire current earnings but also growth opportunities. You get more for your bucks. So, you have to pay more.
But the expected returns for both shares are identical!
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20. 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
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21. FCFM: example Current situation # shares: 100m Project
Euro m
Current situation
# shares: 100m
Market value of company (r = 10%) V0 = 100/0.10 = €1,000m Price per share P0 = €1,000m / 100m = €10
Project
The purpose of this example is to illustrate the use of FCF to value a company.
The initial value of the company is calculated without taking into account the project.
Note are all earnings are paid out as a dividend. Investment each year is equal to depreciation.
The company has identified a project requiring a large investment spread over the next 2 years. The amounts to invest could be funded with retained earnings. However, this would mean cutting the dividend.
On the other hand, the company might issue new shares to finance the project. Current shareholders would not have to reduce the dividend but part of the increase of futures earnings would have to be paid to new shareholders. Current shareholders would have to share the cake with newshareholders.
What should they do?
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22. Free Cash Flow Calculation
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23. Self financing – DIV = FCF, no stock issue
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)
NPV = increase in the value of equity due to project
NPV = 1,694 – 1,000 = 694
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24. Outside financing : Dividend = Net Income, SI = Div. – FCF
Market value of equity with project:
(Discount free cash flow, not total dividends)
Same value as before!
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25. 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)
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26. Number of new shares issued = 100 / 17.74 = 5.67
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:
Number of new shares issued = 100 / = 5.67
Similar calculations for year 2 lead to:
Number of old shares =
Price per share P2 = 1,900 / = 17.98
Number of new share issued = 100 / = 5.56
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27. 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 = n0P (n1-n0)P1
Statement of cash flow (no debt, cash constant):
FCF1 = DIV1 – (n1-n0)P → DIV1 + n0P1 = FCF1 + V1
Conclusion:
V0 = (FCF1 + V1) /(1+r)
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