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

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) Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

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 1872-2000 1872-2000 1872-1950 1951-2000 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. 637-659 Rate of capital gain = (50 – 47.27)/47.27 = 5.77% Tfin 04 Stock valuation

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 = 61.45 * Present value of terminal value = 38. 55 The following table illustrates this decomposition for different horizon Horizon PV(Dividends) PV(PT) 5 37.91 62.09 10 61.45 38.55 20 85.13 14.87 50 99.15 0.85 Tfin 04 Stock valuation

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) 1872-2000 1872-1950 1951-2000 Real growth rate 2.08% 2.74% 1.05% of dividends Tfin 04 Stock valuation

DDM with constant growth : example Data Next dividend: 6.00 Div.growth rate: 4% Discount rate: 10% 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/(.10-.04) This example illustrates the Gordon growth model. Note that Price P1 = 6.24/(0.10-.06) = 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 (104 - 100)/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. Tfin 04 Stock valuation

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 = (3.02 + 60.48) / (1.10) = 57.60 The current price is: P0 = 2/(1.10) + 2.40/(1.10)² + 57.60/(1.10)² = 51.40 Tfin 04 Stock valuation

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 1 100 30 30% 2 100 20 20% 3 100 10 10% 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) Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

Review: dividend discount model (DDM) g = (Return on Retained Earnings) (Retention Ratio) Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

Growth model: example Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

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. Tfin 04 Stock valuation

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 = -4 + 0.60/0.10 = +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 Tfin 04 Stock valuation

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 5 8.45 10 15.48 20 26.16 25 30.19 ∞ 50.00 Tfin 04 Stock valuation

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 10 100% 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! Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

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? Tfin 04 Stock valuation

Free Cash Flow Calculation Tfin 04 Stock valuation

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 Tfin 04 Stock valuation

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! Tfin 04 Stock valuation

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) Tfin 04 Stock valuation

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 / 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 Tfin 04 Stock valuation

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) Tfin 04 Stock valuation