# Stock Valuation Common Stock Valuation Some Features of Common and Preferred Stocks The Stock Markets.

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Stock Valuation Common Stock Valuation Some Features of Common and Preferred Stocks The Stock Markets

T2 Common Stock Cash Flows and the Fundamental Theory of Valuation In 1938, John Burr Williams postulated what has become the fundamental theory of valuation: The value of any financial asset equals the present value of all of its future cash flows. For common stocks, this implies the following: D 1 P 1 D 2 P 2 P 0 = +and P 1 = + (1 + r) 1 (1 + r) 1 (1 + r) 1 (1 + r) 1 substituting for P 1 gives D 1 D 2 P 2 P 0 = + +. Continuing to substitute, we obtain (1 + r) 1 (1 + r) 2 (1 + r) 2 D 1 D 2 D 3 D 4 P 0 = + + + + … (1 + r) 1 (1 + r) 2 (1 + r) 3 (1 + r) 4

T3 Common Stock Valuation: The Zero Growth Case According to the fundamental theory of value, the value of a financial asset at any point in time equals the present value of all future dividends. If all future dividends are the same, the present value of the dividend stream constitutes a perpetuity. The present value of a perpetuity is equal to C/r or, in this case, D 1 /r. Question:Cooper, Inc. common stock currently pays a \$1.00 dividend, which is expected to remain constant forever. If the required return on Cooper stock is 10%, what should the stock sell for today? Answer:P 0 = \$1/.10 = \$10. Question:Given no change in the variables, what will the stock be worth in one year?

T4 Common Stock Valuation: The Zero Growth Case (concluded) Answer:One year from now, the value of the stock, P 1, must be equal to the present value of all remaining future dividends. Since the dividend is constant, D 2 = D 1, and P 1 = D 2 /r = \$1/.10 = \$10. In other words, in the absence of any changes in expected cash flows (and given a constant discount rate), the price of a no- growth stock will never change. Put another way, there is no reason to expect capital gains income from this stock.

T5 Common Stock Valuation: The Constant Growth Case In reality, investors generally expect the firm (and the dividends it pays) to grow over time. How do we value a stock when each dividend differs than the one preceding it? As long as the rate of change from one period to the next, g, is constant, we can apply the growing perpetuity model: D 1 D 2 D 3 D 0 (1+g) 1 D 0 (1+g) 2 D 0 (1+g) 3 P 0 = + + + … = + + +... (1 + r) 1 (1 + r) 2 (1 + r) 3 (1 + r) 1 (1 + r) 2 (1 + r) 3 D 0 (1 + g) D 1 P 0 = =. r - g r - g Now assume that D 1 = \$1.00, r = 10%, but dividends are expected to increase by 5% annually. What should the stock sell for today?

T6 Common Stock Valuation: The Constant Growth Case (concluded) Answer:The equilibrium value of this constant-growth stock is D 1 \$1.00 = = \$20 r - g.10 -.05 Question:What would the value of the stock be if the growth rate were only 3%? Answer: D 1 \$1.00 = = \$14.29. r - g.10 -.03 Why does a lower growth rate result in a lower value? Stay tuned.

T7 Stock Price Sensitivity to Dividend Growth, g 0 2% 4% 6% 8% 10% 50 45 40 35 30 25 20 Stock price (\$) Dividend growth rate, g D 1 = \$1 Required return, r, = 12% 15 10 5

T8 Stock Price Sensitivity to Required Return, r 6% 8% 10% 12% 14% 100 90 80 70 60 50 40 Stock price (\$) Required return, r D 1 = \$1 Dividend growth rate, g, = 5% 30 20 10

T9 Common Stock Valuation - The Nonconstant Growth Case For many firms (especially those in new or high-tech industries), dividends are low and expected to grow rapidly. As product markets mature, dividends are then expected to slow to some “steady state” rate. How should stocks such as these be valued? Answer: We return to the fundamental theory of value - the value today equals the present value of all future cash flows. Put another way, the nonconstant growth model suggests that P 0 =present value of dividends in the nonconstant growth period(s) + present value of dividends in the “steady state” period.

T10 Quick Quiz -- Part 1 of 3 Suppose a stock has just paid a \$5 per share dividend. The dividend is projected to grow at 5% per year indefinitely. If the required return is 9%, then the price today is _______ ? P 0 = D 1 /(r - g) = \$5 (______ )/(______ -______ ) = \$5.25/.04 = \$131.25 per share What will the price be in a year? P t = D t+1 /(r - g) P 1 = D___ /(r - g) = (\$______ 1.05)/(.09 -.05) = \$137.8125 By what percentage does P 1 exceed P 0 ? Why?

T11 Quick Quiz -- Part 2 of 3 Find the required return: Suppose a stock has just paid a \$5 per share dividend. The dividend is projected to grow at 5% per year indefinitely. If the stock sells today for \$65 5/8, what is the required return? P 0 = D 1 /(r - g) (r - g) = D 1 /P 0 r = D 1 /P 0 + g = \$5.25/\$65.625 +.05 = dividend yield (_____) + capital gain yield (_____) = ____

T12 Summary of Stock Valuation I. The General Case In general, the price today of a share of stock, P 0, is the present value of all its future dividends, D 1, D 2, D 3,... D 1 D 2 D 3 P 0 = + + + … (1 + r) 1 (1 + r) 2 (1 + r) 3 where r is the required return. II. Constant Growth Case If the dividend grows at a steady rate, g, then the price can be written as: P 0 = D 1 /(r - g) This result is the dividend growth model. III. The Required Return The required return, r, can be written as the sum of two things: r = D 1 /P 0 + g where D 1 /P 0 is the dividend yield and g is the capital gain yield.

New York Stock Exchange Composite Transactions Quotations as of 5 p.m. Eastern Time Monday, March 28, 1994 52 WeeksYldVol Net HiLoStockSymDiv%PE100sHiLoCloseChg. 28 1/2 25 1/2 MellonBk pfJ2.138.3…272625 3/4 25 3/4 + 1/8 28 1/4 25 5/8 MellonBk pfK2.057.8…5526 1/8 2626 1/8 … 48 1/4 35 3/4 MelvilleMES1.524.01314393837 2/8 37 5/8 + 1/4 12 1/8 9 3/8 MentorIncoFdMRF.969.5…14210 1/4 10 1/8 10 1/8 … 56 1/2 43 5/8 MercBcpMOMTL1.683.4102195049 1/2 49 3/4 - 1/8 41 1/8 29 7/8 MercStrsMST1.022.61763640 5/8 39 5/8 39 7/8 -1 1/4 39 3/8 28 5/8 MerckMRK1.123.7162159530 3/8 3030 3/8 + 1/4 20 3/8 12 1/8 MercuryFinMFN.28f1.7305171716 3/4 16 3/4 … T13 Sample Stock Quotation from The Wall Street Journal Source: Reprinted by permission of The Wall Street Journal, ©1991 Dow Jones & Company, Inc., March 28, 1994. All Rights Reserved Worldwide.

Suppose a stock has just paid a \$5 per share dividend. The dividend is projected to grow at 10% for the next two years, the 8% for one year, and then 6% indefinitely. The required return is 12%. What is the stock’s value? TimeDividend 0\$ 5.00 1\$ ____(10% growth) 2\$ ____(10% growth) 3\$6.534( __% growth) 4\$6.926( __% growth) T14 Quick Quiz -- Part 3 of 3

T15 Quick Quiz -- Part 3 of 3 (concluded) At time 3, the value of the stock will be: P 3 = D 4 /(r - g) = \$_____ /(.12 -.06) = \$115.434 The value today of the stock is thus: P 0 = D 1 /(1 + r) + D 2 /(1 + r) 2 + D 3 /(1 + r) 3 + P 3 /(1 + r )3 = \$5.5/1.12 + \$6.05/1.12 2 + \$6.534/1.12 3 + \$115.434/1.12--- = \$96.55

T16 Solution to Problem MegaCapital, Inc. just paid a dividend of \$2.00 per share on its stock. The dividends are expected to grow at a constant 6 percent per year indefinitely. If investors require a 13 percent return on MegaCapital stock, what is the current price? What will the price be in 3 years? In 15 years? According to the constant growth model, P 0 = D 1 /(r - g) = \$2.00(1.06)/(.13 -.06) = \$30.29 If the constant growth model holds, the price of the stock will grow at g percent per year, so P 3 = P 0 x (1 + g) 3 = \$30.29 x (1.06) 3 = \$36.07, and P 15 = P 0 x (1 + g) 15 = \$30.29 x (1.06) 15 = \$72.58.

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