One Period Valuation Model • To value a stock, you first find the present discounted value of the expected cash flows. • P0 = Div1/(1 + ke) + P1/(1 + ke) where • P0 = the current price of the stock • Div = the dividend paid at the end of year 1 • ke = required return on equity investments • P1 = the price at the end of period one
One Period Valuation Model • P0 = Div1/(1 + ke) + P1/(1 + ke) • Let ke = 0.12, Div = 0.16, and P1 = $60. • P0 = 0.16/1.12 + $60/1.12 • P0 = $0.14285 + $53.57 • P0 = $53.71 • If the stock was selling for $53.71 or less, you would purchase it based on this analysis.
Generalized Dividend Valuation Model • The one period model can be extended to any number of periods. • P0 = D1/(1+ke)1 + D2/(1+ke)2 +…+ Dn/(1+ke)n + Pn/(1+ke)n • If Pn is far in the future, it will not affect P0 • Therefore, the model can be rewritten as: • P0 = S Dt/(1 + ke)t ∞ t=1
Generalized Dividend Valuation Model • The model says that the price of a stock is determined only by the present value of the dividends. • If a stock does not currently pay dividends, it is assumed that it will someday after the rapid growth phase of its life cycle is over. • Computing the present value of an infinite stream of dividends can be difficult. • Simplified models have been developed to make the calculations easier.
Gordon Growth Model • Assumptions: • Dividends continue to grow at a constant rate for an extended period of time. • The growth rate is assumed to be less than the required return on equity, ke. • Gordon demonstrated that if this were not so, in the long run the firm would grow impossibly large.
The Gordon Growth Model Firms try to increase their dividends at a constant rate. P0 = D0(1+g)1 + D0(1+g)2 +…..+ D0(1+g)∞ (1+ke)1 (1+ke)2 (1+ke)∞ D0 = the most recent dividend paid g = the expected growth rate in dividends ke = the required return on equity investments The model can be simplified algebraically to read: P0 = D0(1 + g) D1 (ke– g) (ke – g) =
Gordon Model: Example • Find the current price of Coca Cola stock assuming the following: • g = 10.95% • D0 = $1.00 • ke = 13%. P0 = D0(1 + g)/ke – g P0 = $1.00(1.1095)/0.13 - 0.1095 P0 = $1.1095/0.0205 = $54.12
Gordon Model: Conclusions • Theoretically, the best method of stock valuation is the dividend valuation approach. • But, if a firm is not paying dividends or has an erratic growth rate, the simple model will not work. • Consequently, other methods are required.
Non-constant Growth • Firms typically go through life cycles. • Early in the cycle, their growth is much faster than that of the economy as a whole. • Later in the cycle, their growth matches the economy’s growth. • Finally, their growth is less than the economy’s. • Non-constant or supernormal growth occurs during that part of the life cycle when the firm grows faster than the economy as a whole.
Dividend Growth Rates Div($) Normal growth = 8% Supernormal growth = 30% Normal growth = 8% 1.15 Zero growth = 0% Declining growth = -8% Years 0 1 2 3 4 5
Firm Valuation with Non-constant Growth • The value of the firm equals the present value of its expected future dividends. • Process: • Find the present value of the dividends during the period of non-constant growth. • Find the price of the stock at the end of the non-constant growth period and discount this price back to the present. • Add the two components to find the value of the stock, P0.
Firm Valuation with Non-constant Growth: Problem • Assume the following: • k = 13.4% (required rate of return • N = 3 (years of supernormal growth) • gs = 30% (rate of growth in supernormal period) • gn = 8% (rate of growth in normal period) • D0 = $1.15 (last dividend paid) • Find the value of the stock.
Firm Valuation with Non-constant Growth: Problem • Step 1: • Calculate the dividends expected at the end of each year during the supernormal period. • Dn = Dn-1(1 + gs) • D1 = $1.15(1 + .3) = $1.495 • D2 = $1.495(1 + .3) = $1.9435 • D3 = $1.9435(1 + .3) = $2.5265
Firm Valuation with Non-constant Growth: Problem • Step 2: • Calculate the price of the stock during the normal growth period using the Gordon model. • Calculate the dividend in the fourth period. • Use the constant growth formula to find P3. • D4 = $2.5265(1 + 0.08) = $2.7286 • P3 = $2.7286/0.134 – 0.08 = $50.53 • If the stockholder sold the stock in period 3, he would receive $50.53. Total cash flow at time 3 equals D3 + P3 = $53.0576.
Firm Valuation with Non-constant Growth: Problem • Step 3: • Discount the cash flows found in steps 1 and 2 and sum the amounts to find the value of the supernormal growth stock. • D1 = $1.4950/(1.134) = $1.3183 • D2 = $1.9435/1.2859 = $1.5113 • D3 = $2.5265/1.4583 = $1.7325 • P3 = $50.5310/1.4583 = $34.65 • Value of growth stock = $39.21
Errors in Valuation • Problems with Estimating Growth • Growth can be estimated by computing historical growth rates in dividends, sales, or net profits. • But, this approach fails to consider any changes in the firm or economy that may affect the growth rate. • Competition, for example, will prevent high growth firms from being able to maintain their historical growth rate. • Nevertheless, stock prices of historically high growth firms tend to reflect a continuation of the high growth rate. • As a result, investors receive lower returns than they would by investing in mature firms.
Estimating Growth: Table 1 Stock Prices for a Security with D0 = $2.00, ke = 15%, and Constant Growth Rates as Listed Growth(%) Price 1 $14.43 3 17.17 5 21.00 10 44.00 11 55.50 12 74.67 13 113.00 14 228.00
Errors in Valuation • Problems with Estimating Risk • The dividend valuation model requires the analyst to estimate the required return for the firms equity. • However, a share of stock offering a $2 dividend and a 5% growth rate changes with different estimates of the required return.
Estimating Risk: Table 2 Stock Prices for a Security with D0 = $2.00, g = 5%, and Required Returns as Listed Required Return(%) Price 10 $42.00 11 35.00 12 30.00 13 26.25 14 23.33 15 21.00
Errors in Valuation • Problems with Forecasting Dividends • Many factors can influence the dividend payout ratio. They include: • The firm’s future growth opportunities • Management’s concern over future cash flows • Conclusion: • Analysts are seldom certain that the stock price projections are accurate. • This is why stock prices fluctuate widely on news reports.
Price Earnings Valuation Method • The price earnings ratio (PE) is a widely watched measure of how much the market is willing to pay for $1 of earnings from a firm. • A high PE has two interpretations: • A higher than average PE may mean that the market expects earnings to rise in the future. • A high PE may indicate that the market thinks the firm’s earnings are very low risk and is therefore willing to pay a premium for them.
Price Earnings Valuation Method • The PE ratio can be used to estimate the value of a firm’s stock. • Firms in the same industry are expected to have similar PE ratios in the long run. • The value of a firm’s stock can be found by multiplying the average industry PE times the expected earnings per share. P/E x E = P
Price Earnings Model: Example • The average industry PE ratio for restaurants similar to Applebee’s is 23. What is the current price of Applebee’s if earnings per share are projected to be $1.13? • P0 = P/E x E • P0 + 23 x $1.13 = $26.
Price Earnings Valuation Method • Advantages: • Useful for valuing privately held firms and firms that do not pay dividends. • Disadvantages: • By using an industry average PE ratio, firm-specific factors that might contribute to a long-term PE ratio above or below the average are ignored.
Non-constant Growth Model • The non-constant growth model can be used to estimate the value of a stock that does not pay dividends during its early years, if it is expected to pay dividends in the future.
Non-constant Growth Model • Process: • Estimate the following: • when dividend will be paid • the amount of the first dividend • the growth rate during the supernormal period • the length of the supernormal period • the long-run constant growth rate • the rate of return required by investors. • Use the constant growth model to determine the price of the stock after the firm reaches stable growth. • Find all the cash flows, take the present value of each and sum.
Setting Security Prices • Stock prices are set by the buyer willing to pay the highest price. • The price is not necessarily the highest price that the stock could get, but it is incrementally greater than what any other buyer is willing to pay. • The market price is set by the buyer who can take best advantage of the asset.
Setting Security Prices • Superior information about an asset can increase its value by reducing its risk. • The buyer who has the best information about future cash flows will discount them at a lower interest rate than a buyer who is uncertain.