Question #73

The implied long-term rate is the rate that will cause the present value of expected dividends to equal its current market value. Since Ancis provides specific growth rates for Turbo over the next three years, we can use a multi-stage dividend discount model and solve for the long-term growth rate that makes the present value equal to the current market value. First, we calculate Turbo's expected dividends. D0 = $10.00 current EPS times the dividend payout ratio of 40% D0 = $4.00 dividend per share in year 0. Note that the 19% historical dividend growth rate is irrelevant to the current value of the firm. Since the dividend payout ratio is expected to remain constant at 40%, we can use the expected growth rate in earnings to estimate future dividends. EPS growth is forecast at 20% in year 1, 15% in year 2, and 10% in year 3. Multiplying each year's expected dividend times the relevant forecast growth rate, we calculate: D1 = ($4.00 dividend in year 0) × (1.20) = $4.80 D2 = ($4.80 dividend in year 1) × (1.15) = $5.52 D3 = ($5.52 dividend in year 2) × (1.10) = $6.07 Discounting these back to their present value in year 0 using the cost of equity (the WACC is irrelevant), we find: Present Value (D1 + D2 + D3) = ($4.80 / 1.141) + ($5.52 / 1.142) + ($6.07 / 1.143) = $4.21 + $4.25 + $4.10 = $12.56 Thus, we know that $12.56 of the current $55.18 market value represents the present value of the expected dividends in years 1, 2 and 3. Therefore, the present value of the firm's dividends for years 4 and beyond must equal ($55.18 - $12.56) = $42.62. Since the present value of the firm's dividends beginning in year 4 equals $42.62, the future value in year four will equal ($42.62 × 1.143) = $63.14. Now that we know the value in year 4 of the future stream of steady-growth dividends, we can solve for the growth rate using the Gordon Growth Model: P3 = [($6.07)(1 + x)] / (0.14 − x ) = $63.14 63.14 (0.14 − x) = 6.07 (1+x) 8.84 − 63.14x = 6.07 + 6.07x 2.77 = 69.21x x = 0.04 The long-term growth rate that makes Turbo fairly valued is 4% per year. We can check our calculation by plugging the 4% growth rate we just solved for into the Gordon Growth Model and then plugging that result into the basic multi-stage dividend discount model: P3 = [($6.07)(1 + 0.04)] / (0.14 − 0.04) P3 = 6.313 / (.10) P3 = 63.13 (Note that this value varies from the previous calculation by 0.01 because of rounding error.) P0 = ($4.80 / 1.141) + ($5.52 / 1.142) + ($6.07 / 1.143) + ($63.13 / 1.143) = $55.18, which is the current market value. At a 4% growth rate, Turbo is fairly valued. Note that on the exam, it may be faster to plug each growth rate into the Gordon Growth Model and then plug each of those terminal values into the basic multi-stage formula than to solve for the growth rate. This trial and error method is especially effective if you start with the "middle" growth rate and then decide which value to test next depending on the results of the first calculation. For example, if the first growth rate gives a value for the firm that is too high, you can eliminate all the higher growth rates and try the next lower one.