What is the value of a firm with initial dividend Div 1 , growing for n years (i. e., until year n plus 1 ) at rate g 1 and after that at rate g 2 forever, when the equity cost of capital is r ? (Hint : Find the present value of the n -year dividend stream which grows at g 1 per year and add the present value of the continuation value found at year n .)

stock price = (Div 1 / r - g1) x {1 - [ (1 + g1) / (1 + r) ]ⁿ} + (Div 1 / r - g2) x [ (1 + g1) / (1 + r) ]ⁿ⁻¹

Explanation:

since the company will first grow at g1 for n years, and then at g2 forever, we need to first determine the present value of the dividends growing at g1 for n years:

present value of the dividends during n = (Div 1 / r - g1) x {1 - [ (1 + g1) / (1 + r) ]ⁿ}

e. g. div = $2, n = 5 years, g1 = 8%, r = 12%

(2 / 12% - 8%) x {1 - [ (1 + 8%) / (1 + 12%) ]⁵} = 50 x 0.166263 = $8.31

now we find the formula to calculate the present value for the growing perpetuity g2 at n - 1 years:

= (Div 1 / r - g2) x [ (1 + g1) / (1 + r) ]ⁿ⁻¹

following the same example but changing g1 for g2, and g2 = 5%

= (2 / 12% - 5%) x [ (1 + 5%) / (1 + 12%) ]⁵⁻¹ = 28.5714 x 0.772476 = $22.07

we now add both parts to finish our example = $8.31 + $22.07 = $30.38