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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​.)

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jepessoa

Answer:

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
  • The Answer will be:  (Div 1 / r - g1) x {1 - [(1 + g1) / (1 + r)]ⁿ}    +    (Div 1 / r - g2) x [(1 + g1) / (1 + r)]ⁿ⁻¹ =stock price
  • since, The company will take first grow at g1 for n years, and also, then at g2 forever, we need to first determine the present value of the dividends growing at g1 for n years:

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

For 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

  • Thus, 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  step according to g1 for g2, and g2 = 5%

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

  • Then, we add both parts to finish our example = $8.31 + $22.07 = $30.38

Learn more:

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