Question

Carnes Cosmetics Co.'s stock price is $40, and it recently paid a $1.00 dividend. This dividend is expected to grow by 20% for the next 3 years, then grow forever at a constant rate, g; and r_s = 11%. At what constant rate is the stock expected to grow after Year 3? Do not round intermediate calculations. Round your answer to two decimal places.

Answer 1

Given,
Current Dividend = D0 = $1
Growth Rate for Next 3 Years = 20%
Constant Growth Rate = g = ?
Current Stock Price = P0 = $40
rs = Cost of Equity = 11%
Now,
Dividend at the end of year 1 = D1
= D0(1+Growth Rate)
= $1(1+20%)
= $1(1.20)
= $1.20
Dividend at the end of year 2 = D2
= D1(1+Growth Rate)
= $1.20(1+20%)
= $1.20(1.20)
= $1.44
Dividend at the end of year 3 = D3
= D2(1+Growth Rate)
= $1.44(1+20%)
= $1.44(1.20)
= $1.728
Dividend at the end of year 4 = D4
= D3(1+Constant Growth Rate)
= $1.728(1+g)
Terminal Value of Dividend as it grows Constantly after 3 years
= D4 / (rs - g)
= $1.728*(1+g) / (11% - g)
So,
Current Price of Stock
= Present Value Dividends + Present Value of Terminal Value of Dividend
$40 = 1.20/(1+11%)1 + 1.44/(1+11%)2 + 1.728/(1+11%)3 + $1.728*(1+g)/[(11%-g)*(1+11%)3]
$40 = 1.20/(1.11)1 + 1.44/(1.11)2 + 1.728/(1.11)3 + $1.728*(1+g)/[(11%-g)*(1.11)3]
$40 = 1.20/(1.11) + 1.44/(1.2321) + 1.728/(1.367631) + $1.728*(1+g)/[(11%-g)*(1.367631)]
$40 = 1.081081 + 1.168736 + 1.263499 + $1.728*(1+g)/[(11%-g)*(1.367631)]
$40 = $3.513316 + $1.728*(1+g)/[(11%-g)*(1.367631)]
$40 - $3.513316 = $1.728*(1+g)/[(11%-g)*(1.367631)]
$36.486684 = $1.728*(1+g)/[(11%-g)*(1.367631)]
$36.486684*1.367631 = ($1.728 + $1.728*g) / (11%-g)
$49.900320 = ($1.728 + $1.728*g) / (11%-g)
$49.900320*(11%-g) = $1.728 + $1.728*g
$49.900320*11% - $49.900320*g = $1.728 + $1.728*g
$5.4890022 - $49.900320g = $1.728 + $1.728g
$49.900320g+$1.728g = $5.4890022-$1.728
$51.62832g = $3.7610022
g = $3.7610022/$51.62832
g = 0.0728
i.e. 7.28%