Question

A stock just paid an annual dividend of $1.4. The dividend is expected to grow by 10% per year for the next 4 years. The growth rate of dividends will then fall steadily from 10% after 4 years to 4% in year 8.

The required rate of return is 12%.

Question:

1. What is the stock price if the dividend growth rate will stay 4% forever after 8 years?

2. In 8 years, the P/E ratio is expected to be 22 and the payout ratio to be 80%. What should be the current stock price when using the P/E ratio?

Answer 1

A stock just paid an annual dividend of $1.4. The dividend is expected to grow by 10% per year for the next 4 years. The growth rate of dividends will then fall steadily from 10% after 4 years to 4% in year 8.

The required rate of return is 12%.

1. What is the stock price if the dividend growth rate will stay 4% forever after 8 years?

stock price  = $ 22.31

We can calculate the price of share using non constant or two stage dividend growth model.

Under this model the price is calculated the following way

The model is adjusted by dividing the projected dividend cash flow stream into two parts: (1) the initial fast growth period, and (2) the next period, when normal and sustainable but lower growth is expected

Calculate the present values of the dividends to be received during the first four years and sum the results. Each year’s dividend increases by 10% over the previous year’s dividend

We start with next year dividend

End of year	Dividend	PV factor @ 12%	PV of dividend
1	1.4 + 10% = 1.54	0.893	1.37522
2	1.54 + 10% = 1.694	0.797	1.35
3	1.694+10%= 1.8634	0.712	1.33
4	1.8634+10% = 2.05	0.636	1.303
PV of future dividend – year 1 through 4	5.36

Project the dividend for Year 5 by multiplying the Year 4 dividend by (1 + the growth rate for Year 5 ), which is 1.04, since growth is expected to slow down to 4% in Year 5. The Year 5 dividend is therefore projected to be $2.05  × 1.04, or $2.132 = D5

Pretend that Year 5 is Year 1 and so the end of Year 4 is Year 0. Use the Constant Growth Model to calculate the value of the stock at the end of Year 4 , assuming a required rate of return of 12% and an annual growth in dividends of 4% going forward from the end of Year 4, beginning with Year 5:

P4    = D5  / R –G

P22 = 2.132 / 12% – 4%   = 2.132 / 8% = $ 26.65

The $ 26.65  present value calculated is occurs at the end of Year 4, not at Year 0. Therefore, $ 26.65 must be discounted back 4 years to Year 0

= $ 26.65  * ( 1 / 1 + 12%)^4  = 26.65  *0.636  =   $ 16.95

To calculate the fair value at Year 0 for a share of this stock, sum the present value of the future dividends for Years 1 through 4 and the present value of the dividends to be received from Year 5 to infinity

= $ 5.36 +   $ 16.95  =   $ 22.31

stock price  = $ 22.31  

2. In 8 years, the P/E ratio is expected to be 22 and the payout ratio to be 80%. What should be the current stock price when using the P/E ratio?

P/E = Market price / EPS = 22

Payout ratio = Dividend / EPS = 80%

In 8th year end dividend is = 2.132 + 10% + 10% + 10% =

End of year	Dividend	PV factor @ 12%	PV of dividend
5	2.132	(1/1+12%)^5 =0.5674	1.2096
6	2.132 + 4% = 2.217	(1/1+12%)^5 =0.5066	1.1231
7	2.217+4%=2.306	(1/1+12%)^5 =0.4523	1.043
8	2.306+4%=2.398	(1/1+12%)^5 =0.404	0.9687
PV of future dividend – year 5 through 8	4.34

In the 8th year the price is calculated using P/E ratio

The 8th year dividend = 2.398

In the 8th year

Payout ratio = 2.398 / EPS = 80%

So, the EPS = 2.398 / 80% = 2.9975

In the 8th year EPS is = 2.9975

P/E = Market price / 2.9975 = 22

So, the Market price expected on 8th years = 2.9975 * 22 = $ 65.945

This price is expected at end of 8th year. Therefore, $ 65.945 must be discounted back 8 years to Year 0

= 65.945 * ( 1 / 12%)^8  = $ 65.945 * 0.404 = $ 26.64

To calculate the price at Year 0 for a share of this stock, sum the present value of the future dividends for Years 1 through 4 ( calculated in question 1 ) and present value of the future dividends for Years 5 through 8 ( calculated in question 2 ) and the present value of the stock price expected according to P/E ratio

= $ 5.36 + 4.34 + $ 26.64  =   $ 36.34

current stock price = $ 36.34