Question

Google has paid $2 in dividends one year ago and this year has just paid $4 yesterday. In the next three years the dividends are expected to be $1, $5, and $4 at the end of year three. From there on, the dividend will grow with a yearly growth rate g. What is this implied growth rate that shareholders expect if the stock price today is $40? (The required rate of return for this stock is 10%.)

Select one:

a. 4%

b. 3%

c. 2%

d. 1%

e. 5%

Answer 1

Let D1, D2, D3, D4 be the expected dividends in the next year years respectively.
D1 = $1
D2 = $5
D3 = $4
After 3 years dividend will grow at a growth rate of g.
So D4 = D3 (1+g)

Let S0 be the stock price today.
S0 = $40
R = 10%

As per the multiperiod dividend discount model,
V0 = (D1/(1+R)1) + (D2/(1+R)2) + ……….. + (Dn/(1+R)n) + (Pn/(1+R)n)
where,
V0 = Value of stock today
Dn = Dividend payment for nth period
Pn = Stock price for nth period from now
R = Required rate of return.

Pn is calculated at [Dn * (1+g)]/R – g
P3 = [D3 * (1+g)]/R – g
P3 = [4 * (1+g)]/.10 – g

Using the formula
V0 = ($1/(1+0.10)1) + ($5/(1+0.10)2) + ($4/(1+0.10)3) + {[4 * (1+g)]/.10 – g}/(1+0.10)3

40 = ($1/(1.10)1) + ($5/(1.10)2) + ($4/(1.10)3) + {[4 * (1+g)]/.10 – g}/(1+0.10)3
40 = ($1/1.10) + ($5/1.21) + ($4/1.331) + {[4 * (1+g)]/.10 – g}/1.331
40 = $0.9091 + $4.1322 + $3.0053 + {[4 * (1+g)]/.10 – g}/1.331
40- $0.9091 - $4.1322 - $3.0053 = {[4 * (1+g)]/.10 – g}/1.331
31.9534 ={[4 * (1+g)]/.10 – g}/1.331
31.9534 * 1.331 ={[4 * (1+g)]/.10 – g}
42.53 = 4 + 4g/.10 – g
42.53 (.10 – g) = 4 + 4g
4.253 – 42.53g = 4 + 4g
4.253 – 4 = 4g + 42.53g
0.253 = 46.53g
g = 0.54%

Note : On cross calculating the price of stock taking g = 0.54% we are calculation V0 as 40.