A common stock is just paid an annual dividend of $2 yesterday. The dividend is expeccted to grow at 8% annually for the next 3 years, after which is will grow at 4% in perpetuity. The appropriate discount rate is 12%. What is the priceof the stock?

Question

A common stock is just paid an annual dividend of $2 yesterday. The dividend is expeccted to grow at 8% annually for the next 3 years, after which is will grow at 4% in perpetuity. The appropriate discount rate is 12%. What is the priceof the stock?

What I know:
Differential growth
DIV=2
g1=.08
g2=.04
r=.12
yr.end 5=Div4(1+g2)=2.52(1.04)=2.6208
p4=DIV5/r-g2=2.6208/.12-.04=32.76 FV
p4/(1+r)^4=32.76/1.12^4=20.82
Present Value of all Div as of yr end 0 is 20.82 + 32.76 =
??? not sure about this step

Answer 1

Perhaps my reply to your previous question would help. On the other hand perhaps I am way off base not being a financial type at all.

by the way, my g for geometric series is starting value.
my r is ratio of value at year (n+1) to value at year n.
in that case sum of infinite series is g/(1-r)

by the way, for just n terms
sum = g + g r + g r^2 +...+ g r^(n-1)
= g (1-r^n)/(1-r)

Answer 2

let's see what the dividends are worth after 3 years (n = 4 because we start at the beginning of year one and the end of the other years)

V3 = 2 + 2(1.08/1.12) + 2(1.08/1.12)^2...
g = 2
r = 1.08/1.12 = .9463
so
sum after three years = 2(1-.9463^3)/(1-.9463) = 2(.1526/.0537) = 5.683
now we go on forever at 4%
g = 5.683
r = 1.04/1.12 = .9286
sum to infinity = 5.683/(1-.9286) = 79.56

Answer 3

wait a minute, we have to subtract the first three years at 4% off the infinite 4% years because we did those years separately with 8%

sum at 4% after three = 2(1-.9286^3)/(1-.9286)
= 2*(.1993/.0714) = 5.583
so it is
79.56 - 5.58 = 73.98

Answer 4

To calculate the price of the stock, you need to find the present value of all future dividends.

Here's how you can calculate it step by step:

1. Calculate the present value of dividends for the first three years using the formula:
PV1 = DIV1 / (1 + r) + DIV2 / (1 + r)^2 + DIV3 / (1 + r)^3

Since the annual dividend is $2 and it is expected to grow at 8% each year for the next 3 years, the dividends for year 1, 2, and 3 would be:
DIV1 = $2
DIV2 = DIV1 * (1 + g1) = $2 * (1 + 0.08) = $2.16
DIV3 = DIV2 * (1 + g1) = $2.16 * (1 + 0.08) = $2.3328

Using the appropriate discount rate of 12%, you can calculate the present value of these dividends:
PV1 = $2 / (1 + 0.12) + $2.16 / (1 + 0.12)^2 + $2.3328 / (1 + 0.12)^3

2. Calculate the present value of the dividend at year 4 using the formula:
PV4 = DIV4 / (r - g2)

Using the formula, you can calculate the dividend for year 4:
DIV4 = DIV3 * (1 + g2) = $2.3328 * (1 + 0.04) = $2.424192

Then calculate the present value:
PV4 = $2.424192 / (0.12 - 0.04)

3. Calculate the present value of all future dividends from year 5 onwards (perpetuity) using the formula:
PV5 = DIV5 / (r - g2)

Since the dividend in perpetuity is expected to grow at a rate of 4%, you can calculate the dividend at year 5:
DIV5 = DIV4 * (1 + g2) = $2.424192 * (1 + 0.04) = $2.52226

And then calculate the present value:
PV5 = $2.52226 / (0.12 - 0.04)

4. Sum up the present values of all the dividends:
Price of the stock = PV1 + PV4 + PV5

So the final step is to sum up all the present values you calculated in step 1, 2, and 3 to get the price of the stock.