ABC has been undergoing rapid growth for the last few years. The
current dividend of $2 a share should continue to grow at the rate of
10% per year for the next two years. After that time the earnings are
expected to slow down, with the dividend growing at a more normal rate
of 6% a year for the indefinite future. Because of the risk involved
in such a rapid growth, the required rate of return on this stock is
14%. What is the theoretical price of ABC?

Hello.

Based on the information given, the theoretical price of ABC would be
$28.46.

Since the problem doesn't say otherwise, I'm going to assume that the
dividend is an annual dividend and is paid at the end of the year.

The way to solve a problem like this is to break it down into parts.

The theoretical price, or present value, of the stock is the combined
value of:
(1) the present value of the dividends for first two years; and
(2) the present value of the stream of dividends beginning in the
third year.


(1) 
Since the current dividend is $2, and the dividend is growing at 10%
per year, the dividend one year from now will be 10% higher than it is
now:
$2 * (1 + .10) = $2.20. 
The dividend two years from now will be 10% higher than that:
$2.20 * (1 + .10) = $2.42.

We must now discount those two dividend payments to present value.
The basic formula for that is: 
Present Value = C1/(1+r) + C2/(1+r)^2
C1 is the dividend at the end of the first year and C2 is the dividend
at the end of the second year. r is the discount rate (14% here).
PV = C1/(1+r) + C2/(1+r)^2
PV = (2.20)/(1.14) + (2.42)/(1.14)^2
PV = 1.93 + 1.86 = 3.79

Thus, the present value for the first two payments is $3.79.

---------
(2)

We must now calculate the present value of the perpetuity that begins
at the end of the third year.

From the information given, the growth rate drops to 6%. Thus, the
dividend at the end of the third year will be 6% higher than the
previous payment of $2.42, so:
$2.42 * (1 + .06) = $2.5652.

From that point on, the dividend will continue to grow at 6%.  Thus,
we treat this as a growing perpetuity that begins at the end of year
3.

The formula for present value of a delayed growing perpetuity is
PV = [C1 / r - g ] *  [(1 / 1 + r)^(t - 1)]
where C1 is the first cash payment ($2.5652 here), r is the discount
rate, g is the growth rate (6% here), and t is the time period (year 3
here).

Formula source: Econ 134 problem set, Answer 1(iii), from UC Davis
Economics Department:
http://www.econ.ucdavis.edu/faculty/nehring/teaching/econ134/econ134-key3-f02.pdf

So:
PV = [C1 / r - g ] *  [(1 / 1 + r)^(t - 1)]
PV = [2.5652 / .14 - .06 ] * [(1 / 1.14)^(3 - 1)]
PV = [2.5652 / .08 ] * [(.87719)^2]
PV = [32.065] * [.7694675] = 24.67

------

Thus, when we combine our two results, we get:
Theoretical price of ABC = $ 3.79 + $ 24.67 = $ 28.46


search strategy: "delayed growing perpetuity"

I hope this helps. If anything is unclear, please use the "request
clarification" feature. Thanks.

Thanks, this matches my calculations.