I plan to buy a stock and my analyst has told me that it will pay an
annual dividend of $4 per year for ever starting at the end of the 5th
year. The analyst also expects the stock to pay equal divident each
year in year-end 1,2,3 and 4. He recommends that I pay no more than
$20 for this stock now. If the discount rate for this stock is 15%,
what is the annual dividend of the first four years I he has in mind.

Hello.

The annual dividend of the first four years is $1.66.

The way to solve this problem is to break it down into two parts: the
perpetuity that begins in year 5 and the annuity that consists of
years 1, 2, 3 and 4.

The present value of the stock ($20) is the combined value of the
present value of the annuity and the present value of the perpetuity.
PVstock = PVannuity + PVperpetuity

We start with the perpetuity.

A delayed perpetuity is involved because the stock will pay an annual
dividend of $4 per year forever beginning in year 5.

The present value of a delayed perpetuity is calculated using the
following formula:
PVperpetuity = C * (1 / r) * ((1 + r)^t)
C is the cash flow and r is the discount rate. Payments begin in year
t+1 .
formal source: Univ. Minnesota: "Time Value of Money" Section 4.1
http://www.sls.csom.umn.edu/users/phd/vyerramilli/fina3001/notes3.pdf

Here: C = 4, r = .15, and t = 4 because payments begin in year t+1
(i.e., 4 + 1 = 5).

PVperpetuity = C * (1 / r) * (1/(1 + r)^t)
PVperpetuity = 4 * (1 /.15) * (1/(1 + .15)^4)
PVperpetuity = 4 * 6.66667 * (1/(1.749))
PVperpetuity = 15.2468


Now, we move on to the annuity. 

Since we know PVperpetuity now we can calculate PVannuity.
PVstock = PVannuity + PVperpetuity.
20 = PVannuity + 15.2468 
PVannuity = 20 - 15.2468 = 4.7532

Now, using the value of the PVannuity, we can calculate the dividend
rate of years 1, 2, 3, and 4.
The present value of the perpetuity is equal to the following:
PVannuity = C * (1 / r) * ( 1 - 1/((1+r)^t))
C is the cash flow and r is the discount rate. The length of the
annuity is t.
source: Univ. Minnesota: "Time Value of Money" Section 4.2
http://www.sls.csom.umn.edu/users/phd/vyerramilli/fina3001/notes3.pdf

Here: C is unknown, r = .15, and t = 4.

PVannuity = C * (1 / r) * ( 1 - 1/((1+r)^t))
4.7532 = C * (1/.15) * (1 - 1/((1.15)^4))
4.7532 = C * 6.66667 * (1 - 0.57175)
4.7532 = C * 6.66667 * 0.428825
4.7532 = C * 2.854978363
C = (4.7532 / 2.854978363) 
C = 1.66

Thus, the cash flow (i.e., dividend) for the first four years is $
1.66.


search strategy: "time value of money", delayed, perpetuity, annuity,
"discount rate"

I hope this helps. If any part of this answer is unclear or if you
believe that any additional information is needed, please request
clarification.
Thank you.