00:01
All right, for this problem, we are considering a corporation, which is not going to begin paying dividends until three years from now.
00:10
So we would have d1 equals d2 equals d.
00:15
Or rather, d1 equals d2 equals zero.
00:20
And we are given that the dividend that they will pay three years from today will be $1.
00:26
We are told that the growth rate for years four and five, so we'll write that as g4 equals g5, will be 50 % per year, and then after year five, the company will grow at a rate of 8%.
00:48
So growth, i'll just write g plus or later on will be 0 .08.
00:57
If the required return on the stock, which is r sub s there, is 15%, or 0 .15, we're asked, what is the value of the stock today? so that would be p -hat 0 equals a question mark.
01:14
So this is going to be a problem, obviously, where we have non -constant growth.
01:20
If we were to plot it out, actually i have a little plot up here, if we plug it out, we're essentially told that because the company is expanding rapidly, it will not pay any dividends because it needs to retain all of its earnings for now.
01:35
But basically, we would have no dividends until year three, when we have dividend of one, then a period of rapid growth from year three up to year five, and then afterwards a year of more normal growth.
01:49
So the way that we approach a non -constant growth problem like this, or the way that we approach finding the present value, is by considering the value essentially in two pieces.
02:05
We have the general equation that the present value is going to be the sum from t equals 1 up to infinity of the tith, the t if, t with a t, h, dividend or the dividend at time step t divided by 1 plus r sub s that required rate of return to the power of t in this case we have excuse me we have a specific equation that we can use so actually i'll write that's general in the case of a stock that is growing with a constant rate of growth we have p hat not is equal to d1 over rs minus g, where g is the growth rate, that's constant, using a formula that allows us to sum up that series from t equals 1 up to infinity, but that requires a constant unchanging growth.
03:15
So what we can do here is essentially split up our period into a portion of non -conforming growth.
03:26
Constant growth and a portion of constant growth.
03:31
So we'd have p hat not is going to equal, we have d1 plus d2 plus d3 over 1 plus r s, 1 plus r s the power of 2, 1 plus r s the power of 3.
03:53
Then, so that is going to be, or the first three terms there are no dividends.
04:00
Those will actually be zero.
04:07
Then we have d3 plus d4, it's d5 over, excuse my messy writing here, 1 plus rs to the power of 4, 1 plus rs to the power of 5, where from d4 to d5 we have rapid growth and then afterwards we are going to add on what is called the horizon value, p hat n.
04:39
So basically, that is going to be the value for all of the subsequent terms.
04:46
So you'll have p hat zero.
04:50
It's going to equal.
04:51
Actually, let me take a step back here.
04:52
So we have that the third dividend is going to be $1.
04:56
And we know that we'll have 50 % growth from 3 to 4 to 5.
05:02
So that means then that d4 is going to be 1 .5.
05:12
That's our 50 % growth.
05:15
And then d5, we need to, i'll write this out a little bit more formally here.
05:21
That's going to be 1 .5 times 1 plus 0 .5 .5.
05:26
This is going to be 1 .5 squared, first of all.
05:32
And 1 .5 squared is going to be a number that i'll get after throwing it into the calculator quickly here...