Well, let's rewrite the formula that we had before. Remember when we put this expected return, we said that it consists of dividend yield and capital gain. But this formula can be easily rewritten in the following form. That P sub 0 = dividend 1 + P1 /1 + r. Here for r, I would put this expected return. Look, this looks very much like discounting. What' this? 0, 1, this is cash flow at between 1, discounted over the expected return, r, and when we arrive at P sub 0. Well, the good news is that we finally came close to discounting. The bad news is that we expressed the known and observed P sub 0 through the expected dividend 1, P1, and r. Well, strictly speaking, this is r1. But from now on, we will put that these r's are, I wouldn't say constant, we'll say they are the same for all securities of equivalent risk. Why is that so? Because, for example, if there are two equivalent risk, and one produces greater return than the other, that would mean that everyone will be happy to invest in this very security. So people line up, the price of the entry ticket goes up, and the return is brought down. So that is the idea. And if we replace this r here with the expected return on the securities of that kind of risk then this formula does indeed become the formula for the calculation of the PV. Well, We can easily proceed, and we took one step forward and said P1 is equal to dividend 2 plus P2 divided by 1 + r. Again, we pretend that the expected return does not change because nothing happens to the risks of the stock, which is an assumption, but oftentimes people use that. See what happened. We just took this formula and moved it one period forward. Now, we can put this whole formula here and calculate. Now, from now on, we put that r is constant for all periods of time. And then we come up with a general formula that says, P sub 0 is equal to the sum from k1 to T of dividends k divided by 1 plus r to the Kth power, plus PT divided by 1 plus r to the Tth power. So this is a set of dividends, from one to T, and this is the price at point T. Well clearly, we know that the price of point T can be expressed through the future prices of the stock. So oftentimes, this is called tail. Now, if T goes to infinity, then clearly the tail disappears. And then we get the formula that basically says, this is general formula for finding the present value or the price of the stock. That says basically that. The price of the stock is the present value of all its future dividends. Now, there are some observations here. Some people say, well, some stocks do not pay dividends. Well, the result is that if you never paid dividends then the price of the stock is equal to zero. So stocks do not pay dividends only in the hope of being able to distribute more cash to stockholders later. So that may be a long period of time, but this is not an infinite period of time. So therefore, for example, if the dividends are zero here, we can move further, further, further, and then find that the growth will be high. We will see that in the next episodes of this week. And then later on, we'll be able to get much more cash. And even after having been discounted, 2.0, that will be more beneficial for the stockholders. But now, I would like to flip it and then analyze this formula in some, they're not exactly shortcuts, but they're simplifying assumptions. Well first of all, what about the growing stock? We know that growth has value. And now, we would say, what if, let's say dividend K+1 is equal to dividend K(1+G). And if you put that this g is constant then we are back to the growing perpetuity formula that we studied in the very first week. And we can say that then P sub 0 is equal to dividend 1 divided by r- g. This is a very well known Gordon's formula for constant growth. Well, for example, if g is 4%, which is a lot, r is 12%, and dividend 1 is $6, then P sub 0 will be $75 if you check that. Well, So this is a case of constant growth. Well clearly, what happens if there is no growth? Well, we have an idea that g disappears. But the question is, what becomes dividend 1? And in order to analyze that in some greater detail, we will take a pause here. And in the next episode, we'll analyze this mechanism of growth. We will see that the general idea is as follows. When you make investments, then you make your cash flows potentially higher in the next period. And then there's always this investment choice. You can either pay it out in the form of a dividend, or you can reinvest. And this game of balancing reinvestment and dividend will actually show us what is the a no growth case, and what is a growth case, and what exactly is the so called present value of growth opportunities, or namely, the value of this growth. That's all we will discuss in the next episode.
