Welcome. As I said, a stock or a share or equity, is such a new thing for most people that even people who actually trade it, don't pause to see the power of it. What I'm going to try to do now in the next 15-20 minutes, is give you a sense of how you would think about the pricing of a stock, how would you value it. The first thing I want to introduce is some ideas about the pricing. The main goal is, how is a stock priced? I'm going to give you some concepts or more some ways of thinking about it. P naught is the price of a stock today. Quick question. Do you know this? If you go to buy a stock, should you know this? Answer is absolutely. You should know how much you're paying for something. In that sense is like a price of banana, but it's really fascinating though. The price of a stock today we'll call P0. The price tomorrow we'll call P1. No big deal. However, the word expected. Remember now I'm pulling risk into the main part of our brains a little bit more active. Next week, we'll start talking about it pretty actively. The expected stock price, why? Because I do not know it today, and if you do, you don't need this class. Everybody is trying to figure it out, and we'll talk about that too. But let's call that P1. Let's assume that the expected dividend at the end of year is DIV1. Remember, I told you, dividend is like a coupon. The only difference is what? It's not a contract so if a company doesn't pay dividend, doesn't mean anything. Whereas if you promise to pay a coupon and you don't, it's a contract and you're effectively in default. Again, why expected? Because timeline hasn't happened as yet. Now, I'm going to show you a timeline again, but a stock is expected to live a very long time. What I'm going to do is, I'm going to do one period at a time. The reason I'm doing it is because it's very reasonable to think of something one period at a time, then a whole long stretched timeline. Having said that, let's draw the timeline because it's important to understand what the heck is going on. Very clearly, I'm standing p naught. The goal is to try to figure p-naught, i.e, how would I value a stock? Stock goes on what? Hopefully, when you issue stock, you don't want it to stop. You're the entrepreneur. By the way, every entrepreneur should think like that. That they live forever, their idea will live forever. You go on, what have I done now? Instead of trying to look forward and figuring it all out once, I'm putting it into bite-size pieces. I'm forcing you to think of only this first period. For the time being, just ignore the future, but keep this timeline is the back of your mind. That the future is there, but I'm just ignoring it for the time being, but actually I'm not. You'll see the key to that in a second. Let's see if we get this right. Turns out, if you're thinking one period at a time, the price today will be dividend at the end of one year plus what? The price at the end of the one year. Remember what I told you, what you could do is, you could say, ''Man, I don't want to deal with this long period of time. I'm going to sell the stock after one year, and try to figure out what price today should be.'' Now turns out, does this make sense? Sure. Imagine this is the last year or last six months of your coupon paying bond. What is the first thing? The coupon. What is the second thing? The face value. The difference is that you do not know the price of your stock six months from now. Whereas if you had a government bond, or a corporate bond and you had only six months left, you know that P1 is what? Thousand bucks or a 100 bucks, whatever the face value is. Does this help? You are now in a zone where you know that you have to somehow figure out, what the value of dividend and price is in the future. Not an easy challenge, but as I said, today we'll focus on what the heck is going on. Then we'll start using numbers. What is expected P_1? Remember, and I'm going to write this, that P naught is equal to DIV_1 plus P_1. But in which period is this? If I hold it for one period, I have to discount it by 1 plus r. Let's just talk about cash flows and discount rates. Do you think I've gotten both in the formula? Answer is yes. Dividend is one form of cash flow. What is the other form of cash flow? Me selling the stock. What is r? Remember, r is what? The best alternative investment of the same risk. If you're evaluating, say, Kmart, you cannot use the stock of Apple to figure out what the rate of return should be. They're different animals. We'll explicitly get into it when we talk about risk. The question I'm asking is, what is this P_1? Imagine thinking in the head of P_1. What is P_1 thinking? It's one period forward, what would P_1 be? I'll let you pause and think about it. Again, time travel to time one, what would P_1 be? Go one period at a time. Think about it. What would P_1 be? P_1 will be very similar to P naught, but removed how many periods? One period. P_1 should be this, and I'm going to write it and you see if you agree. Have you seen comic books? There is, what is Snoopy thinking? If P_1 is a comic character, what is it thinking? Who am I? It's googling, who the heck am I? A P_1 has to be the present value of what's expected at the end of the second year. We're going logically. What is DIV_2? What dividend will be at the end of period two? What is price 2? What will the price if I sell it at the end of the second year? But now what do I have to do? I have to bring it back to year one, so I discount by 1 plus r. Does this make sense? I hope it does, because this is key to understanding stock pricing. In some senses, it's a building block approach to what we already know. Whenever you have a tough problem, break it up into bite-size pieces. The key element here is please do not forget that this is all expected. We are ignoring risk explicitly, but the fact that something is expected means this is not known today. Let me ask you this, standing today, do you know DIV_1? No, it's one year from now, and unlike a coupon, it's not promised. Do you know P_1? No. In fact, P_1 is even more complicated than DIV_1. Why? Because when you think about P_1, what is P1? P1 is what DIV_2 plus P_2 would be discounted one period back to year one. I apologize, I used a cap r, let me use a lowercase r to be consistent, and while I'm doing it, what is the assumption I made? That the per period rate of return built into the pricing of this stock, if it's IBM share or Google's, is not changing. That's a strong assumption. But imagine your standing today, you don't even know for sure what's the rate of return for one year. Figuring it out, how will it be different in year two, is a little bit tricky. We're going to assume that the rate of return, which is largely based on the riskiness of this animal, is roughly stable, and the same. Let us see what is P naught then. You just saw that I've given you a flavor of thinking. Tell me what will P naught be? Let's modify P naught. By the way, I'm going slow here simply because this example, simple "derivation" is the most famous formula ever in finance, and finance is the most awesome thing. It has to be the most famous formula, even more famous than E equals MC squared. Take it or leave it. I believe it. P naught will be what? Let's write it. Is DIV_1 going to stay the same? Yeah. Because I'm just staring at the numerator at the top part of this equation. Remember I'm substituting for P_1. Let's substitute for P_1 plus DIV_2. I'm forgetting the I in this, divided by 1 plus r. I've substituted for P^2. But what do I have to do? I have now to discount the whole thing by 1 plus r. What have I done? I've just taken the value imagined about what P_1 would be and substituted it. Can I expand? Sure, I can. Let's do it. I'm a little cramped for space, but we'll manage. Dividend in period 1 will be divided by how much? Notice I'm dividing by 1 plus r. 1 plus r. Why not r? Because this is a one-period discounting. That was easy. But now what do I have? I have DIV_2 divided by 1 plus r and P_2 divided by 1 plus r to bring it to P_1. But now I have to discount it twice. Why? Because DIV_2 is how many periods from now? Two periods from now. Look how cool this formula is. It makes so much sense. It's DIV_2 divided by 1 plus r, divided by 1 plus r, which is 1 plus r squared plus P_2 divided by 1 plus r squared. Isn't this so logical? Think about it. Think about it now in the following way. What would be the price of the stock if I suddenly changed and added one more period in my mind? Earlier we had started off with what? In this specific formula, we had done the third experiment that we are going to sell the stock after one year. Let's do the third experiment that we're going to keep the stock for two years. What would be the price today? Well, it has to be the present value of what's happening in the two years. In the first year, the stock will give a dividend, DIV1, and I'll discounted for only one period. In the second year, it'll give dividend 2 discounted by two periods. Therefore, 1 plus r squared. Also I'll sell the stock for P_2, 1 plus r square because it's two periods away. This is so logical. That's what I love about finance. There's no bringing in some other factors suddenly that we never heard of before. It's tough, it's difficult, but it's very logical. The difficulty and the toughness of this example is not coming because of some profound mathematical thing. It's coming from a simple fact. Quick question. Do I know DIV_1? Remember I'm standing today, P naught. Do I know DIV_1? No, I don't know DIV_1. Do I know DIV_2? No, I don't know DIV_2. Do I know P_2? Heck, I don't know P_2. Actually, if you look about it, I should do this. What does E stand for? The expectation that I have today. Similarly this and similarly this. I am expecting to get paid, or if I get paid, this is my expected value. This is so important that it is not for sure. Therefore, talking about stocks without talking about risk is pointless. But I'm not going to be explicit about risk except the fact that these are not real things. These are what I expect. Quick question. During the technology boom, something dramatic happened. It was the following. People firms used to pay dividends regularly in the firms that are existing in the world at that point. Suddenly, lot of technology firms and lot of dividends are not being paid. We'll see why. But look what has happened as a result. Would do you call a stock any kind of contract? No. But what's fascinating about it is, is that if you really understand what a stock is, you understand what value creation is. Because the tragedy of a bond is it cannot be a participant in value creation. I mean a beneficiary because it has agreed to give money only with a contract in hand. Who gets to gain from the value creation? Stockholders. Who gets to lose from lack of value creation of an idea? Again, the stockholders and who's the ultimate stockholder? The entrepreneur who starts it's all. I'm going to show you some further developments of this formula, but I want to make sure that we are all on the same page. I have done this. This is what the formula continues if you keep substituting in the future and go from 2 to 3 to 4 to 5 to n. This is what the formula looks like. Hopefully, you can do it on your own. In fact, this is the one time I would encourage you to take a long, long break to mimic and understand what's written over there. Create your own notes. Think through this. Be careful, but not too careful where you're not getting the non-expressible beauty of what a stock is. What is the stock? Imagine now you're holding the stock for n periods. What is the coupon? Not for sure, but that it could be paid as DIV. How many of them? n of them. Are they being discounted appropriately? Yes. Because if the period is two, you're discounting by 1 plus r square. If the period is n, you're discounting by 1 plus rn. What the heck is P_n? P_n is the price at which you expect to sell the stock after n periods. I've separated the two out, and by the way, don't get too worried about that summation sign, the first is just summing up all the dividends and discounting by the period. If n is 1, that means you're discounting one period and so on. I've separated out P_n from this whole thing. The reason is very simple. If you think about this, and this is 20 years or 20 periods, and it's the bond we talked about, what is DIV_1? In our example, the coupon was 30 bucks. Coupon, known? Yes. What was P_n for a bond? It was $1,000, known in advance, and we called it the face value. Why? Because it's written on the face of the thing, right on the I owe you $1,000. Now what's the difference between this and this? Even if there are 26 month periods left, the first is I don't know whether the dividend will be paid or not. If so, by how much? I certainly don't know what price I'll be selling it at. Let me ask you this, what will P_n be? Will P_n be anything to do with the past or the future? Well, P_n itself, if you have finance in your blood, which I'm sure you do, will be the present value at that time of everything beyond period n. Let me ask you this. Let's keep expanding, expanding, expanding. Lets come to the first most important concept or formula plus concept plus beauty of a stock. What is the price of a stock if n is extremely large? In other words, the stock is expected to live for a fairly long period of time. Imagine, how long has Ford been around? A long time. When n is large, what will happen to the value of P_n today? Let's stare at this and we'll be done in a second. What will happen to this as n becomes close to this guy? What is this guy called in math? Infinity, meaning very far away. What will happen to this? This guy will turn to 0. Why? Because of, pause, compounding. What are you left with? You're left with finite parts of this? That's what this formula is called. We are going to take a break in a second. But I want to show you what this amounts to as n goes to infinity. This, by the way, is the most famous formula I have ever seen, and it's ingrained in my being, and that is the price of a stock today is the present value of a bunch of dividends way out into the future. The reason there is no finite price in the end is because its present value is essentially going to be 0. Does everybody get this? Please spend time on this. Please think about it. I have gone through the concept and now I'll go through the example upon example upon example. The reason I'm going to do examples is twofold. You need to internalize this. Also, stocks are the most fascinating way of showing value generation people could have thought of. Because there's no contract, there's only expectations. There's risk-taking, and there's value creation. See you in a little while or a long while, depending on how you feel. I would encourage you to read, think, before doing the examples we do together. See you soon.