Welcome, as I said, a stock or a share, or equity, is such a new thing for most people, that even people actually trade it don't pause to see the power of it. So 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. So the first thing I want to introduce is some ideas about the pricing. So the been call is how is a stock priced. I'm going to give you some concepts, or more some ways of thinking about it, so P naut 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 absolutely. You should know how much you're paying for something. So it's, in that sense it's like a price of a banana but it's really fascinating though. So the price of a stock today, we'll call it 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 more active. Next week we'll start talking about it pretty actively. So the expected stock price, why? Because I do not know it today and if you do, you don't need this class. Everybody's 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, it doesn't mean anything. Whereas if you promise to pay a coupon and you don't, it's a contract, and you are effectively in default. Okay, again why expect, because time one hasn't happened as yet. Now, I'm going to show you a time line again. But a stock is expected to live a very long time. So what I'm going to do is, I'm going to do one period at a time. And 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. So very clearly I am standing P naut and the goal is to try to figure P nait, i.e, how would I value stock? Stock goes on what? Hopefully when you issue stock you dont 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. Okay, so you go on, whatever I had done now, instead of trying to look forward and figuring it all out once, I'm putting it into bite size pieces. So I'm forcing you to think of only this first period. Okay, so for the time being, just ignore the future. But keep this timeline in 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, and you'll see the key to that in a second, okay. So 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 have to deal with this long period of time. I want to sell this stock of the one year and try to figure out what price today should be. No turns out, does this makes sense? Sure imagine this is the last year or last six months of a coupon paying bond. What is the first thing, the coupon. What is the second? The face value. The difference is that you do not know the price of your stock one year from now, or sorry, 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? 1,000 bucks or 100 bucks whatever the face value is. Does this help, right. So 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 and then we'll start using numbers. Okay, so what is expected P1, right? So, remember, and I'm going to write this, that P0 = DIV1 plus P1. But in which period is this? If I hold it for one period, I have to discount it by one 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. Okay, r, what is r? Remember, r is what? The best alternative investment of the same risk. So if you're evaluating say, K-Mart, you cannot use the stock of Apple to figure out what the rate of return should be. They are different animals. And we'll explicitly get into it when we talk about risk. So, the question I'm asking is, what is this P1? Imagine, thinking in the head of P1, what is P1 thinking? If it is one period forward what will P1 be? I'll let you pause and think about it. Again, time traveled to time one, what would P1 be? Go one period at a time, think about it, what would P1 be? P1 will be very similar to P naut but removed how many periods? One period. So P1 should be this and I'm going to write it and you see if you agree. So have you seen comic books, right? And there is, what is Snoopy thinking? So if P1 is a comic character, what is it thinking? Who am I? It's Googling, who the heck am I? P1 has to be the present value of what's expected at the end of the second period, right? We're going logically. So what is DIV2? What dividend will be at the end of period two? What is price two? 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 just count by 1 + 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-sized 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. So let me ask you this. Standing today, do you know div one? No. It's one year from now. And unlike a coupon, it's not promised. Do you know p of one? No. In fact, p of one is even more complicated than div one. Why? Because when you think about p of 1, what is p of 1? P of 1 is what div 2 + p 2 would be discounted one period back to year one. And I apologize, I used a cap R, let me use a lower case r to be consistent. And while I'm doing it, what is the assumption I make? The current period rate of return built into the pricing of this stock, if it's IBM share or Google's, is not changing. And that's a strong assumption but imagine you're 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. So, we are going to assume that the rate of return, which is largely based on the riskiness of this animal, is roughly stable and the same, okay. So, let us see what is p naught then. So you just saw that I've given you a flavor of thinking. So tell me what will p naught be. Let's modify P naught. And by the way, I'm going slow here simply because this example, simple, quote-unquote, derivation is the most famous formula ever, ever in finance. And finance is the most awesome thing. It has to be the most famous formula, even more famous than e = MC squared. Okay. Take it or leave it. I believe it. So P naught will be what? Let's write it. Is Div 1 will stay the same? Yep. Because I'm just staring at the numerator att the top part of this equation. So remember I'm substituting for P1. So let's substitute for P1 + DV2 sorry I'm forgetting the I in this, divided by 1 + r. So I've substituted for P2, but what do I have to do? I have now to discount the whole thing by 1 + r. So, what have I done? I've just taken the value imagined about what P1 would be and substituted it. Can I expand? Sure I can. So, let's do it. I'm a little cramped for space, but we'll manage Dividend in period one will be divided by how much? Notice I'm dividing by 1+r. 1+r, why? Why not r? Because this is a one period discounting. That was easy. But now what do I have? I have DIV2 divided by 1 + r and P2 divided by 1 + r to bring it to P1, right? But now I have to discount it twice, right? So why, because DIV2 is how many periods from now? Two periods from now. So look how cool this formula is. It makes so much sense. It's DIV2 divided by 1+r divided by 1+r, which is 1+r squared. + P2 divided by 1 + r squared. Isn't this so logical? So 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? So earlier we had started off with, what? In this specific formula, we had done the thought experiment that we are going to sell the stock after one year. Let's do the thought 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 discount it for only one period. In the second year, it'll give dividend two, discounted by two periods, therefore 1+r squared. And also, I'll sell the stock for P2. 1 + r squared because it's two periods of A. 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. And 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 DIV1? Remember, I'm starting, standing today P naught. Do I know DIV1? No, I don't know DIV1. Do I know DIV2? No, I don't know DIV2. Do I know P2? Heck, I don't know P2, right. So 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's not for sure, and therefore talking about stocks without talking about risks is kind of 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 traumatic happened and it was the following. People, firm used to pay dividends regularly in these firms that are existing in the world at that point and suddenly lot of technology firms and lot of dividends are not being paid. And we'll see why. But look what has happened as a result. Would 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. So who gets to gain from the value creation? Stockholders. Who gets to lose from value, lack of value creation of an idea? Again, the stockholder. And who's the ultimate stockholder? The entrepreneur who starts it all. So 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. So have done this. And this is what the formula continues if you keep substituting in the future and go from two to three to four to five 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 is 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. So 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 are discounting by 1+R squared. If the period is n, you are discounting by 1+ Rn. What the heck is Pn? Pn is the price at which you expect to sell the stock. After n periods. Right? So I've separated the two out. The first, 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. So if n is one, that means you're discounting one period and so on. And I've separated out Pn from this whole thing. And 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 DIV1? In our example the coupon was $30. The coupon. Known? Yes. What was Pn for a bond? It was $1,000, known in advance and we called it face value. Why? Because it's written on the face of the thing. Income, I owe you. $1,000. Now, what's the difference between this and this? So, 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 Pn be? Will Pn be anything to do with the past or the future? Well, Pn 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. So let me ask you this, let's keep expanding, expanding, expanding. Let's come to the first most important concept or formulas, 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? Long time. So when n is large, what will happen to the value Pn today? Let's stare and 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. Right? What will happen to this? This guy will turn to 0. Why? Because of post compounding. So, what are you left with? You're left with finite, parts of this and that's what this formula is called and 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 and this by the way is the most famous formula I have ever seen. And it's engrained in my being. And that is, the price for stock today is the present value of a bunch of dividends way out into the future. And the reason there is no finite price in the end is because its present value is essentially going to be zero. 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. And the reason I'm going to do examples is two-fold. You need to internalize this. And 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.