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# Pricing a Stock – Details

Pricing a Stock - Details
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So hopefully you took my challenge, and let me just start, now, having fun. Okay, so I'm going to write that the price of a stock, and we have red today, P1 has to be equal to D1 + P1 / (1 + r). You see how simple this is. The key here though is that we have very little idea of what P1 is. Why? Because we know P naught but P1 will happen only tomorrow. And everybody thinks they can predict it, and everybody's kinda wrong. For simplicity, or for once, I'm going to put expectation in c and bulk expectation. So the fact that we don't know it, we usually have an expectation sign.
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And I'm not going to write all of that out. Okay, so this is the formula. However, let's think a little bit more about it. This is relatively, perhaps easy to say what it's likely to be, right? Because you know the pattern of the past, and there are patterns sometimes. If a stock hasn't paid dividend ever, chances of it paying dividend next month, low, and so on. However, this animal here, expected P1, is very tough. So let's think about if P1 were a person, what does he or she think? What is she thinking I am? It's like Shakespeare, to be or not to be. So P1 is thinking, who am I?
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It's got to be the case, and let's start working on this, it's got to be the case that P1 go 1 period of time, has to be what? D2, I'm dropping the expectation sign, plus P2 divided by (1 + r). The assumption of course I made, which we make very often because we don't know the future, is that 1 + r is not changing. If it were changing, you would know about it first, and therefore stock markets would adjust, and so on and so forth. So now substitute. What do you get? You get D1 by itself. Plus, and write this with me, and what would be even better is just pause and do it on your own.
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So let's substitute for P1. It will be D2 + P2 over (1 + r). And this whole thing will be divided by (1+ r).
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I'm going to keep going, and we'll take a break only after we've done the whole formula. So even if this is a little bit tedious and slow let's do it. So let's rewrite this. It will be D1 over (1 + r) + D2 over what? (1 + r) by (1 + r), so (1 + r) squared + P2 over (1 + r) square as well. So this is going to make a lot of sense, and that's why I'm in love with finance, is that it basically makes sense, right? Because if you're holding it for 2 period, the first dividend is discounted at 1 plus r. The second dividend at 1 plus r squared, and imagine you're selling the stock.
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It'll be 1 plus r squared divided by 2, exactly what time value of money tells you. Okay, so let keep going. Can you substitute for P2? And I'm going to say yes you can, and I will do that one time, and then we'll kinda try to see what the formula is. So substitute for P2, so it will become D1 over (1 +r ) + D2 over 1 + r squared + D3 over (1 + r) cubed + P3 over (1 + r) cubed. And that is because I'm repeatedly now substituting. So, first I substituted for P1, then P2 and so on. Let's keep going. Nothing to stop. Why? Because I said initially, a stock doesn't have a life.
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In fact, it doesn't promise you anything. You buy it for love, and you expect something of course, right? And also, I call it love and fresh air, it's the most fascinating thing ever. It reflects our belief in the future, right? I mean, right? It's cool. So, let's keep writing, and I'm going to do it a little bit faster now. (1 + r) + D2, (1+r) squared + D3, (1 + r) cubed + I'm going to write dot dot dot, which means on and on and on. Similar terms. Plus DN, (1 + r) raised to power n. N stands for some number far into the future, but could be ten, could be for, right. + Pn over (1 + r) n.
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Now, to make life simple, let me ask you this. As n goes to a very large number, infinity, what happens to this? If you take a price, even if it is hugely high. Remember when I said the power of compounding and the reverse of that is power of discounting? So if a stock wants to live forever, the price way into the future, it's present value is going to be close to 0. Of course, some of these very last dividends also will be, but making the 0 makes you get rid of that P part. And this is one of the most famous formulas of finance. It says the following.
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I going from 1 to n Di OVer (1 + r) raised to power.
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I. So what is this? This is called dividend discount model. This is the present value of dividends.
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What I'm going to do is I'm going to pause here and come back and do some good examples, but I want you to just think through this. Let's think through this together and you can come back. If I substitute i through n, it means the first i has to be? Mr.Gothom, explained, i = 1, when you substitute 1 you have D1 over 1 + r raised to power 1 which is substitute two, you get this. Substitute three, you get this. So this thingy over here, is a short form of writing something that's very long, and turns out it's a geometric kind of series, and it's very cool to show what it look like in the certain circumstances.
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And it's a very, very powerful thing. Remember we talked about and I say like differently than most people, and so be it. When we come back, we'll do some examples. We'll do some formulas and examples with them. Initially, simply. But I want to do a mega problem at the end to show you the beauty of what value creation can look like. It'll have to be set up properly, but I'll do it in ways that hopefully you can relate to, and you can see how fascinating it is to study stocks.

Do you have any questions? What was your key takeaway?