Hi. I hope you have had a chance to, kind of, carefully review what we did just before this break I took. And I wanted to remind you that these breaks are more to kind of give you my sense of where you probably need to , need to stop but remember this is you in many different ways of learning. There are lots of you there, so it's up to you too, to always take a break, and if you don't understand something, you have other resources. So you don't have to go through these videos till you feel comfortable moving forward. I'm emphasizing this right now because the idea of a stock is not easy to comprehend. So let's just quickly look at this formula. What this formula says is basically what we know about cash flows is that if you have cash flows going on for a long time you'll have to discount them at an increasing rate because of the gains impounding. So, that's all that it's saying. Why dividend? That's just a name for the way the stock pays stuff to you, right? Like for example for a bond, it was coupon, and then face value. Why is no face value here? Because, you expect when you buy a stock. That the stock, even if you are going to sell it, is going to have value. And when you sell it, the only way it will have value is, if it is expected to move on, not just die. So this inherent built in value coming from looking forward is extremely important to internalize. Okay. So at any point, a stock unless you know for sure is not going to last very long. Has value because of the longevity, that is, so that's a, that's a point that you need to understand and really emphasize. What I'm going to do now is I'm going to start by giving you examples and I will, I'll stop, take breaks, but you take breaks as I said whenever it's convenient for you. Okay. The special case of a stock, which is I call a dividend stock. And I think many people in the real world would probably call it an income stock. This is the kind of stock that kinda gives you a steady flow of "Income." and the income paid by stocks is called div idends. So let' s look at a special case. Suppose dividends are expected to remain approximately constant. What would be the price of a stock? So the notion here is, that if you remember I talked about, a little while ago I think it was, but when I was doing annuities and so on, I just introduced the concept of perpetuity. So what is this saying? Generically first it is saying, that you're standing here and you're getting approximately the same C, cash flow and for a very long period of time, okay? The only thing twist here is that we'll call this cash flow dividends. So the question is what is the formula for this? Turns out and if you have the time you can do these calculations for yourself and derivations actually for yourself. But I, I think it's very cool to derive this, but imagine the formula is the simple as possible. C / r. I actually, I apologize, because I sometimes use R, r but R unqualified, means the discount rate or cost of capital. And quickly, one more time, where does it come from? It comes from your competition. So whenever your thinking r or your cost of capital, you are thinking market. If there was no market, it would very tough to figure out your value, because values are always relative, okay. So see in this case it becomes DIV / R. So this is the formula. What I'm going to do is I'm now going to try to derive it for you. And this is where I think you need to put in some work based on the level of curiosity you have. And you can look at the books I have recommended to you or you can sit down, if you're curious, just derive it. And remember, this is equal to what? Div one, (one + R) + DIV two (one + R) square going on for a long time. In fact, perpetuity is almost infinity. The only constraint I'm putting on this is these two guys are the same. Approximately. And why are, why do we use these formulas? Because remember, a stock, you don't even know what the differential is going to be in the first year, second year, third. So getting too precise can be actually hurtful to your thinking. You are doin g a ve ry detailed calculation of stocks, doesn't make that much sense. So we will use formulas like these because they kind of capture both what's going on and in some sense what happens in the real world, okay. So, this is basically the formula. And, let's go back. So, we know that if it's dividend is, stock is a constant level of dividend whatever it is, let's do this. Let's, let's spend five minutes and you do it with me. And I am assuming when the problem is relatively straight forward, we kind of do it with each other. Otherwise, just take a break, do it and we'll come back to it. And I will try my best to make sure that I'm making a good judgement about what is doable together and what is, you want to do a little bit of a break and test yourself. And remember, you have always the opportunity to go to the assessments and assignments to do similar problems and then come back. Okay? So this problem is relatively easy. It says, suppose Green Utility is expected to be a dividend. To pay a dividend of 50 cents. Not to be a dividend, but to pay a dividend of 50 cents per share for the foreseeable future. And the return on the business is ten%. What does this mean, return on the business? It means another way of saying that the cost of capital belongs not to you, not to anyone, but the type of business you are in. And that return is a function of demand, supply, and everything put together. What should be the price of the stock? Now, just take a pause, think about it. I'm going to start scribbling stuff on the board. So what is it, time line wise? Very straightforward. I'm getting 50 cents dividend for a long, long time. And I'm asking you, what would the price of this stock be? And I'm calling it the utility, because it turns out utilities are regulated. And it's very common to view them as income stocks. As opposed to another example I get into, which is at the heart of the rest of this session this week, it's called growth stocks. And I just love that stuff because it will convey to you what really is going on. And h ow growth is good, how growth could be bad, and so on. But let's stick right now with the stock that's not planning to grow, but is planning to pay 50 cents. You know the formula for this is what? .50 / .10 which is DIV / R. We just did. Right? What is that? Point five over point one is the same as what? Multiplying by ten. Why? Because one / .1 is same as multiplying by ten10 when you're dividing, right? So this is five bucks. So this was pretty straightforward right? What I would encourage you to do is think about how easy this is to value. Right? So. You took 50 cents and you just multiplied by ten and you got five bucks. And it's basically that, and people do this when, when you're comfortable with this level of, the assumptions behind the ease of the formula. You just suddenly realize how, how cool it is. You know, how people are so comfortable with numbers, mainly in the financial world. It's because they use formulas like this. That's what. It's ingrained at the back of my head, and therefore I can feel very comfortable. It's not that I'm very comfortable calculating complicated formulas with numbers in Excel. In fact, I shouldn't be. You know, I have better things to do. Okay, so let's just see. What does forever mean? Now, many people get caught up in, nothing is forever. How could it be forever? Let's just, this is not quite real, okay? So let, let me ask you the following question. Let me assume that forever means 30 years. And by the way, that's, that's not terribly long, right? A lot of companies do survive 30 years. That's not the important point though. We are trying to price a stock that is not expected to die tomorrow, because there's no point in doing that, right? So let's take this example, same example, and say okay, I got them I mean forget about this perpetuity stuff. It doesn't make sense. So let's just assume it lasts for 30 years. And the dividend is .5 so what am I doing and what is this called? Have you done this before? I think we have, right? What is this called? This is called an, an nuity of 30 years. The only difference between an annuity of 30 years and a perpetuity is the perpetuity is going to go beyond the year 30. Okay? So just recall, what was the value of the perpetuity? It was 50 cents / .1, very simple. So, 50 cents multiplied by ten was five bucks. Keep that at the back of your mind. Now, let's do this on a calculator, you see, what is going to happen. You can't do this in your head and that's the part of the proposition, value proposition I was talking about. So let's go to the next sum. Let's do equal sign and what are we figuring out PV. I'm actually much slower than you probably by now, and you guys are just rolling along with this stuff, and saying Gautam come on, go fast. I don't type very fast, that's, that's the way I am. Well anyways, so the rate is .5, that can't change. And, how much am I, sorry, rate is .1, right? There you go, I'm talking and I'm messing up numbers. The rate was ten%. I think I got it right, .1. The number of P that was what? Not infinity, I don't like that, but 30 is fine. And how much was my money? 50 cents And I hope my fingers haven't done anything bizarre. What's the answer or what's the value? Look, it's $4.71. Why did I do this? Let's go back. So what's the value? If I use an annuity, value is 4.71. Let me ask you this. Are you sure for the next 30 years you'll get that dividend? Because if it was exactly true that you expected it and you got it, there's something really magical about you. Oh, the real world. The real world doesn't operate like that. It's approximately that, right? So getting very precise about five bucks for 30, I mean, 50 cents for 30 years would make sense if you were exactly sure that's going to happen. But why be so precise about something that you kinda feel will happen? So this is a very powerful way of showing you the value of a perpetuity. How much did we calculate the actual perpetuity? And this is, by the way, seemingly, a very simple example. But it's, it's, it's a very deep issue, right? So, so this shows you why finance is bo th art and science. All your numbers are wrong, so what's the point getting very precise about being wrong, right? So let's compare these two. Am I close when I do the perpetuity of five bucks? Yes, I'm pretty close. Which was easier? Heck, I could do the five bucks in my head just multiply 50 cents by ten, right? Okay, now annuity of 4.71. It's for thirty years, right? So tell me what is 29 cents? Five - 4.79 is 29 cents. If you could think for a minute, tell me what that is. And it will kind of make you pause about what's going on. If the present value of the 50 cents in year 31. After that forever, so getting 50 cents forever, on the face of it should be what? Infinity. But you see now the power of pause again, compounding? At a ten% rate of return, the money that you get after thirty-first year, thirty-first year is almost trivial. It's only 29 cents even though it's forever. So recognizing that the interest rate is positive, and for stocks. Stocks are risky. Relative to bonds, they're likely to be high. You know that formulas like perpetuity's will bring you so close, that you don't need to necessarily be very precise. I am not saying don't use Excel. I am saying most of value of the framework comes from your thinking not from your answers. They are all wrong. I hope you found this little example very useful because formulas like C / R are used all the time. They are bases of what are called multiples in finance. Venture capitalists, people in iBank, in iBanking. People who value stocks don't try to get too precise. On the other hand, bond pricing. I just touched upon a little bit can get very, very technical and precise. And the reason is, there is uncertainty only in one thing. Fundamentally, right? And government bonds for example. If you expect cash flow to be paid and uncertainty in interest rates is driving everything, so you can get really precise in trying to model that. But anyways, here in stocks, everything is uncertain. Right? So what do you, what's the point in getting too precise about pretty much everything because you can't, okay. Now let me move on to something that's much more interesting in my book. Suppose dividends I expect it to grow at a rate of G per year. What is the price of this stock? And this is called a growth stock. And I'm sure you've seen examples of these and things happening as we speak. Which is the biggest company in the world, right now? In terms of value of the stock. Remember you, when I say value for company, I could mean many things. The first thing I could mean is, being a finance guy, what is the cap, market cap, which is the value of their stocks? But companies also have debt. So when I say value of the whole company, people would want to include debt, which makes sense, okay? So anyways, so which is the company whose stock value is most in the world? It's Apple. Apple has almost gone and survived. Almost died and survived and then grown rapidly in different phases. So, I would call it a growth stock but it depends on where is it in it's life cycle or new idea generation, regeneration, or whatever. Okay? So, let's see how this would be priced. Look what I'm saying here. Your P-naught here where there first dividend is DIV one. You can also call this, generically, C1. You see, that's, I told you, the nice thing about finance is, once you know time, value, and money then we'll do risk at a fundamental level, you can do any problem because the beauty is the same framework, same tools. Just the symbols change, the names change. Okay, so the first is C1. What is the second one? Div two so why am I labeling them now with one and two and I didn't do so before? Because the numbers are changing right? And in this case, DIV two is DIV one, one + G and so on. Now obviously we'll see and I'll show you the formula again. It looks very similar to the previous one. Let me first show you the formula but remember genetically what will DIV two be called? C2 and so on. So let me write the formula down and then we'll, by the way this is the only form, formula that I haven't d erived it for you and the reason it will just take up too much time. The generic formula is C1 / R - G. What is G? It's the growth rate in your cash flows. In this case, expect a dividend. What is C1? The first cash flow. So this is very important. First cash flow is C1. G is the growth in, the cash flow not an R, R is already a percentage, right? So these are both percentages. R and G both are percentages. So I am not subtracting dollars from a percentage, okay. And what is to use this formula, it makes sense to do what? If R is greater than G. If R is not greater than G, what will you have to do? You have to go the long way. So do C1 / one + R. C2 / by one + R squared, until such a point that R is greater than G. And that will always happen. Think about it. If G was greater than R, you would own the world. It's, it's, it's not possible in a steady state. But those are things that you'll get to practice in your assessment. So let's, let's call this the formula, and replace it by DIV one / R - G. Okay? And I'm going to use examples where R is greater than G. But as I said again, don't use a formula when it doesn't make sense to use. For example, if R = G what are you going to do? Sit there stare at the formula? You have something divided by zero, go figure. We use the long method. Okay. And that's what, another thing I don't like about the way we get taught math and algebra is, we are never taught in a context. I shouldn't say never. I was lucky in high school. I was taught math always in a context. And I benefited so much by it. Instead, many times, we are taught stuff and all we see is the back side of the person teaching. And that's not very interesting, anyways. Okay. So, let's get started with the formula. And I'm going to now let you take a little while to do it. We'll, this is a good time to take a break but let's first read the formula. Suppose Moogle Inc. And apologize, my sense of humor is limited, is expected to pay, Ryan is laughing with me. Is expected to pay dividends of twenty per share next year. You see how I have to specify the first dividend? And the dividend s are expected to grow indefinitely at a rate of five% per year. Again, the word, indefinitely doesn't literally mean forever, remember? It's, we're using formulas as an approximation. Stocks of similar firms are earning an expected rate of return of fifteen% per year. Why am I saying this? Because I want to repeatedly, repeatedly remind you that the fifteen% is not owed by, owned by you. In fact, it doesn't belong to anybody. It belongs to the marketplace. Businesses get different rates of return due to a fundamentally different set of risks. And we are going to get to that next week. What should be the price of a share of Moogle Inc? Okay, so just take a minute. Try to do it in your head. When we come back. We'll do it in a, with together and see how easy it is. Okay? Take a break. See you soon.