Portfolio Theory and the Capital Asset Pricing Model

My name is Rag Rao, right? I'm the Sir Evan, the Rothchild professor of finance at University of Cambridge, most of the Gresham professor business here at Gresham College. As a lot of people know this already because I see some familiar faces around the room. Um, this is the second in my series of lectures this year, which is on the big ideas of finance, right? So what I'm gonna talk about today is the second of the big ideas, and this is one of the first idea we're gonna talk about that actually won a Nobel Prize in economics. So there will be some maths involved here, but I'll try to make it as intuitive as I can, right? Especially late evening. You don't wanna get hit with a whole blizzard of equations. So let's try to make it intuitive. So all these, um, ideas are actually in my book, which is Short Introduction to Corporate Finance. Six chapters tells you one chapter per idea. So if you want to go back, read all about it, that's it. It's also in Chinese, if you prefer to read it in Chinese. Okay? So what are these six basic ideas? The first idea is the idea I talked about last time, the concept of net present value, right? But net present value as we see today, requires a discount rate. Requires an interest rate. So that is a topic of today's lecture, is portfolio theory and the capital Asset pricing model. And two people, Harry Markowitz and William Sharp got a Nobel Prize for this in 1990. The other five, the other four ideas, capital Structure Theory, also won a Nobel Prize for Ian Miller in separate years. Um, option pricing got a Nobel Prize for Shoals and Merton in 1997. Asymetic information, three people, Olof, Spence, and Sitz won a Nobel Prize in 2001. And finally the last topic, which will be in June next year, is a bit confusing because several people won the Nobel Prize in this area, including people who said completely the opposite things, but won the Nobel Prize in the same year. That's far and Shiller in particular, but we'll come back to that in June next year. So what is finance all about? Right? At its core, finance is very simple. It's basically a set of promises. Basically, somebody is saying, you know what, gimme some money now and in return I'll give you some fantastically large amount of money at some point in the future. The question which we have to answer as a finance person is, what are these promises worth, right? How much would you pay somebody who's promising to pay you a huge amount of money in the future? And last time you saw that, the answer to this question was about net present value. That means we have to compute the present value of those future cash flows to today and compared to the amount we're supposed to spend today. So if somebody says, I'll give you $500, five years from now, the question is what's the value today? How much are we willing to pay for that $500 worth today? And unfortunately to get that, we need one equation. And that is the basic equation in finance. That concept of net present value. So what is that equation? The equation is straightforward, it looks complicated, but all it's saying is the present value of a bunch of cash flows noted by C one, C two, and C three and so on, is just divided by one plus one plus squared, one plus of the power three. We discussed that in the NPV, um, NPV lecture, but the I see ideas, each of those cash flows in the future is worth c divided by one plus R today. Okay? The problem which we didn't answer there is that where does that interest rate come from, right? We have everything we need except we don't know what that discount rate, what's the appropriate R In that equation? We can figure out what the cash flows are. For example, if you're valuing a company, the cash flows are the earnings. If you're valuing a stock, the cash flows are the dividends. If you're valuing a valuing a bond, you're getting interest payments and the final phase value. So those are the cash flows. You know what the cash flows are, but how do we figure out the interest rate? That's the subject of today's lecture. Okay? So how do we think about this? Well, somebody comes to you and says, I want you to invest in my company, right? What you are thinking about is, okay, if I don't invest in the company, what else can I do with my money? Right? What you want is to find out what is the opportunity cost of investing in this particular asset. Because the moment you invest in this guy's company, the money's locked up. You can't take it out and use it for something else. You can't spend it on a good restaurant, you can't spend it on a movie, right? It's locked up in this guy's investment. So the question is, what is my next best opportunity? If I don't invest in this company, where do I put my money? But of course, that opportunity has to be equivalent, right? You can't say, okay, invest in my high tech company. The alternative is put my money in Barclays, right? The interest rate at Barclays is pretty low. And so it's definitely not comparable to anything which you might get from investing, for example, in open ai, right? So that means the opportunity should be equivalent in terms of risk to what the actual asset is you're being asked to invest in. Okay? But then the question becomes what is risk, right? How do we measure risk? So does it depend on how risk averse you are? Right? Maybe someone who's risk averse says, you know what? This is extremely risky. I don't want to invest in this, so I'm not willing to pay very much for this very risky investment. Or does it, if you hate risk, for example, will you pay less? That means, in other words, will your discount rate be higher than someone who doesn't care about risk? So basically, someone who doesn't care about risk will say, I'm willing to pay more for this highly risky investment. Someone who hates risk says, I'm not willing to pay very much for this investment. So how do you decide what is the right level of the opportunity cost for investing in a particular asset? Alright, so let's ask ourselves, first question is, what does history tell us about opportunities? So this graph here is taken from Eon and Synco fell. Basically they've been calculating every year, what happens if you invest money over time? And so what you have here, you go back to 1926 and invest $1 in small cap stocks. Basically small companies by the year 2022 and end of last year, you'd end up with $49,000 today, $1 in 1926, you'd end up with $49,000 today. However, if you invest in large cap stocks, big companies, companies like gm, Tesco, large companies, you'd end up only with $11,535. If you're invested in long-term government bonds issued by the UK, government issued by, you know, the US government, whatever, you'd end up with $130. If you had invested in short-term bonds, bonds of less than one year.$1 in 1926 would end up with $22 today. Right? And finally, inflation, what was worth $1 in 1926 is worth $16 and 58 cents today. Okay? So taking this graph, you've all seen this graph. Now imagine you've invented a time machine. You can go back to 1926 and you have the opportunity to talk to your granddad or your great granddad depending on how old you are. And so your granddad says, you go to your granddad's, introduce yourself and say, look, I'd like you to take $1 and put it into something. What is that something you will put it into? Let's go and ask that question. So basically scan that code in and you can vote for any of those opportunities. Which one would you pick? So small stocks give you 49,000, $52, $1 large cap, 11,535 long-term government bonds, $131, US treasury bills $22. What would you invest in? Okay, this, you know, this information, by the way, you know all this, right? So you've gone back in time, you've gone into the past knowing what you already know.<silence> Somebody here is super risk averse.<laugh>. I mean, given that they already know this, they're still choosing to invest in treasury bills. I, I would really like to meet that person after this lecture.<laugh>, two people, <laugh>, maybe they're online. This is also open to anyone online as well, right? Okay, so it looks like majority of people would say, yeah, makes sense, right? So invest in small stocks.$1 in 1926 ends up with $49,052. Today it's almost a no brainer, right? Knowing what you know, of course the problem is keep that open. By the way, I'll keep asking more questions. Um, the problem is You've persuaded your granddad to invest in small stocks and then one year later it's 1926, right? And great depression, his $1 has lost 92% of its value in one year. What's your granddad going to say? I say my good for nothing. Grandkid came back from the future claims, he knows everything. And I've lost most of my money. What a kid, right? I'm not, I wish the kid had never been born. So you have to have nerves of steel to say No, hold on, keep, keep investing. Don't sell. At that point, and you know, eventually it'll get to 49,000, but he's like, I have lost 92% of my money in one year. He's not gonna be very happy. Okay? So obviously then the answer depends on how long your horizon is. Okay? So this graph here shows you what happens if you invest in different types of assets. If you invest for one year, what you see is there are lots of, so the red lines there are for a portfolio of small stocks, the blue lines, the the green line there. The blue line is the s and p 500. That is a portfolio of large stocks. Corporate bonds are the yellow line and the green line at the bottom is treasury bills. Right? Notice that treasury bills, you almost never lose any money, right? It's flat. However, if you invest in small stocks for one year, there's a possibility of making a lot of money. Look at 1927 or 1929, sorry. But there's also a possibility of losing a lot of money, right? So you can either make a lot of money or lose a lot of money. So over a one year horizon, small stocks are not better, okay? Large stocks aren't better. Either you can make or lose a lot of money. What about five years? Well, there are still periods of time when you hold small stocks for a long period of time that you see that it's better off. With treasury bills, you have to go to 10 years before the numbers start moving up. So in over here, what you see is there are still periods of time when you're better off with treasury bills, right? Even over a 10 year horizon, stocks are not always better. It's only when you go to 20 years that the green line is perpetually below the red and the blue line. However, it's still, there are points in time when you have been better off in bonds than in stocks. So clearly the only way you can get that answer invest in small stocks is if you know the future for 120 years, right? If you don't, you never know going to know whether you're gonna make money or not, okay? In fact, we don't even know what the future is going to be in the next five years. So if you look at, if the future is gonna be like the past, well 1987, we had a high in October, 1987 where the volatility jumped by 150% on one day, the 19th of October, 1987. And similarly different crisis, the global financial crisis, the Greek crisis, the Euro crisis, the C Ovid 19 oil shock, all have these spikes in risk. So how do we predict the future? We don't even know one year ahead what's going to hit us, okay? In fact, that particular day, October 19th, 1987, the entire Dow fell by 23% in one day. So why did that happen? What do you think? What do you think? I have no clue. That is actually a brilliant answer. The answer is nobody has a clue. Right? 45 years later, we still don't know why it happened. Could it happen again tomorrow? Maybe we don't know. Nobody knows why it happened. Okay? Biggest loss in history, okay? The other ones are more explainable. 2020 March Covid, uh, October, 1929, great depression. Those are sort of explainable, but the biggest drop ever. We still don't know why it happened. Okay? Tough to figure out what risk is. I saw another documentary say that it was the dawn of agreement trading. That is absolutely, there are lots of explanations, but there's no convincing explanation. Nobody has said this is what triggered it. What happened? What happened on that day that started it falling? We don't know. We may think something made it worse. Portfolio insurance, other things like that. But nobody knows why it started. Okay, fine. Good answer. Okay, so what can we tell from the past? Not very much, right? It's not a very good guide to predicting the future. But what we can tell is if you look at the historical returns from 1926 or 2017, these are the frequency with which we see different annual returns. So the green one at the top is treasury bills. So you can notice that with treasury bills, what happens is you don't lose much, you don't lose any money, but you make a little bit of money, right? With AAA corporate bonds, you see there's a slightly wider dispersion over here, right? There's some possibility you lose money, but not much money. Some possibility make money, but again, not much money. Portfolio of large stocks, well some possibility. You lose a lot of money, some possibility you make money. And of course with small stocks, there's a possibility you lose a lot of money. But it's also a possibility you make a lot of money, right? So one way to think about risk may be you're looking at the width of that distribution. How dispersed are those returns over time? Are you gonna make money? Are you gonna lose money? There's a wide dispersion, maybe this is riskier. Okay, fine. Can we quantify this? Well, let's take treasury bills, right? We call treasury bills risk free. We call government bonds risk free. So taking government bonds, the question is, if I wanna buy a corporate bond, which has some possibility of going bankrupt the company, going bankrupt and not paying back its money, people say, okay, you want me to invest in that? I need a little more money than The amount I spend in treasury bills. How much more money? About 2.9% more. That's called the risk premium, right? It's the additional return which you ask for in order to invest in small stocks or s and p 500 or corporate bonds and so on. So what you see over here is what we call the risk premium. How much additional return do I want for investing in something? Okay? Now one question at this stage is, what is that risk-free rate that's crucial to our analysis, right? So let me ask you at this stage to go back and ask the question, if I have these three government bonds, which of these is the least risky? That would be number one. Which one is the most risky? That'll be number three. So go ahead and use your phones to move it around. What do you think is the least risky? And what is the most risky? So you can slide it up and down. Excellent. What this basically saying, German government, born least risky, Indian, somewhere in the middle. And argentian government born definitely the most risky, right? This is common sense, right? This is what most people would say is indeed correct. Unfortunately, according to finance, this is wrong, right? Why? Well, what does risk free mean? Risk free means what's the likelihood you're gonna get back the money? So let's go to the bond here, which we say is the second riskiest bond, the Indian government born. Suppose you go to the Indian government and you say you issued a bond, time to pay up. I want my money, but the Indian government has no money. What can they do? They can actually just print the money. It's just piece of paper, right? Print the money, give it to you, have to print too much paper. It's inflation. And that's why no one trusts the Argentinian government because you know, they started issuing pesos, but every time a peso bond was issued, they would always print more money, which means hyperinflation or inflation or whatever. So, but as long as the government is responsible, it's completely risk free because you will always get your money back. You will always get a piece of paper back saying this is the debt to the government. Okay? The German government, interestingly, bonds are issued in euros, however, euros are not under direct control of the German government is European central bank. And so the chance of this is almost close to zero, but there's a tiny, tiny, tiny chance of the German government says we want to issue bonds and euros. The ECB will say no, very, very small chance. But because of that very, very small chance, the safest bond is indeed your Indian government bond. The second safest is your German government bond. You may say the chance of that is zero. It isn't. It's slight possibility, right? And that's why we say it's not completely risk free. Okay? So now risk free is a very simple definition. Do you get your money back? And the answer is yes. If the government issues bonds in its own currency, you will get your money back because the government will just print the money if it needs to. Okay? So that's the definition of a risk-free rate, which essentially means there is only one risk-free rate around the world. You can think of American risk-free rate, you can think of English risk-free rate, doesn't matter because one of the principles of finance is if there were two different risk free rates, you would borrow it a cheaper amount, you would sell it the more expensive amount and you could make money. And as we'll see, as we saw last time, there's no such thing as a free lunch in finance. This is the basis of something called the carry trade. We'll talk about that in the last lecture. But even the carrier trade carries risks. It's not risk free, it's not arbitrage. But at the moment, let's leave that out, we are gonna assume risk free rate is a bond issued by the government in its own currency. Okay? So if you plot these returns, this one we just saw against risk. So you've got the volatility here, you've got the right return here, and what we see is a beautiful straight line and you say, oh wow, this is pretty cool. I've solved the problem. Because what this is telling you is high volatility, the higher the width of that distribution, the higher the return. And I've got a line, the line explains it. So if you tell me what the volatility is, I can put it on the line and I can tell you what the return should be. And that's what I'm looking for. Remember the return is the discount rate in that equation, which I'm trying to find out. The NPV equation, the R is a denominator at the bottom. And this graph seems to tell me everything. Well, no, the problem is it works for portfolios, but it doesn't work for individual securities. So if you look at IBM, Tesla, general Motors, Tesco, doesn't matter, they're scattered all over the place. There is no relationship between risk and return for all the individual securities. But that's what you're being asked to invest in. You're not being asked to invest in a portfolio. You're saying if I invest in something, one of those is what you're doing, okay? This is the problem we have. It doesn't work for individual stocks, alright? So what we have established so far is history is useless, right? We've also established that it doesn't for outside anything. Individual securities doesn't tell us very much. So we need to step back, right? What does theory say? What is risk? According to theory, the answer is we don't know, right? We really as finance people, we don't know very much, right? But honestly the good news is we get paid a lot of money for not knowing anything, which is a good thing. But anyway, the point of this is we can't, we finance people think about risk in one particular context. We focus on something called the variance and the standard deviation. What? Why do we focus on that? To understand that the variance is just the width of a distribution. How wide that distribution is, that's the average number. So how much do you see numbers on the left and how much you see numbers on the, like that's the width of the distribution. So you want a quantitative measure trying to find that width. Okay, fine. Why do we focus on variance? The good news is if you, if stocks are drawn according to a normal distribution, we only need two things to know everything about that distribution. We need to know the number in the middle, where the peak of that distribution is and the dis standard deviation, how wide that deviation can be. Now what's interesting about this, what's interesting about this is you can appear incredibly smart as a finance person, right? So let's assume, and normal distributions are everywhere, right? If you take all the heights, so people in this room, it'll be a normal distribution, most people will be in the middle, some people will be short, some people will be tall, weights of people, normal distribution, IQs of people, normal distribution, everything in nature is a normal distribution. So we assume that stocks follow a normal distribution. And the advantage now is, let's say your client comes to you and says, I'd like to invest in something. And you say, Hmm, I think I should invest in large company stocks. Really? How much money can I make? Say okay, I think you're gonna make about 13%. Why? Because history tells you that's a normal distribution. So okay, that's fine, but I'm also worried about losing my money. What's the chance I'm gonna lose my money again? Normal distribution tells you there's a 68% chance that a return drawn from a normal distribution will be within one standard deviation of the mean. So basically you say, you know what, you're not gonna lose. There's a two third chance, you're not gonna lose more than 7%, but you're not gonna make more than 33%. I said, well that still leaves one third. What about the other one? Third, no worries. There's a 99% chance you won't lose more than half your money and a 99% chance you won't more make more than 75%. That makes you sound super smart, right? I'm giving you numbers. A 99% chunk. People are like, wow, this guy is an amazing financial advisor, but all you're doing, pulling the numbers out of a normal distribution, so our stocks distributed normally answer is not quite right. What happens is bad stuff and good stuff happens a little more nor a little more than predicted by the normal distribution. Why? Because portfolio theory assume that everybody is acting by themselves. All of us are making individual decisions, but reality we don't. We look at each other. So if everybody's panicking, what happens to us? We panic too. So bad stuff happens a little more frequently than predicted. Good stuff. Fear missing out. They're like, oh my God, everybody else is making money. How am I not making money? So we buy more shares than we can. So, but for most of us it sort of looks like a normal distribution. So I'm gonna pretend it's a normal distribution. Okay? Alright, so let's restate this problem. Now, the problem we have is we want the opportunity cost. If you don't invest in a particular investment, okay, fine. That is the return, the next best opportunity. That's the return we need for the NPV formula we discussed in the last lecture, okay? But every individual security will have its own expected return and its own standard deviation. We'll see that in a minute, but, and in addition, every investor has his or her own risk preferences. So how do we come up with one number that everybody agrees is the only interest rate for that particular security? This is the problem that Markowitz and Sharp were faced with. Nobody had solved this and that solution to this question got them both a Nobel Prize. Okay? So let's see how that works. Let's start with the intuition. You have three possibilities, right? You can invest in an ice cream company in sunny weather, everybody eats ice cream. So your a thousand dollars investment becomes $1,200. That is a 20% return. Rainy weather, nobody eats ice cream. So the a hundred thousand dollars investment becomes 920. You have lost 8% of your money. Umbrella company the other way around. Sunny weather, nobody uh, buys umbrellas. Rainy weather, everybody buys umbrellas, the numbers are flipped. Okay? You make 20% or you lose 8%, okay? Uh, the probabilities of sunny and rain running weather are 50 50. This slide was not written in England, right? Obviously 'cause you'd say 80 20, but no, that's fine. So the expected profit, sometimes you're gonna make 1200, sometimes you're gonna make nine 20. The average of the two is 1 0, 6 0. That's the $60 profit you make. Okay? You've invested a thousand, that's 6% expected return. And the standard deviation is sometimes you make $140 more than 1 0, 6 0, that's a 1200. Sometimes you make a hundred and uh, $40 less than 1 0 6 0. That's nine 20, that's 14% here and there, right? Fine. Umbrella company, same thing. Opposite directions. T-bills don't depend on the weather. So regardless of whether it's sunny or rainy weather, you get back 1 0, 3 0, that's a 3% return and no standard deviation. You're getting the same amount every period. Okay? So if you have a choice between these three things, right? What would you choose to invest in the ice cream company, the umbrella company or treasury bottles? To put that into perspective, let me show you again what I'm actually looking for, right? So you have a treasury bill, zero risk, and 3% return. Ice cream umbrella, 14% standard deviation and 6% return. Well that's what you, what you are looking at. Excellent, right? Key part that you can see here is there's no pattern. You can see it. People are a little, they like ice creams a little more than umbrellas, but you know, there's nothing wrong with these answers, right? I mean all of the answers are totally fine, right? So you can't say that anybody's wrong on this basis. You like umbrellas fine, that's you over there, you like ice cream fine. That's you over here, you like to be safe, that's you over here. No big deal, okay? Nothing wrong. Okay? Now let's change the question a little bit. I'm gonna add a portfolio. Half my money in the ice cream company and half the money in the umbrella company. So in rainy weather, in sunny weather, the ice cream company makes money. So my $500 investment becomes $600. But the umbrella company loses money. The 900, uh, the thousand, the $500 investment becomes four 60. Add the two together. 1 0 6 0. Okay? Same thing over here. Opposite, right? The umbrella scheme company is making money. The umbrella company's losing money. And so you make $60 on average, which is a 6% return, but with no standard deviation, again, you're basically saying I'm going to get 1 0 6 0 because what you make in one company, you lose the other, they cancel each other out, put it together. What you have is a portfolio investment, which has the same returns as the tbi, but higher returns, right? Same risk as the tbi, but higher return. It has a lower risk than the ICM umbrella company and the same returns. So the question now is, if you have a choice between these four investments, which one do you choose to invest in? Somebody who's still obsessed with ice cream <laugh>, Right? And that's what I mean by the intuition behind portfolio theory. What you have over here is individually, when you're given all these things, it's very difficult to choose, but once you start combining things into a portfolio, some of that risk disappears. And so there is an overwhelming consensus that you should invest in the portfolio company. It is giving you the highest return per unit of risk, right? That's the idea. That's what we're going for. Okay? But to get there, we need a couple of additional steps. Let's start with a pair of securities. And I'm trying to find, this is the way the math comes in, so I'm gonna spend a little bit of time talking about that, right? So we are gonna compute the expected return, the variance and the standard deviation of a stock fund and a bond fund. And what I'm gonna point out is there are three states of the world, a recession, normal times, and a boom in a recession. Probability is 20%. Stocks do really badly. You earn, you lose 7%, but bonds do well. So you earn 17%, normal times, most of the time stocks earn about 12 bonds, own about seven. Boom. 30% of the time stocks own 28%. Bonds own lose 3% of the value. Okay? So this is what you have and you have to compute the same thing. Expected return, standard deviation or variance. Why normal distribution? Those are the only things you need for a normal distribution. Okay, let's do that. So how do you do the expected return? The answer is simple. You just say what are the different states of the world? And multiply it by the probability. So for example, the expected return to a stock fund is 20% of the time you get minus seven, 20% times minus seven. 50% of the time you get 12, 50% times 12. 30% of the time you get 28, 30% times 28. The expected return is 13%. Okay? It's just a weighted probability of those numbers. What about the bond fund? Same thing. 20% chance of getting 17, 50% chance of getting seven 30% chance of getting minus three. That's this equation over here. 6%, right? So that's the expected return. Okay, what about the variance? Well, for the variance, what you notice, you never actually get 13%, right? Sometimes you get more, sometimes you get less. So what you want is a measure that tells you how far away are you from that 13%. So in a in a recession, how far away are you from 13%, you're 20% less than 13 minus seven minus 13, 20% below 13, right? Normal times your 1% below 13, and then boom, you're 15% above 13. But if you just add 'em up, what do you get? First case you get minus 22nd case, you get minus one. Third case you get plus 15. If you just add 'em up, the negative number will cancel the positive number and it'll look like, oh, this is very safe security, right? So how do you convert a negative number to a positive number? Simple square it, right? So that's what we are gonna do here. So what we're gonna say is minus seven, minus 13, how far away are you from 13%. That's 20% below square it 0.04, same thing. How do you get your 1% below 13% square it, you get 0.01, same thing. 28% minus 13%, 15% 0.0 2, 2 5, right? And once you have those deviations, square deviations multiply by the probabilities, 20% time 0.04, 50% time 0.01, same thing, which you just did. You get the variance and of course take the square root to the variance, you get the standard deviation, you're going back, you're taking out the square, which you just did, right? So this is pretty much high school statistics, okay, next. So this is okay, we can do this, right? But what about now putting these things into a portfolio, what's the same thing? The expected return and the standard deviation of a portfolio. Half my money in the stock fund and half in the bond fund. How do I do that? Well, one way you can do that is to treat it just like another security. You say, you know, 50% of my money is in the stock fund, so 50% times seven plus 50% times 17, add them together, I get 5%, right? So that's it. And they do the same thing 50% times 12, 50% times seven. That's 9.5 50% times 48 plus 50% times minus three, 12.5. And that's a new security run. The whole process all over again, okay? However, in real life we don't have those probabilities. So I can't do this because I don't know what the probabilities are for a recession are what a boom is, what a normal time is. I have do something else. I want to go directly from here and here to here. And that's easy. Turns out if I do it this way, the PO expected return of portfolio is a weighted average with the amount of money I've invested in each of these. So in this case, 50% of my money is in the stock fund. That's earning me 13% on average, 50% in the bond fund, that's earning me 6% on average. Put the two together, I get my 9.5 straight. I don't have to go through the probabilities, right? Okay, so expected return is easy, I just wait all the numbers and put it together. Alright? But now I want to go to the variance. I've got these two variances. I wanna figure out what's the variance of my portfolio directly. I don't want to go through all those intermediate steps so I understand what the variance is. I need a new term called how these assets move together. Because remember the ice cream company and the umbrella company, they were moving in opposite directions. That's why I could reduce my risk. So I wanna figure out, get a measure that tells me whether they go in the same direction or in opposite directions. What do you think over here? Do they go in the same direction or opposite directions in a recession? One is losing money, the other's making money, normal times making money, sort of making money. This is making money that's losing money. So essentially they're going in opposite directions, right? So you wanna measure that's high when they're going in the same direction and low when they're going in opposite directions. And that's all the core variance of two securities. So this is like core variances over time. And usually this is the case. Stocks and bonds go in opposite directions except this year, right? So where everything crashed at the same time, but you know, that's bad luck. Okay? So how do we figure out what the core variance is? What it turns out it's actually surprisingly easy. So what we have is when security one, the stock fund is doing really badly, it's making 20% less than what we expected. The bond fund is doing really well. It's earning 5% more than what you expected. So what you're doing is multiplying minus 20 with 11 to get minus 0.02. So you're multiplying the two deviations together, one's doing well, the other's doing badly. Positive number multiply by negative number, negative number, both are doing badly at the same time. Negative number multiply by negative number, positive number, right? So basically if you're just multiplying the deviation, you should get all these answers right? So 0.05 in this case is a core variance here and we call it we sigma times. sp by the way, finance involves a lot of Greek. So if you sit through all these lectures, you'll end up with working knowledge of Greek. We'll talk about alphas, we'll talk about gammas, we'll talk about deltas. So when you go to Greece on a holiday, you'll be able to read the street science for sure. Okay? Alright. What's the relation between the variance in the cos? We've already done it. If he replaced the bond fund with a stock fund, the you product of the two deviations is a square of the product. So the variance is a special form of the co variance. The co variance of something with itself is its variance. Okay? So let's leave that. Now we get to a really complicated looking formula. This one says that the variance of portfolio is a weighted average of the variances, the individual parts plus a product of the two core variances. How do we understand this formula? Let's take an example. Imagine that you have a giant sack with two animals on it, right? An elephant and a mouse. So what's the, how much is the sack moving? That's a question you want to answer, right? So how much is the sack moves? Depends on two things. One is how much the elephant moves. That's the standard deviation of the elephant, right? You square it, the elephant is big, so it'll really dominated. The mouse is relatively small. So regardless of how much the mouse moves, the bag is not gonna move very much, okay? However, it also depends on how they react to each other, right? Obviously with the elephant and the mouse, the elephant dominates completely. So you're really not gonna see much of a difference. So no problem. Let's say put in two elephants into the back. Okay? So what you have here, same thing, how much does the sac move? But now it depends on where the elephant's moving in the same or opposite directions. If they're moving in the same direction, the back will move. That means high core variance. The back will move a lot. You can move in opposite directions. Negative Covance, the bag won't move at all. That's the idea here. Okay? So putting that into that formula, we're talking about a stock fund and a bond fund or an elephant and a mouse, okay? How many combinations can you have? Turns out you might think I only have two combinations, elephant or mouse. Actually there are four. What are the four combinations? Well, the elephant by itself, the mouse with itself, right? The elephant first and then the mouse. The mouse first, and then the elephant. The all four different combinations. Okay? So to put that, what you're going to do is four different boxes. So what you have is how much is the elephant moving the elephant by itself? How much is the elephant moving relative to the mouse? How much is the mouse moving relative to the elephant? How much is the mouse moving by itself, right? So that is essentially our formula here kind of looks complicated, but essentially that's all it is. So plugging all these things in, you get a variance, right? Which is looks like complicated thing, but we already got the numbers. That's the core variance. Those are the two variances, and over here are the weights. Okay, fine. So that's the ultimate. When you take all this, what do you conclude? Not really so much about being calculating the expected portfolio. The key is, if you look at that portfolio, that portfolio has a lower risk, lower standard deviation and a higher return than the bond funder by itself, right? So in fact, you can do this for any length of combinations. So 0% in stocks all the way to a hundred percent in stocks. You just keep running that formula again, and you get a graph that looks like this. The top line there a hundred percent in stocks, the bottom line point there is a hundred percent in bonds. That's 95% bonds, that's 90% bonds and so on. So the numbers are, you know, go on this. But looking at that one here, this is a hundred percent bonds, right? That's 7% standard deviation and a really low 6% return. So a question for you is, would you choose to buy a portfolio consisting of 100% bonds? Okay? Right? So again, pretty much of a no brainer. If you look at this, almost a no brainer, if you look at this, what you will see is if you invest a hundred percent in bonds, you end up right there, right? But if you go up straight here from, oops, sorry, if you go up, If you go straight up from the a hundred percent bonds to a hundred percent stock for the same risk, you get a higher return. Nobody should pick any of those combinations at the bottom. Forget a hundred percent bond. You should not even go for 95% bond. You shouldn't go for 90% bond. You shouldn't go for 85% bond because they're all at the bottom. So essentially what you have is what we call the efficient frontier. So this is basically saying these are some portfolio combinations which give you the highest return for the minimum amount of risk. Nobody will pick a portfolio that gives you a low return for a high amount of risk. That's pure and simple statistics, right? There's nothing beyond statistics at this stage, okay? You can generalize this to many securities, I won't go through this, but literally what it's saying is the weighted average of the expected returns, the individual parts, and this is every potential pair of co variances multiplied by each other. And I promise you more Greek symbols. So that is a double summation sign. I'm just adding 'em all up. Okay? Alright, so with three securities, what do you get? You get nine combinations. So one with itself, one with two, one with three, two with one, two with itself, two with three, three with one, three with two, three with itself, right? So with four you get 16 combinations. When I was working at BGI, we had about 3000 securities in our portfolio. That meant we had to deal with 9 million portfolio combinations. It would take us two days to, no, we wouldn't do it. The computer would do it, right? But it'll still take the computer two days to do it, right? But that's it. This is the formula, right? So it looks really complicated, but literally all you're doing is taking every possible pair and just adding 'em up, fine. So the veins of a portfolio is basically a matrix that looks like this. So what you have over here is the stocks. Every diagonal term is a variance, and every off diagonal term is a core variance. Okay? Let's keep this in the back of our head and now let's say in a many security world. So we sort of get the intuition that you put these securities together, you can come up with a portfolio frontier, right? So we take all these securities, add 'em together, you have every possible combination of every security in the world. You'll end up with curve connecting to curve connect. All this. And what you see is the outermost convex hull of all the securities is called our efficient frontier, right? So anything above the minimum variance portfolio is what we call the efficient frontier, right? So that's why Markovitz stopped. So Markovitz said, okay, you know what? We have so many combinations we worry about that people will hold, but we don't need to worry about most of them. People will only hold efficient portfolios, only portfolios on the frontier. They will not hold any of the internal portfolios because the efficient portfolios dominate them. That's a brilliant conclusion. Also completely useless. Why? Because there is an infinite number of efficient portfolios, right? Because that curve goes on all the way. So what Markowitz essentially said was infinite number of portfolio combinations, get rid of the infinite number of inefficient portfolio combination and you end up with infinite number of efficient combinations. So that's why you only got half a Nobel price, right? So what did William S. Sharp add to this? William S. Sharp said, well, there's something you've left out here and the one thing he had left out was a risk asset. Risk-free asset will be there on the Y axis, right? That is not there. So William Sharp said, fine, what happens if you put in a risk-free asset? How does it change what we are gonna do? It turns out that every combination of a risk-free asset and any other asset lies on a straight line. Why? Because the core variance of anything, how much does the risk-free asset move? It doesn't move, doesn't depend on the weather, doesn't depend on the economy, gives you the same amount, the core variance of anything with the risk-free asset, zero variance, the risk-free asset zero. So every combination of anything, the risk-free asset will lie on a straight line, right? If you have half your money in the risk-free asset, half money in the risk is halfway there and so on and so forth, fine. So what he then said was, alright, let's draw a tangent line between any of the intermediate points to the efficient frontier. Now, if you don't like risk at all, you can invest all your money in rf. If you like a moderate amount of risk, you can invest on the white line. The white line is always above every efficient portfolio on the curve. So what he's basically saying is a white line is a combination of RF and a balanced fund, which is a ency portfolio to that curve, right? That's all we need. So he says, everybody in the world should invest only in two assets. A risk-free asset, and a broad based market fund, which covers every security in the world. If you can get it, if you can't get it, take as broad based an index as possible. So the s and p 500 index or the MSC global MSCI global index take as broad based an index as possible. That's the efficient way to proceed, okay? But that means we can now derive an equation. Now this goes back to high school trigonometry, and the equation of a line is very straightforward. Y the number on the Y axis is equal to the intercept plus slope times X axis. Everybody presumably remembers that from high school. Okay, fine. What is, we'll drop that. That's Sigma M. That's the expected return on M, that's rf and that's the distance, the perpendicular and the base. So the slope of a line perpendicular divided by the base, right? So what do we have the expected return to a portfolio number in the Y axis is the risk free rate, the intercept plus the perpendicular RMM minus RF divided by the base sigma M times the standard deviation of the portfolio. That is a version of the CAPM. What that tells you is if you give me the expected, the standard deviation of the portfolio, I can plug it in here and tell you what the expected return of the portfolio is. How useful is this equation? Totally useless. Why? Well, it's because we already know that all these lines lie on a straight line. We don't need all those calculations to figure out what is the equation of that line. We already had it from history. What we want is the individual assets. How do the individual assets, what's the formula for that? And this formula doesn't work for that. So how do we go from a portfolio to an individual asset to understand that? Let's go back to this one here, right? I've now established that everybody in the whole world is holding the risk-free asset and the market portfolio M the balanced portfolio, right? The global portfolio. That means if I say invest in my company, that person is gonna say, okay, how much does my risk go up if I add you to what I'm already holding? But what am I holding the market portfolio? So the measure of risk is relative to what I'm already holding the market portfolio, okay? But going back to this line here, if I add one new asset to my, to my portfolio over there, I'm adding only one variance term the diagonal, but I'm adding two NN here and N there core variance terms, which is more important. The core variance or the variance, obviously two end core variances only one core variance term. The core variance is way more important than the variance. Put that into another way of thinking about it is you have a bag full of animals, right? They're all in the bag. You stick a tiger into the bag, the movement of the tiger isn't as important as the movement of all the other animals reacting to the tiger, right? That's why we say the core variance is how the other animals in the bag react to the introduction of the tiger. But the tiger itself moves a little bit, not going to really affect how much a bag moves may affect it a little bit, but really not very much. Okay? So the idea therefore is if everybody's holding the market portfolio, what we want is the core variance with the market. That means how much does the market move? If you add your security to that market, how close are you to that market? Okay, that is called the beta, another Greek letter. Okay, so what is the beta? Well, that's a core variance, but the core variance changes depending on how much the market's moving. If the market's moving a lot, the core variance will be high, purely artificially, nothing to do with the stock. The core variance is high. So you need to standardize it by dividing it by the variance in the market to control for points in time when the variance is high and the core variance, uh, or when the variance is low, right? You're controlling for all of that. So we also know if everyone's holding the market portfolio, the beta of the risk-free asset must be zero. Why? Because the co variance of anything with a risk-free asset is zero. We also know the beta, the market must be one. Why? Because the core variance of the market with itself is its variance. Variance divided by variance is one. So putting that into context, this is the final equation we are going to look at. What this says is the risk-free asset has a beta zero, the market has a beta one. So what is the equation of this line? Well, again, the same thing. Y is intercept plus slope times x, the intercept, the slope is perpendicular divided by base RM minus RF divided by one. The intercept is just rf. So that's a formula, right? Expected return is RF plus beta times RM minus rf, right? That number, that equation is called the capital asset pricing model. What that basically says is if you can compute the beta, which is the core variance of anything with the market, that tells you how risky that asset is, okay? In this particular case, expected return is RF plus beta time RM minus rf, right? That formula got William Sharp, his half of the Nobel price, right? So two parts, one part, getting to the fact that only people would hold efficient portfolios. The other one going one step beyond introducing a risk-free asset, which would give you this formula here. So everybody okay with this? With risk sort of, okay, let's see if you really understood this. Let's say we have two companies, solar mat in Madagascar where the environment, lots of degradation, deforestation over overgrazing and so on. Lots of major infectious diseases, high food insecurity, and it has repeated bouts of political instability including coups, violent unrest, and disputed elections. That's Madagascar. Okay? The other side, Tesco in the uk, while the UK has a whole bunch of environmental problems, right? We have health, we have obesity issues, ozempic, all these things. We also have high inflation, right? Um, and we have high political uncertainty, right? So all these things together, the question is right, which one is riskier? Okay? So that's what we want to establish, right? Which should have a higher discount rate. So what we have is the Madagascar central bank rate is nine 8.9. The Bank of England bank rate is 5.25 GDP growth rate 4.4. This is about 0.2 inflation rate 9.31, inflation rate 6.5. And the Madagascar market is about four times as volatile as the UK market. So for you guys, final question, what do you think should be the risk-free rates. Uh, sorry, the discount rates. So if you think Tesco is low risk, that should be here. If you think solar matter is low risk, it should be there. If you think solar matter is high risk, it should be here and so on. So just drag your pointer on your phone. Brilliant. Right? And this is common sense, right? Solar MAD is riskier than Tesco. Also completely wrong, right? Why? Well think about what the CAPM is telling you. It says how much risk do you take on when you add solar MAD or Tesco to what you're already holding. But we here in the uk, what are we holding the UK stock market. So if something happens in Madagascar, yes, solar mat is incredibly risky, but it's a small part of a portfolio. Madagascar is uncorrelated with the restaurant portfolio, the risk doesn't matter. Tesco is a big part of the UK market. If something happens to Tesco, our entire portfolio is affected because we are already holding the UK market. If you're a global investor, the UK is the bigger part of the world than Madagascar is, right? The key part is what are you holding? If you're holding a big global diversified index, which you should be, Tesco is way riskier than Solared, okay? That's the takeaway which we have over here. That's the counterintuitive part. We don't care about risk, which is unique to Madagascar. That's risk. You can diversify away from. You can just buy as many securities as you can and you don't care about Madagascar. It's a tiny part of your portfolio. But some stuff you cannot get away from. Even if you hold every stock in the world, the price of oil will affect you. War geopolitical risk will affect you. Climate change will affect you. How much it affects you depends on the beta. And that's what the CAPM says. The higher the beta, the higher the level of risk and therefore the more returns you ask for. Okay? And that's it. Thank you very much Professor Rao. That was great. I've got a few questions on here, then I'm gonna come to the audience in the room. So first question is, how do we account for the fact the correlations we input into the model are only historically relevant? We don't know how they're gonna work as forecasts. Um, good question. Good question. Yes, the numbers are all historical numbers and ideally we would like forward looking numbers. But with all due respect, if you know the forward correlations, you should not be here in this lecture. You should have your own private island somewhere in the Caribbean. Maybe not island anymore because of climate change, but whatever, right? You should have your own layer. So the best we can do indeed, is to look at historical stuff, but look at it for a very long period of time. So for example, when we compute things like the risk premium, we go back all the way to 1925. Why? Because we have that data. And why do we need to go so far back? Well think of 1925. We've had great depression. We have had world wars. We've had multiple local conflicts, we've had disease outbreaks, we've had literally everything. So the number is going to be there. Something history repeats itself. So the idea is long periods of history should indeed be work for future we hope. Right? Of course, we hope. I can't guarantee anything <laugh>. We hope. And and also we have to bear in mind that they don't affect the individual stocks. We have to be careful. We don't put a whole theory onto buying one stock, which 'cause basically people are gamblers. They, they're, they're optimists. That's why more people wanted ice cream than umbrellas. Excellent. Actually that is an interesting point. I'm gonna build on that a little bit. One of the things which a lot of people say is, I'm not diversified myself, right? I'm only holding Tesco. I'm, I'm not really holding the entire market. Why does this risk, why am I not taking solar Mat is very risky compared to Tesco. The answer is if you're holding, if you're not diversified, you are taking on a lot of risk. That means you're not willing to pay so much per share. Who is the person who pays a lot per share? The guy who's diversified. Because the risk doesn't matter that much to that person. A big mutual fund, a big pension fund, they're highly diversified. So they set the price. If you're not diversified, you're not in the market. You're not willing to pay high enough a price to get there. So it doesn't matter. So the people who are setting the prices are indeed the diversified investors. That's why all this works. Okay, Thank you. Now, anyone here would like to ask a question? I gentleman down here, just wait one moment for the microphone. Going back to the very early part of your talk, you didn't build in several factors. One, the inflation rate going at the current time in any market, which, which actually influences the way trading happens in markets. Um, which infl again impacts on the rates you get on various, uh, uh, on various investments. Um, and the other thing, you didn't actually on the, on the umbrella and ice cream model, you didn't allow us to have shares in ice cream for the six months in the warmer period and uh, six months for umbrellas in the winter period, which is where you get make the profit. That's a very good point. The second point in particular, you can switch back and forth between in the different parts of the year, warmer months, cooler month, fine. Right? But that doesn't affect the basic model. Even if you do that, allow that you can actually still run through exactly the same calculations to get the idea that you're diversifying. Right. And again, think climate change, right? Are you really gonna have warm winters or warm summers here? We don't know. Could be. But you couldn't. But we don't, can't predict the future, right? Um, your other question, um, was about things like inflation and other factors, expectations built in. We are assuming they're built into the rates which people are asking for. So they take that into account when deciding, number one, how much do government bonds to buy, right? Because they have to worry about inflation. How much will the bond actually by me? So those expectations are built into the numbers already. Lovely. Thank you. It's past, oh sorry, one more question and then that's the last one.'cause we are past seven o'clock. Sorry. I just realized Thanks for a great talk. Um, it seems like the core variance between one stock to all the other stocks in your portfolio is very important. How do you exactly calculate that? It seems like you assume that you need to have the expected, like the performance in three different categories, like normal time, refresh, uh, recession, and um Right. Good time. Okay. So the background to everything we did was to assume that we knew different states of the world and we assumed the probabilities. But later on, once we know what the returns are, right, based on historical factors goes back to a little bit what Lucy was talking about. We kind of say, okay, over the past a hundred years, that's the expected return for the stock. We are using historical data. But once you have the returns to each individual security, you don't need the probabilities anymore. You assume that built into the numbers. So it doesn't apply to new stocks that you don't have that historical, uh, true. But then think of the fact here that these are market, make, uh, market participants who are taking this into account. If you think a recession is now much higher probability than before, do you buy stocks or bonds? You buy bonds because you know that stocks are gonna do badly in a recession, right? So the numbers, the prices will adjust based on people's estimates of the true probability. Okay? These are assumptions and we'll see in the last lecture on, in June next year, that people sometimes go crazy and the market behave in very weird ways, right? But that's a subject for next year. Okay. On that note, can you please join me in? Thank you Professor . Thank you.