# Lecture 4 Portfolio Diversification and Supporting Financial Institutions

## Overview

In this lecture, Professor Shiller introduces mean-variance portfolio analysis, as originally outlined by Harry Markowitz, and the capital asset pricing model (CAPM) that has been the cornerstone of modern financial theory. Professor Shiller commences with the history of the first publicly traded company, the United East India Company, founded in 1602. Incorporating also the more recent history of stock markets all over the world, he elaborates on the puzzling size of the equity premium and the very high historical return of stock market investments. After introducing the notion of an Efficient Portfolio Frontier, he covers the concept of the Tangency Portfolio, which leads him to the Mutual Fund Theorem. Finally, the consideration of equilibrium in the stock market leads him to the Capital Asset Pricing Model, which emphasizes market risk as the determinant of a stock’s return.

Lecture Chapters

- Introduction [00:00:00]
- United East India Company and Amsterdam Stock Exchange [00:01:14]
- The Equity Premium Puzzle [00:16:19]
- Harry Markowitz and the Origins of Portfolio Analysis [00:21:09]
- Leverage and the Trade-Off between Risk and Return [00:29:41]
- Efficient Portfolio Frontiers [00:39:55]
- Tangency Portfolio and Mutual Fund Theorem [01:00:21]
- Capital Asset Pricing Model (CAPM) [01:09:20]

## 4.1 Introduction

Professor Robert Shiller: OK, so this is the fourth lecture for Economics 252, Financial Markets. And I wanted today to talk about some really basic concepts, about portfolios. A portfolio is a collection of investments. And I want to talk about risk and return, and eventually get into the core theory, which is the Capital Asset Pricing Model, in finance.

But first I wanted to say something about last lecture. Last time, I talked about innovation in finance. And I presented finance as a sort of branch of engineering in a way. We invent financial devices. And the devices serve certain functions, and, in order to serve those functions, they have a number of details that have to be gotten right. Moreover, there’s a process of invention, and the process of invention involves experimentation, and when an experiment doesn’t work, we forget about it and we move on, but when it does work it gets copied all over the world.

## 4.2 United East India Company and Amsterdam Stock Exchange

So, I thought a nice way to transition to today’s lecture would be to talk about one very important moment in the history of finance: when the first real important stock was invented. And it was, see if I can spell it right, Vereenigde Oost-Indische, I might be misspelling this, Compagnie 1602. This was the first–Did I get that all right? I think got it right! That’s Dutch for the United East India Company. It was founded in that year. It was a time, when Holland was at war, and the government was worried about the economy and willing to experiment with raising capital to keep the economy prospering. And someone had this idea. Let’s start a company with shares in it, and let’s trade them.

And in the same year, and I can’t write this in Dutch, but they created the Amsterdam Stock Exchange. And initially it had only one stock. And so this is called VOC. OK? And it was a trading company. They were going to set them up, and they were going to buy ships, and they’re going to sail all over the world, and they were going to trade in various commodities. So, it sounds pretty basic. But no one had ever done this before. So, it’s interesting how much got invented in this one year, 1602.

First of all, they invented a corporate logo. I don’t know if I have it right. It was something like V-O-C, I don’t know if I did that right, just like we would put on, you know, advertisements for a company today. Maybe that’s not very important. But what’s really important, also, is that this was a long-term venture. There were lots of ventures already in Europe, where a group of merchants would get together and they would pool their money for one trip. They would send ships out, and these ships would trade and come back, and then they’d dissolved the whole thing. But this was different. This one was going to go indefinitely. And in their initial announcement, they said, we’re going to set up operations all over the world. We’re going to have an office in India, and another one, I don’t know, in Indonesia, and, it’s a big thing. And in the New World, in America, but primarily East Indies, from the name.

But the interesting thing is they set up a stock exchange to trade shares in it. And the stock exchange arranged that you could trade every day. So, there was lively trading. This was part of the idea. Because when they set up a company in those days, you could get your shares when they founded the company, and that was it, right? I mean, you couldn’t trade them, or maybe you could infrequently. The company might open its books once a year, and they would take new shareholders in. But someone had–hey, this is an idea. Even though the VOC doesn’t open its books regularly, we can trade them every day. What difference does it make? You know, so we’re going to have stockbrokers on the Amsterdam stock exchange. And maybe they’ll own some shares in VOC, OK? And then they’ll have an inventory of shares. And then somebody wants to buy some, you buy them from the broker. You don’t have to contact the company. And then the broker will, you know, maybe at the year-end, will report to the company that you own the shares. But doesn’t have to, right? The broker does it. All right.

So, the broker says you own these shares. You trust the Amsterdam Stock Exchange, because they have rules and code of ethics, so you think you own, well, you do own VOC shares if you buy them from a broker. The VOC doesn’t know it yet, but you’ve got the shares, because the broker is a member of the Amsterdam Stock Exchange and says that you have the shares. Then, almost immediately after 1602, a funny thing happened. Can you guess what it was? The brokers started selling more shares than they had, right? What’s to stop them from doing that? Or some of them did that, right? So, the broker maybe owns some shares in VOC and he gets lots of buyers and the broker ends up selling more, more shares than he has. And he thinks, well, I’ll get them later, you know. And he says, what do my customers care, if they own shares, because I don’t report to the company right away anyway. I’ll make good on this. I’m a broker. I know what to do. I’ll buy them later and I’ll get them.

So, you see what starts to happen? There are more shares out there being traded than there exists in the company, because the broker is selling shares that he doesn’t own. And so there began in, way back, even in this time, there began what we call short-sales, or short-interest. It happens when you set up a stock market and you allow, well, we would call it today street name, owning a stock in street name. We have stock exchanges, many stock exchanges in the world today, and they, including–by the way, the Amsterdam Stock Exchange is the oldest stock exchange in the world and it’s still trading. But it has merged. First, it merged with Brussels, and Paris, and now they’re called Euronext Amsterdam. But they’re still doing this. Nothing stopped them in over 400 years. They keep doing it.

But this stock exchange, and many others like it, allows brokers to sell you stocks in what’s called street name, OK? And what that means is that when you buy shares, the broker puts in your account that you own these shares, but the company doesn’t know it, because the actual ownership is registered in the name of the broker, OK. And so, the broker is selling you shares. And it’s only through the broker that you know that you have shares. So, the broker on the stock exchange may be short. May have sold more shares then he or she has. That’s all right, OK? It happened as long ago as right from the beginning of the stock market.

There was a scandal. I was reading the history of this, and, who was the guy? Isaac La Maire, a Dutchman, in 1609. He was not a broker, he was a businessman. He was able to sell more shares than he had. So, he had negative–a broker allowed him to do that. And so, he had a short-interest in VOC, massive short-interest. And people who own VOC shares started thinking, what’s going on here? Someone is selling, he’s borrowing shares from a broker and selling them. That tends to bring down the price. And there was a downward movement in the Amsterdam stock market. And this guy was blamed for having shorted the stock, and forcing down the price. And so, the Amsterdam Stock Exchange, for two years from 1609 to 1611, banned short-selling, but then they decided to let it go again.

The point of all this discussion–I’m telling you a story about Holland 400 years ago, but the reason I’m telling you the story is to try to emphasize how certain things just happen naturally. Once you set the framework up, you set up a big company, and it’s a company that lasts a long time, OK. It’s very valuable. Anybody can buy shares in it, OK, so it’s democratic. And the value is very uncertain, because this company is going to be in business into the far future. And it’s building a whole arrangement, an empire of trading posts and ships, and who knows what it’s worth. So, the price is very uncertain, and buying it is a sort of a gamble. And so, the price starts fluctuating wildly. And it attracts all kinds of interest. And some people think it’s going to go up, and some people think it’s going to go down, and they start debating about this, and wondering about this.

And some guy like Isaac La Maire thinks it’s going to go down, so he wants to short the stock. He wants to sell, he doesn’t want to buy it, he wants to short it, so that he can have a negative quantity. Other people are really positive and excited about it, and they want to buy all they can get. And they want to even buy more than–they want to borrow money to buy the stock. So, you have this tension between the shorts like Isaac La Maire and the gung-ho traders who want to buy it. And it creates a lot of volatility in the market. But the whole effect of this is to create interest in the stock. So, it brings in money. And it ultimately made the VOC very successful, because so many people wanted to give money to this trading company. So, they were able to build hundreds of ships, and set up big outposts all over. And it became very valuable.

And it was an invention, kind of a social invention. I’m thinking, it’s kind of analogous. We have recent inventions that we think about, the social media. We have, you know, Facebook and other recent inventions. This was an invention like that. It was an invention that got people together and communicating and excited about something. And it created a sort of a game that people were playing that turned out to be productive. That’s why it was copied all over the world. So, the core concepts, which began in Holland in 1609, are everywhere now. Every country of the world has this.

I should also add, by the way, that the VOC was a limited liability corporation. Amazing. When I told you that limited liability came in in 1811 in New York. I think I qualified that. It used to be that some companies had in their charter an agreement with the government, that the stockholders had limited liability. What came in in 1811 in New York was a law that said all companies are limited liability. And moreover, anybody in the world can start a–well, anyone in New York can start a company, and it will always be limited liability. So don’t worry, you can invest in any company. And you can not worry about being sued for the debts of the company.

Well, back then, Holland didn’t go that far, but they did create one company that did have limited liability. So what that meant was, you could invest in this company, and it’s just a game, you know? I can’t lose more than I put into it. And if these guys turn out to be crooks and some of them are hanged for their crimes, no problem with me, because I’m an innocent investor. The law doesn’t require that I investigate, you know, whether the guys who run the company are really honest. Let’s protect investors. So, all you can lose is the money you put in.

So, it created a tremendous opportunity. It was talked about, because the stock price went up and up and up, and it made people rich who invested in it. But it was also very volatile. It went up and down. People had never seen anything like this before, because nothing was so actively traded, and had such an interesting story that you could change your mind about from one day to the next. Anyway, I didn’t want just tell stories. This is a story, though, that illustrates our last lecture. It was a breakthrough innovation. It was a kind of gambling, but not gambling. It was gambling on real things. And so, you know, people like to gamble, but, you know, it’s usually a waste of their time. This is not a waste time. This was setting up trading around the world. And so, it was important, it was a very important innovation.

## 4.4 Harry Markowitz and the Origins of Portfolio Analysis

So, that’s an interesting question. How can it be that some asset, namely stocks, outperforms all other assets? That comes up then to, what is the standard answer? Why is it? And standard answer is risk. Stocks are riskier. The price jumps up and down from day to day, so the extra return is a risk premium. That is what I want to pursue today in this lecture. Does that explain the equity premium? How should we think about the equity premium? So, what I’m going to do is feature the theory that was originally invented by Harry Markowitz when he was a graduate student at the University of Chicago. And shortly after he was a student, in 1952, he published a classic article in the Journal of Finance that really changed the way we think about risk in finance, changed it forever.

It gets back at this core idea, you know, people looking at, going back to the days of the VOC, people had the idea, “you know I think stocks are the best investment.” OK, I’m writing that down, and I’m putting it in quotation marks, because it’s not a term that I would use. What is a best investment? Well they say, look, the VOC is just returning tremendous amounts. Any smart person would just put as much as he can into that investment. Something seems wrong about that. I mean, it can’t be true that–so what Markowitz–when I went back and read his Journal of Finance article in 1952, it’s kind of remarkable to me that what he was talking about wasn’t known yet in 1952. He was getting at this core idea of what’s the best investment. And how do you judge what’s the best investment. And judging from his article, to me it sounded so basic and simple. Of course, I’ve studied finance. But it seemed odd to me that everyone didn’t know that in 1952. So let me–the question is–I’ll kind of paraphrase what Markowitz said.

Let’s imagine that you’ve got a job as a portfolio manager, OK? And you’re kind of mathematically inclined. And you know numbers, and statistics, and you know how to compute standard deviations and variances, things like that. So, what is the first thing you do? You’re a numbers person, OK, you’re a math person. But now imagine you’ve been entrusted with managing a portfolio for some investor. And the investor gives you a horizon, you know, let’s say you’re managing it for one year. OK? And you’re thinking, all right, what should I do? Well, I want to collect data on every possible investment I could make. Not just stocks and bonds, but real estate, commodities, whatever, OK? And I can for each of these–I can compute what the average return was on those investments. OK? And I can compute the variance, and I can compute the covariance and the correlation, right? So Markowitz, do you see it? I’ve got all the data. Now I could say, I don’t believe these data are relevant to the future, because I’m smarter, or I can predict that some company’s going to do better than it did in the past, or some asset class will do better than it did in the past.

But let’s step back. Let’s do it basic. Let’s just think like a mathematician here, all right? Let’s just take as given all the historical average returns and variances and co-variances. So Markowitz says, well what’s the best portfolio given that? Ok? I could compute all these numbers. What’s the best assembly of all these things? And you know, he realized that nobody had ever thought like that. Isn’t that a well-defined problem? I give you all the variances, I give you all the covariances, I give you all the average returns. And I say, let’s just assume that this is going to continue like this, what should I do as an investor?

And it’s funny, Markowitz said, he was reminiscing. He won the Nobel Prize later. And deservedly, I think. This was a breakthrough idea. He said as a graduate student he was chatting with someone in the hallway and thinking about this. And he said, it suddenly hit me as an epiphany. If I have these statistics, I ought to be able to compute the optimal portfolio. It’s mathematical, right? It’s just one thing. What is the optimal portfolio? It took him like two or three days to figure the whole thing out.

You know, it’s almost like, haven’t I set it up in your mind? You see the problem. You could figure this out too, right? If you put your ingenuity onto it. The funny thing is, nobody thought about it before Markowitz. So actually, I was intrigued by that. So, I went back trying to find, what people were talking about before 1952. And we have a new thing on the web, relatively new, called–you ever play with this? It’s called ngrams.googlelabs.com. And what you can do is, you can put in any phrase you want and search it for the, it goes back like 400 years. In English, you can’t do Dutch. I don’t think. Maybe you can do that, too. I didn’t try.

And you can start to see what people were talking about in books. They have all these books scanned-in now. Now you can search for key words. And so, I did a search on ”portfolio analysis.” That’s what this is all about, right? Figuring out what the optimal portfolio of stocks, bonds, commodities is. Hardly anyone even used the term before 1952. I guess, it didn’t exist. There was no theory of–you kind of imagine. How can that be? I mean, you had all these sophisticated banks in finance. They had no theory of portfolio analysis. And I looked at portfolio variance, portfolio return. It all started with Harry Markowitz. Again, this is another testimony to how there are sudden breakthroughs. It should’ve been obvious. But somehow people didn’t think of it.

Then, I found one thing though. I did a search on Ngrams on “eggs in one basket.” There’s an old adage, “Don’t put all your eggs in one basket.” And that’s kind of what we’re coming to with Markowitz here. He’s got a whole theory of it, but I found an investment manual from 1874–I can’t find it here–this is from a book, 1874, about investing. “There is an old saying that is inadvisable to put all your eggs in one basket.” So, it was already in there and he says diversify. OK? And then he’s done. He doesn’t tell you, how do you diversify? How do I know what I should do? It just stops there.

## 4.5 Leverage and the Trade-Off between Risk and Return

There was no theory of risk until 1952. So, let’s think about that. You see the concept I have? You know all the variances. This isn’t a judgment thing. You know all the covariances. What should I do? Well, the first thing I want to talk about is the very simple case of pure leverage. Let’s go back to 1602. OK. And there’s only one stock, that’s VOC. OK, and there has to be something else, otherwise there’s nothing – the other thing I’m going to say is there’s an interest rate. Riskless interest rate. So, I can invest in, let’s say, Dutch government bonds, which are completely safe. Of course, you might say they’re not completely safe, but they’re much safer than VOC. VOC was wild. The price was going all over the place. So let’s, as an approximation, say there’s an interest rate. You can borrow and lend at the interest rate. We’ll call the riskless rate rf. OK? And let’s say, that’s 5% a year. OK? We’re investing for one year. So, I can invest at the interest rate and this is a boring investment. It’s just getting interest. It’s 5%. But I can also borrow at the interest rate. There’s a market rate, and I can borrow at 5%.

You know in practice, I would probably have to pay a little bit more as a borrower than I could get as an investor. But let’s assume that away. There’s just an interest rate, and anybody who wants to can borrow and lend at the interest rate. I’ll make it 5% just for a round number. OK. And let’s say VOC, the Dutch East India Company, has had a historic average return of 20%. This is spectacular investment, right? But let’s say it is, so that’s its mean, its μ, the mean of the investment. But let’s say, it’s really risky, so the standard deviation is 40%. All right? So, what can I do? Suppose I have only–this is the first–let’s do the simple problem first, OK? I have only one asset, VOC, and I have riskless debt. I’m going to draw a chart here showing–I’m going to do σ on this axis, and r on this axis. So, σ is the standard deviation of my portfolio, OK? And r is the expected return on the portfolio, OK? All I’m going to do is choose mixtures of the stock and the riskless rate. So, for a couple of points, I’m going to plot what the available options are. I can see right here that I can invest at 5%, right, the riskless rate, and then I’ll have no risk. So, do you see this? This is rf. This is 0, OK, and these are positive numbers. See what I’ve plotted here? This is just the most boring investment. Because there’s no risk at all, and I’m earning 5%.

I can also plot this one, right? So, here is VOC. Is this big enough for you to see back there? OK. So, VOC is up here, and this is a risk of 40, and a standard deviation of 20. Sorry, an expected return of 20 and a risk of 40, right?

So, those are two points, but I can do other things, too. What if I borrowed money to buy–let’s say I have 100 guilders. I’m talking Dutch. OK, that was the currency of the time, the guilder. I’ll write it for you. Guilder. OK. So, I have 100 guilders to invest. I could put it all on VOC stock, and I would expect to get 20 guilders profit, and I’d have a standard deviation of 40 guilders, right? But what if I said, I’m going to actually borrow another 100 guilders. I only own 100 guilders, but I’m going to borrow another 100 guilders and put it in VOC stock. That means I’ll own 200 guilders of VOC stock. And I’m going to have a debt of 100 guilders. So, what’s my expected return then? Well, my expected return is going to be 35%. Because, look, I’m owner of 200 guilders worth of VOC stock. The expected return is 20%, so I’m going to get 40 guilders out of that. But then I have a debt. I’ve got to pay five guilders to my lender. So, 35 is what I’ve got. And as a percent of my initial investment, that’s 35%.

So, I’ve got another point out here. This is 35 and down here is 80. See my standard deviation is 80 guilders now, right? Because I have 200 dollars and the standard deviation was 40%. All right? Here, I am 2-for-1 leveraged. I have $100 but I’ve put $200 in the stock. OK, it’s easy to do. You know, you could do this in 1602. So, you can see, obviously, this is a straight line here. I can do anything along this straight line. Here would be putting half of my money in the riskless asset and half into VOC. This would be putting one-and-a-half, 150 guilders, in VOC and borrowing 50 guilders. See, I can go out as far as I want. Then, there’s another branch to this. What if I short 200 guilders of VOC stock, OK. So, I go to the broker, and I say, I want to sell VOC stock. I don’t own any. And the broker would say, all right, fine, I’ll lend you some shares and then you can sell them. But you owe me the shares, all right? So, then I have minus 200 guilders worth of VOC stock. So, what is my expected return then? Meanwhile, by the way, the broker says, after you sell the shares, I will get 200 guilders from the person who bought them from you, and I’ll hold that, and I’ll pay you interest on that.

OK. So, what do I get? I expect to lose 40 guilders, because I’ve got $200, 200 guilders of the stock. But meanwhile, I’ve got my original 100 guilders, and now I’ve got another 200, and they’re all there earning interest at 5%. So, I will get 15 guilders. So, the expected return is 15 minus 40, or minus 25. So, that’s this point down here. But you can see that you can also do anywhere you like on that line. So, what we have here is a broken straight line. I can get anything I want, right? This is kind of obvious right now. Anywhere I want on that line, on that broken straight line. And I can do that.

So, here’s where you got saying, what is the optimal portfolio anyway? I can get any return I want. You know, my client, who’s asking me to invest, says, I want 100% return, expected. You say, got it. I’m no genius, right? I’m just doing the most obvious thing. Anyone who wants a 100% return can get it. I’m just going to leverage. So, then I create an investment. If I have an investment company that merely buys VOC stock and leverages it, my investment company can have any expected return that you want.

So, this is what Markowitz was wondering about. What does it mean to have the optimal investment, anyway? And the core thing that he talked about in 1952 is, there is no best investment. There’s only a trade-off between risk and return. And we have to think about the best trade-off. In this case, I’ve shown the trade-off here. This is what you can get. Any one of those points is available. And so, anyone who wants to invest with you has to choose between risk and return. There’s no optimal portfolio in that fundamental sense. It’s a matter of an optimal trade-off. And, you know, nobody knew that before 1952.

## 4.6 Efficient Portfolio Frontiers

So, let me just show formally this–what I just did on the blackboard. I’ve switched to dollars from guilders. Now we’re in the USA. And so, put x dollars in a risky asset, 1-x dollars in the riskless asset. The expected value of the return on the portfolio is r. That’s equal to

all right? It’s linear, that’s because that’s how expected values work. The variance is x2 times the variance of the return. And so, if I want to write the portfolio standard deviation as a function of the expected return, I solve for x. Taking this equation for x, solve for x in terms of r. So,

And then, I substitute that in to this equation. Well, I want to take the square root of it, because this is σ2. And so, I’ve got

[Correction: This fraction is multiplied by the standard deviation of portfolio return 1]

Well actually, I have these absolute value marks. If that’s negative, I switch sign and make it positive. So, that gives the formula for this broken straight line right here. So that’s pretty simple. That’s the expected value [correction: portfolio standard deviation].

So now, I want to move ahead, move on from this simple idea to–we haven’t really gotten fully into Markowitz yet. Because this is a very simple story. By the way, this broken straight line is what we call a degenerate case of a hyperbola. You know the, remember in math, hyperbola is a curve, a certain mathematical curve. And we’re seeing a hyperbola here, but I’m going to show you other hyperbolas in a minute. What Markowitz really said–well OK, this is simple. This is all, just pure leverage is a simple thing to understand. By the way, it’s also called gearing in the United Kingdom.

But let’s think about now–suppose I have more than one risky asset. Let’s get past the year 1602. And let’s think about assets in a more modern context. I want to move to another example, which is two risky assets. We’ve moved past 1602 and now we have two stocks. And for the moment, I’m going to forget about leverage. And let’s just say you can put x1 in the first risky asset, that’s stock number one. And I can put 1-x1 in the second risky asset. That’s stock number two. OK? So, what do I get here? The portfolio expected return is just the linear combination of the two expected returns. So, r1 is the expected return on the first stock, and r2 is the expected return on the second stock.

Well actually, I’m assuming you have $1 to invest in this example. I’m sorry. I was assuming you had 100 guilders over there. Now, I just made it $1. Unrealistically small amount, but I just wanted a nice number, OK? So, let’s say $1 is 100 guilders, and then I haven’t changed anything. OK. So, I start out with $1, so if I put x1 dollars in the first one, I have 1-x1 left for the other one. So it’s very simple. And this is the formula for the variance of the portfolio, which we saw–essentially we saw that in the second lecture.

So, what I can do is, go through the same sort of exercise I did there with two risky assets, all right? And so, what I want to do is, draw a curve something like this. But I’ll solve for x1 in terms of r, just like I did for the riskless asset. And I’ll plug it into the equation for the variance. I’ll have to take the square root of that, and I can plot that, OK? And you might think it would look something like that. Well, it’s not going to look exactly like that, because it’s risky. Something’s risky. So, I did that. And incidentally on your problem set, you’re going to have to think about issues like this.

But what I did is, I took data on the average return for the U.S. stock market, as measured by the S&P 500, and the variance. And then, the alternative investment I took was 10-year treasuries for the United States government. Long term, because they’re 10 years, but we’re only investing for one year. So, they’re risky, because the market price goes up and down. They’re not riskless. I call those bonds. There’s other kinds of bonds. And I computed the relationship between the standard deviation of the portfolio and the expected return, just as I showed you. Again, using data from 1983 to 2006. And it kind of looks like this curve, doesn’t it? Except, this is a degenerate parabola, but it looks like this. I’m sorry, parabola. I said it wrong. Hyperbola.

You know how hyperbola–remember this from math? Hyperbolas, well they look like that, and they approach asymptotes, which are straight lines. So, here is the hyperbola for stocks and bonds. So, just as I had a point here which represented 100% VOC, I can have over here a point which represents 100% U.S. stocks. OK? And I can take another point which is 100% bonds, that’s here. This point is 25% stocks, 75% bonds. This point is 50% stocks, 50% bonds. OK? This is the choice set that I as an investor have between stocks and bonds. So, is that clear?

You see, all these are different portfolios. If you’re just going to do stocks and bonds and nothing else, what you choose to do depends on your taste, on your risk tolerance. I could go 100% stocks, but I’m going to have a lot of risk. I’m going to have a nice expected return, it looks like it’s about 13%, 14%. But I’m going to have a high variance. Looks like it’s about 18%. This is the S&P 500 stock market. And so, it has a lot of variance. I could be safe, and I could go all in bonds. I could be here. Then I’d have, you know, a lower, much lower return, but I’d have a lower variance. So, what should I do?

Well, what do you learn from it? First of all, you learn there isn’t any single optimal portfolio, but there is something. Let’s talk about being 100% bond investor. What do you think of that? Is that a good idea? You’d get this point right here. Well, you definitely should not be a 100% bond investor. That’s one thing we just learned. Why’s that? Because if I go up here, I have no more–see that’s the same standard deviation, the same risk, but I have higher return, right? Higher expected return. So, what did we just learn? We learned that if you just stay in this space of stocks and bonds, maybe you could be 100% stock investor, but never in a million years should you ever even think of being 100% bond investor. OK, because, look, it’s just simple math. I can figure it out. I can figure out that I get a higher expected return and no more risk.

So, this is lesson number one that Markowitz showed us. Amazing. It’s so simple and obvious, right? It’s not so simple, because, at the time Markowitz wrote, Yale University was probably 100% bond investor, believe it or not. They couldn’t figure it out in those days. So, we’ve made progress. That’s why I think Markowitz is among the most deserving of the Nobel Prize winners in economics. This is really basic. It actually intrigues me. I don’t know how much you like math, but going back to my childhood I was interested in geometry. These simple mathematical curiosities like hyperbolas are just fascinating to me.

It goes back to Apollonius of Perga, writing in around 200 BC, wrote a book on conic sections. And he invented the word hyperbola, parabola, ellipse. So, I was thinking of looking back at his book. I think it still survives, and seeing what he says about finance. But I can be sure he had no idea that his theory would apply to finance. I wish I could go back in a time machine and talk to him. He would be so happy to know that his theory of conic sections–it, you know, it ended up applied to astronomy by Kepler and Newton. And now it hits into finance. Isn’t it amazing how there’s a unity of thought? And this simple diagram has just taught us something about investing. That’s not obvious. Not obvious until you think about [it]. I’ve just told you, never invest only in bonds. But it doesn’t tell you how much stocks and how much bonds. You know, once you’re above this point, it seems to be a matter of taste. There isn’t any single decision that you can make.

So, now, I want to move to a more complicated world where we have three assets. OK. We’re starting from–we had one risky asset, then we had two, now let’s go even further. Let’s say three risky assets. Well, the expected return is the same, it’s the weighted average. Now we have three weights, x1, x2, and x3, and they have to sum to $1. I could have written x3 as 1-x1-x2. I wrote it differently here. It looked messy to write it the other way.

And this is the formula for the portfolio variance. It’s the (x1)2, times the variance of the return on the first risky asset, plus (x2)2, times the variance of the return on the second risky asset, plus (x3)2, times the variance of the return on the third risky asset. And then you have three more terms representing covariances. You have to take account of the covariances of the assets. Because if they move together, if they all go in the same direction at the same time, that’s going to make your portfolio riskier. And so, that’s the portfolio variance. And the portfolio expected return is just–why didn’t I write it there? It’s x1Ÿr1 + x2Ÿr2 + x3Ÿr3, where the sum of the x’s is 1, $1.

So, it’s something that you can do to calculate what is the optimal portfolio. So, I decided to add a third asset to my diagram. The pink line up here is the same. We call that an Efficient Portfolio Frontier. I have that in the title of the slide, for stocks and bonds. That’s the pink line here. But I’ve added the Efficient Portfolio Frontier for three assets. Stocks, bonds, and oil. Oil is an important investment, because our economy runs on it. And the total value of oil in the ground is comparable to the value of the stock markets of the world. It’s big and important. So, let’s put that in. And what I have actually here is the minimum variance mixture for any given expected return for the three assets. And you can see that it’s possible, when you add a third asset, oil, to bring the Efficient Portfolio Frontier to the left. OK? Because we’ve got another asset. And it’s also paying a good return. And it’s not correlated. Oil doesn’t correlate very much with the stock market. So, we’re spreading the risk out over more assets.

We’re putting more eggs in our basket. [Correction: We are providing more different kinds of eggs for our basket.] OK? And so, we have a better choice set now, right? We can pick any point on that blue line. And so, we shouldn’t just have stocks and bonds. We’ve learned we should have stocks, bonds, and oil. We’re leading toward a fundamental insight, which is due to Markowitz, which is, the more the merrier. The more different kinds of assets you can put in, the lower you can get the standard deviation of your return, for any given expected return. So, the better off you are, this is diversification. So, while diversification was applauded in the nineteenth century, no one had ever done the math like this before. And now we can see, that when you do the math, you want to have all three in your portfolio. And yet people don’t know that. They don’t, there’s an emotional resistance to this implication.

I once went to the government of Norway. I gave a talk at their Norges Bank, which is the central bank of Norway. I told them in my talk, I calculate that Norway has something like 70% of its portfolio in oil. I don’t remember the exact number, but it was something close to that. Why do they have so much in oil? Well, because they have the North Sea Oil. And so, I asked them at the bank, don’t you realize, where are you on this portfolio? You’re not on the frontier. What Norway should be doing is something like 15%, they could pick this point right, that would be reasonable. 15% oil, 53% stocks, 32% bonds, which would give them that point. Or they could pick this one. I have a point labeled up here. That’s 21% oil, 79% stocks, no bonds. All right, those are all choices depending on your risk tolerance.

But they’re not going to just pick 100% oil. That would be way over here, much higher risk. So, I asked them about that. Why do you do this? And I don’t know if I got a good answer from them, but basically it was, well, we don’t want to sell the oil [correction: oil fields], because it’s our national heritage, you know, we own it. And I said, well you don’t have to sell it, you can just do a derivative transaction. You can short the futures market for oil and reduce your exposure. And then, they said, well, some people have mentioned that, but it’s politically difficult. So, they’re not doing it. Maybe next year. Maybe the next government will do that. So, they still aren’t there yet. They’re not managing their risks well. So, it’s a powerful and important thing, because if the market for oil collapses, Norway is in big trouble. They’re not diversifying enough. I don’t mean to put them down. They’re smart people, but like in any country –

I did the same thing with Mexico. I went to the Banco de Mexico. And I talked to Guillermo Ortiz when he was director of it. Same thing in Mexico. It’s not as dependent on oil, not as dependent on oil as Norway is. It’s not so obvious for Mexico. But it is politics that came in. The question of, are you really saying Mexico should go into the futures market and take a massive short-position of billions of dollars of oil? Again, it was like, this isn’t reality. We’re not going to do that. But I tried to make the point.

Now, another thing is, I have shown here three assets. The pink line is irrelevant once we realize we have three assets, we have stocks, bonds, and oil. So, you should choose on this curve. And, of course, you should never take down here, even though that’s possible. In other words, you could say, what portfolio would give me 9% return with the least risk? Well it turns out it’s 100% bonds. But I just told you, never do 100% bonds, because you can go up to this point, all right? So, you never do down here. So, the Efficient Portfolio Frontier is really the part of the hyperbola that’s above the minimum variance.

And you don’t want to do minimum variance either, right? This is the lowest possible risk portfolio. You can’t get down to zero risk if all of your assets are risky. So, you’re stuck here. That’s not necessarily the best thing, because people allow some risk. This is having the minimum risk, but I can get my return up much higher without taking much risk, so I’d probably do that. OK. Now, I can do this with more than three assets. I can do it with 1,000 assets, now that we have computers. Back in 1952, I erased it, but I had 1952 here, Markowitz had do it all by hand. But now that we have computers, it’s so easy. You know, there are all kinds of programs. In fact, on your problem set, we have Wolfram Alpha, which will do all these calculations for you for its own data. So, these are easy to do now.

## 4.7 Tangency Portfolio and Mutual Fund Theorem

But what I want to do now is add the riskless asset. So, what we’ve done, the blue line takes three risky assets. It looks only at assets with a standard deviation greater than zero. Now I want to do the optimal portfolio when there are four assets. I’ve got stocks, bonds, oil, all risky. And now I have the thing that isn’t risky, would be your 1-year governments [correction: government bonds]. Right? It’s not risky, because the maturity matches my investment horizon. I know exactly what I’m going to get. It’s 5%, let’s say. So, what can I do investing in these four assets?

Well, here it goes back to what I did over here with this simple diagram. I can pick any portfolio on the Efficient Portfolio Frontier and consider that as if that were VOC, right? And then, I can compute just how leverage allows me to combine that with the riskless asset and that portfolio. So, I can pick a point, like I can pick this point here. And then, I could achieve, by combining that portfolio, which is 15% oil, 53% stocks, and 32% bonds–I could combine that with any amount of risky debt. And I would get a straight line going between–actually, this diagram doesn’t show zero, I should have maybe done it differently–but between 5%. So, actually that point right here, I can do it on this diagram. That point here is like 12% expected return and 8% variance. So, it would be some–well, it would be here, except this would be 12% and this would be 8%. I can pick any point. And this would be 5%. Any point along the straight line connecting those points is possible.

So, what do I want to do? I want to get the highest expected return for any standard deviation. I want to take a line that goes through 5% on the y-axis and is as high as possible. So, I’m taking a point right over here, at 5%, and trying to get as high as I can. It turns out then, that I want to pick the point which has a tangency with the Efficient Portfolio Frontier. And so, that means the highest straight line that touches the Efficient Portfolio Frontier. And so, now I can achieve any point on that line. And that’s again Markowitz’s insight. So, if I were to pick that point, I would be–what does it look like? I don’t have it indicated. Probably something like 11% oil, 30% stocks, 50% [government bonds], something like that, it doesn’t add up. [Clarification: these guesses do not add up to 100%, but the actual numbers do.] And that would be holding no debt, right? [Clarification: No debt, in the sense of shorting the risk-free rate.]

But I could get even higher returns, if my client wants that, by leveraging. I would borrow and buy even more of this risky portfolio. So, this portfolio here is called the Tangency Portfolio. And what Markowitz’s theory shows is that, once you add the risky asset [correction: riskless asset], the relevant Efficient Portfolio Frontier is now really this tangency line. And so, I want to do a mixture of the riskless asset and the Tangency Portfolio that accords with my risk preferences. But I don’t want to ever just move to one of these other portfolios. Because, [each of] these other portfolios, like 15% oil, 53% stock, 32% bonds, is dominated, has a higher expected return for the same risk, by a portfolio of the Tangency Portfolio, leveraged up a little bit by borrowing.

I don’t know if it was clear in Markowitz’s paper, but it became clear soon after. There really is, in a sense, an optimal portfolio. It’s the Tangency Portfolio. Because everyone wants to invest on this line, and any point on this line is a mixture of the riskless asset and the Tangency Portfolio. And so, everyone wants to invest in the same portfolio. So, there is an optimal portfolio, in a sense. It’s in a sense that everybody wants to do the same risky investments. People will differ in their risk preferences, and so some of them will want to do a riskier, a more leveraged version, and some of them will do a less leveraged version of the Tangency Portfolio, but everyone wants to do the Tangency Portfolio. So, that is the key idea of Markowitz’s portfolio management. And it’s been expressed by some as the Mutual Fund Theorem.

First of all, I have to just to define for you, what is a mutual fund? You may not know that. A mutual fund is a certain kind of investment company aimed at a retail audience. They could have just called this the investment company theorem, but history–I can’t tell you the history of thought on this. A mutual fund is a certain kind of investment company that is mutual. That means that the owners of the shares in the fund are–there’s no other owners. There’s just one class of investors. You’re all equal, so it’s mutual. But that’s irrelevant to you. The idea is that, all we need is one mutual fund. There’s thousands of mutual funds to serve investors. Because I had that everyone is investing in the Tangency Portfolio. So, they should call their fund the Tangency Portfolio fund. And our fund is the optimal mix of stock, bonds, oil and whatever else. And then, you don’t necessarily want to own only that mutual fund, but you want to own mixtures of that mutual fund and the riskless asset.

So, you only need one investment company. See, I told you this story. I said, imagine that you were mathematically inclined, and you have all the statistics, and you’re going to figure it out. What’s the best thing to do? We’ve just figured it out. I haven’t gone through all the math details. There is a best thing to do. You should offer, as your investment product, the Tangency Portfolio. And that’s it. Once you’ve figured it out, there’s nothing more to do. There’s no need to hire any more finance people. You’ve figured it out, according to Markowitz’s theory. And all the investors in the world will just invest in this one. And that’s [it]. Case closed. We don’t need thousands of mutual funds. Under the assumptions of Markowitz, which is that we’re agreed on the variances, and covariances, and expected returns, there’s a single optimal risky portfolio. And then the instructions to investors are very simple. All you need is two assets in your portfolio. The mutual fund that owns the Tangency Portfolio, and whatever amount of debt you want. So, if you’re footloose and fancy free, you can even leverage it. You can borrow and [do] 2-to-1, 3-to-1, it’s up to your tastes. But you don’t need to look at anything other than the mutual fund.

## 4.8 Capital Asset Pricing Model (CAPM)

So, that’s an important insight. And what it means, then–that leads to something else. Markowitz didn’t get this idea. It came out later. Someone was thinking, well, if everyone should be investing in the same portfolio, it doesn’t add up unless that portfolio is equal to the total assets out there in the world, right? If there’s twice as much oil as there is stock, then there has to be twice as much oil as stock in the Tangency Portfolio. Otherwise, it doesn’t add up, right? Everyone has to own everything. Supply and demand have to equal. So it means, the Mutual Fund Theory implies that the market portfolio equals the Tangency Portfolio, OK. And now I’ve pretty much finished the theory. I should say, it implies, if investors follow this model that we’re having, that they all want to–that Markowitz’s model–if all investors think like Markowitz says, they all want to do the same thing. They all want to invest in the same best portfolio. So, that has to be proportional to the market portfolio. So, the Tangency Portfolio equals the market portfolio.

So, I was saying earlier, why is it that everyone doesn’t invest in VOC stock? How does it add up, right? If VOC stock is just better than something else, then that suggests everyone wants to put all their money in VOC stock. But we’re realizing, they don’t. Because they’re concerned about it. They see this trade-off between risk and return, and they want to hold some proportion of VOC stock and the riskless asset. It has to add up, so that the market is cleared and all the VOC stock is owned. And more generally, if there are many assets, all the assets have to end up owned by someone. So, the cardinal implication of this theory is that the market portfolio, which is everything that’s out there in the world to invest in, has to be proportional to the Tangency Portfolio [correction: has to be equal to the Tangency Portfolio]. And so, one of the implications is, if that’s true–and I have just a couple more slides here.

Now it’s called the Capital Asset Pricing Model in finance. So, that’s Capital Asset–which is [a] pricing model, which was not invented by Markowitz, but was invented by Sharpe and Lintner somewhat shortly after Markowitz. The Capital Asset Pricing Model–and I’m not going to derive this equation. But it says the expected return on any asset, the i-th asset, equals the risk free rate, plus the β of that asset times the difference between the expected return on the market and the expected return on the riskless asset.

I was just going to try to explain this intuitively, and then I’ll be done. I have one more slide about the Sharpe ratio. But the intuitive idea–let me just say, everything should have a very simple explanation. And the intuitive idea is this: Starting from Markowitz, we got an understanding of what risk is. People didn’t clearly appreciate that. People used to think that risk was uncertainty, right, in finance. If a stock has a lot of uncertainty, that uncertainty means that it’s a dangerous stock, and people will demand a high expected return, otherwise they won’t hold the stuff. But the CAPM says no, people don’t care about the uncertainty of the stock. Because if it’s one stock out of many, they’ll put it in their portfolio, and if it’s independent of everything else, it all gets averaged out, and so, who cares? So, people don’t care about variance.

Well, what is it that people care about? People care about covariance. This is the basic insight that followed from Markowitz. People care about how much a stock moves with the market, because that’s what costs me something. I don’t care, I can own a million little stocks that all have independent risk. It all averages out, doesn’t mean anything to me. I’ll put them in tiny quantities in my portfolio. But if they correlate with the market, I can’t get rid of the risk, because it’s the big picture risk. That’s what insurance companies, that’s what everyone cares about. It’s this market risk. The big risk. You only care about how much a stock correlates with the big picture in its risk.

So, that’s measured by β. The β is the regression–the slope coefficient, when you regress the return on the i-th asset on the return of the market. So, high β stocks are stocks that go with the market. We found out that Apple has a β of 1.5, or roughly that. That means, they respond in an exaggerated way. It’s not 1, it’s greater than 1, they more than move with the market. And so, investors will demand a higher return on Apple stock, because its β is greater than for other stocks. That’s the core idea that underlies it. Do you see that intuitively? So, you have to change your idea of what risk is. Risk is covariance. It’s co-movements.

I have just one more slide here. It’s named after William Sharpe, who is the inventor, with Lintner, of the Capital Asset Pricing Model. The Sharpe ratio is, for any portfolio, the average return on the portfolio minus the risk-free rate, divided by the standard deviation of the portfolio. And if you take the CAPM model, the Sharpe ratio is constant along the tangency line.

This is a way of correcting the average return from some investment for leverage. The idea is–some companies used to advertise, we’ve had a 15% average return, and then investors would say, but wait a minute. You didn’t tell me what your leverage is. That’s the first thing you should learn from this course. Someone advertises that they had 15% return, you say, ha, I want to know what your leverage was. I want to know–you were just leveraging it up and taking big risks, and on average you’ll do well, but it’s risky.

So, this is the correction you make. How do you correct for leverage? You might say, well I want to look at what fraction of the investment portfolio is in the risky asset, and what fraction is in the riskless asset. But it’s not so easy to do that, because the company can cover up its tracks. It can invest in a company that’s leveraged, right? And so, you have to go one step further and undo the leverage for that company. It’s hard to do that. But the easy thing to do is just calculate the Sharpe ratio for the investment company. So, if some guy is investing and claiming to have done 15% of return per year on his portfolio, well, I’m going to look at the standard deviation of the portfolio. That’s evidence of how leveraged this guy was. And I compute the Sharpe ratio. And unless it’s bigger than the Sharpe ratio for the, you know, the typical stock, I’m not impressed.

Anyway, so I think, I’ve come to the end of this lecture. So, what you should have gotten from this lecture is a concept of risk return trade-off, a concept of optimal portfolio as being something subtle and related to Apollonius of Perga in difficult ways. But there’s also very simple things about how to evaluate portfolios and portfolio managers that comes out of this.

## Assignment

Fabozzi, Foundations of Financial Markets, Ch. 12

Siegel, Stocks for the Long Run, Ch. 1, 2