# 4. Portfolio Diversification and Supporting Financial Institutions

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 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 was 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 India, 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? A 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 Well, 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, 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 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 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 can 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 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 do not need to 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 can 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 But now I want to use it as a lead-in to the main theme of this lecture, which is about portfolio management and risk

And the first concept I wanted to talk about is leverage Well, and also let me add the equity premium These are the two main concepts Maybe I'll do equity premium first Here's the conundrum

That people were presented with And I'll stay on the VOC story, but it's much more general than that VOC, after a few years out, people thought, you know, this company is amazing It's just growing so fast, it's making so much money, it

Might have a really high return, like unbelievably high, like 20% a year, or even more, but let's say 20% a year And that's what generated excitement But some people wondered, well, how can it be? Maybe it's earned 20%, but how can it consistently do that?

So, let me put ''puzzle'' We've gone through 400 years of history since the VOC was established And since then, it seems to be remaining true that companies' shares do extremely well And that's a puzzle

Because you know, if you can make a high return on some investment, wouldn't you think that enough people would flock into the investment, so that it no longer — you know, too many people trying to do this, so it's no longer performing so well? But in fact, it seems like the average return on stocks has

Been very high This is a theme in Jeremy Siegel's book Stocks for the Long Run, which I have on the reading list Siegel has data, doesn't go back to 1602, but it goes back to the nineteenth century And he says that the geometric average return, annual return,

On the United States stock market from 1871 to 2006 was 68% a year, corrected for inflation That's 65% a year after inflation So, if you're at 3% or 4% inflation, right, that's 10% a year Let's compare that with short-term governments, which

Are the safest thing in the United States The average real return on them was only 28% a year So, the difference is 4% a year So, for well over 100 years in the United States, stocks performed extremely well Moreover, he points out there was no 30-year period since

1831 to 1861 when stocks under-performed either short-term or long-term bonds So, the stocks have been good investments What do we make of that? Don't people learn? You'd think if people learn, they would all want to do the

Good thing Why does anyone invest in something else? That was the puzzle here It's not just a United States phenomenon The London Business School professors Dimson, Marsh, and

Staunton wrote a book called The Triumph of the Optimists That is, optimists about the stock market And they looked at the equity premium in many different countries around the world And they found that all of the countries, and this is looking over much of the twentieth century, all of the countries

Had an equity premium, that the stocks did better than the bonds of that country The lowest of the countries they studied was Belgium, which had an equity premium of only 3%, and the highest was Sweden, which had an equity premium of 6% So, that's an interesting question

How can it be that some asset, namely stocks, outperform 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 the 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 says 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, or a math person But now imagine you've been entrusted with managing a

Portfolio for some investor And the investor gives you a horizon, and 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 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 And 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? 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 ngramsgooglelabscom

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 try to 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 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? 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

Some other person 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 r sub f 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 a spectacular investment, right? But let's say it is, so that's its mean, its mu, 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 sigma on this axis, and r on this axis So, sigma is the standard deviation of my portfolio, OK?

And r is the expected return on the portfolio, OK? And 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 there that I can invest at 5%, right, the riskless rate, and then I'll have no risk So, do you see this? This is r sub f 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 guilder's 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%

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 the VOC This would be putting 1 1/2, 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? And 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 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 minus x dollars in the riskless asset The expected value of the return on the portfolio is r That's equal to x r1 plus 1 minus x times rf, all right? It's linear, that's because that's how

Expected values work The variance is x squared 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, x equals r minus rf all over r1 minus rf And then, I substitute that in to this equation Well, I want to take the square root of it, because this is sigma squared And so, I've got sigma equals r minus rf times r1 minus rf

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 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 minus 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 minus 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 US stock market, as measured by the S&P

500, and the variance And then, the alternate 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% US stocks

OK? And I can take another point which is 100% bonds, that's here This point is 25% stock, 75% bonds This point is 50% stock, 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 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 words hyperbola, parabola, ellipse So, I was thinking of looking back at his book

I think it still survives And see what it 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, 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 minus x1 minus 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 squared, times the variance of the return on the first risky asset, plus x2 squared, times the variance of the return on the second risky asset, plus x3 squared, 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 x1r1 plus x2r2 plus x3r3, 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 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, 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, 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 its 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 a person 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 These are easy to do now

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 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 for 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 You 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% bonds, something like that, it doesn't add up And that would be holding no debt, right?

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 ,

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 You see, 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 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, if everyone should be investing in the same portfolio, it doesn't add up unless that 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 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 beta 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 beta The beta is the regression — the slope coefficient, when you regress the return on the i-th asset on the return of the market

So, high beta stocks are stocks that go with the market We found out that Apple has a beta of 15, 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 beta is greater than for other stocks That's the core idea that underlies it Do you see that intuitively? 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 is 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