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 6.8% a year, corrected
for inflation. That's 6.5% 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 2.8% 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
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 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 [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 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 U.S. 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% U.S. 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. [Correction: We are providing
more baskets for the eggs.] 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? [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. 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 [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 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 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 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.
