Professor Robert Shiller:
Today's lecture is about portfolio diversification and
about supporting financial institutions,
notably mutual funds. It's actually kind of a crusade
of mine--I believe that the world needs more portfolio
diversification. That might sound to you a
little bit odd, but I think it's absolutely
true that the same kind of cause that Emmett Thompson goes
through, which is to help the poor
people of the world, can be advanced through
portfolio diversification--I seriously mean that.
There are a lot of human hardships that can be solved by
diversifying portfolios. What I'm going to talk about
today applies not just to comfortable wealthy people,
but it applies to everyone. It's really about risk.
When there's a bad outcome for anyone, that's the outcome of
some random draw. When people get into real
trouble in their lives, it's because of a sequence of
bad events that push them into unfortunate positions and,
very often, financial risk management is part of the thing
that prevents that from happening.
The first--let me go--I want to start this lecture with some
mathematics. It's a continuation of the
second lecture, where I talked about the
principle of dispersal of risk. I want now to carry that
forward into something a little bit more focused on the
portfolio problem. I'm going to start this lecture
with a discussion of how one constructs a portfolio and what
are the mathematics of it. That will lead us into the
capital asset pricing model, which is the cornerstone of a
lot of thinking in finance. I'm going to go through this
rather quickly because there are other courses at Yale that will
cover this more thoroughly, notably, John Geanakoplos's
Econ 251. I think we can get the basic
points here. Let's start with the basic idea.
I want to just say it in the simplest possible terms.
What is it that--First of all, a portfolio,
let's define that. A portfolio is the collection
of assets that you have--financial assets,
tangible assets--it's your wealth.
The first and fundamental principle is:
you care only about the total portfolio.
You don't want to be someone like the fisherman who boasts
about one big fish that he caught because it's not--we're
talking about livelihoods. It's all the fish that you
caught, so there's nothing to be proud of if you had one big
success. That's the first very basic
principle. Do you agree with me on that?
So, when we say portfolio management, we mean managing
everything that gives you economic benefit.
Now, underlying our theory is the idea that we measure the
outcome of your investment in your portfolio by the mean of
the return on the portfolio and the variance of the return on
the portfolio. The return, of course,
in any given time period is the percentage increase in the
portfolio; or, it could be a negative
number, it could be a decrease. The principle is that you want
the expected value of the return to be as high as possible given
its variance and you want the variance of the return on the
portfolio to be as low as possible given the return,
because high expected return is a good thing.
You could say, I think my portfolio has an
expected return of 12%--that would be better than if it had
an expected return of 10%. But, on the other hand,
you don't want high variance because that's risk;
so, both of those matter. In fact, different people might
make different choices about how much risk they're willing to
bear to get a higher expected return.
But ultimately, everyone agrees I--that's the
premise here, that for the--if you're
comparing two portfolios with the same variance,
then you want the one with the higher expected return.
If you're comparing two portfolios with the same
expected return, then you want the one with the
lower variance. All right is that clear
and--okay. So let's talk about--why don't
I just give it in a very intuitive term.
Suppose we had a lot of different stocks that we could
put into a portfolio, and suppose they're all
independent of each other--that means there's no correlation.
We talked about that in Lecture 2.
There's no correlation between them and that means that the
variance--and I want to talk about equally-weighted
portfolio. So, we're going to have
n independent assets; they could be stocks.
Each one has a standard deviation of return,
call that σ. Let's suppose that all of them
are the same--they all have the same standard deviation.
We're going to call r the expected return of these
assets. Then, we have something called
the square root rule, which says that the standard
deviation of the portfolio equals the standard deviation of
one of the assets, divided by the square root of
n. Can you read this in the back?
Am I making that big enough? Just barely, okay.
This is a special case, though, because I've assumed
that the assets are independent of each other,
which isn't usually the case. It's like an insurance where
people imagine they're insuring people's lives and they think
that their deaths are all independent.
I'm transferring this to the portfolio management problem and
you can see it's the same idea. I've made a very special case
that this is the case of an equally-weighted portfolio.
It's a very important point, if you see the very simple math
that I'm showing up here. The return on the portfolio is
r, but the standard deviation of the portfolio is
σ/√(n). So, the optimal thing to do if
you live in a world like this is to get n as large
possible and you can reduce the standard deviation of the
portfolio very much and there's no cost in terms of expected
return. In this simple world,
you'd want to make n 100 or 1,000 or whatever you could.
Suppose you could find 10,000 independent assets,
then you could drive the uncertainty about the portfolio
practically to 0. Because the square root of
10,000 is 100, whatever the standard deviation
of the portfolio is, you would divide it by 100 and
it would become really small. If you can find assets that all
have--that are all independent of each other,
you can reduce the variance of the portfolio very far.
That's the basic principle of portfolio diversification.
That's what portfolio managers are supposed to be doing all the
time. Now, I want to be more general
than this and talk about the real case.
In the real world we don't have the problem that assets are
independent. The different stocks tend to
move up and down together. We don't have the ideal world
that I just described, but to some extent we do,
so we want to think about diversifying in this world. Now, I want to talk about
forming a portfolio where the assets are not independent of
each other, but are correlated with each other.
What I'm going to do now--let's start out with the case
where--now it's going to get a little bit more complicated if
we drop the independence assumption.
I'm going to drop more than the independence assumption,
I'm going to assume that the assets don't have the same
expected return and they don't have the same expected variance.
I'm going to--let's do the two-asset case. There's n = 2,
but not independent or not necessarily independent.
Asset 1 has expected return r_1.
This is different--I was assuming a minute ago that
they're all the same--it has standard--this is the
expectation of the return of Asset 1 and r_2
is the expectation of the return--I'm sorry,
σ_1 is the standard deviation of the return
on Asset 1. We have the same for Asset 2; it has an expected return of
r_2, it has a standard deviation of
return of σ_2. Those are the inputs into our
analysis. One more thing,
I said they're not independent, so we have to talk about the
covariance between the returns. So, we're going to have the
covariance between r_1 and
r_2, which you can also call
σ_12 and those are the inputs to our analysis.
What we want to do now is compute the mean and variance of
the portfolio--or the mean and standard deviation,
since standard deviation is the square root of the variance--for
different combinations of the portfolios.
I'm going to generalize from our simple story even more by
saying that, let's not assume that we have equally-weighted.
We're going to put x_1
dollars--let's say we have $1 to invest, we can scale it up and
down, it doesn't matter. Let's say it's $1 and we're
going to put x_1 in asset 1
and that leaves behind 1-x_1 in asset
2, because we have $1 total.
We're not going to restrict x_1 to be a
positive number because, as you know or you should know,
you can hold negative quantities of assets,
that's called shorting them. You can call your broker and
say, I'd like to short stock number one and what the broker
will do is borrow the shares on your behalf and sell them and
then you own negative shares. So, we're not going
to--x_1 can be anything and x--this is
x_2 = 1-x_1,
so x_1 + x_2 = 1. Now, we just want to compute
what is the mean and variance of the portfolio and that's simple
arithmetic, based on what we talked about before.
I'm going to erase this. The portfolio mean variance
will depend on x_1 in the way
that if you put--if you made x_1 = 1,
it would be asset 1 and if you made x_1 = 0,
then it would be the same as asset 2 returns.
But, in between, if some other number,
it'll be some blend of the--mean and variance of--the
portfolio will be some blend of the mean and variance of the two
assets. The portfolio expected return
is going to be given by the summation i = 1 to
n, of x_i*r _i,.
In this case, since n = 2 that's
x_1 r_1 +
x_2 r_2,
or that's x_1 r_1 + (1 -
x_1) r_2; that's the expected return on
the portfolio. The variance of the portfolio
σ²--this is the portfolio variance--is
σ² = x_1²
σ_1² + x_2²
σ_2² + 2x_1
x_2 σ_12;
that's just the formula for the variance of the portfolio as a
function of--Now, since they have to sum to 1,
I can write this as x_1²
σ_1² + (1 -
x_1)² σ_2²
+ 2x_1 (1 - x_1)
σ_12 and so that together traces out--I can
choose any value of x_1 I want,
it can be number from minus infinity to plus infinity.
That shows me then for any value of x_1,
I can compute what r is and what σ²
is and I can then describe the opportunities I have from
investing that depend on these. Now, one thing to do is to
solve the equation for r and x_1 and I
can then recast the variance in terms of r ;
that gives us the variance of the portfolio as a function of
the expected return of the portfolio.
Let me just solve this for--let's solve
x_1 for r. I've got--this should
be x--did I make a mistake there--so it says that
r_1 - r_2 =
x_1 r_1 -
r_2, so x_1 =
(r - r_2)/(
r_1 - r_2) and I can
substitute this into this equation and I get the portfolio
variance as a function of the portfolio expected return r.
That's all the basic math that we need.
If I do that, then I get what's called the
frontier for the portfolio. I have an example on the screen
here, but it shows other things. Let me just--rather than--maybe
I'm showing too many things at once.
Let me just draw it. I'll leave that up for now but
we're moving to that. What we're doing here is
the--with two assets, if I plot the expected annual
return r on this axis and I plot the variance of the
portfolio on this axis, what we have--I'm sorry,
the standard deviation of the portfolio return. It tends to look--it looks
something like this--it's a hyperbola;
there's a minimum variance portfolio where this σ
is as small as possible and there are many other possible
portfolios that lie along this curve.
The curve includes points on it, which would represent the
initial assets. For example,
we might have--this is asset 1--and we might have something
here--this could be asset 2. Depending on where the assets
expected returns are and the assets' standard deviations,
we can see that we might be able to do better than--have a
lower variance than either asset.
The equally-weighted case that I gave a minute ago was one
where the two assets had--were at the same--had the same
expected return and the same variance;
but this is quite a bit more general.
So that's the expected return and efficient portfolio frontier
problem. I wanted to show an example
with real data that I computed and that's what's up on the
screen. The pink line takes two assets,
one is stocks and the other is bonds, actually government
bonds. I computed the efficient
portfolio frontier for various--it's the efficient
portfolio frontier using the formula I just gave you.
The pink line here is the efficient portfolio frontier
when we have only stocks and bonds to invest.
You can see the different points--I've calculated this
using data from 1983 until 2006--and I computed all of the
inputs to those equations that we just saw.
I computed the average return on stocks over that time period
and I computed the average return on bonds over that time
period. These are long-term government
bonds and I--now these are--since they're long-term,
they have some uncertainty and variability to them.
I computed the σ_1,
σ_2, r_1,
and r_2 for those and I plugged it into that
formula, which we just showed. That's the curve that I got out.
It shows the standard deviation of the return on the portfolio
as a function of the expected return on the portfolio.
I can achieve any combination--I can achieve any
point on that by choosing an allocation of my portfolio.
This point right here is, on the pink line,
is a portfolio 100% bonds. Over this time period,
that portfolio had an expected return of something like a
little over 9% and it had a standard deviation of a little
over 9%. This is a portfolio,
which is 100% stocks, and that portfolio had a much
higher average return or expected return--13%--but it
also had a much higher standard deviation of return--it was
about 16%. So, you can see that those are
the two raw portfolios. That could be investor only in
bonds or an investor only in stocks, but I also show on here
what some other returns are that are available.
The minimum variance portfolio is down here.
That's got the lowest possible standard deviation of expected
return and that's 25% stocks and 75% bonds with this sample
period. I can try other portfolios;
this one right here--I'm pointing to a point on the pink
line--that point right there, 50% stocks, 50% bonds.
You can see–You can also go up here, you can go beyond
100% stocks, you can have 150% stocks in your portfolio.
That means you'd have a leveraged portfolio,
you would be borrowing. If you had $1 to invest you can
borrow $.50 and invest in a $1.50 worth of stocks.
That would put you out here; you would have very much more
return, but you'd have more risk.
Borrowing to buy stocks is going to be risky.
You could also pick a point down here, which is more than
100% bonds--how would you do that?
Well, you could short in the stock market,
you could short $.50 worth of stocks and buy $1.50 worth of
bonds and that would put you down here.
Any one of those things is possible it's just the simple
math that I just showed you. Do you have any idea what you
would like to do, assuming this?
Well if you're an investor, you don't like variance.
So, you probably don't want to pick any point down here,
because you're not getting anything by picking a point down
there because you could have a better point by just moving it
up here. You'd have a higher expected
return with no more variance. It's getting kind of
complicated, isn't it? We started out with just a
simple idea: that you don't want to put all your eggs in one
basket and if you had a lot of independent stocks you would
want to just weight them equally.
But now, you see there are a lot of possibilities and the
outcome of your portfolio choice can be anything along this line.
I'm not going to tell you what you want to do except to say,
you would never pick a point below the minimum variance
portfolio, right? Because, if you did,
then you would always be dominated.
You could always find a portfolio that had a higher
expected return for the same standard deviation.
But beyond that, if you were confined to just
stocks and bonds, it would be a matter of taste
where along this frontier you would be.
You would call it an efficient portfolio frontier.
It would be anywhere from here to here, depending on how much
you're afraid of risk and how much you want expected return.
Now, we can also move to three assets and, in fact,
to any number of assets. The same formula extends to
more assets. In fact, I have it--suppose we
have three assets and we want to compute the efficient portfolio
frontier, the mean and variance of the portfolio.
What I have up there on the diagram are calculations I made
for the efficient portfolio frontier with three assets.
So, now we have n = 3 and in the chart are stocks,
bonds, and oil. Oil is a very important asset,
so we want to compute what that--so now we have lots of
inputs. Let's put the
inputs--r_1, r_2,
and r_3 are the expected returns on the
three assets. Then, we have the standard
deviations of the returns of the three assets and we have the
covariance between the returns on the three assets.
There are three of them-- σ_12,
σ_13, and σ_23.
That's what we have to know to compute the efficient portfolio
frontier with three assets. To make this picture,
I did that. I computed the returns on the
stocks, bonds, and oil for every year from
1983 and I computed the average returns,
which I take as the expected returns, I took the standard
deviations, and I took the covariance.
These are all formulas, I just plugged it into formulas
that we did in the second lecture.
What is the portfolio expected return?
The portfolio expected return--we have to choose three
things now: x_1,
x_2, and x_3.
x_1 is the amount that I put into the first
asset, x_2 is the amount that I put into the
second asset, and x3 is the amount I
put into the third asset. I'm going to constrain them to
sum to one. The return on the portfolio is
x_1 r_1 +
x_2 r_2 +
x_3 r_3.
The variance of the portfolio, σ²,
is x_1² σ_1^(2) +
x_2² σ_2²
+ x_3² σ_3²
-- then we have to the count of all of the covariance terms -- +
2x_1x _2
σ_12 + 2x_1x_3
σ_13 + 2x_2x_3
σ_23. Is that clear enough?
It seems like a logical extension of that formula to
three assets and you can easily see how to extend it to four or
more assets, it's just the logical extension
of that. What I did in this diagram is I
computed the efficient portfolio frontier--now it's the blue line
with three assets. Now, once you have more than
three--more than two assets--it might be possible to get points
inside the frontier. But I'm talking here--this is
the actually the frontier--the best possible portfolio
consisting of three assets. You can see that it dominates
the pink line. When you add another asset,
you do better when you have three assets,
you do better than if you just had two because there's more
diversification possible with three assets than with two.
Oil, bonds, and stocks are all independent--somewhat
independent--they're not perfectly independent,
but they're somewhat independent and,
to the extent that they are, it lowers the variance.
You should see that the blue line is better than the pink
line because, for any expected return,
the blue line is to the left of the pink line,
right? So, for example,
at an annual expected return of 12% if I have a portfolio of
stocks, bonds, and oil I can get a
standard deviation of something like 8% on my portfolio.
But if I would confine myself just to stocks and bonds,
then I would get a much higher standard deviation.
Are you following this? The general principle of
portfolio management is: you want to include as many
assets as you can. You want to get it--if you keep
adding assets, you can do better and better on
your portfolio standard deviation.
You can see some of the points I've made along the blue line
here. This is--let's see if I have it.
This is a portfolio, which has all oil and stocks
and it has no bonds. This portfolio,
the minimum variance portfolio, is 9% oil, 27% stocks,
and 64% bonds and most of the--many choices you can make.
The first--you see, the idea here is that in order
to manage portfolios what we want to do is calculate these
statistics, which are the expected returns
on the various assets, the standard deviations of the
various assets, and you've got to know their
covariances because that affects the variance of the portfolio.
The more they covary--they move together--the less they cancel
out. So, the higher the covariance
is, generally, the higher--you can see from
here--the higher the σ² of the portfolio. Is that clear?
There's one more thing that we can do.
I have three assets shown here. I have stocks,
bonds, and oil but I want also to add one more final asset,
we'll call it the riskless asset,
which is the asset–long-term bonds are
somewhat uncertain and variable because they're long-term.
If we have an annual return that we're looking at,
we can find a completely riskless asset with an annual
return--it would be a government bond that matures in one year.
Now, assuming that we trust the government--I think the U.S.
Government has never defaulted on its debt--we'd take that as a
riskless return. It probably has some risk,
but the way we approximate things in finance,
we take the government as riskless.
With the government expected return, we want to make that
expected return as a fourth asset--we could call it
r_4 but I'll call it
r_f--it's a special asset.
So r_f is the riskless asset. So, for it, σ_f
= 0. It's like a fourth asset but
we're using a special feature of this asset: that it has no risk.
Moreover, the correlation--the covariance between any of these
σ_1f = 0, etc.
It just doesn't have any risk to it, it has no variability.
If we want to add that asset to the portfolio,
what it does is it produces an efficient portfolio frontier
that is now a straight line; I show that on the diagram.
The best possible portfolio that you can get would be points
along this straight line. That is the final aspect of the
efficient portfolio calculations.
Again, I'm not able to give this as much of a discussion as
I would like because I don't want to spend too much time on
this. In the review sections I'm
hopeful that your T.A.'s can elaborate on this more.
There's a very important principle that finally comes out
here, it is that you always want to reduce the variance of your
portfolio as much as you can. That means that you want to
pick, ultimately, a point on this--this line is
tangent to the efficient portfolio frontier with all the
other assets in it. Tangent means that it has the
same slope, it just touches the efficient portfolio frontier for
risky assets at one point, and the slope of the efficient
portfolio frontier, including the riskless asset,
is a straight line that goes through the tangency point,
here. That is the--I think that's the
end of my mathematics. What I have shown here is how
you calculate your portfolio management.
The way you would go about it, if you're a portfolio manager,
is you have to come up with estimates of the inputs to these
formulas--that means the expected returns,
the standard deviations, and the covariances.
You take all the risky assets and you analyze them first to
get their--you have to do a statistical analysis to get
their expected returns, their variances,
and their covariances. Once you've got them together,
then you can compute the efficient portfolio frontier
without the riskless asset. Then finally,
the final step is to find what is the tangency line that goes
through the riskless rate. It doesn't show it on this
chart, it goes through 5% at a standard deviation of 0,
then it touches the risky asset efficient portfolio frontier at
one point, then from there it goes up above in terms of higher
expected returns for the same variance.
That's the theory of efficient portfolio calculation.
There's something that--a fundamental principle--and this
is leading us now to the institutional topic of this
course--is that there's only one tangency portfolio and that
portfolio is called the tangency portfolio,
where a line drawn from the risk--from the x-axis at the
riskless rate is tangent to the efficient portfolio frontier.
The tangency portfolio is the portfolio that one should hold.
The tangency portfolio gives rise to what's called the mutual
fund theorem in finance, which says that all investors
need is a single mutual fund. Now, I haven't defined mutual
fund yet. A mutual fund is an investment
vehicle that allows investors to hold a portfolio.
The theory of mutual funds is: nobody is supposed to be
holding anything other than--the ideal theory of mutual funds is
holding something other than this tangency portfolio.
So, why don't we set up a company that creates a portfolio
like that and investors can buy into that portfolio.
What this--if my analysis is right--namely,
if I've got all of the right estimates of the expected
returns on stocks, bonds, and oil,
and the standard deviations and covariances--and assuming the
interest rate is 5%, which is what I've assumed
here, that this line, if you go to 0 it hits that 5%.
It hits the vertical access at 5%.
Then everyone should be holding the tangency portfolio.
What is the tangency portfolio in this case?
It's 12% oil, 36% stocks, and 52% bonds.
That's what I got using this sample period.
Some people might disagree with that, they might not take my
estimates. They might say my sample period
was off, but that's what the theory--using my data for the
sample period that I computed--the expected returns
and co-variances says one should do.
The theory says everybody should be investing in these
proportions and this theory then--it doesn't leave any
latitude for individual choice except that you can choose which
mixture of the mutual fund and the riskless asset you want.
Somebody who is very risk averse could say,
I want to hold only the riskless assets because I just
don't like any risk at all. That person--I should have
maybe included that in the diagram--that person could get
5% return with no risk. Somebody else might say,
well I want to just hold this point, I want to hold the
tangency portfolio. That's attractive to me because
I could then get a bigger expected return,
I could get almost 12% return per year,
and I'd sacrifice--I'd have some standard deviation of like
8%; but, if that's what I want,
if I have different tastes about risk, then--and that's
what I want--then that's the optimal thing to do.
Other people might say, well you know,
I'm really an adventurer--I don't care too much about
risk--I want the much higher return.
Such a person might pick a point up here and that would be
a portfolio with--a leveraged portfolio.
That would be a portfolio where you borrowed at the riskless
rate and you put more than 100% of your money into the tangency
portfolio. What you could do is,
say, borrow $.50 on your $1 and put $1.50 into a portfolio,
which consisted of 9% oil, 27% stocks, and 64% bonds.
Everyone would do that, no one would ever hold some
other portfolio because you can see that this line is the
lowest--it's far--you want to get to the left as far as
possible. You want to,
for any given expected return, you want to minimize the
standard deviation, so it's the left-most line and
that means that everyone will be holding the same portfolio.
I don't find that my analysis is profound in the final answer,
I just took some estimates using my data and,
again, we could--if someone wanted to argue with us they
could argue with my estimates of the expected returns of the
standard deviations and the covariances,
but not with this theory. This theory is very rigorous.
If you agree with my estimates, then you should do this as an
investor, you should hold only some mixture of this tangency
portfolio, which is 9% oil,
27% stocks, and 64% bonds. You see what we've got here?
I started out with the equally-weighted--I was talking
about stocks--about n stocks that all have the same
variance and are all independent of each other.
But, I've dropped that assumption and now I'm going on
to assuming that they're taking account of their dependence on
each other, taking account of their
different expected returns, and taking account of their
different covariances and variances;
so that's what we've got. This is a famous framework.
This diagram is, I think, the most famous
diagram in all of financial theory and it's actually the
first theoretical diagram. I did it myself using my data,
but it would always look more or less like this--slightly
different positions if people use different estimates.
I actually showed this diagram--I went to Norway with
my colleague--I actually have a couple more pictures here. That's my colleague,
Ronit Walny, and I, we're posing in front of
the Parliament Building in Oslo. We went to Norway to discuss
the--with the Norwegian Government--their portfolio.
This is a slide that I showed them.
I showed them the slide that I just showed you,
showing the optimal portfolio, and then I looked at the
Norwegian Government's position. The Norwegian Government has
pension fund assets in the amount of, as of 2006,
just under two trillion Norwegian Kroner;
but they also own North Sea Oil. If you know that,
it's kind of divided between the UK and Norway.
Norway has a much smaller population than the UK and they
have a lot of oil up there in the North Sea.
As of then, I calculated the value of their oil in the North
Sea and that's what I got--it's worth about 3.5 billion
Norwegian Kroner. Do you see the difference?
In fact, the assets that the Norwegian Government owns is
about two-thirds oil and one-third government pension
fund assets. This government pension fund,
I guess, in dollars, is about $200 billion.
It's a huge amount of money that they're managing,
but I was trying to convince them that they should do
something to manage their oil risks because they're way
over-invested in oil. Where are they on the efficient
portfolio frontier? They have 64% oil in their
portfolio. Where does that put them?
Well it's not--it's really off the diagram.
The furthest point that I recorded was 28% oil--that puts
them there--so if they--if you wanted to, where would it be?
It would be somewhere over there off the charts.
What the Norwegian Government is doing wrong is--it's a little
bit controversial, my pointing this out to them.
I ended up in the newspaper the next day for having argued that
they are way off on their investment opportunities because
oil is such a volatile thing. They've got so much of their
assets tied up in oil. I got a good hearing.
I went to the Ministry of Finance and we went to the
Norges Bank, which is their central bank,
and I think the answer I got from them was,
yes you're right. I never got it quite like
that--something like that. There was a conditional
agreement: yes, Norway should manage this oil
risk but it's politically difficult and that's the
problem--is that we're not able to do our optimal management.
Maybe there are a number of structural problems that prevent
them from doing that and they think,
maybe, that--I think Norway may be moving that way,
so we'll see in the future. I went to the Bank of Mexico;
I tried to convince the--I met the president of the Bank of
Mexico and tried to tell them that Mexico is too reliant on
oil, too much oil--they have to get
rid of their risk. I'm going to the Russian
Stabilization Fund in--I think I've got an arrangement to meet
with them in Moscow in March--that would be during the
semester. I'll tell you what reaction I
hope I get from the Russians. Oil is very important to the
Russian economy and are they managing the risk well?
I bet not. I'm going to do a diagram like
this for Russia. The countries that really
matter are the ones that--the Arabian--are the Persian Gulf
countries; I was just talking to people at
the World Economic Forum about that.
Some of those countries are really reliant on oil,
so they really have to do--they really should do a calculation
of efficient portfolio frontiers.
Well, one of the lessons of this course is we have a
wonderful theory, but we don't manage it well,
mostly. And I don't mean to be
criticizing foreign countries. The same criticism applies to
the United States as well. We're in a different position.
Where are we on this frontier regarding oil in the United
States? Well, we don't have much oil
relative to the size of our portfolio.
I don't know what percent; our oil reserves in the U.S.
are pretty small, so we're kind of lying
somewhere inside--maybe on this pink line.
So, people in the U.S. don't have the optimal
portfolio either. I set up a theoretical
framework here and I wanted to give you--I mentioned oil
because it seems to make it, to me, so clear what we're
talking about here. It's talking about not getting
tied up in risks and so--I was talking to someone at the World
Economic Forum from a Persian Gulf country and I said,
aren't you worried about reliance on oil?
He said, of course we're worried on reliance,
so much of our GDP and of our government revenues is
oil-related. We've seen the price of oil,
lately, move all over the map. It went up to $100 recently and
it was just as late as late 1990s that it was under $20 and
people just don't know where it's going to go.
I think that these countries are somewhat trying to manage
this oil risk but they can't yet get onto the frontier.
That's a sign to me that we're not there yet and we have a lot
more to do in finance. There's one more equation that
I wanted to write down and I'm going to not--I'm not going to
spend a lot of time explaining this because it's going to take
awhile. This is the equation that
relates the expected return on an asset.
It's so called capital asset pricing model.
In the capital asset pricing model, in finance--this is the
most famous model in finance. It's abbreviated as the CAPM
and I'm not going to do it justice here,
I'm sorry, but there are so many ins and outs of this.
You should really take ECON 251 to learn this more.
It was--the model divided by James Tobin here at Yale,
was the original--who got the original precursors,
but it was more invented by William Sharpe,
John Lintner, Harry Markowitz. Everyone of these,
except Lintner I think, won the Nobel Prize.
I think he died too young--that's one of the
misfortunes of living. Did I say that right?
One of the misfortunes of scholarship, you have to live
long enough to get your accolades.
The asset pricing model--and this is critical--assumes
everyone is rational and holds the tangency portfolio. That is a wild assumption,
but it's fun to make, because I know pretty well--I
know very well--that people aren't doing this.
They have lots of maybe good--maybe it's not because
they're irrational, it's that they're political or
they're constrained by tradition or laws or regulations,
all sorts of things; but, they're not holding the
tangency portfolio. It's a beautiful theory to
assume, to see what would happen if they did.
That would mean that everybody is holding that same portfolio
of risky assets and nobody is different,
they're only different in how--what proportions they hold
the risky--the tangency portfolio.
It implies that the tangency portfolio has to equal the
actual market portfolio and that means then--it's a very simple
implication of the theory. In my diagram,
I said that the tangency portfolio--I estimated that the
tangency portfolio is 9% oil, 27% stocks, and 64% bonds.
If we're all doing that, then that has to be what the
outstanding is. If we're all holding the same
portfolio, that has to be the total, so that would mean that
9% of all wealth is oil, 9% of all--is oil--27% of all
wealth is stocks, and 64% is bonds.
If you accept my estimates and you accept the capital asset
pricing model, that would have to be true.
Again, I don't want to make too much of my estimates because
different people would estimate these things in different ways,
but that's the theory. The theory says that the
tangency portfolio equals the optimal portfolio and that gives
us the famous equation--I'm not going to derive this,
but there is the most famous equation in finance--says--can
you read that? That r_i,
the expected return on the ith asset,
is equal to the riskless rate plus something called the beta
for the ith asset times the expected return on the market
minus the riskless rate. Again, I'm not going to spend
much time on this, but the β
of the ith asset is the regression coefficient when you
regress the return on the ith asset on the return
of the market portfolio. r_m is the
expected return on the market portfolio, which is the
portfolio of all assets. The market portfolio is,
if you took all the stocks and bonds, and oil,
and real estate, anything that's available to
invest in, in the whole world, put them all together in one
portfolio. It's the world portfolio,
it's everything and we compute the expected return on that
portfolio, that's r_m.
Also, we need to know how much individual stocks are correlated
with r_m; we measure that by the
regression coefficient. The β of a stock is how
much it reacts to movements in the market portfolio.
If β = 1, it means that if the market
portfolio goes up 10% in value then this asset also goes up 10%
in value. If β is two,
it means that if the market goes up 10% in value,
the stock tends to go up 20% in value and so on.
These are the basic theoretical structures that you incidentally
need for the problem set, the first problem set.
The first problem set--I guess you've--you can turn in your
first problem set here before you leave because it was due
today. The second problem set is about
this model. I realize I've given you some
difficult mathematics, but--it's not that difficult
actually--but I kind of went through it quickly.
So, we are setting up our--you have already gotten email and
you've talked about setting up your review sessions--so I have
a few more minutes here. What I want to do,
I want to talk about Jeremy Siegel's book and the equity
premium puzzle. Underlying this analysis,
we have estimates of the expected returns on assets,
notably, the expected returns on stocks and bonds. Jeremy Siegel,
in his book, which is assigned for this
course, is really emphasizing this capital asset pricing
model, emphasizing the kind of
efficient portfolio frontier calculations that I've done.
What Siegel emphasized--the book is really about this--he
talks about what is the expected return of stocks and what is the
expected return of bonds and so on.
We're going to call asset one stocks in the U.S.
and we're going to call asset two bonds.
He estimates, for his purposes,
and he shows you calculations of the efficient portfolio
frontier. I want to just talk a little
bit about his estimate. He has data for a very long
time period, 1802 to 2006, for the U.S. Over that very long time
interval, the expected return that we got for stocks was 6.8%
a year in real terms, this is real inflation
corrected; whereas, for bonds,
it was, over this whole time period, 2.8% a year,
real. Then he also computes
σ_1 and σ_2 and
σ_12 but I'm not going to talk about that right
now. Then he computes the efficient
portfolio frontier. Now, he's using a much longer
sample than I did, so he's not going to get this
tangency portfolio that I did. The thing that is very
interesting that he finds is the difference, 4%,
between the historical real return on stocks and the
historical real return on bonds. This is called the equity
premium. Actually, I should take that
back, this is really r_f--he shows
all three. r_f that I'm
reporting is the riskless rate. There's also--if you look on
Table One in Chapter One, it shows r_1
stocks, r_2 long-term
bonds, and these are short-term that I'm recording here.
The equity premium is the--this short-term 2.8% is the riskless
return, historically, for a period of almost 200
years. This is the return on stocks
for a period of almost 200 years.
Stocks have paid 4% more a year over this incredibly long time
period than short-term bonds. Also, they've paid much more
than long-term--well it's not so different.
I don't have the data in front of me, so let's emphasize the
difference between r_1 and
r_f. It surprises many people that
stocks have paid such an enormous premium over the return
on short-term debt. The theme of Siegel's book is,
can we believe this? I mean, do you really--you
might wonder, aren't people missing something
if the excess return is so high of stocks over short-term bonds?
Why would anyone not just hold a lot of--a really large number
of stocks? That's the theme of his book
and his book is at--his conclusion is that he largely
believes that this is true, that the returns that we have
seen in the U.S. in the stock market have
exceeded those of other assets by quite a substantial margin.
That means--his calculations are very different than the ones
I have because he has a much longer sample period.
I was using only from 1983 to the present.
For him, the optimal portfolio should be very heavily into
stocks. Now, this is a controversial
view, but that is the view that he advances in the book and I
think it's a very interesting analysis.
So, what it means then is that the optimal portfolio should not
be the one I show there, but should be one that's very
heavily into stocks. This is--I'm showing here U.S.
data, but Siegel also argues in the latest edition that the
equity premium is also high for advanced countries over the
whole world. If you look at his book,
again in the first chapter, he gives a list of
countries--it's based on an analysis that some others--that
Professors Dimson, Marsh, and Stanton used in
their 2002 book. They show the expected return
on stocks versus bonds or short-term debt for a whole
range of countries. In every one of these
countries, since 1901, there has been a very high
equity premium. I'll let you read Siegel and
his discussion of this possibility, but I think
that--what I want to get now is not necessarily any agreement on
whether you believe that there's such a high excess return for
stocks. but I just want you to
understand the basic framework. The first problem set asked you
to manipulate the model that I just presented--the model of how
you form portfolios and the model of the capital asset
pricing model. I have one final question,
just about the mutual fund industry.
The mutual fund industry is supposedly, and according to
theory, doing this for you. The ideal thing would be that
the mutual fund does these calculations and it puts it all
together for you. At least in some approximate
sense that's what they are doing.
I asked you in the final question to just get on a couple
of websites, The Federal Reserve and the ICI,
which is a mutual industry website, and write a little bit,
a couple paragraphs about what this industry is really
achieving or how it's trending. Next Wednesday–well,
there are three lectures this week, Wednesday will be about
the insurance industry and I'll see you in two days then.
