# Ses 16: The CAPM and APT II

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visit MIT OpenCourseWare at ocw.mit.edu. ANDREW LO: OK. What I'd like to do today is
to continue where we left off last time in talking about this
risk-reward trade-off, which ultimately will allow us
to be able to figure out how to calculate the proper
discount rate for any project under the sun. Now, where we left off last time
was this equation and the one after it. This equation we
actually derived. I showed you how to
get this equation from this particular bullet
and the tangency line. And today I am going to
give you names for them. The bullet, as we've said
before, is the frontier. It's the set of
frontier portfolios. And the upper arc
of that bullet is called the efficient frontier.

Now the tangency line
has a special name too. That tangency line is known
as the capital market line because it represents what
all efficient capital markets should represent in terms
of risk-reward trade-off. So if you are an efficient
portfolio manager, you want to be on that line. OK? So the capital market
line, the equation for that tangency
line, is given by this. The expected rate of return
is equal to the risk-free rate plus some kind of risk
premium, where the risk premium is given by a multiple of
the market's risk premium, or the market excess return. And the multiple
is simply the ratio of the riskiness of
your efficient portfolio relative to the
market portfolio, or the tangency portfolio. If it's twice as
risky, you're going to get twice the risk premium. If it's half as
risky, you're going to get half the risk premium. And we said last
time that, while this is helpful and interesting
and even possibly useful, it's going to be of
limited applicability because not everything is
an efficient portfolio.

What we mean by an
efficient portfolio, the intuition for what an
efficient portfolio is, is a portfolio where
you cannot do better. By cannot do better, I mean you
can't get less risk for that same level of return, or you
can't get more expected return for that same level of risk. That's what we mean by
an efficient portfolio. It's the best you can do. Now, most investments are,
frankly, not efficient. If you pick an arbitrary
stock, like IBM, that's not an efficient portfolio. It doesn't mean it's no good. It doesn't mean you
don't want to hold it. But it means that
you would never want to hold just IBM because if
you mixed IBM with other stuff, you can always do better. By do better, again
I'm going to reiterate, I mean you can have
higher expected return for the same level of risk or
lower risk for the same level of expected return.

That's what I mean by do better. So you would never want
to hold IBM just by itself because you can
do better, right? You can do better in getting
up to that Northwest quadrant from the IBM point. But even though IBM
is not efficient, you might still want to hold it. And more importantly,
you might still want to know what the
appropriate discount rate is for companies
that are like IBM. That's what we're
going to do next. Where I left off last time was
not the capital market line, but this equation, which I did
not derive, but which I argued comes from the
equilibrium argument that Bill Sharpe
made 50 years ago.

And this really
relies on the fact that, if markets
are in equilibrium, there is a relationship
between risk and expected return for
all securities, not just efficient portfolios. But any arbitrary security
has to satisfy this equation if supply equals demand,
if there's an equilibrium. If everybody holds the
tangency portfolio, and the tangency
portfolio therefore is the market portfolio,
in that situation, this equation has to hold. So where we left off was to
try to interpret this equation. This equation is almost
identical to the capital market line. There's only one difference. The only difference is that
the multiplier, the thing that multiplies the
market risk premium, is not sigma p over sigma m. It is beta. And beta, we said, was the
ratio of the covariance of an asset with
the market divided by the variance of the market. It is a measure of that
particular asset's riskiness.

It's not variance anymore,
or standard deviation, it's something else. And so I want to spend
this class talking about what that something else
is and why it makes sense. First of all, let's make sure
we understand the equation. I want to do a
few special cases, and then I'm going to take
apart this notion of beta as being the right measure
of risk in all circumstances. So the first thing I want to
do is to look at some examples. Let's take an example where
the beta is equal to 1. If the beta is equal to
1, what that's saying is that the covariance between
the asset and the market divided by the
variance of the market, that number is equal to 1.

If that's the case,
then it turns out that the expected rate of return
of this asset with a beta of 1 is going to be just equal
to the market risk premium. Or the expected return is
equal to the market's expected rate of return. When beta is equal to
1, the Rf's cancel out, and the expected rate
of return of the asset is equal to the expected
rate of return on the market.

Now, what about if the
beta is equal to 0? If the beta is equal to 0, then
you get the risk-free rate. It's important to realize
that if the beta of an asset is equal to 0, it doesn't
mean that the asset has no volatility. In the past, when
we've looked at mixing the risk-free asset,
T-bills, with the market, we know that when the
risk-free asset is included, you get that straight line. And it's because the risk-free
asset has no covariance. It has no risk at all. But now, with an asset
that has a beta of 0, you're getting the
risk-free return, even though an asset
with a beta of 0 still may have some volatility. It's not the risk-free asset. It's any asset with a 0 beta. The third observation
that I want to make is when a beta is negative. With the beta that's negative,
the expected rate of return is actually less than
the risk-free rate. Now that's really odd. I've got an asset that is risky. But just because it
has a negative beta, the expected rate of return is
less than the risk-free rate.

That means that you are willing
to take a lower expected rate of return than the risk-free
rate for an asset that has this weird property
of negative beta. And the question I want to
answer today is, why is that? Why is it the case
that you might be willing to take such
a low rate of return? And actually, if the
beta is negative enough, if this beta is
negative enough, it could be the case that the
expected rate of return is negative. In other words, you might
be willing to pay somebody for the privilege of
bearing that risk. That seems completely
counterintuitive. Why would you be willing
to pay to take risk? You should be getting
paid to take risk, right? That's the standard
hypothesis that we come into the
financial markets with. So yeah, Leah? AUDIENCE: [INAUDIBLE] ANDREW LO: That's right. AUDIENCE: [INAUDIBLE] ANDREW LO: Exactly. That's exactly the intuition
for all three of these cases. It turns out that the
beta, remember, has this covariance term in there. And it is going to
turn out that, if you can find an asset that
is negatively correlated to the tangency portfolio,
that is going to be of tremendous value to you.

It's very, very valuable. Now, OK. Let me try to explain
the logic behind it. And we're going to
do a few examples. So let's go through the basic
logic of what's going on. The tangency portfolio
plays a central role in that everybody
in the world is going to want to have
a portfolio that's a combination of
the risk-free asset and the tangency portfolio. That's fact number one. Fact number two, the tangency
portfolio is a portfolio that has the aggregate measure
of the total amount of risk in the economy that cannot be
diversified beyond that point. In other words, in
order to get lower risk than this particular
portfolio, you have to decrease your
expected rate of return. There's no way to get lower
risk and keep that same level of expected return. You can't go this way. You have to go down this line. OK? So if you're going to
hold a portfolio of purely risky securities,
then basically this is the best that you can do. This is the best
trade-off that you can get in terms of risk-reward. So right away you know
that this market portfolio plays a very special role.

It is really the representation
of the aggregate risk in the stock market. And that's why it can serve as
a kind of a benchmark for what the stock market is doing. We're going to come back
to that benchmark idea in a few minutes. So, if you have a security
that is very highly correlated to that
market return, then that's not going
to help you in terms of your own diversification. If, on the other hand, you have
a security that is negatively correlated with that
market portfolio, that's going to help you a lot. And if it's going
to help you a lot, you're willing to pay for it. When you're willing to pay
for it, what does that mean? You drive the price today high. And therefore, the
expected rate of return, which is the return between
today and next period, that becomes lower. So an asset that
helps you hedge what is essentially unhedgedgeable,
in other words, the market portfolio, that can
benefit you a great deal. As a result, it's
going to be very hard to find negative
beta securities or assets.

But in any case, this
relationship really tells you that, given a
particular covariance, you can measure the
expected rate of return that you ought to be getting. And this relationship
is so important that we give it a separate name. It's called the security market
line, not the capital market line, but the
security market line because this applies to
every single security in your entire universe.

Yeah, Dennis? AUDIENCE: You say it's
hard to find something with a negative beta. But when you short
something, does that mean you're
getting a negative beta? ANDREW LO: It is. You do. But the problem with
shorting something where you get a
negative beta, you also get a negative expected
rate of return typically. So what you want to
have is an asset that's got a positive beta–
or negative beta, but a positive expected
rate of return. That's what's very rare. But you can manufacture a
negative beta security very easily by just shorting it. The problem is that
when you short a stock, you're going to also get a
very negative expected rate of return. And that doesn't
help you in terms of producing an attractive
investment opportunity. Ken, question? AUDIENCE: So, what's an example
of a negative beta security? ANDREW LO: Well, it's
very hard to come by. But the closest thing that
exists in markets today is stocks that are involved in
gold production, gold mining stocks.

That has a beta of around 0,
but sometimes it's negative. Sometimes it's positive. But it's small. So that's an example, but
that's about the only example that we can come up
with in the data that looks slightly negative. By and large, most of the betas
in the data sets are positive. And they're actually
clustered around 1. So the typical beta is in the
neighborhood of 0.5 to 1.5. All right. Now, before we start
looking at the data– I'm going to show you
some data in a minute– I want to take the security
market line and apply it. So the security market
line I did not derive. I want to make that clear. And for those of you who are
interested in the derivation, you can take a look at the
appendix in the Brealey, Meyers, and Allen.

They provide the derivation. It's a little bit messy,
but with a little bit of matrix algebra you
can work through it. But the implications
are extremely important, so I want all of you
to know how to use it. So it may look pretty
simple, but I'll show you a few applications that
you might not have thought of. For example, suppose this is
true for every single security. Well, if it's true for
every single security, it turns out that this works
for portfolios as well.

And let me show you why. Suppose you've got
a portfolio that's a weighted average
of the returns for the individual
components securities. Then if I calculate
the covariance between the portfolio return
and the tangency portfolio, or the market, the
covariance is going to look like this,
which is actually just going to look
like a weighted average of the covariances. This is a mathematical identity,
from here down to here.

And therefore, when
I divide both sides by the variance of the market,
I get something pretty neat, which is that the
beta of my portfolio– where beta is defined
as the covariance between the portfolio
and the market divided by the variance– the
beta of the portfolio is just equal to the
weighted average of the betas of my component securities. That's really neat
because what that says is that, when I want to measure
the risk of a collection of securities, as long
as I know the betas of each individual
security, I can calculate a weighted
average of those betas. And that is the beta
of my portfolio. So if you think of beta
as a measure of risk, this measure of risk
is actually linear, unlike volatility,
which is not linear.

The variance of a
portfolio is not simply equal to the sum of
the variances weighted by their portfolio weights. It's that complicated
expression where you're adding up all of
those cross products as well. So we get an enormous
simplification with the security market line. It says that we can measure
the risk of a portfolio using this concept called beta. And beta happens to
be linear in the sense that, when you take
a weighted average, the beta is equal to
the weighted average of the individual asset betas. So therefore, if you
know that the betas are going to be a weighted
average, then, in fact, the expected rate of
return on the portfolio now is equal to
the risk-free rate plus this weighted average beta
times the market risk premium. Do you see the power of this? This now allows you to
analyze the expected return on anything, any
collection of assets. If you know what the betas are
for the individual components, you know what the betas
are for the whole thing. So now you can calculate the
appropriate rate of return for virtually anything. And this is not just
limited to stocks.

You can apply this to projects. For example, if you
want to know what the expected rate of return
is on an oil drilling project, well, measure the beta of oil
drilling stocks, use that beta, and that will be
appropriate discount rate for that particular
oil drilling project. It's a really remarkable result. So we have an expression for
the required rate of return, opportunity cost of capital,
risk-adjusted discount rate, for all the various different
kind of examples and cases that we looked at up until now. And the last point I want
to make about this equation is, how do you actually
take it out for a spin? How do you estimate the
expected rate of return on the market and
the risk-free rate? Well, that comes from the data. That comes from the marketplace. We observe it in
the marketplace. And we can actually see it. OK. So let's do some examples, just
to make sure that we all get this and know how to apply it. Using returns from
1990 to 2001, we estimate that Microsoft's beta
during that period of time is 1.49.

And if you do the same
thing for Gillette, you get that Gillette's
beta is 0.81. Now, let's not even look
at the next set of numbers for a moment, and just talk
about those two numbers, 1.49 and 0.81. Does that make sense to you? Let's think about
what that's saying. 1.49 says that the
covariance between Microsoft and the market
portfolio is actually a lot higher than the
variance of the market itself. So let me ask you to
think about whether or not adding Microsoft
to your portfolio is going to make it less
risky or more risky. And here's how I want
you to think about it. Remember what we said
about diversification. When you hold a
collection of securities, what matters more, the
variances or the covariances? AUDIENCE: Covariances. ANDREW LO: Right.

Why are the covariances
more important? What's a quick and dirty way
of arguing that the covariances matter more? AUDIENCE: Because there
are n squared minus n. ANDREW LO: Exactly. There are a heck of a
lot more covariances than there are variances. You only got n variances
to worry about, but you got n squared
minus n covariances. And if they all line up
in the same direction, you get the subprime
crisis problem, right? So covariances matter
more than variances. Well, if that's the case,
then when we look at a stock and think about bringing
it into our portfolio, we want to ask
the question, what is it doing in terms of adding
or subtracting covariances to our portfolio? And one way to
measure whether or not it's adding or subtracting
is to ask the question, what is the covariance between
Microsoft and my stock holdings? Now what are your
stock holdings? If everybody is a
rational investor– by rational I mean
you like return and you don't like risk– then you know you're going to
be holding the risk-free rate and the market portfolio.

You're going to be on
that capital market line. So if you're a
rational investor, the only stock holding you
have is that mutual fund, m, the tangency portfolio. So therefore, the most
important thing in your mind is, when you think
about buying a new stock and putting into your
portfolio, is this going to be highly correlated
with my market portfolio? Well, that's what beta measures. Beta is a relative measure that
says, OK, the total variance that you're holding in
risky securities, that's sigma m squared. That's the variance of
the market portfolio. How does Microsoft
compare to that in terms of what
it will contribute, in terms of its covariance
with your holding? So you're holding
one mutual fund, and you're thinking
about adding Microsoft. The only covariance that
you should care about is the covariance
between Microsoft and what you're holding.

Well, that's what beta measures. If the number is greater
than 1, what it's saying is that, when you bring
Microsoft into your portfolio, you're going to be
increasing the variance because the covariance,
which is what we care about, is greater than the variance
of what you're holding. If, on the other hand,
the beta is less than 1, then, presumably,
that's helping you because that's lowering the
variance relative to what you're holding. But helping or
hurting, that only can be answered
directly if you explain what you're getting in terms
of the expected rate of return. So looking at beta by
itself is not enough. Beta is a measure of risk. It measures this
covariance divided by the variance, or
covariance per unit variance in the marketplace. But you want to know what
the expected rate of return is as well. That's what the security
market line gives you. OK. Now let's get back
to the example. Microsoft is a lot more
risky than the market. It's about 49% more risky
according to this measure.

On the other hand,
Gillette is actually less risky than the market. Now do you guys buy that? Does that pass the smell test? Does that make sense? Why? What's the intuition for that? Courtney? AUDIENCE: Well, people don't
necessarily need computers. And the technology is variable,
but Gillette sells razor products and deodorant, which
is kind of a staple in a lot of people's– ANDREW LO: Exactly.

That's right. If you make the argument that,
from 1990 to 2001, if there are economic downturns,
what's the first to go, razorblades or Windows? Thankfully Windows. [LAUGHTER] Nowadays, I don't know the
answer to that actually. Because nowadays, we depend
so much on the internet, that actually it
could be different. So I haven't updated
this analysis to see what the beta
is from 2001 to 2008, but it could be different. So now we may have
unshaven geeks that, you know,
during downturns– and maybe it's flipped around. Eduard? AUDIENCE: Could you
give us an intuition? Because beta allows
us to compute the appropriate return
for a certain risk. ANDREW LO: Yes. AUDIENCE: But what
is the intuition of, how far away am I from
the efficiency portfolio? So, how bad is this portfolio
compared to the efficient one? ANDREW LO: Yeah. So, that's a good question. Let me go back to this
equation and take a look at it, and try to provide even
more intuition for this before we go on.

So the idea behind
this equation is that it tells you that
this is the rate of return that you should have if
you have a certain beta. Now you can actually measure
the deviation from that very simply by
asking the question. For a portfolio manager
or an investment project that yields an expected
rate of return that's different from this,
that difference is actually what we call alpha. And alpha could be
positive or negative. So when you say, how far
away are you from efficiency? This gives you a direct measure
of how far away you are. It's basically the difference
between the expected rate of return you
have versus what you're supposed to have given
the beta of the security. But let me add one
more thing to that, which is that beta is a measure
of a particular kind of risk that a particular security has.

And the kind of risk,
as I said before, is this covariance
between the rate of return on a particular
asset and the rate of return on the market portfolio. This kind of risk is
not the total risk of a particular security. In fact, it is called
the systematic risk. The systematic
risk is the portion of the risk that is related
to the market portfolio. So how far away you are
from efficiency really depends upon how much
risk you have that is not necessarily systematic risk. Now, I don't expect you
to understand all of it yet because I need to develop
a little bit more machinery. But I'm going to get
back to that intuition in just a few minutes, OK? So I'm going to
give you a better answer to your question than
what I just did because I'll explain the difference
between systematic risk and idiosyncratic risk.

And I think then it'll make
this completely transparent. So give me another 15 minutes. Yeah? AUDIENCE: So you
mentioned that gold stocks can reduce the volatility. They may have– ANDREW LO: Historically it has. AUDIENCE: Yeah. So there are some
currencies that are indexed to gold prices. So let's say you're company x. If you are listed in the Dow
Jones versus this country, whose currency is
indexed to gold, do you expect the company
that's listed in the gold index currency to have a lower beta? ANDREW LO: Well, it could. But on the other
hand, the question is, what are they doing to
try to hedge that currency exposure? In other words, if they end up
hedging all of that exposure, then it doesn't
matter anymore, right? So it depends.

But the idea is that, if it
has exposure to a 0 beta asset, then you will find that the
ultimate fluctuations are going to have less correlation
to the market portfolio. Slomi? AUDIENCE: Maybe Microsoft
has a bigger beta because during this
period, the 10 years, Microsoft was a [? growth ?]
company, and [INAUDIBLE].. So this is the reason
why it has a bigger beta. ANDREW LO: You know,
that's possible. But remember that when
we estimate the beta, we're estimating it
using monthly returns. And so we're measuring
the covariance on a month-to-month
basis, not just the trend. We're measuring fluctuations
around that trend. So the trend alone won't
necessarily explain all of it.

It has to also be the
fluctuations are actually going both up and down,
higher than the variability of the market portfolio. Let me continue on
with the example. And we're going to come back. I'm going to show
how to estimate this, and then you'll develop
more intuition from it. So this makes sense from
the smell test in the sense that Microsoft, at least
during that period of time, was not necessarily something
that you would expect would do well in good times and bad. But something like razor
blades and shaving cream we need to use regardless. So that sort of tells us that
these betas look about right. OK. If you agree with
the betas, then it turns out that we can
actually calculate the required rate of return
for each of these two stocks.

So if you assume that
the risk-free rate is 5%, which is what it was
about in that period– not today obviously, but back
in that period, so about 5%. And if, historically,
the risk premium, as I told you last
time, is about 6%, then when you do the
calculations using the security market line, you get
a very sharp answer to the question, what are the
appropriate discount rates or costs of capital for
these two companies? The answer, for Gillette, it's
about 9.86%, for Microsoft, 13.94%. So now if you're sitting
in these companies and you're asking
the question, we're going to expand our operations,
but in order to do that, we have to do an NPV calculation
to see whether it's worthwhile. We have to compute the expected
net present value of expansion. And in order to do that,
we've got estimates of what our cash flows are
going to be for our expansion, but we don't know what
the cost of capital is.

Well here's the answer for you. You've actually got hard
numbers to plug in to your NPV calculations now. Now, there are a bunch of
assumptions that we've made. So we're going to have
to go back and justify those assumptions each and every
time you use this technology. It's not physics. This is not mathematics. You're applying a set of
theories and approximations to a much, much more
complex reality. Every time you
apply it, you've got to go back and ask the
question, does it make sense? Do these assumptions hold? And if so, great,
go ahead and use it. If not, you've got to
go back and rederive some of these analytics. OK, so the security
market line is now a line that describes
the expected return, or required rate of return,
on an asset or a project as a function of the riskiness,
where the riskiness is now measured by beta, not by sigma.

It's not variance or
standard deviation that measures the appropriate
risk for most projects. Most projects, the way
you measure their risk is not by sigma. It turns out that the way
you measure their risk, for the purposes of calculating
the required rate of return, you measure it by beta. OK? That's a very deep insight. It changes the way
we think about risk and expected rate of return. It's not to say that risk,
in terms of volatility, doesn't matter. Of course it does. That is the basis of
this entire framework. We started out by saying
that people don't like sigma, and they do like mu. They don't like variants. They do like expected
rate of return. That still holds.

But in doing so, when we derive
all of these implications, what we find is that,
as an investor, you don't get rewarded for taking
larger and larger amounts of volatility necessarily. You do get rewarded for taking
larger and larger amounts of beta. That's the sense in which beta
is a better measure of risk. For the purposes of computing
the required rate of return, beta is the right
measurement, not sigma. The only cases where sigma is
the right measure of risk– when I say right, I mean where
increases in sigma must imply increases in the
required rate of return– is when? When is sigma the
right measure of risk for the purposes of computing
the required rate of return? For what kind of
securities or portfolios? Yeah? AUDIENCE: Is it for
efficient portfolios? ANDREW LO: Exactly. For efficient portfolios only. Efficient portfolios meaning
these guys, meaning these guys. Anything on this line, then
sigma is the right measure.

Sigma p over sigma m, that is
the right measure for any kind of portfolio that is efficient. But we know that the
typical security, the typical project, the typical
division, is not efficient. Efficient, again,
meaning you can get the– you can't get any better
expected rate of return for the same risk, or you can't
get a lower amount of risk for the same expected
rate of return. So for all of the inefficient
securities, portfolios, or projects, you've got
to use this relationship, and this relationship tells
you beta is what matters, not sigma. Sigma is not the same as
beta, except if you happen to be an efficient portfolio. OK. Question– do you
have a question? No? OK. All right. So here's an example of the
security market line at work. And the slope of this
security market line is, of course, the expected
rate of return on the market minus the risk-free rate.

And the idea behind the
security market line is that, no matter
what your beta is, you've got a required
rate of return that's determined by this slope. And now, to answer Eduard's
question about deviations and how far away you
are from efficiency if you deviate from this line,
then the vertical distance is the alpha of your portfolio
or project or investment opportunity. If markets are working
exactly the way they should, then you're on this line. You're always on this line. And if you're Warren Buffett,
you're off of this line. You've got a very
large, positive alpha. So if you've got skill, if
you can forecast markets, then you will do
better than this.

But what this framework tells
you is that even if you cannot forecast markets, even if
you don't know what's going to happen next year
to stock prices, you should still do as well as
what this line suggests that you can do. OK? On average, this line should
be achievable by everybody that understands the
basics of portfolio theory. Now, as I said, the
performance evaluation approach to using the
security market line is just a measure of
the vertical distances. And it can lead to some
interesting results. For example, here
are three managers. All three of these managers have
a 15% expected rate of return. But they have different betas. And so the question
is, if you had money to put into
these managers, which would you choose? Well, clearly you
would choose manager A because the manager
is only supposed to have a 6% rate of return,
but, in fact, he's offering 15 for that level of risk. Manager B is just
basically doing what you would expect the
manager should be doing.

And manager C is
actually underperforming. Given the risk that manager
C is exposing you to, manager C should be doing
much better than he is. And by the way, notice that
I've said that the same– all three managers have
the same volatility, 20%. You can have the same volatility
but have different betas. Betas and volatilities do not
necessarily go hand in hand. There is actually a relationship
between beta and volatility. We'll talk about that
in a little while. But that relationship is not
nearly as straightforward as you might think. Ingrid? AUDIENCE: Can you estimate a
future return of a mutual fund by [INAUDIBLE]. I mean, I understand
it's the best you can do, but how realistic is it? ANDREW LO: Well, so it
depends on who you are. If you are a typical
index fund manager, you would argue that
mutual funds are basically going to provide you with a
relatively stable expected rate of return over time.

So in other words, it'll
fluctuate up and down because it's got some
variance, but there is a baseline expected
rate of return that a mutual fund offers you. And that's what
people are buying. When you put your money
in an emerging market equity mutual fund, you're
going to have a higher return on average, on average,
than if you put your money in a S&P 500 index fund. Why are you going to have
a higher rate of return on average? Because you're going be
bearing more risk on average. The only way to convince you to
put your money in an emerging market fund is if it does
have that higher expected rate of return on average.

So what you're basing these
kinds of calculations on is not that I can forecast
what mutual funds are going to do next year, but
rather, mutual funds offer expected rate of returns
that are stable over time. So what happened last year and
the year before and the year before that, when you
average it all together, it's about what you're going to
get over the next five years. That's it. That's the argument. AUDIENCE: [INAUDIBLE] ANDREW LO: Yes. We don't know– right, exactly. So there is a very large
idiosyncratic component that fluctuates year
by year, and who knows what that could be. OK. Yes? AUDIENCE: Last
time, when you were describing more [INAUDIBLE].

ANDREW LO: Yes. AUDIENCE: You had mentioned
that for more risk people will expect more return,
but not necessarily proportional to the– so you end up having a line
that just goes up exponentially, right? Because you had given
the analogy that, OK, now to get investors
to be in more risk, to take on more risk, you have
to offer a lot more than just [INAUDIBLE]. ANDREW LO: Well, you may. But the equilibrium
theory that we've argued has to hold actually says
that the risk-reward trade-off is, in fact, linear for
efficient portfolios. If it's not for efficient
portfolios, then who knows? But the theory of the
capital asset pricing model, or the security
market line, which is what we derive– what we
what we talked about today, this says that, in
fact, it is linear.

So in other words, this
result, as I told you, we didn't derive it,
but it's a major result that shows that the relationship
between risk and expected return, where risk is measured
by beta now, is linear. This is a linear relationship. So this is a major advance
that we didn't expect. And, in fact, so what I showed
before was the preferences. So in other words, we talked
last time about the situation where, suppose that
you're an investor, and I start you off
at a point like this. And I ask the
question, if I want you to tell me where
you're going to be such that you're just as well off. You have the same
level of utility. That curve, that
indifference curve, is going to look
something like this. That's going to be curved. That's what you're
talking about. But that's the behavior
of one individual. The point about the
CAPM is that if you aggregate all of the individuals
together and ask the question, what does the expected rate
of return and volatility or expected rate of
return and beta look like? How are they related? In fact, it's magical that
it actually is linear.

So it's exactly the fact that
we didn't expect linearity. Given that there are
diminishing marginal returns to risk and reward, you
wouldn't expect linearity. But, in fact, it drops out. I mean, this drops out of this
tangency portfolio argument, right? Nothing up my sleeve,
this was an argument that we all did together. And we derived this curve
from first principles. So this is really an
astounding result, but it's even more
astonishing that you get this result
for all securities, not just for
efficient portfolios.

OK, other questions? All right. Let's do another example. So here I want to show you how
you can use the security market line, also called
the CAPM, C-A-P-M, for Capital Asset Pricing Model. I want to show you
how you can use it to do performance attribution. This is the data from
a real live hedge fund manager, who will go nameless
since I don't want to get sued by him for any reason. Hedge fund managers are both
very wealthy, typically, and also very
litigious, so you want to be careful when you
talk about them in public. Hedge fund XYZ had an average
annualized return of 12.5% and a return standard
deviation of 5.5% from January '85 to December 2002, and the
estimated beta over this period was minus 0.028. Now, somebody asked
about a negative beta. Well here's an example
of a negative beta asset. Positive expected rate
of return, negative beta. So if it's got a
positive expected return and a negative beta, you
know something can't exactly be right because that doesn't
sound like it makes sense in terms of the CAPM framework.

And, in fact, it doesn't. It doesn't make sense. Well, let's do the math. The expected the
rate of return is equal to Rf plus beta times
the market risk premium. Market risk premium of
6%, risk-free rate of 5%, plug that in with the
beta of minus 0.028, and you get that
this manager should have earned 4.83% per year. That's what the manager
should have earned. In fact, the manager earned
a rate of return of 12.% per year. So that's an alpha. If you define the alpha as
what the manager did earn minus what the manager
should have earned, you've got an alpha of
771 basis points per year. That's a humongous alpha. Very, very big amount
of excess performance. This is why people are
excited about hedge funds. Now, we're not going to talk
in great detail about it in this course because it goes
beyond the scope of Finance 401. But in an investments
course, the next level of sophistication would
be to take a look at this and say, OK, is this
alpha really alpha, or is it due to other factors,
other risks that we're not measuring? Right now the only
risk we're measuring is this tangency portfolio
risk, this beta risk.

But maybe there are
multiple betas out there. We're not going to talk
about it in this course, but in 433 you will discuss it. And it will turn out
that hedge funds actually do have multiple betas. So you shouldn't go out and put
all your money in hedge funds right away because this
extra performance, some of it is due to true genius and
insight and unique skill. But part of it is
also due to the fact that you're bearing risks that
you had no idea you're bearing.

And so you've got to be careful
about getting on the bandwagon and saying, yeah, give
me some hedge funds. I want some of this alpha. Anon? AUDIENCE: [INAUDIBLE]
But how do we know this is good for the investor? Because they could have invested
in the asset [INAUDIBLE].. ANDREW LO: So the
question is, how do we know that this is actually
good for the investor? Because they could have gotten
some returns from the S&P.

The way we know that
is because we're measuring the expected rate
of return relative to the S&P. So in other words, the
way I got this number, this is the excess
return on the S&P. That's what the market
risk premium is. So in fact, given the
beta of this manager, it should have only
given you 4.83% return relative to what the
S&P would have given you, which is a 6% excess
rate of return. And, in fact, what we see is
that this manager produced a 12% rate of
return, or 7% above and beyond what it was
supposed to have done. So this takes that into account. What it doesn't
take into account is how much liquidity risk the
hedge fund manager is taking, how much currency risk,
how much commodity risk, and a bunch of other risks
that are not represented by the tangency portfolio. Megan? AUDIENCE:
[? When you hear about ?] looking out for beta
dressed up as alpha, it's really because there
are multiple sources of beta that aren't getting
wrapped into that. ANDREW LO: Exactly. That's exactly right. Recently, a lot of
institutional investors have become skeptical
of hedge funds because they say,
hedge funds alpha is really dressed up data.

In other words,
hedge fund managers are taking risks
that are not captured by this very simple framework. And so when you run
these kind of regressions and do this analysis, you're
getting tremendous alpha, but in fact, it's not all alpha. There's other kinds
of betas in there. And so there's a
whole literature that has developed
about multiple, what are called exotic,
betas or alternative betas. Again, not part of the
scope of introductory finance, but it is something
that's covered in investments. Just another illustration of
what this hedge fund has done. Take a look at the
growth of \$1 invested in the hedge fund over
the last 20 years, and you'll see that the
blue line is the hedge fund. The red line is the S&P 500. So to your point, Anon, the
S&P 500 gave you a wild ride. And so for a while
you were doing better than the hedge fund, but,
in fact, now this hedge fund has done quite a bit better.

Now this ends in 2002. I'll give you a little
bit of an update. I don't have it
here in the graph, but you can use
your imagination. It turns out that up
to 2007, the blue line is way ahead of the red line. That's actually changed in 2008. This hedge fund has done
very badly this year. Of course, the S&P
has done even worse. So the gap is not as
wide as it used to be. There's still a gap, but the
gap is actually narrowed a bit. Yeah? AUDIENCE: [INAUDIBLE] The
risk-free rate is changing. The market rate
is changing, so– ANDREW LO: No. In fact, everything is changing.

So if you want to
take seriously change, you've basically got to figure
out how the risk-free rate is changing, the expected
rate of return is changing, and the betas actually
are also changing. But what I'm trying to do
with a simple illustration is use a long period and
say, over that entire period, let's average across
all of these changes.

Justin? AUDIENCE: [INAUDIBLE]
If you look at the graph and it seems like the
differences of the graph were just higher
with the market. ANDREW LO: It is, but you
don't adjust for the beta. That's the key, right? The S&P has a beta of 1. This guy, this
hedge fund manager, has a beta of 0 or
slightly negative. That's the difference. That's why looking at
volatility can be misleading. If you look at
volatility, you'd say, well, you know, obviously
the S&P has done better. But keep in mind that look
how smooth the blue line is. And the lesson of the
CAPM is that investors pay for smoothness, but only
a certain kind of smoothness. In other words, smoothness
means low volatility, right? The smoothest line,
of course, is T-bills. That's a straight line. That will go sort of like this. And investors are not
going to pay a lot for that because that doesn't
really help them generate expected rate of return. If you've got expected rate of
return and smoothness together, you get a really big, big alpha. And that's exactly what we see
here, an alpha of 771 basis points.

Now, here I talk about
these multiple sources of systematic risk. I don't want to focus
on that for this course because, as I said, it's going
to be much more complicated and requires more machinery. But the basic
intuition is the same. Instead of just one
source of systematic risk, you may have multiple sources. And so therefore, you're
going to have multiple risk premia as opposed to just one. But for now, let's
not focus on that. And we're going to
focus our attention just on this simple equation, and
make sure we understand it and know how to apply it.

Now, I want to go– I want to give you one
more intuition for why it is that beta is the
appropriate measure of risk, and not sigma, for arbitrary
inefficient portfolios. And the idea is
actually pretty simple. When you think of an
investment like Microsoft or like Gillette, you can think
of the risk of that portfolio as being– the risk of that investment
in those individual securities as having two components.

So when you think of the
volatility of Gillette, you can think of the
volatility of Gillette coming from two sources. One source is the aggregate
risk that affects all companies. And the second source
of risk is the risk unique to Gillette,
the fact that they've got a particular manufacturing
plant in a particular location of the country, the
fact that they've got a specific set of managers
that are either good or bad, the fact that they
are subject to a very specific set of requirements
for producing their blades, who knows, but very
specific to that company. When you think about
that kind of risk, let me ask you a question from
a purely business perspective. You're the investor. I'm a representative
of Gillette. I'm trying to sell you
my company's stock. I want you to invest
in my company. And therefore, I have to pay you
to take the risk of Gillette. If I tell you that Gillette
has these two pieces of risk– so I'm the representative from
Gillette and I tell you that our company is subject to
economy-wide fluctuations that will help or hurt our business
and unique fluctuations that are specific and
special to Gillette– which of these two
risks are you going to be more concerned about
from your investment portfolio perspective? Rami? AUDIENCE: I think you'd
be more concerned about– as a portfolio as a whole,
you look at the economy, but I think, for this
specific purpose, [INAUDIBLE] you obviously
will pick Gillette-specific– ANDREW LO: OK.

But I want– you hurried
through the first point, and I want you to expand
on that a little bit. You said that if you're worried
about your portfolio, then obviously the economy-wide risk. AUDIENCE: Absolutely. If you're worried about
the economy-wide risk, and for example, over the
next two, three years, you don't think the economy
is going to recover, then you're going to just
avoid that type of investment as a whole. ANDREW LO: OK. OK, fine. But on the other hand, you
also said something else, which is that if you're
comparing between two stocks, then what you're focusing
on is the risks that are unique to Gillette. AUDIENCE: You also might look at
how much the economic downturn would affect a
company like Gillette. ANDREW LO: Right. AUDIENCE: So if the economic
downturn, as you said before, affects Microsoft greater
than it would Gillette, and you suspect something
is going to happen, you'd go for Gillette. If, on the other hand,
you didn't suspect that, you might go for– ANDREW LO: OK.

But let's now talk
about the negotiations between you and me. I'm a representative
of Gillette. I'm trying to get you
to invest with us. And I've got two
sources of risk that you might be concerned about, market
risk or Gillette-specific risk. From your portfolio
perspective, since you just care about maximizing the
value of your portfolio, you're not worried about
Gillette in particular. You're not management. You're an investor. I'm management. I'm worried about Gillette. I couldn't care less about your
portfolio, I'm sorry to say.

What I care about is my
company, but what you care about is your portfolio. From your portfolio
perspective, what are you going to care more
about, Gillette-specific risk or the macroeconomic risk that
I represent to your portfolio? AUDIENCE: When I'm
talking to you, I care more about
your specific risks. ANDREW LO: Well, I'm
asking a question though about your portfolio. What you care about
is your portfolio. I understand that when
you're talking to me, you're going to be asking me
about my company-specific risk to try to get a handle on it. But is that what
you ultimately are going to be concerned about? AUDIENCE: I would be
concerned about, obviously, the macro portion of it
and how you fit into that.

But I'd look at
your beta and see– ANDREW LO: You don't
know the CAPM, though. So now you're cheating
because you now know the CAPM. But suppose you didn't. What I'm trying to get
at is the intuition, a businessman's
intuition, for what you would care
about more in terms of what I do to your portfolio. Yeah, Sema? AUDIENCE: Didn't you
say, two lectures ago, that the idiosyncratic risk
is [? their survival? ?] ANDREW LO: Yes.

AUDIENCE: So, you care more
about systematic risk– ANDREW LO: Why? AUDIENCE: Because only one
[INAUDIBLE] in my portfolio. ANDREW LO: That's exactly right. The idea is that if
it's Gillette-specific risk, then by
definition, if you're holding a well-diversified
portfolio, then you're not going to care
about that because that's going to average out to nothing. Now, of course, we have
to think about the case where you're not holding
a diversified portfolio, but let me get back
to that in a minute.

I'm assuming that all of you
are good business folks, which means that you're
going to be holding a diversified portfolio. You're not going to
concentrate all your bets on one particular kind
of investment, right? So if you are already holding
a very well-diversified portfolio, then when
you interview me as a potential
investment opportunity, do you really care about
the idiosyncratic risk? Because that risk is going
to be diversified away.

What you care about from a
portfolio perspective is, how much am I going
to be contributing to your overall risk? And now that you know the CAPM,
you understand the logic of it. It says, you care about my beta
because my beta is a measure of the amount of risk I'm going
to be adding to your portfolio that you cannot get rid of. How do you know you
cannot get rid of it? Well, by definition it's
the market portfolio. Everybody's holding it. Nobody wants to get
rid of it completely. You're on the capital
market's line. That's the best you can do. So this notion of
firm-specific risk versus economy-wide
risk, that distinction is a really important one. And I'll show you the
mathematics of it in a minute, but I want to give
you the intuition. As a result, think
about this conversation happening not just for
Gillette, but for every company in the economy. If it's the case that portfolio
managers that are buying stocks only care about the systematic
risk, about the market risk, about the risk that they cannot
get rid of, and then you have to reward them for that, then
what that means is that you don't have to reward them
for idiosyncratic risk.

Why? Because that's not risk
that you are forced to bear. There's nothing
that says you have to bear idiosyncratic risk. How do you get rid
of idiosyncratic risk if you don't
want to bear it? Diversify. Exactly. Just buy 10 stocks
instead of one, and then you're diversified. 20 stocks is better than 10. And mathematically,
after 50 stocks, you're basically diversified. You're done. So nobody should be holding
two or three stocks. Or if they do, they are bearing
risk that they need not bear. And they may want to do
it for other reasons. For example, as a
manager of Gillette, I believe in the company. I want to demonstrate
to my shareholders that I'm tied to
the company, so I'm going to hold a lot of my
wealth in Gillette stock. That's not well diversified. I'm holding a lot of
Gillette-specific risk. That's not a smart thing
to do from an investment point of view, but
that is a smart thing to do from a management
point of view because I'm tying my fate
to the fate of the company.

It's definitely not a smart
thing from an investments perspective. If you're in a financial
services sector, you should not be buying
financial services stocks. If you're in the
pharmaceutical sector, you should not be
buying biotech stocks. And yet we do that for
reasons other than portfolio management. But given that I'm teaching
you about portfolio management, I'm not going to focus
on those other reasons. If this were an
organizational studies course, you'd be getting a
different perspective. And you should get a
different perspective. But for the purposes of
building financial wealth, what you want to do is
to focus on how much the systematic component
is contributing to your risk, because the
idiosyncratic component you don't have to bear. Yeah? AUDIENCE: So like an employee
purchase plan, where, if you're in financial
services, you're working for a
mutual fund company, and they offer you 10%,
15% you put your salary in, and you can buy stock– ANDREW LO: Yes.

AUDIENCE: Would you
recommend not doing that? ANDREW LO: Well, I
recommend not doing it from the financial
perspective, but I may recommend doing it
from the management, or managerial
incentives, perspective. The reason that companies
do that is very simple. They're trying to suck you in. They're trying to get you
to be more intimately tied to the company so you'll act
like an owner of the company, as opposed to an employee. And if you act
like an owner, you will engage in behavior
that is much more productive for building
the company's wealth, rather than as an employee.

But from your
personal perspective, you're bearing risk
that you don't need to. So you know, the
analogy that I give– I've given before– it
may work for some of you. It may not. Let me explain. Anybody know how much window
washers in midtown Manhattan get paid on an annual basis? You know what I'm talking about? These are the folks that climb
up on these two-foot catwalks that are 40 stories high,
and they wash the windows of these skyscrapers. That's a pretty risky job.

Anybody know what
their annual salary is, when you annualize it? I actually decided
to find out one day. I was kind of curious about
that because, you know, there's a trade-off
of risk and return. And that's really risky. You know, there was one day
when I was staying at a hotel. I think it was the Millennium,
and I was on the 30th floor, and it was a windy winter day. And, sure enough, there was
somebody there pulling up the thing, cleaning the window,
and looked happy as can be. You know, no problem. And I was thinking, boy, this
guy's taking a lot of risk, you know? And I hope he's
getting paid for it. And these salaries
are determined by supply and demand.

What do you think it would be? Anybody have a guess? Yeah? AUDIENCE: About \$70,000. ANDREW LO: \$70,000, OK. Anybody else? Justin? AUDIENCE: \$125,000. ANDREW LO: \$125,000? [LAUGHTER] That's higher than some
NBA starting salaries. [LAUGHTER] OK. Leah? AUDIENCE: \$30,000. ANDREW LO: \$30,000. AUDIENCE: Do they take interns? ANDREW LO: Insurance? Interns. Interns. I don't know about that. [LAUGHTER] Well, so when I looked
last time, which is about four years
ago, it turns out that the typical window
washer for these skyscrapers gets paid about \$60,000
a year, annual salary.

\$60,000. Now, you know, I
don't know whether you think that's a lot or a little. But, seems to me that
that compensation reflects the kind of risk that
we're talking about. And you know, you have no
educational requirements, no degrees, no certifications. You just show up and,
you know, up you go. Now let me ask you a question. Suppose that a
window washer comes to the job who happens
to really enjoy dancing while he washes windows. In particular, he
dances that, you know, the Irish jig or whatever. [LAUGHTER] You know what I'm talking about? You know, that
dancing, the very– and he just likes to do that
while he's washing windows on the 40th floor. You agree that that's
more risky, right? Do you think that that
particular individual gets paid more than \$60,000 a year? Why? He's taking more risk.

Why not? Why isn't he getting paid more? Exactly. He doesn't have
to take that risk. That's not part of the job. He can choose to take
that risk, but he's not going to get compensated for
it because it's not necessary. And there are 100,000 people
behind him waiting in line to get that job that
won't necessarily need to take that risk. Yeah? AUDIENCE: I thought
of it differently. If you had a long,
short portfolio, would it be the
other way around? Because you wouldn't care
about the market risk. You can hedge that out. You're thinking only
about specific risk. So like in that case, I would
short the guy who likes to jig. And I would invest in
the guy who didn't. ANDREW LO: Well, that
depends on whether or not doing the Irish
jig actually makes you wash windows
better or worse, in other words, where there's
an alpha to that risk.

It may be the case that
dancing the Irish jig actually helps you
scrape off dirt that much more effectively. In which case, you may
not want to short him because then he will earn a
premium in certain markets. Those with lots of pigeon
poop on the windows. [LAUGHTER] So we can get into this analogy
more deeply than we should. [LAUGHTER] But the point is that, when
you think about the CAPM, all it's saying is that
you get what you pay for.

And you pay for what you get. In other words, if there
is a certain amount of risk in a particular investment
that is risk that nobody can get rid of easily– in other words, you
have to bear it– then you have to pay for it. Because otherwise people
aren't going to do it. However, if there's risk
in a particular company that you don't have to pay for,
that you don't have to take, then you don't
have to pay for it. That's all that
the CAPM is saying. Beta is a measure of that
hard little pellet of risk that you can't get rid of. And the variance is the
measure of the entire risk in a particular portfolio.

The only case where variance
and beta are the same is for what kind of portfolio? An efficient portfolio. What is an efficient portfolio? It's one that is already
maximally diversified. By adding more
securities, you are not going to do any better
than that straight line. All of these portfolios have
been completely diversified. How do I know that? Because you're at the
tangency portfolio. There is nowhere to
go in the Northwest region off of that line. So for all of these securities,
the sigma and the beta are literally numerically
identical because there is no more extra risk
in the portfolio. It's been diversified away. But for Gillette, for Microsoft,
for IBM, General Motors, and Motorola, each of these
contain both beta risk and non-beta risk. The non-beta risk
is that Irish jig that you don't have to do
while you're washing windows. And you're not going to get
paid for it, I'm sorry to say. So the relationship that
you want to focus on is the capital asset pricing
model's security market line.

Measure the stuff that
you're going to get paid for. And this is what you're
going to get paid for it. Yeah, Ingrid? AUDIENCE: When you go back
into the sort of the real world and include transaction
costs, how do they enter here? ANDREW LO: Well, so there
have been versions of the CAPM with transactions cost. And it turns out that it
doesn't change things too much.

If you impose transactions
cost on all securities, and say that
there's a percentage cost for going in and
out, you can derive a net of fee transactions costs. It won't affect the
pricing necessarily. What it will affect is
the dynamics, the trading. What it will mean is that you
will rebalance your portfolios less frequently than
you otherwise might. But there is still going
to be a relationship between the net of
fee transactions costs of these securities. So you can actually look
at the transactions cost as a way to deduct the
expected rate of return from each of these securities. So transactions cost
is not a big issue, but there are other
issues that I'll come to that will be
a problem for this. AUDIENCE: Looking at this,
[INAUDIBLE] one of the best, efficient portfolio that I can
have is basically the index.

If I want to allocate
my money globally, should I buy indexes
according to the market capital of the market? ANDREW LO: Yeah. AUDIENCE: This is the most
efficient portfolio I would– ANDREW LO: Right, so
that's a great question. And we actually have a separate
course on international finance that deals just with
those kind of issues because they're so tricky. But I'll give you
the short answer. According to the theory,
this tangency portfolio is not just for the
US stock market. It should be for the world
stock market, everything. So this tangency
portfolio should not be the S&P 500 or
the Russell 2000, it should be the
MSCI, or EAFE, index that has all of the assets
in the world weighted relative to their market cap
in that particular currency of the investor that
you happen to be.

So if you're a US investor,
it'll be in dollars. If you're a Brazilian
investor, it'll be in real. It'll be in whatever
currency you trade in. But that presupposes
that there's capital market integration
throughout the world. So in other words, if I call
this the world portfolio, implicitly I'm
assuming that you're free to trade stocks in any
part of the world freely. There are no
barriers to trading. And we know that
that's not the case. There are barriers, in fact. So what that means is
that the CAPM, applied to international stocks, is an
approximation that may actually be worse than applying
it country by country and then looking to see
whether there are any distances or discrepancies across
those different countries. But people have come up
with international versions of the CAPM. And they don't work very well. At least, they didn't
as of 10 years ago. Within the last 10
years, a lot has changed. So it could be that
capital market integration has made the world
CAPM look a lot better in terms of the data.

OK. Other questions? All right. So now I'm going to talk
about implementing it. And we're going to deal
with all of the messy issues that I tried to put
off a lecture ago. How do you take this
thing out for a spin? Well, one thing you
could do is to try to test the CAPM
to see if it works. And one way to test it
is to ask the question, if we assume that all
securities are priced according to this equation, then another
way to write the equation that doesn't rely on expected
returns, but relies on realized returns, is to write
it as a regression equation, as you did in DMD.

The regression equation is the
return, the actual realized return on security i, is
given by the risk-free rate plus beta times the realized
return on the market minus the risk-free
rate plus epsilon. Epsilon is the error term,
the disturbance, the residual, that is giving you
the fluctuations around the expected value. So when we remove the expected
values from this equation, we have to stick in
this epsilon term that sort of bounces around. By the way, when you
look at this equation, you now have an
explicit representation for systematic risk
and idiosyncratic risk. For a given return,
it's comprised of three pieces, the risk-free
rate, beta times the market return that bounces around, and
this epsilon is the Irish jig. That is the idiosyncratic
bouncing around that you don't get
any reward for. How do you know you don't
get any reward for it? Because on average, the expected
value of this is equal to 0. How do I know that? By definition that's how I got
from this equation down here.

If you take the expected
value of this equation, the only way that the
expected value of this gives you this
equation on the top is if that epsilon has a 0 mean. So you don't get paid
for bearing epsilon risk. It's there. And for some stocks it's huge. But you don't get paid
for it because you don't have to bear it. And the reason you
don't have to bear it is, if you take
50 of these stocks and stick them in a portfolio,
the epsilons average out to 0. How do I know that? Well, this relies on
a piece of mathematics that's known as the
law of large numbers.

You may have heard that term
used in casual conversation, but it's actually
a real theorem. What it says is that
when you have large, large numbers of
fluctuations that are not correlated with each other– and by definition,
idiosyncratic risks for Gillette and Microsoft
and other companies are not correlated because
they're idiosyncratic. They're unique to those firms. That when you get a large
number of these uncorrelated fluctuations, that in the
limit, they actually go to 0. You can disregard them. So the law of large
numbers is what tells you that this
idiosyncratic risk is not going to be something
you will get paid for. So this is the CAPM
relationship using actual data. And if we stick in an alpha term
to represent deviations from the CAPM, and I subtract the
risk-free rate from both sides, just to have everything
in excess returns, then the CAPM reduces to the
hypothesis that the alpha– across all stocks, all
managers, all projects, the alpha is equal to 0.

That's what the CAPM says. And if you want to
formulate it strictly in terms of total
rates of return, it says that the alpha has to
be equal to the risk-free rate times 1 minus beta. This is a different
alpha than this alpha. This alpha represents the
excess rate of return. OK, so let's do it. Let's see whether
or not it's true. Let's take a bunch of stocks,
subtract the risk-free rate from the stock returns, run
a regression of that stock's return on a constant and
the market excess return, and let's see whether the
intercept is equal to 0. Well, if you do this for two
stocks, Biogen and Motorola– I've done this from
1988 to I think 2006.

When you run that regression,
here's what you get. For Biogen, the beta is
1.43, the intercept 1.61%, and the standard error is 1.1%. And then Rf times 1
minus beta is minus 2.1%. This should be equal to the
alpha in that previous equation that I gave you. So in other words, the alpha
that we've estimated for Biogen using this CAPM
regression is 3.7%, or on a monthly basis 45% alpha. Biogen is an incredibly good
buy according to the CAPM, if you believe the CAPM. All right. Now that's Biogen.
What about Motorola? During the same period,
when you estimate Motorola, it's got a beta of 1.42. And we're estimating an alpha
not quite as big, but 23.5% percent on an annualized basis. That's still pretty big. So if you run this
regression and analyze it, this is what you would
conclude, that these two stocks are wonderful buys. What we're going to
talk about next time is whether this interpretation
really makes sense, or whether we've got
some missing factors, or whether we're measuring
things improperly.

We're going to need
to do a bit more work. But we're very close to
being able to figure out exactly how all of
these pricing models work in tandem with the kinds
of risk budgeting calculations that we're going to need to
do for the rest of the course. So I'll see you on Monday..

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