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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..