Commons license.

Your support will help MIT OpenCourseWare

continue to offer high quality educational resources for free.

To make a donation, or to view additional materials

from hundreds of MIT courses, visit MIT OpenCourseWare

at ocw.mit.edu.

ANDREW LO: Well, let me pick up where we left off last time

and give you just a very quick overview of where we're at now,

because we're on the brink of a very important set of results

that I think will change your perspective permanently

on risk and expected return.

Last time, remember, we looked at this trade-off

between expected return and volatility.

And we made the argument that when

you combined a bunch of different securities

that are not all perfectly correlated,

what you get is this bullet-shaped curve in terms

of the possible trade-offs between that expected return

and riskiness of various different portfolios.

So every single dot on this bullet-shaped curve

corresponds to a specific portfolio, or weighting,

or vector of portfolio weights, omega.

So now what I want to ask you to do for the next lecture or two

is to exhibit a little bit of a split personality

kind of a perspective.

I'm going to ask you to look at the geometry of risk

and expected return, but at the same time,

in the back of your brain, I want

you to keep in mind the analytics

of that set of geometries.

In other words, I want you to keep in mind how we

got this bullet-shaped curve.

The way we got it was from taking

different weighted averages of the securities

that we have access to as investments.

So every one of these points on the bullet

corresponds to a specific weighting.

As you change those weightings, you

change the risk and return characteristics

of your portfolio.

So the example that I gave after showing you this

curve where I argued that the upper branch of this bullet

is where any rational person would want to be.

And by rational, I've defined that as somebody

who prefers more expected return to less, and somebody

who prefers less risk to more, other things equal.

So if you've got those kind of preferences,

then you want to be in the Northeast.

You want to be as north, sorry, Northwest as possible.

And you would never want to be down in this lower branch when

you could be in the upper branch because you'd

have a higher expected return for the same level of risk.

So after we developed this basic idea,

I gave you this numerical example

where you've got three stocks in your universe.

General Motors, IBM, and Motorola.

And these are the parameters that we've estimated

using historical data.

Now there's going to be a question,

and we've already raised that question, of how stable

are these parameters.

Are they really parameters, or do they change over time.

And I told you, in reality of course, they change over time.

But for now, let's play the game and assume

that they are constant over time,

and see what we can do with those parameters.

So with the means, the standard deviations,

and most importantly, the covariance matrix--

So this is the matrix of variances and covariances--

With these data as inputs, we can now

construct that bullet-shaped curve.

The way we do it is of course, to recognize

that the expected return of the portfolio

is just a weighted average of the expected returns

of the component securities, where the weights are

our choice variables.

That's what we are getting to pick,

is how we allocate the 100% of our wealth

to these three different securities.

And the variance, of course, is going

to be given by a somewhat more complicated expression where

you have the individual security variances entering here

from the diagonals.

But you also have the off diagonal terms

entering in that same equation for that variance

of the portfolio.

And when we put these two equations together,

the mean and the variance, and we take the square root

the variance to get the standard deviation,

and we plot it on a graph, we get this.

This is the curve, the bullet-shaped curve,

that we generate just from three securities,

and from their covariances.

And where we left off last time is

that I pointed out a couple of things that was

interesting about this curve.

One is that unlike the two asset example, where when you start

with two assets, the curve, the bullet

goes through the two assets.

In this case, with three or more assets,

it's going to turn out that the bullet is actually

going to include these assets as special cases,

but they won't be on the curve.

In other words, what this curve suggests

is that any rational person is going to want

to be on this upper branch.

What that means is that it never makes sense

to put all your money in one single security.

You see that?

In other words, if we agree that any rational investor is going

to want to be on that efficient frontier, that upper branch,

why would you ever want to be off of that branch?

You'd like to be Northwest of that, but you can't.

You'd never want to be below that branch,

or to the right of that branch because you could do better

by being on that branch.

So what this suggests is that we never

are going to want to hold 100% of IBM,

or 100% of General Motors, or 100% of Motorola.

If we did, we'd be on those dots,

and those dots would lie on that efficient frontier.

But in fact, they don't.

So right away, we have now departed

from Warren Buffett's world of, I want to pick a few stocks

and watch them very, very carefully.

Yeah, Brian?

AUDIENCE: Would you expand that to say that you'd never

want to invest in less than three stocks at a given time?

ANDREW LO: That's not necessarily true.

There are points on this line where--

and they may be pathological, so in other words,

they may be very rare--

but there may be points on the line

where you are holding two stocks, but not the third.

So you've got to be careful about that.

But those are exceptions.

As a generic statement, you're absolutely right.

The typical portfolio is going to have some of all three

of them.

And if you had four stocks, the typical portfolio

would have some of all four.

Yeah, [INAUDIBLE].

AUDIENCE: You answered my question,

which is if you take one more stock,

you'll always have your package [INAUDIBLE] n stocks include

all the [INAUDIBLE], all the n stocks so, at the limit

you should have an infinite number of stocks [INAUDIBLE]

ANDREW LO: Well, let me put it another way that may

be a little bit more intuitive.

What this diagram suggests-- you guys are already groping

towards--

is the insight that the more, the merrier.

As you add more stocks, you cannot make this investor worse

off.

So in other words, I've now shown you an example with three

stocks, we used to do two.

Is it possible that by giving you an extra stock

to invest in, I've made you worse off?

Yeah?

AUDIENCE: No

ANDREW LO: Why

AUDIENCE: Because you can just not invest in that stock.

ANDREW LO: Exactly.

I can never make you worse off in a world

where you're free to choose, that is.

Because you always have the option of getting

rid of the stock that you don't like.

You can always put 0 on it.

So to your point, [INAUDIBLE], as I add more stocks,

first of all my risk-reward trade-off

curve will get better.