Yeah, capital as a pricing model is based on rational markets and rational investors,
right? What does it mean to be rational?
And maybe you could explain what
the saying is "The market can stay irrational longer than you can stay solvent".
What does it mean to be rational?
I'm married to a psychologist but you can ask my wife.
We're economists.
We have a very simple model of people.
But what does it mean to be rash- Sometimes I think rational people,
they don't pay attention to financial markets at all.
They're in the important things like making friends.
But on the other hand, there is some basic simple notion of human rationality
that is I think it's partly correct.
People are partly rational.
But they often make mistakes.
This is something we're going to talk about in the semester.
Behavioral finance is an important field.
And we mostly think of ours- Do you think of yourself as rational?
Sometimes.
I don't know if we have agreement on this about
who's- I never thought of myself as particularly rational.
I have trouble with self-discipline. Do you ever get lazy?
Yeah.
Well, that's part of irrationality.
Or do you ever find that you're just following entertainment?
You're just bored with getting the facts.
You want to hear entertainment.
So these are human traits.
The capital asset pricing model,
the heroic assumption that is often taken
uncritically is that everybody is investing according to that model.
Now, on the face of it, that is absolutely absurd because if you
asked the population to describe the capital asset pricing model,
it wouldn't be more than one in a hundred who would know what you're talking about.
To assume that they're just doing it, it's pretty extreme.
On the other hand, I still like to go through the mathematics and see what would it
be if everybody were extremely rational and logical,
and how would the markets behave. It's interesting.
So, let's just think about one risky asset and one risk-less asset.
Suppose I put X dollars into a risky asset one.
Now I'm going to invest $1 total.
I'm just normalizing it on one dollar.
So the money I have left over after I invest X is 1-X,
to help with that in the second asset,
which is earning a sure but low return of rf.
So what is the expected value of the return on my portfolio?
Well, I have X dollars in the first asset so it's going to be X times
the expected return on the first asset. The risky asset.
And I've got 1-X dollars in the risk-less rate.
And so, the total expected return is Xr1+1-X*rf.
And so, what is the variance of my return?
Or the variance is just equal to X
squared times the variance of the return on the first asset.
So, if X is one,
that means I put it all in the first asset,
then it's just the portfolio variance is the variance of the return on the first asset.
What if I shorted?
I put minus one.
If I shorted, that wouldn't be- It doesn't sound like a smart move,
generally, on average shorting
the stock market and investing in the risk-less asset. But I could do that.
I could make X= -1, and then I would have $2
invested in the risk-less asset.
My portfolio variance would be the same,
but I'd be on the wrong end of it, right?
Assuming that I've got my numbers right,
I would be shorting the high return asset and investing in the low return asset.
So, and having the same risk anyway.
You can compute the portfolio of standard deviation which is
just the square root of this portfolio variance.
So it's linear. The standard deviation of the portfolio is
linear in the expected return on the portfolio.
See, the real answer here as,
I tried to convey that last time.
You want an expected return,
I can give it to you on stocks and anything you want,
but I'll do it by exposing you to risk.
I can leverage you up.
You want a 100 percent expected return next year?
Great. I'll leverage it to the hilt.
And then you'll have that expected return which will probably get wiped
out because you've taken on to leverage than investment.
Let's illustrate the idea of how levering up to the hilt
can give you any expected portfolio return you want.
Let's say you start with $1 and there are only two investable assets in the world.
Let's look at a risky asset which offers an expected return of
20% and a return standard deviation of 5%.
And the other asset is the risk free asset which guarantees a return of 10% with no risk.
If you want 100% expected return on your $1 portfolio.
First, borrow, AKA "short",
$8 from the risk free market,
and now you have $9 to play with.
Invest all $9 in the risky asset.
This provides you with an expected return next period of 20%.
So your portfolio now has $10.80 on average.
Pay back what you owe which is 8*1.1 or $8.80,
you're left with $2 in your portfolio and you
have thus doubled your initial investment on average.
Remember though, you took on an 8 to 1 leverage ratio to get here.
Using the formula the standard deviation of your portfolio return was 9*5%, or 45%.
If the risky asset realized any return less than
-2.2% which is half of one standard deviation away from the mean,
you would have to file for bankruptcy.
But we're not going to worry about being wiped out here,
we're just worrying about what your return will be.
Now, suppose we move from just one risky asset to two risky assets.
Now, I want to put $X1 in risky asset one,
and 1-$X1 in risky asset two.
What is the portfolio expected return?
And I'm not putting anything into the riskless asset.
So now I have two risky assets.
The expected value of the return on my portfolio
is equal to X1 which is the dollars I put in asset 1,
times the expected return on an asset one,
plus one minus X1 dollars.
Now what is the variance of this portfolio?
It turns out this is the formula.
The variance of the portfolio is X1 squared
times the variance of the return on the first one,
plus 1-X1 squared times the variance of the return on the second risky asset,
plus 2X1*1-X1 times the covariance of the returns.
Now the covariances matter because if they covariate positively,
that makes them interact in a positive way increasing variance.
So, positive covariance is bad for your portfolio.
It raises the variance of your portfolio.
Negative covariance is good.
If you can find two risky assets that move
opposite each other or tend to move opposite each other,
then there covariance is negative and this thing reduces the portfolio variance.
What we're doing with the CAPM is that we're moving beyond Mr. Crumb's day.
We now have statistics.
And so, I'm imagining here that you are managing a portfolio and you
have historical data on the returns of
the different assets that you can put into the portfolio.
So you've calculated what the expected return is for the first asset,
the average historically for its return.
You've calculated what the expected return on the asset, second asset is.
You've taken an average.
You've also computed the variances of the returns.
So you know how variable they are.
And you've even calculated the covariance of the returns.
Now, you might question whether covariances, variances,
expected values that I estimated historically will continue to be true in the future.
Maybe things are different now.
But at least as a first exercise,
it sounds like this is something we should understand, doesn't it?
It's a first guess.
We'll assume that these variances are going to be stable through time.
So I want to know, what I should do if I assume they are stable through time?
That's the key idea.
And that had not been worked out until Harry Markowitz
developed the capital asset pricing model in the early 1950s.