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Yeah, capital as a pricing model is based on rational markets and rational investors,

Â right? What does it mean to be rational?

Â And maybe you could explain what

Â the saying is "The market can stay irrational longer than you can stay solvent".

Â What does it mean to be rational?

Â I'm married to a psychologist but you can ask my wife.

Â We're economists.

Â We have a very simple model of people.

Â But what does it mean to be rash- Sometimes I think rational people,

Â they don't pay attention to financial markets at all.

Â They're in the important things like making friends.

Â But on the other hand, there is some basic simple notion of human rationality

Â that is I think it's partly correct.

Â People are partly rational.

Â But they often make mistakes.

Â This is something we're going to talk about in the semester.

Â Behavioral finance is an important field.

Â And we mostly think of ours- Do you think of yourself as rational?

Â Sometimes.

Â I don't know if we have agreement on this about

Â who's- I never thought of myself as particularly rational.

Â I have trouble with self-discipline. Do you ever get lazy?

Â Yeah.

Â Well, that's part of irrationality.

Â Or do you ever find that you're just following entertainment?

Â You're just bored with getting the facts.

Â You want to hear entertainment.

Â [inaudible]

Â So these are human traits.

Â The capital asset pricing model,

Â the heroic assumption that is often taken

Â uncritically is that everybody is investing according to that model.

Â Now, on the face of it, that is absolutely absurd because if you

Â asked the population to describe the capital asset pricing model,

Â it wouldn't be more than one in a hundred who would know what you're talking about.

Â To assume that they're just doing it, it's pretty extreme.

Â On the other hand, I still like to go through the mathematics and see what would it

Â be if everybody were extremely rational and logical,

Â and how would the markets behave. It's interesting.

Â So, let's just think about one risky asset and one risk-less asset.

Â Suppose I put X dollars into a risky asset one.

Â Now I'm going to invest $1 total.

Â I'm just normalizing it on one dollar.

Â So the money I have left over after I invest X is 1-X,

Â to help with that in the second asset,

Â which is earning a sure but low return of rf.

Â So what is the expected value of the return on my portfolio?

Â Well, I have X dollars in the first asset so it's going to be X times

Â the expected return on the first asset. The risky asset.

Â And I've got 1-X dollars in the risk-less rate.

Â And so, the total expected return is Xr1+1-X*rf.

Â And so, what is the variance of my return?

Â Or the variance is just equal to X

Â squared times the variance of the return on the first asset.

Â So, if X is one,

Â that means I put it all in the first asset,

Â then it's just the portfolio variance is the variance of the return on the first asset.

Â What if I shorted?

Â I put minus one.

Â If I shorted, that wouldn't be- It doesn't sound like a smart move,

Â generally, on average shorting

Â the stock market and investing in the risk-less asset. But I could do that.

Â I could make X= -1, and then I would have $2

Â invested in the risk-less asset.

Â My portfolio variance would be the same,

Â but I'd be on the wrong end of it, right?

Â Assuming that I've got my numbers right,

Â I would be shorting the high return asset and investing in the low return asset.

Â So, and having the same risk anyway.

Â You can compute the portfolio of standard deviation which is

Â just the square root of this portfolio variance.

Â So it's linear. The standard deviation of the portfolio is

Â linear in the expected return on the portfolio.

Â See, the real answer here as,

Â I tried to convey that last time.

Â You want an expected return,

Â I can give it to you on stocks and anything you want,

Â but I'll do it by exposing you to risk.

Â I can leverage you up.

Â You want a 100 percent expected return next year?

Â Great. I'll leverage it to the hilt.

Â And then you'll have that expected return which will probably get wiped

Â out because you've taken on to leverage than investment.

Â Let's illustrate the idea of how levering up to the hilt

Â can give you any expected portfolio return you want.

Â Let's say you start with $1 and there are only two investable assets in the world.

Â Let's look at a risky asset which offers an expected return of

Â 20% and a return standard deviation of 5%.

Â And the other asset is the risk free asset which guarantees a return of 10% with no risk.

Â If you want 100% expected return on your $1 portfolio.

Â First, borrow, AKA "short",

Â $8 from the risk free market,

Â and now you have $9 to play with.

Â Invest all $9 in the risky asset.

Â This provides you with an expected return next period of 20%.

Â So your portfolio now has $10.80 on average.

Â Pay back what you owe which is 8*1.1 or $8.80,

Â you're left with $2 in your portfolio and you

Â have thus doubled your initial investment on average.

Â Remember though, you took on an 8 to 1 leverage ratio to get here.

Â Using the formula the standard deviation of your portfolio return was 9*5%, or 45%.

Â If the risky asset realized any return less than

Â -2.2% which is half of one standard deviation away from the mean,

Â you would have to file for bankruptcy.

Â But we're not going to worry about being wiped out here,

Â we're just worrying about what your return will be.

Â Now, suppose we move from just one risky asset to two risky assets.

Â Now, I want to put $X1 in risky asset one,

Â and 1-$X1 in risky asset two.

Â What is the portfolio expected return?

Â And I'm not putting anything into the riskless asset.

Â So now I have two risky assets.

Â The expected value of the return on my portfolio

Â is equal to X1 which is the dollars I put in asset 1,

Â times the expected return on an asset one,

Â plus one minus X1 dollars.

Â Now what is the variance of this portfolio?

Â It turns out this is the formula.

Â The variance of the portfolio is X1 squared

Â times the variance of the return on the first one,

Â plus 1-X1 squared times the variance of the return on the second risky asset,

Â plus 2X1*1-X1 times the covariance of the returns.

Â Now the covariances matter because if they covariate positively,

Â that makes them interact in a positive way increasing variance.

Â So, positive covariance is bad for your portfolio.

Â It raises the variance of your portfolio.

Â Negative covariance is good.

Â If you can find two risky assets that move

Â opposite each other or tend to move opposite each other,

Â then there covariance is negative and this thing reduces the portfolio variance.

Â What we're doing with the CAPM is that we're moving beyond Mr. Crumb's day.

Â We now have statistics.

Â And so, I'm imagining here that you are managing a portfolio and you

Â have historical data on the returns of

Â the different assets that you can put into the portfolio.

Â So you've calculated what the expected return is for the first asset,

Â the average historically for its return.

Â You've calculated what the expected return on the asset, second asset is.

Â You've taken an average.

Â You've also computed the variances of the returns.

Â So you know how variable they are.

Â And you've even calculated the covariance of the returns.

Â Now, you might question whether covariances, variances,

Â expected values that I estimated historically will continue to be true in the future.

Â Maybe things are different now.

Â But at least as a first exercise,

Â it sounds like this is something we should understand, doesn't it?

Â It's a first guess.

Â We'll assume that these variances are going to be stable through time.

Â So I want to know, what I should do if I assume they are stable through time?

Â That's the key idea.

Â And that had not been worked out until Harry Markowitz

Â developed the capital asset pricing model in the early 1950s.

Â