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In this module, we're going to bring together all the results that we have

generated from mean variance portfolio selection from market with a risk-free

asset. And use that to define a model that

constructs prices for assets. And this model is going to be called a

Capital Asset Pricing Model. So, in order to connect the Sharp optimal

portfolio. There's something that's happening in the

market. Let's define a new portfolio.

And the portfolio that I'm going to define is something called a market portfolio.

The market portfolio is defined as the portfolio where you take the Ci.

And then you normalize it by the sum of all the capitalization.

So, the i-th component of the market portfolio is simply Ci divided by the sum

of all the Cj's, so these all add up to 1. In fact, they are all greater than equal

to 00 as well. Let mu m denote the expected net return on

the market portfolio. It's simply mu I times XMI summed, of I

going from 1 to D. And let sigma M denote the volatility of

the market portfolio, as before it's this quadratic function, take the square root.

Now let's connect up how this market portfolio relates to what investors are

doing. Suppose all investors in the market had

mean variance optimizers. And all of them will invest in the Sharp

optimal portfolio as starred. Let w super k denote the wealth of the

k-th investor. Let x0 k denote the fraction of the wealth

that the k-th investor puts into the risk free asset.

Then the total capitalization of the i-th risky asset is simply going to be the wk,

the wealth of the k-th Investor 1 minus x0 k.

This is the fraction that is going into the risky assets.

Times s star i, why s star i? Because I'm looking at the capitalization

over of just the i at asset. And what are the summation k over?

This is over all investors. The thing that I want to do focus on, is

the fact that this, this summation here, over all investors, doesn't depend on the

s star. So, if I write it differently.

I can simply take that s star i, and pull it out of the bracket.

So, this s star i, I can pull it out of the bracket and just write it over there.

What does that mean? That means that if I do this calculation

to compute the market portfolio, I will get that the market portfolio is nothing

but the Sharpe optimal portfolio. This should not be a surprise.

Everybody's investing in the same Sharpe optimal portfolio, and therefore the

capitalization should be completely related to the Sharpe optimal portfolio.

If I re-normalize the capitalization to get the portfolio, I should get back the

Sharpe optimal popular portfolio. This is great, why is this important?

This is important because the sharpe optimal portfolio depends upon expended

returns, depends on covariances. These are quantities that are hard to

compute. On the other hand xm is a market

portfolio. They're relatively easier to computer.

I can go and calculate out the values of the various capitalization.

And it'll give me a portfolio that will have, will have some insight about the

efficient frontiers. Okay, now we want to go further and see

what that means. So, capital market line is another name

for the efficient frontier. Now what we're going to do is remember in

the last module we showed that the efficient frontier was a line that goes

from the risk free asset through the shop optimal portfolio and all the way through.

Now, it's a line that goes form the risk free asset through the market portfolio

and all the way through. This efficient frontier is going to be the

excess return on the market portfolio, divided by the volatility of the market

portfolio. Previously, we had computed this slope as

the excess return on the sharp optimum portfolio, divided by the volatility of

the sharp optimum portfolio. It's also the maximum achievable shopping

operation. This quantity is also frequently called

the price of risk. Everything that is efficient, must lie on

this line. And if you take on a certain amount of

risk, this capital market line tells you what is the, that you must demand.

It can be used to compared projects. Here's a simple example, suppose the price

of a share of an oil pipeline venture is $875 right now.

It's expected to yield $1000 in one year, but the volatility is 40%.

It's very high volatility. The current interest rate is 5%, the

expected return, rate of return or the net return on the market portfolio is going to

be mu M equal to 17%. The volatility of the market portfolio

sigma M equals 12%. The question is, is this oil pipeline

worth considering? Should one invest in it?

Another way of asking the same question is, is this oil pipeline venture on the

efficient frontier? Because if it's not, I should not be

interested in it. So what's the return that I get, what is

the net return that I get on this oil pipeline venture?

It's 1000 divided by 875 minus 1, it's approximately 14%, what is the return that

I should demand for taking on the volatility?

Sigma is the volatility of this oil venture, this is the slope of the capital

market line. The capital market line starts off from

the point RF. Just to remind you, it's a straight line,

starts from the point RF and has a slope M.

So if I look at some point sigma, I should demand this return, which I'm labeling as

R bar in order for it to be efficient. Plug in the numbers.

Rf is given. R, r, this shouldn't be on rm.

But mu m, mu m is given. Sigma n is given.

If you plug in all of this, then in order to take on the risk of 40%.

You should be compensated at a rate of return or a net rate of return of 45%,

which is way higher than the 14% that the oil venture is giving.

Therefore, the oil venture is not efficient and should not be considered.

Now what we want to do is take this idea that the market portfolio is there and try

to inffer asset returns from the market returns.

So the way we going to do it is think about the fact that every asset is in fact

a portfolio. If you look at the j at asset, it's a

portfolio. It just corresponds to investing the one

dollar in the j at asset, and nothing everywhere else.

For now let's consider, the set of portfolios that I could generate by

diversifying between the j at asset. Okay, and the market portfolio.

So I put an amount gamma into the j at asset and amount one minus gamma into the

market portfolio. The return on this portfolio is going to

be mu gamma, is an expression that gives you gamma here, is the volatility of the

portfolio, gamma squared. This is the, this is the volatility that's

coming from asset j, this is from the market.

This is of course volatility that's coming from the market and the portfolio.

So what have I plotted on this first? I plotted the green line, which is the

efficient frontier with the risky asset. The blue line, which is the efficient

frontier, frontier with only risky asset. And then this dotted black line is the

efficient frontier. It's the frontier that is generated by

diversifying between the J asset and the market portfolio.

So this dotted line must lie below the efficient frontier of just a risky asset

because the market portfolio is just a risky asset, the j at asset is just a

risky asset, so everything that's going to be here is going to be below that.

But now what happens? Notice that for gamma equal to 0, this,

the dot, the red dot belongs to the efficient frontier corresponding to this

black line, because gamma equal to 0 gives you the market portfolio.

The market portfolio is an efficient front, an efficient portfolio.

It also lies on the efficient frontier of a market with a risk-free asset.

So all of these three curves are tangent at the same point.

I know the slope of the green line. That I, also know that the same slope must

be equal to the slope of the blue line must also be equal to the slope of this

dotted black line and I am going to use that to compute what the returns are going

to be. The firmer the market, the capital market

line is clearly noted; just mu r divided by r j divided by sigma.

Now we want to compute the slope of the frontier generated by the asset j in the

market portfolio sigma M. So that's the slope of this, this black

dotted line that I want to compute. So D mu over D sigma.

But I don't know how to directly compute it, so I take, take this chain rule and

write it as d mu gamma divided by D gamma, D sigma gamma divided by B gamma.

Why do I put this mu gamma and sigma gamma here because these are functions of gamma.

The D gammas cancel, I get back the same thing, but this is something I can

compute. This expression is exactly equal to this.

This denominator is a much more complicated expression, and this is the

expression that you end up getting. You're not going to be responsible for

computing out what that expression is. This is only for the derivation that I

want here. If you compute all these expressions and

substitute gamma equal to zero, which is the point that responds to the market

portfolio. You'll get an expression that this ratio

is just mu J minus mu N, divided by sigma JM minus, sigma M squared over sigma M.

If you equate the slope, this slope that you get, to that slope, you end up getting

and rearrange the results. You end up getting that the excess return

on the j at asset, this is nothing but mu hat j.

Must be equal to the excess return on the market, mu hat m.

And they are related by a quantity which is the covariance.

This is the covariance of the return on the j at asset, with the return on the

market divided by the median of the return of the market.

And that is called a beta of asset j and this full pricing formula is called the

Capital Asset Pricing Model. Towards the end of the last module, I said

that returns in this market should be determined by just one return because

there's only one portfolio everybody is investing.

And this theorem says exactly that return on any asset, the excess return on any

asset is determined by just an excess return on the market portfolio.

This one thing determines everything. Now let's connect this story, because

there's a beta floating around. It seems like it should be connected to

linear regression. So let's connect it to linear regression

and see what we end up getting. So suppose we take the random excess

return. So Rj here will denote the random excess

return on asset g, not the expected return and regress it on the excess market

return, which is Rm minus Rf. Here's the regression formula.

On the left hand side is the even variable that I'm trying to regress.

On the right hand side is the variable r minus r f on which I'm trying to regress

it. Alpha is the intercept, beta is the

coefficient, and this is the residual noise.

The coefficient beta is exactly the beta that we computed sigma i m square divided

by sigma m squared. The in, intercept alpha j, there's an

expression for it. It's simply going to be the expected

return on the asset to get expectation on this side, to get expectation on that

side, subtract it, whatever you get, that's going to be the alpha, because the

expectation on the residual noise is zero. So mu j minus rf minus beta mu m minus rf.

Residual sigma epsilon j and r m minus r f are uncorrelated.

The correlation between them is 0, so all of this is just regression theory.

I've not added any financial economics or financial engineering there.

Now I'm going to add it in. I know that capital asset pricing model is

true. What does that mean?

That means that this difference is exactly equal to 0, which means that the alpha j

on every asset is equal to zero. And the effective relations here that I

end up getting is at the random return on asset j, excess return on random J is beta

times the random excess on the market plus epsilon j.

So now let's look at the variance. The variance of this quantity is nothing

but the variance of just the f of j because Rf is a constant.

Epsilon j and the excess return are uncorrelated, so therefore you end up

getting beta squared, variance of rm minus Rf, which is nothing but the variance of

the market, plus the variance of the residual.

Now let's, for a moment, compare this expression.

So here's the expression for risk. The expression for return says mu hat j,

which expected return on asset j must be beta times new hat m.

You are taking two components of risk here.

The total risk to your portfolio has two components, has one component which

correlates with the market because of the beta And that's called, market risk.

The rest, the leftover risk, is called residual risk.

And if you look at what return that you're getting for that amount of risk it just

depends on the market risk, just the beta part.

The residual part you don't get compensated for it.

What does that tell you? It tell you that this residual risk should

somehow be diversifiable by looking at a, a properly diversified asset we should be

able to get rid of this. Because if we couldn't get rid of it and

we had to take on this risk. The market should have compensated us,

for, for taking on this risk. But the market does not, which means that

this residual risk is diversifiable. The market risk is not, here's another way

of looking at what CAPM tells you. We did that in the capital asset pricing

model, via the capital market line. Here's another way of looking at the

inside accounts from the capital asset pricing model.

This is called a security market. This says that if I plot the historical

returns of an asset, with respect to this other quantity.

I should get a straight line, why? Because we just said that mu j, must be

equal to rf plus the beta of that particular asset, times mu m minus rf.

So this shouldn't be rm, but mu m and therefore we expect to get a straight

line. And if you dig the 8 asset classes that

were there in the spreadsheet that was given to you and [unknown] out the

quantity over here. And in a separate module, I'm going to

show you how to compute this line using the data in the spreadsheet.

You'll end up getting that this is exactly, that those assets fall pretty

much under straight line. The line there's one asset that falls

below. What does this insight tell you?

You might want to pose the video here for a moment to try to understand what is,

what is the information over here? The story is, all of these assets are

going to be on the straight line, because they are all efficient.

The 7th asset, and, to some extent, the 1st asset, is inefficient.

And therefore, it falls below the security market line.

You only want to hold those assets that fall on the line, or sometimes above the

line. So why the discrepancy?

Because mean, cap m and mean variance is not always true.

So the security market line can be used to identify inefficient assets, and assets

that might be mispriced. The assumption that are underlying CAPM

are all investors have identical information.

Not true, all investors are mean variant optimizers or at least their returns are

normal. Not true, the markets are in equilibrium.

Again, not true, all of these assumptions are not true, and therefore you don't

expect that all the asset should fall in the security market line.

Many of them will, but some might lie below, some might lie above.

And so, how does one leverage deviations from the security market line?

There are two ways of looking at it. One way is to compute the Alpha for

particular asset. Remember We said a few slides before that

if CAPM is true, the alpha is equal to 0. If alpha is positive, it means that, that

asset is, has been mispriced low. And therefore, we should buy it.

Because we expect to get higher than expected returns from that asset.

Alpha is negative means that mis, asset that as it has been mispriced high, and we

must short sell it. Because later on the return will catch up

and will make a difference. So alpha, alpha positive will correspond

to this space. Alpha negative will correspond to this

space. So alpha positive here alpha negative

here, we don't like the asset down here we need to short them.

The asset over here we like them we have to hold them long.

Another way of getting to the same concept of alpha is to look at the sharp ratio of

a stock. We remember a few slides before we have

said that the market portfolio gives you the highest Sharpe ratio and therefore the

securities market line corresponds to the highest slope.

Now, if the assumptions behind CAPM are not true the there might be instances,

short periods of time where things are not in equilibrium for example where some

asset might be mispriced and may actually have a higher sharpe ratio In the market.

Those assets we want to hold long. Those assets with sharp ratios below, we

want to short. The final slide here I want to show you

how to use CAPM as a pricing formula. Suppose there is payoff from an investment

in one year, is some random quantity x. And I want to compute what the fair price

of this investment is, then there net return from this investment is simply x

over p. The beta of x is the covariance of rx with

rm and sigma m squared. If you plug in the expression, you will

get, there is a covariance of the pay of x with rm divided by the variance of the

market times 1 over p. So this p is actually unknown, we are

trying to compute out what the price is going to be.

Supposed CAPM holds, then mu x, which is the expected return must lie on the

security market, that line. So mu X must be equal to RF, plus beta of

that payoff X, RF minus RF. I plug in the formula.

I have one unknown P. I have one equation.

I solve for it by rearranging terms, I end up getting that P must be equal to the

expected value of the payoff, discounted back, which kind of makes sense.

This is what you would have said, even if you didn't know CAPM, take the expected

value of the payoff, discount it one period before, one year before, at the

risk free rate. Plus another term which correlates with

returns on the market. And this again, this shouldn't be Rm but

Mu M. It's going to be the expected return on

the market. But the thing that I want you to focus on

is the fact that the price goes up, when the correlation, of the payoff is negative

with the, with respect to the market return.

Which means that if the payoff is such, that, it pays on, in, in situations where

the market is low. Then the price that somebody can demand

for that payoff turns out to be high. Does that make sense?

You might want to pause and think for a moment.

So in situations where the market return gives you a very high return, there's very

low demand for this particular asset. If it's positively correlated with the

market, the asset will not have a very high demand because I could simply invest

in the market portfolio and then do not take on any of the residual risks.

So it's efficient for me to just put my money into the market portfolio.

On the other hand, if there is a particular asset that gives me returns in

situations where the market gives me low returns, which means that the payoff from

partic-, this particular asset is negatively correlated with the market,

then I might want to hold that asset, which means that the demand for that asset

is going to be high, which means that the price that the seller could demand is

going to be higher. And that explains why negative correlation

with the market results in a higher price for that particular asset stop here.