0:29

When all investor follows such as strategies

Â there is an important result that we have illustrated in the previews video,

Â which is that the tangency portfolio is actually the market portfolio.

Â So the market, the relative weight of each stock in the total market capitalization

Â corresponds exactly to the portfolio weight of the tangency portfolio.

Â This is depicted in this graph.

Â Where the tangency point between the green line and

Â the red line is the market portfolio.

Â 1:48

So for example, we could consider that a stock index like the SNB

Â 500 is a good representation of what the market actually is.

Â From that observation, we can say something about the red line.

Â The characteristic of the red line.

Â Like any fine functions straight line, it is characterized by it's

Â intercept where it starts, and here it starts at the level of the risk free rate.

Â The second element that characterize the equation of a line is the slope.

Â The slope here is measured by how much you move up when you move to the right.

Â How much additional return you get by taking additional level of risk.

Â And for the red line the efficient frontier, we can use the coordinate

Â of the market portfolio to completely describe the slope of the red line.

Â The slope is going to be defined by the difference between

Â the expected return on the market and the risk free rate.

Â This is the height of the point along the y axis

Â of the market point minus the height of the risk free rate.

Â So here it will be a little bit more than 6% minus the 2% of the risk free rate.

Â And if we look now at how much you move to the right when moving along the red line,

Â the horizontal movement for the market coordinate is

Â 10% to the right minus the level of the origin which is 0% for the risk free rate.

Â So the slope is 6% minus 2% divided by 10%.

Â This is the slope of the red line.

Â Now, any portfolio which is on the red line can be identified by the intercept,

Â the origin, the risk-free rate.

Â The level of risk it is exposed to and the slope of the red line.

Â This relation between the expected return of an efficient portfolio and

Â it's level of risk is called the Capital Market Line.

Â And it's just a reinterpretation of the efficient frontier

Â including the risk free rate.

Â I'm going to display now the equation of this capital market line.

Â And this is precisely what we've just said, right,

Â the expected return here of one particular efficient portfolio,

Â which we write E for expectation of Ri, the return

Â of one particular efficient portfolio, so one portfolio on the red line.

Â Satisfies the equation of the straight line,

Â this straight line start at the risk free rate which we write here RF,

Â and then there is a level of risk for that portfolio which is sigma I which

Â multiplies the slope of the red line, how is the slope of the red line defined?

Â It is defined relative to the coordinate of the market portfolio, so

Â we have expected return on the market portfolio minus RF,

Â this is E[RM] expected return of the market,

Â hence the M, divided by the level of risk of the market, sigma M.

Â So this ratio here E of RM minus RF divided by sigma M.

Â This is the slope of the red line.

Â This equation links risk and return, but not for all asset in the market.

Â It links risk and return only for

Â those portfolio that are optimally diversified and are on this red line.

Â 5:17

The next question that we will like to address is whether we can write something

Â similar for an asset, a stock for example, that is not on the red line.

Â For example we could take stock A or stock C.

Â Is there a way to link expected return to

Â the level of risk of a particular security?

Â And intuitively the first thing we can say is that the risk of asset C for

Â example, this 20%, some of it should not be compensated by expected return.

Â Why?

Â Because we know we can choose a portfolio with the same level of risk using

Â the effect of diversification but same level of expected return.

Â Sorry, but with a lower level of risk.

Â So part of the risk of asset C can be diversified away and

Â we shouldn't expect to receive an expected return, a reward for

Â bearing that risk that can be diversified away.

Â So the correct measure of risk for

Â an asset that is not on the efficient frontier.

Â It's not the total level of risk but another measure.

Â A measure of risk that we cannot diversify.

Â 7:15

And then we have the difference between the expected return on the market and

Â RF similarly to what we had before for the capital market line, but

Â instead of having the level of risk of the assets we have this measure

Â of none diversifiable risk, which is the beta.

Â This quantity measures the amount of risk that we cannot

Â get away from by just combining the assets with other asset in the economy.

Â Some of the risk of each asset can be diversified away

Â by combination with other financial security.

Â That part is diversifiable.

Â What's left and

Â what should be rewarded by an increase in return is measured by the beta.

Â How is this beta computed?

Â Well, you have the equation here on the second line.

Â It is the ratio between two quantities.

Â The first one, measures the dependence between the asset return and

Â the market return, and we know that we can measure this dependence using

Â correlation or covariance.

Â Here the beta is defined with covariance.

Â 8:28

This measure of covariance between Ri, the return of the financial security

Â we're interested in, and RM, is the first element in the definition of the beta.

Â The second quantity is the variance of the market return itself.

Â Remember that the variance is a measure of risk, a measure of dispersion.

Â It's the square of the standard deviation.

Â Okay, so how do we actually use a formula like this one to construct some

Â expectation of what we expect to see in terms of return for

Â a particular financial security.

Â Well, first we have to measure the better of the security.

Â This can be done using statistical measures, co-variants and variants.

Â Historical data can be used to measure the level of co-variants and

Â to measure the level of variants of the market.

Â Now if we look at the link between better and

Â return we can see that again they should align on a straight line.

Â So different level of beta correspond to different level of return and

Â the link between the beta and the return is linear.

Â This relation between beta and

Â return along a particular line, also has a particular name.

Â It's called the Security Market Line, this is depicted here in a new example.

Â 10:11

And, one particular level is important is the level of the beta of the market.

Â And what is the beta of the market?

Â Well, if we come back to the definition of the beta, the covariance

Â of the market with himself, which would be the denominator in this equation to

Â compute the beta of the market, correspond actually to the variance.

Â The covariance of a random variable with itself is the variance.

Â So the beta of the market is exactly equal to 1.

Â So on the security market line, the intersection here

Â of the straight blue line with better equals to 1.

Â This is the expected return on the market.

Â So this is equal to RF, so level of the risk free rate, the intercept.

Â Plus beta times

Â expected return on the market minus expected return on the risk free, okay.

Â So the security market line describes all

Â the expected return we would like to see for

Â the financial securities available in the market as they differ in terms of beta.

Â So let's look at one example.

Â 11:27

A stock with a beta of 1.4, what do we expect its return to be?

Â Well, if the measure of beta corresponds to the non-diversifiable risk,

Â we can use the previous equation to compute it's expected return.

Â So let's just use these inputs in the previous form that

Â I compute the expected return.

Â So the expected return should be equal to the risk free rate RF

Â plus the product of the beta 1.4 and the difference between

Â the expected return on the market and the risk free rate.

Â So here 8%- 2%.

Â If we compute this equation we obtain that the expected return for

Â these assets with a beta of 1.4 is exactly equal to 10.4.

Â 12:18

What's important here about the security market line is that we can

Â observe different securities located at the same point with the same beater.

Â Which did not necessarily have the same level of total risk.

Â Let me go back to this slide,

Â this graph that takes the efficient frontier and the individual security.

Â We see that asset C is located here at the total risk level of 20%,

Â and it's expected return is 8%.

Â All the securities if we had other financial asset in this market.

Â Located on the horizontal line starting at the origin,

Â at the 8% of the y axis and crossing though C.

Â All the points located on this line would generate the same expected return,

Â but different level of total risk.

Â What we can say is that, all these points

Â actually corresponds to financial securities which have the same beta.

Â Why do they have the same beta?

Â Because beta defines the level of expected return.

Â So all these securities align on a horizontal line crossing C

Â will have the same level of expected return.

Â So according to this equation, they will have the same level of beta, okay?

Â So in the Security Market Line here, we could very well have many stock

Â located at the same point with the same beta on the other graph,

Â all these points would be located on the same horizontal line.

Â So to conclude, the measure of risk that we should take into account,

Â according to this model, to measure expected return,

Â the level of risk that really matters is the level of non-diversified risk.

Â And this measure corresponds the beta which is computed by taking the ratio

Â of the co-variance of the return with the market to the variance of the market.

Â [MUSIC]

Â