[MUSIC] We have seen in our previous video that the two fund separation theorem implies that investor are going to hold a portfolio composed of two sub parts, one is the risk free rate or the risk free investments and another one is what we called the tangency portfolio. When all investor follows such as strategies there is an important result that we have illustrated in the previews video, which is that the tangency portfolio is actually the market portfolio. So the market, the relative weight of each stock in the total market capitalization corresponds exactly to the portfolio weight of the tangency portfolio. This is depicted in this graph. Where the tangency point between the green line and the red line is the market portfolio. Now we would like to see whether this has some implication in terms of market equilibrium for the link between risk and return. For a given level of risk, what do we expect a particular financial asset or a particular financial portfolio. What do we expect at the expected level of return to be. To do that, to create this link between risk and return, we start from the results of the two-fund separation theorem, okay. So the tangency portfolio is the market portfolio. So the coordinate of that point, the coordinate of this market point are the expected return on the market and the risk of the market. So for example, we could consider that a stock index like the SNB 500 is a good representation of what the market actually is. From that observation, we can say something about the red line. The characteristic of the red line. Like any fine functions straight line, it is characterized by it's intercept where it starts, and here it starts at the level of the risk free rate. The second element that characterize the equation of a line is the slope. The slope here is measured by how much you move up when you move to the right. How much additional return you get by taking additional level of risk. And for the red line the efficient frontier, we can use the coordinate of the market portfolio to completely describe the slope of the red line. The slope is going to be defined by the difference between the expected return on the market and the risk free rate. This is the height of the point along the y axis of the market point minus the height of the risk free rate. So here it will be a little bit more than 6% minus the 2% of the risk free rate. And if we look now at how much you move to the right when moving along the red line, the horizontal movement for the market coordinate is 10% to the right minus the level of the origin which is 0% for the risk free rate. So the slope is 6% minus 2% divided by 10%. This is the slope of the red line. Now, any portfolio which is on the red line can be identified by the intercept, the origin, the risk-free rate. The level of risk it is exposed to and the slope of the red line. This relation between the expected return of an efficient portfolio and it's level of risk is called the Capital Market Line. And it's just a reinterpretation of the efficient frontier including the risk free rate. I'm going to display now the equation of this capital market line. And this is precisely what we've just said, right, the expected return here of one particular efficient portfolio, which we write E for expectation of Ri, the return of one particular efficient portfolio, so one portfolio on the red line. Satisfies the equation of the straight line, this straight line start at the risk free rate which we write here RF, and then there is a level of risk for that portfolio which is sigma I which multiplies the slope of the red line, how is the slope of the red line defined? It is defined relative to the coordinate of the market portfolio, so we have expected return on the market portfolio minus RF, this is E[RM] expected return of the market, hence the M, divided by the level of risk of the market, sigma M. So this ratio here E of RM minus RF divided by sigma M. This is the slope of the red line. This equation links risk and return, but not for all asset in the market. It links risk and return only for those portfolio that are optimally diversified and are on this red line. The next question that we will like to address is whether we can write something similar for an asset, a stock for example, that is not on the red line. For example we could take stock A or stock C. Is there a way to link expected return to the level of risk of a particular security? And intuitively the first thing we can say is that the risk of asset C for example, this 20%, some of it should not be compensated by expected return. Why? Because we know we can choose a portfolio with the same level of risk using the effect of diversification but same level of expected return. Sorry, but with a lower level of risk. So part of the risk of asset C can be diversified away and we shouldn't expect to receive an expected return, a reward for bearing that risk that can be diversified away. So the correct measure of risk for an asset that is not on the efficient frontier. It's not the total level of risk but another measure. A measure of risk that we cannot diversify. This measure of risk is called the beta. So let's see exactly how this measure is defined. This equation is the relation between risk and return for an arbitrary asset such as asset A, B, or C that is not necessarily on the efficient frontier. This relation is called the capital asset pricing model or CAPM for short. We see that the equation looks a little bit like the one we had before for efficient portfolios. We still have an expected return, E of Ri, this is what we want to explain, the expected return of the stock. We have an intercept at a minimum level which is RF. And then we have the difference between the expected return on the market and RF similarly to what we had before for the capital market line, but instead of having the level of risk of the assets we have this measure of none diversifiable risk, which is the beta. This quantity measures the amount of risk that we cannot get away from by just combining the assets with other asset in the economy. Some of the risk of each asset can be diversified away by combination with other financial security. That part is diversifiable. What's left and what should be rewarded by an increase in return is measured by the beta. How is this beta computed? Well, you have the equation here on the second line. It is the ratio between two quantities. The first one, measures the dependence between the asset return and the market return, and we know that we can measure this dependence using correlation or covariance. Here the beta is defined with covariance. This measure of covariance between Ri, the return of the financial security we're interested in, and RM, is the first element in the definition of the beta. The second quantity is the variance of the market return itself. Remember that the variance is a measure of risk, a measure of dispersion. It's the square of the standard deviation. Okay, so how do we actually use a formula like this one to construct some expectation of what we expect to see in terms of return for a particular financial security. Well, first we have to measure the better of the security. This can be done using statistical measures, co-variants and variants. Historical data can be used to measure the level of co-variants and to measure the level of variants of the market. Now if we look at the link between better and return we can see that again they should align on a straight line. So different level of beta correspond to different level of return and the link between the beta and the return is linear. This relation between beta and return along a particular line, also has a particular name. It's called the Security Market Line, this is depicted here in a new example. So what's important here is that we have not change the y axis, it's still expected return. But we have modified what's on the x axis. Instead of having the measure of total risk, the measure of standard deviation. We now have a new measure, the beta. The measure of non-diversifiable risk. And, one particular level is important is the level of the beta of the market. And what is the beta of the market? Well, if we come back to the definition of the beta, the covariance of the market with himself, which would be the denominator in this equation to compute the beta of the market, correspond actually to the variance. The covariance of a random variable with itself is the variance. So the beta of the market is exactly equal to 1. So on the security market line, the intersection here of the straight blue line with better equals to 1. This is the expected return on the market. So this is equal to RF, so level of the risk free rate, the intercept. Plus beta times expected return on the market minus expected return on the risk free, okay. So the security market line describes all the expected return we would like to see for the financial securities available in the market as they differ in terms of beta. So let's look at one example. A stock with a beta of 1.4, what do we expect its return to be? Well, if the measure of beta corresponds to the non-diversifiable risk, we can use the previous equation to compute it's expected return. So let's just use these inputs in the previous form that I compute the expected return. So the expected return should be equal to the risk free rate RF plus the product of the beta 1.4 and the difference between the expected return on the market and the risk free rate. So here 8%- 2%. If we compute this equation we obtain that the expected return for these assets with a beta of 1.4 is exactly equal to 10.4. What's important here about the security market line is that we can observe different securities located at the same point with the same beater. Which did not necessarily have the same level of total risk. Let me go back to this slide, this graph that takes the efficient frontier and the individual security. We see that asset C is located here at the total risk level of 20%, and it's expected return is 8%. All the securities if we had other financial asset in this market. Located on the horizontal line starting at the origin, at the 8% of the y axis and crossing though C. All the points located on this line would generate the same expected return, but different level of total risk. What we can say is that, all these points actually corresponds to financial securities which have the same beta. Why do they have the same beta? Because beta defines the level of expected return. So all these securities align on a horizontal line crossing C will have the same level of expected return. So according to this equation, they will have the same level of beta, okay? So in the Security Market Line here, we could very well have many stock located at the same point with the same beta on the other graph, all these points would be located on the same horizontal line. So to conclude, the measure of risk that we should take into account, according to this model, to measure expected return, the level of risk that really matters is the level of non-diversified risk. And this measure corresponds the beta which is computed by taking the ratio of the co-variance of the return with the market to the variance of the market. [MUSIC]
