Learning outcomes. After watching this video, you will be able to calculate the covariance between a risky asset's return and the market's returns. Calculate the beta of a risky asset given expected returns, standard deviations and covariances. Calculating the CAPM beta. Last time we saw the mathematical form of the CAPM and the equation to calculate the CAPM beta. Now we will apply this formula to calculate the betas of risky assets. Let's go back to our three risky asset case, but change the correlations a little. Remember, we saw that the portfolio that has negative weights for some risky assets cannot be a market portfolio. In our previous risky asset example, Z has a negative weight in the mean variance efficient portfolio. To solve this problem, we'll change the covariances a little. We continue to assume that there are only three risky and one risk free asset in the entire world. The expected return and standard deviation for X are 10% and 7% respectively. Similarly for Y, we have 20% and 10% respectively and for Z we have 15% and 12% respectively. The covariance between X and Y is 0.0032, that between X and Z, is 0.0013, and that between Y and Z is 0.0054. The risk free rate is still 5%. Using solver in Excel, you can confirm that the weights of X, Y, and Z in the mean radiance efficient portfolio are 0.0546, 0.8424, and 0.1030, respectively. Since all the rates are positive, this mean variance efficient portfolio is a valid market portfolio. Remember how to calculate the expected return and standard deviation of returns for a portfolio. The expected return of the market portfolio is 0.0546 times 10%, plus 0.8424 times 20% plus 0.1030 times 15% which gives us 18.94%. The standard deviation of the market portfolios returns is the square root of 0.0546 squared x 0.07 squared, +0.8424 squared, x 0.10 squared, + 0.1030 squared, x 0.12 squared, + 2 times 0.0546, x 0.8424 x 0.0032 + 2 x 0.0546 x 0.1030 x 0.0013 + 2 x 0.8424 x 0.1030 x 0.0054. The standard deviation of the market portfolio comes out to be 9.22%. To calculate the betas of the three assets, we need to calculate the covariance of each asset's returns with that of the market portfolio. Remember, the market portfolio itself consists of 0.0546 in X, 0.8424 in Y, and 0.1030 in Z. It can be shown that the covariance between Xs returns and the market portfolios returns can be calculated as follows. It is the covariance between X and 0.0546 of X plus the covariance between X and 0.8424 of Y plus the covariance of X and 0.1030 of Z. The covariance between X and X is nothing but the variance of X. So now we have 0.0546 times the variance of X plus 0.8424 times the covariance between X and Y plus 0.1030 time the covariance between X and Z. This works out to be 0.0031. You can similarly calculate the covariances of Ys returns and Zs returns with the market portfolio's returns. They work out to be 0.0092 and 0.0061, respectively. I leave it up to you to do the calculations and verify these two numbers. We can now calculate the betas of the three risky assets. For X, it is the covariance of its returns with the market portfolio's returns, which is 0.0031 divided by the variance of the market's returns, which is 0.0922 squared. The beta comes out to be 0.36. Similar calculations for Y and Z yield betas of 1.08 and 0.72 respectively. Please verify these numbers. Do note that Y has the highest expected return of 20% among the three risky asset and its beta of 1.08, is also the highest. X, on the other hand has the lowest expected return of 10%, and also the lowest paid of 0.36. This is consistent with investors demanding a higher compensation in the form of higher expected returns for holding riskier higher beta assets. Next time we will define a plot of the gap M which is called the security market client and compare it to the capital market client.