Learning outcomes. After watching this video, you will be able to solve for the risk premiums once we have the factor sensitivities, determine the arbitrage free price of an asset given the prices of other risky assets, discuss a key drawback of the APT, the arbitage-free price. Last time, we saw an example where Coke was incorrectly priced relative to Microsoft and Intel. Let's continue with that example. In this first step, we have solved for the for the factor sensitivities of Microsoft and Intel. In the second step now, we're going to solve for the risk premium using the APT. Write the APT for Microsoft and Intel, remember that it says that the expected return equals the risk free rate plus beta times the risk premium. We do not know the risk free rate and the risk premium for the business cycle risk factor. We will have two equations, one each for Microsoft and Intel, and so we can solve for the two unknowns. For Microsoft we have 0.2 equals the risk free rate plus 0.8 times the risk factor premium. For Intel we have 0.5 equals the risk free rate plus three times the risk factor premium. Solving we find that the risk free rate is 9.09 percent and that the business cycle risk factor premium is 13.64 percent. Now for the third step. The APT must hold for all risky assets, provided the no arbitrage condition holds. For coke, the APT is the expected return of Coke equals 9.09% + 13.64% times coke's beta. This is one equation with two unknowns. How do we go about solving this problem? Assume that the correct price of Coke today is P. Let's write the single factor model for Coke in the good and bad states. In the good state, it is 55 / P- 1 equals Coke's expected return plus 0.5 times Cokes beta. In the bad state it is 15 / P- 1 equals Coke's expected return minus 0.5 times Coke's beta. We can solve for Coke's expected return and beta in terms of P. Coke's expected return is now 35 / P- 1, and its beta is 40 / P. Plugging this back into Coke's APT, we have 35 / P- 1 = 9.09%, + 13.64%, times 40 / P. This is now one equation and one unknown. Solve for P to get Coke's arbitrage free price today to be 27.08. Let's revisit our arbitrage portfolio that consists of buying one share of Coke, and shot selling half a share each of Microsoft and Intel. The future payoffs in the good and bad stage are still 5%, but the cost to setup this portfolio today is now 4.58. So, what is the rate of return on this portfolio? Without any calculations we can confidently that it is 9.09%. How is that possible, you wonder. Well, portfolio is risk free because it always pays five after one period. It has no risk or uncertainty which means that it is risk free. So its return must equal the risk free rate, which is 9.09%. If it is not, there will be an arbitrage opportunity. You can verify that 5 / 4.58 -1 is indeed 9.09%. Remember, we started off by assuming Microsoft and Intel were correctly priced, and Coke wasn't. This gave us a risk-free rate of 9.09% and a factor risk premium of 13.64%. The problem of arbitrage pricing is that the risk free rate, and factor risk premium will depend on what we assume are the correctly priced assets, and which one is incorrectly priced. If we assume that Intel and Coke are correctly priced and Microsoft is incorrectly priced, the risk-free rate will be 63.64% and the factor risk premium will be negative 4.55%. Similarly, if we assume that Microsoft and Coke are correctly priced, but Intel isn't, the risk free rate comes out to negative 9.09% and the factor risk premium to 36.36%. You can verify these numbers by following the three steps I have outlined in this example. If you look at these numbers, some of them seem absurd. What does it mean to have a negative risk-free rate? Or a negative factor risk premium? Is it even realistic? The answer is of course no, but this is the problem with the relative pricing idea that APT uses. Different sets of assets will give different values for the risk free asset and the factor risk premiums, some of which may be absurd. We have talked about multiple factors affecting expected returns, but what exactly are these factors. Next time we will see a popular set of factors that researchers and practitioners use.