>> Learning outcomes, after watching this video you will be able to solve for the risk premiums once we have the factor sensitivity. Determine the arbitrage-free price of an asset given the prices of other risky assets. Discuss a key drawback of the APT, the abitrage-free price. Last time, we saw an example where Coke was incorrectly priced relatively to Microsoft and Intel. Let's continue with that example. In the first step, we have solved for the factor sensitivities of Microsoft and Intel. In the second step now, we are going to solve for the risk premium using the APT. Write the APT for Microsoft and Intel, remember, it says that the expected return equals the risk period plus beta times the risk premium. We do not know the is risk free rate and the risk premium for the business cycle risk factor. We will have two equations, one each from Microsoft and Intel and so we can solve for the two unknowns. For Microsoft we have 0.2 = the risk free rate + 0.8 x the risk factor premium. For Intel, we have 0.5 = the risk free rate, plus 3 times the risk factor premium. Solving, we find that the risk free rate is 9.09%, and that the business cycle risk factor premium is 13.64%. 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 = 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 over P- 1 = Coke's expected return + 0.5 times Coke's beta. In the bad state it is 15 over p- 1 = Cokes expected return- 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 over P- 1 and its Beta is 40 over P. Plugging this back into Coke's APT, we have 35 over p- 1 = 9.09% + 13.64% times 40 over P. This is now one equation and one unknown. Solve for P to get Coke's APT price today to be 27.08. Let's revisit our arbitrage portfolio that consists of buying one share of Coke and start selling half share each of Microsoft and Intel. The future payoffs in the good and bad states are still 5%, but the cost to set up this portfolio today is now 4.58. So what is the rate of return on this portfolio? Without any calculations, we can confidently say that it is 9.09%. How is that possible, you wonder, your portfolio is risk-free because it always pays 5 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 over 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 the factor risk premium of 13.64%. The problem with 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 come's 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.
