Learning Outcomes, after watching this video, you will be able to identify arbitrage opportunities. Solve for factor sensitivities given information on asset returns and unanticipated shocks to the risk factors. Identifying arbitrage opportunities. Let's see how the APT works using an example. Suppose we have a world with only two time periods, now that is t=0 and the next period which is t=1. It is uncertain which date the world will be in the next period. We could have two equally likely states, each with a probability of 50% of good and bad. We have data on three stocks, Microsoft stock price today is 25, and will be 40 in the Ggod state, and 20 in the bad state. Intel's stock price today is 20, and will be 60 in the good state, and 0 in the bad state. Coke stock price today is 22.5, and will be 55 in the good state, and 15 in the bad state. Is there any arbitrage opportunity here? The answer is yes. Let's form a portfolio where we buy one share of Coke and short sale half a share each of Microsoft and Intel. The cost of setting up this portfolio is 1 x 22.5- 0.5 x 25- 0.5 x 20 which is 0. In the good state the pay off from this portfolio is 1 x 55- 0.5 x 40- 0.5 x 60 which comes out to be 5. In the bad state, the payoff from this portfolio is 1x15 -0.5 x20 -0.5 x 0, which comes out to also be 5. Regardless of whether the good or bad state occurs next period, the payoff from this portfolio is always 5 and there is no cost of setting this portfolio up today. There is no uncertainty or risk in this future payoff, you're always guaranteed to get 5. This is the idea of arbitrage, guaranteed profits at no cost. One can scale up a strategy to make unlimited profits. Now, which of the three stocks is mispriced? Remember that APT is based on the idea of relative pricing. That is, given that two stocks are correctly priced, one is incorrectly priced. Let's assume that Microsoft and Intel are correctly priced and Coke at 22.5 is mispriced. Given that the prices of Microsoft and Intel are correct, what must be the correct price of Coke? We will use APt to determine Coke's correct price. As a first step, we need to identify the risk factors. To keep things simple we'll assume that there is only one risk factor, namely, the business cycle factor. The business cycle factor takes a value of 1 in the good state and 0 in the bad state. The expected business cycle factor is the probability of the good state, which is 0.5, x 1 plus the probability of the batch date, which is also 0.5,x 0, which comes out to be 0.5. This means the unanticipated shock to the risk factor in the good state is 1-0.5, which is 0.5. And that is the bad state 0-0.5, which is -0.5. These values for the shocks to the business cycle ensure that the expectation of the unanticipated shocks in 0. We will determine the correct price of Coke in the following three steps. One, determine the factor sensitivities or betas for Microsoft and Intel. Two, determine the factor risk premiums using ATP for Microsoft and Intel. Three, use the APT equation to price Coke. We need the returns of Microsoft and Intel in the good and bad states. In the good state Microsoft's return is 40/25-1 which is 60%. In the bad state, its return is 20/25-1, which is -20%. You can similarly calculate the returns of Intel in the good and bad states. And let to verify that these numbers are 200% and -100%, respectively. Remember the single factor model. It says that the return can be decomposed into expected returns plus beta times the unanticipated shock to the risk factor. In this example, there is no unanticipated from specific shock. We can decompose Microsoft's returns in both the good and bad states. In the good state, it is 0.6 equals Microsoft's expected return plus beta sub m times 0.5. In the bad state, it is -0.2 equals Microsoft's expected return plus beta sub m x -0.5. We have two equations and two unknowns. Solving for Microsoft's expected return and beta sub m, we have 20% and 0.08 respectively. We can similarly set up the equations in the good and bad states for Intel and solve for it's expected return and beta. You can see the equations on the slide. They come out to be 50% and 3 respectively. We will conclude this example next time by computing the risk premium for the single risk factor, and then calculating the arbitrage free price of Coke.
