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 risk factors. Identifying arbitrage opportunities. Let's see how the APT works using an example. Supposed 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 good state, and 20 in the bad state. Intel 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 you buy one share of Coke and short sell 0.5 share each of Microsoft and Intel. The cost of setting up this portfolio is 1 times 22.5- 0.5 times 25- 0.5 x 20, which is 0. In the good state, the payoff from this portfolio is 1 times 55- 0.5 times 40- 0.5 times 60, which comes out to be 5. In the bad state the payout from this portfolio is 1 times 15 minus 0.5 times 20 minus 0.5 times 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 its future payoff. You're always guaranteed to get 5. This is the idea of arbitrage, guaranteed profits at no cost. One can scale up the strategy to make unlimited profits. Now which of the three stocks is mispriced? Remember that ABT 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 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 + the probability of the bad state, 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 in the bad state is 0- 0.5, which is -0.5. These values for the shocks to the business cycle ensure that the expectation of the unanticipated shocks is 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 APT 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 over 25- 1, which is 60%. In the bad state its return is 20 over 25- 1, which is -20%. You can similarly calculate the returns of Intel and the good and bad states. I'd like 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 the plus beta times the unanticipated shock of 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 = Microsoft's expected return + betas sub M x 0.5. In the bad state it is -0.2 = Microsoft's expected return + 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 0.8 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 3.0 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.