Learning outcomes. After watching this video, you will be able to understand the arbitrage pricing theory, use the arbitrage pricing theory to calculate expected returns. The arbitrage pricing theory. Asset pricing models relate expected returns and risk through factor betas. The most general asset pricing model, called the arbitrage pricing theory, APT in short, posits that the expected return of asset i, e of r sub i is the risk-free rate r of f plus beta sub I1 times RP sub 1 plus beta sub I2 times RP sub 2 plus 1 until beta sub Ik times RP sub k. Beta sub Ij is the sensitivity of asset I's returns to factor j an R piece of J is factored J's risk premium. The risk premium of each factor is their excess expected return over and above the risk-free rate. That investors expect as compensation for holding one unit of the risk factor. The cap then is under the example of an asset pricing model where the market portfolio is the only risk factor. The Multifactor Models decompose an assets actual returns into an expected part and an unexpected part with the unexpected part being attributable to unanticipated shocks to K risk factors and unanticipated firm-specific shocks. Asset Pricing Models related asset's expected returns to the risk factors through the factors risk premiums and the sensitivity of assets returns to the risk factors. Asset pricing models, such as the APT, are alternatives to the CAPM. They allow for multiple risk factors as opposed to the single one in the CAPM. Like the CAPM, the APT is a theoretical model rather than a data-based one. In APT, asset values are determine by the principle of the law of one price. The law of one price pauses that assets with the same future pay off must have the same current price. Otherwise, one can make infinite profits without putting any money at risk such opportunities to make infinite profits without risking any money on advertise opportunities. Any investor, regardless of her risk preference or wealth will want to take advantage of such opportunities. These opportunities rarely exist and even when they do their last for very short periods of time. How do you determine the risk premium for each factor in the APT? To answer this, let's continue with our Microsoft example. Say we identify two special well diversified portfolio's 'G' and 'I'. G as of beta of 1 with respect to the GDP growth factor and the beta of 0 with respect to the inflation factor. I as of beta is 0 with respect to the GDP growth factor and the beta of 1 with respect to the inflation factor. Such portfolios are called factor portfolios. The track this progression of specific sources of macroeconomic risk but there uncorrelated with other sources of risk. This factor portfolios will served as the benchmark portfolios for a multifactor security market line. Let's say that g has an expected return of 10% and I has an expected return of 13% with the risk free rate being 3%. Recall that Microsoft had a beta of one with respect to the GDP growth factor and a beta of 0.4 with the inflation factor. Let's form a portfolio q that has a weight of 1.0 on g and a weight of 0.4 on i and a weight of -0.4 on the risk free asset. Remember, weights must add to one and hence the weight on the risk free asset is -0.4 By construction, Q has the same factor sensitivities as Microsoft. Q's expected return is 1 times 10 % plus 0.4 times 13% minus 0.4 times 3%, which equals 14%. Given that factor sensitivities are the same for Portfolio Q and for Microsoft, Microsoft must also have an expected return of 14% to prevent arbitrage. We can verify this by using Microsoft's factor sensitivities in the APT. That is the risk free rate 3% + 1 x (10%- 3%) + 0.4 x (13%- 3%) which equals 14%. The APT prices assets in such a way that it prevents arbitrage opportunity. Because we now have two factors, we get a security market plane with the factor securities on two axis and the expected returns on the third axis. Assets that do not lay on the SMP present arbitrage opportunities. One important thing to note about the APT is that arbitrices about relative pricing and not absolute pricing. In our example, we say that if the factor returns a 10% and 13% and the rest filler is 3%, Microsoft must have an expected return of 14%. It does not say anything about whether the factor returns themselves are correct. Next time we will look at a detailed example of how APT works, which illustrates this idea of relative pricing.