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 (r sub i), is the risk-free rate, r sub f + beta sub i,1 x RP sub 1 + beta sub i,2 x RP sub 2 + so on until beta sub i,K x RP sub K. Beta sub i,j is the sensitivity of asset i's returns to factor j. And RP sub j is factor j's risk premium. The risk premium of each factor is the excess expected return over and above the risk-free rate that investors expect as compensation for holding one unit of the risk factor. The CAPM is another example of an asset pricing model where the market portfolio is the only risk factor. The multifactor models decompose an asset's 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 model relate assets' expected returns to the risk factors through the factor's risk premiums and the sensitivity of assets' returns to the risk factor. 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 determined by the principal of the law of one price. The law of one price posits that assets with the same future payoff, 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 are called arbitrage 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, they last for very short periods of time. How do we 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 portfolios, G and I. G has a beta of 1 with respect to the GDP growth factor, and a beta of 0 with respect to the inflation factor. I has a beta of 0 with respect to the GDP growth factor, and a beta of 1 with respect to the inflation factor. Such portfolios are called factor portfolios. They track the progression of specific sources of macroeconomic risk, but are uncorrelated with other sources of risk. These factor portfolios will serve as the benchmark portfolios for a multifactors 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 1 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 rate of 1.01 G and a weight of 0.4 on I and the net weight of -0.4 on the risk free asset. Remember, rates must add to 1 and hence the rate on the risk free asset is -0.4. By construction, Q has the same factor sensitivities as Microsoft. Q's expected return is 1 x 10% + 0.4 x 13% + (-0.4) x 3%, which equals 14%. Given that factor sensitivities are the same for Portfolio Q and 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 prizes assets in such a way that it prevents arbitrate opportunities. Because we now have two factors, we get a security market plane with the factor sensitivities on two axes and the expected returns on the third axis. Assets that do not lie on the SMP present arbitrage opportunities. One important thing to note about the APT is that arbitrage is about relative pricing and not absolute pricing. In our example, we say that if the factor returns are 10% and 13% and the risk free rate 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.
