# Arbitrage Pricing Theory – Part Two

In our previous blog on this subject, we identified Arbitrage Pricing Theory (APT) as a multi-factor model in which a series of input variables, such as macroeconomic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset. Recall from its equation that the expected return of an asset i is a linear function of the assets’ betas to the n factors:

E(ri) = Rf +bi1 RP1+. . .+bin RPn where

• RPn is the premium on factor n
• Rf is the risk-free rate

Let’s explore the theory in a little more depth.  It is based on the concept of arbitrage, which is when a trader makes a risk-free profit when a single asset is priced differently in different markets. By shorting the higher price while simultaneously buying at the lower price, the price differential is locked in.  Unfortunately, in today’s lightning-fast global markets, this is a very rare opportunity.

The theoretical value of the APT is its ability to detect a mispriced asset: if the weighted sum of each factor’s risk premium is different from the asset’s current risk premium (as indicated by the asset price), then an arbitrage opportunity exists. If you think as the expected return on an asset as a discount rate on its future cash flows, a price can be calculated and compared to the actual current price.

We use the term asset here to encompass both single assets and portfolios.  If a single asset is being modeled, then the asset is exposed to each macroeconomic factor to the extent of its beta with that factor. Let’s assume we identify a mispriced asset based on a 5-factor ABT model. An equivalent portfolio would contain a set of 6 correctly-priced assets, one asset per factor plus one extra. Each asset would have the same exposure to a single factor as the mispriced asset. By varying the amount of each asset in the portfolio, you can arrive at a portfolio beta/factor that is the same as that for the mispriced asset. You can now treat the portfolio as a synthetic version of the asset, and take a short or long position in the portfolio to offset the opposite position in the mispriced asset, theoretically capturing a risk-free profit.

Of course, identifying the important factors associated with a particular asset is no small undertaking, and APT does not offer a roadmap.  However in general, one would have to take into account, for each of the factors affecting a particular asset, the expected return of each factor and the asset’s sensitivity to the factor.

We’ll complete our look at APT next time by comparing it to the Capital Asset Pricing Model.