Arbitrage Pricing Theory is a general theory of asset pricing that uses a linear function which includes various macro-economic factors or theoretical market indices to model an asset’s expected return. At the core APT’s main function is to realize that only few systematic factors can affect an asset’s average returns. Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors included within the model. The APT was a revolutionary model used for pricing an asset because it allows users to customize their model to the security being analyzed. As with other pricing models, it helps user to decide whether an asset is undervalued or overvalued and so people can profit from this information. While APT does rely upon static equilibrium arguments, it allows a world where occasional mispricings occurs. Investors constantly seek information about these mispricings and exploit them as they find them. It claims that investors will ‘price’ these factors precisely because they are sources of risk that cannot be diversified away, which means they will demand compensation in terms of expected returns for holding securities exposed to these risks. APT can also bring benefit for asset/portfolio managers when they try to test whether their portfolios are exposed to certain factors. They can choose their own systematic profile of risk and returns by selecting a portfolio with it own peculiar array of betas(to specific factor). In short, the APT allows for an industry of information collectors, risk arbitrageurs and speculators. It allows diversified types of investors as well as evolving types of risks. In other words, it describes a world somewhat closer to the world in which we live right now.

It is hard to forecast the exact events that may change stock’s price in the future, but we can measure the sensitivity towards changes in each factor by β through factor model. The APT assumes that these opportunities will be taken advantage of until prices shift and the arbitrage opportunities disappear. This provides a justification for the use of empirical based factor models in pricing securities; if the model were inconsistent, the arbitrage opportunity will exist and so the prices would adjust.

As I clarified before, it is hard to predict the expected return accurately and consistently, partially because every asset is always exposed to the arbitrage market which implies that if the expected return of the asset is high, then more people will enter into the market for this asset by which the returns will be diluted in the end. So one of the practical uses of APT is to integrate with Long-Short Equity Strategy based on factor model ranking system. Because the expected returns calculated by APT model imply the returns when the market is fully arbitraged. If you know what the expected return of an asset(given that the market is arbitraged) and you assume that the market will be mostly arbitrated over the timeframe on which you are trading, you can construct a ranking system based on it for Long-Short Equity Strategy.

First, we should estimate the expected returns for each asset on the market and rank them. Based on the strategy, we should long the top percentile and short the bottom percentile. You will make money on the difference in returns eventually. Because you expect the high-ranking assets’ value to increase in the future such as x% and the low-ranking asset’s value to decline x% later(The market return is Rm). You can then make total (Rm + X%) — (Rm — X%) = 2X% in the end. By using Long-Short Equity strategy, you actually accept the fact that you can never make the exact prediction on single asset. Instead, you choose a bunch of stocks and make your prediction on them. You can make money through the spread and increase your winning chance dramatically as long as the model is 51% accurate(we can predict the expected returns for a group of 1000 assets as the errors average out.).

Now, I will show the specific steps to compute expected returns for two assets.

The two assets I choose are asset 1=’ECA’(Encana Corporation) and asset2 =’WWAV’(WhiteWave Foods Co.). The benchmark is S&P500 and the risk-free asset I choose here is 3-month T-bills. I intend to use the benchmark and risk-free asset’s return from 2014–11–11 to 2015–11–11 to build a predictive model to predict the returns from 2014–12–11 to 2015–12–11 of asset1(‘ECA’) and asset2(‘WWAV’).

Here I construct two similar OLS regression model on two assets respectively and the relative parameters are shown below. We can see that the p-values of both models are relatively small(<.05), so we can trust the regression outcome to some extent.

But like I have mentioned before in previous articles, these numbers cannot relly tell us much by themselves even though the p-values are small. The assumption behind the p-value is that the data will not change over time. We need to look at the distribution of these coefficients to check whether they’re consistant over time. Here I use the rolling 90-day regression to check the result:

From the test above, it seems that the market betas are stable. So I will continue conducting a test on market(‘SPY’)’s beta(rolling 90 days) on asset2:

From the graph above, you can see that market’s beta changed massively within the scale from 9 to 14 rather than the scale before from -15 to 25. Now, we can use the model we built before to predict future prices(1month ahead the time right now). So the timeframe of S&P500 and T-bill is from 2014–12–11 to 2015–12–11 and the target asset2's prices series is from 2015–01–11 to 2016–01–11( the expected prices shown as dash line in the graph).

But this analysis did not include the quality test of our predictions. We may need to use techniques such as out-of-sample testing or cross-validation(set several training timeframes to generate different parameters within each timeframe). For the purposes of long-short equity ranking systems, the Spearman Correlation provide a good way to check the quality of ranking system.

I would like to simply introduce Regime Changes within the model. In short, a regime change causes future samples to follow a different distribution. We can tell that there is a regime change in the graph below at the end of 2014. The split results in a better fit(in red) than the regression line for the whole dataset(yellow). From the graph, it is clearly that the regression model is no longer predictive since 2014–10 of future data points. The more pieces we break the data set into, the more precisely we can fit to it. But it is important not to fit to noise which is always fluctuating and not predictive.

“Below we use a test from statsmodels which computes the probability of observing the data if there were no breakpoint.”.

**Appendix:**

The APT formula is:

E(rj) = rf + bj1RP1 + bj2RP2 + bj3RP3 + bj4RP4 + … + bjnRPn

where: E(rj) = the asset’s expected rate of return rf = the risk-free rate bj = the sensitivity of the asset’s return to the particular factor RP = the risk premium associated with the particular factor

Arbitrage is the practice of taking positive expected return from overvalued or undervalued securities in the inefficient market without any incremental risk and zero additional investments.

Please feel free to comment on inefficiencies about anything I explained here as usual. I am always open to the mistakes and willing to learn from them as a beginner in this area.

*Reference:*

*Quantopian: Arbitrage Pricing Theory**Quantopian: Regression Analysis**https://en.wikipedia.org/wiki/Arbitrage_pricing_theory**http://www.cfapubs.org/doi/pdf/10.2469/faj.v51.n1.1868**http://viking.som.yale.edu/will/finman540/classnotes/class6.html**http://www.investinganswers.com/financial-dictionary/stock-valuation/arbitrage-pricing-theory-apt-2544*