# statistical arbitrage vs factor trading

I've recently read Avellaneda & Lee which seems to be widely recommended as an introduction to Statistical Arbitrage methods in trading. For those who aren't familiar with the paper, the method in the paper (which I'm assuming is similar to other Statistical Arbitrage strategies) is to use the residuals of some factor model (whether it be by PCA or fundamental factors) as the signals to trade. The paper then fits a mean reverting process to the residuals and uses the estimated halflife to bet on the residuals converging to the mean (when they are overstretched).

My question is two fold - first, I understand that this only works if your factors (PCA or fundamental) span the cross section of returns of the assets in the model (i.e. explain much of the variance). However, if you already knew factors that spanned the cross section of the returns, why wouldn't you trade the actual factors instead of the residual. In other words, if you have a factor model

$\mathrm{PredictedRet}_{i+1} = \alpha_i + \sum_{i=1}^{N} \beta_i * F_{i} + \epsilon_i$

which is reasonably good at explaining the cross section of returns of some universe of securities, why wouldn't you just use your $\mathrm{PredictedRet}$ as the signal instead of some reversion function of the residuals e.g. $\mathrm{Reversion}(\epsilon_{i...N})$.

Next question is on the implementation in the Avellaneda paper. They seem to only use the residual of the mean reverting process to determine the signal (see pg 20), where

$\mathrm{Signal} = \frac{X_i - \mathrm{mean}_i}{\sigma_i}$

which is essentially some type of z-score on the residual mean reverting process. Now, as far as I can tell, Avellaneda just goes on to trade these signals, without hedging the factors of the factor model out. So it could be the case where your factor model is predicting a very bullish outcome however the residual z score happens to be slightly positive, resulting in a negative trade signal (vs the very bullish signal from the factor model prediction). Are you supposed to trade these statistical arbitrage strategies factor neutral (and why isn't that being done in the Avellaneda paper)?

• In practice I believe people do hedge one or more factors that they don't want exposure to (typically the Market Factor, i.e. they Beta neutralize the portfolio), but of course they don't neutralize all the factors, which would defeat the whole purpose. – noob2 Oct 16 '17 at 14:33

1) Why would you trade the error on the residual instead of creating a long/short factor model and trade expected returns?

I would posit that the biggest reason people do this is for orthogonality of return. There are about 2,000 incredibly mature firms trading value, momentum, vol, etc. You would be competing with the likes of AQR, LSV Asset Management, etc. What you are doing when trading the residual of the return is making a bet that the "unexplained" component of the return is mean-reverting without trying to explain the reason for those price movements. Let's unpack this with a thought simulation. On page 39 of Dalio's book, Principles, he states "theoretically...if there was a computer that could hold all of the world's facts and if it was perfectly programmed to mathematically express all of the relationships between all of the world's parts, the future could be perfectly foretold."* For the sake of argument and simplicity, let's assume that there are 5 factors that perfectly explain equity returns for the entire universe in a linear fashion: market, value, momentum, size, and the amount of people currently playing chess in the world (an example of some factor you likely would never find, but might drive the stock for whatever reason - if I recall correctly, Jim Simons has publicly stated that he regressed weather in Milan to the S&P return). Let's say you have data for the first 4, but not the last. In this case, you run your linear regression and are left with some error term driven by your lack of knowledge / data about the number of people playing chess. By constructing an engine to trade this mean reversion, you are making a bet (without knowing the following specifics) that the number of chess players world-wide is a stationary process around some mean and you want to bet that it will snap back to that mean whenever it gets out of wack.** In the real world, there are many factors that likely nobody has found that drive stock return in a non-linear fashion, and this process would bet on them all together instead of just one. People say all the time "yesterday's alpha is today's beta," so it's critical to develop new signals that other people haven't found to add value. Absolute Return allocators (i.e. endowments, pensions, etc.) will often run a correlation between a new investment and their current investments to understand how uncorrelated their alphas are.

The other limitation is the type of factor you are using. Take a look at Quantitative Equity Portfolio Management by Chincarini and Kim. You have fundamental factors like div. yield, size, number of holders, and separately, economic factors like market, inflation, interest rates. Fundamental factor values are known at t0 and any historical regression is to find expected return associated with that fundamental factor level. For example, you expect an incremental 0.05% return for every additional mutual fund that holds the stock. At t0, you know there are 100 holders, so you expect 5% return associated with this level. Economic factors like market, or in this case, sector ETFs are only known at t1 (not t0), so you are actually detailing more of a risk model than a prediction model. The paper trades the overstretched nature of the residual at t0 on t1, which makes this okay, but you wouldn't be able to use this is a prediction model unless you had some idea of what tomorrow's IBB return would be in addition to the IBB beta that you have calculated on Dynavax, for example.

2) Are you supposed to trade mean reversion on error terms factor neutral and why isn't this being done in the Avellaneda paper?

The only way I know to get pure exposure to the error of the regression is to purchase one unit of the stock and to buy and sell commensurate units of the factors within the regression. In this way, you will be factor neutral because the stock's implicit factor exposure will be offset with purchases and sales of those factors / ETFs. I believe that is what the paper is saying / implementing on page 16:

Entering a trade, e.g. buy to open, means buying one dollar of the corresponding stock and selling βi dollars of its sector ETF or, in the case of using multiple factors, βi1 dollars of ETF #1, βi2 dollars of ETF #2, ..., βim dollars of ETF #m. Similarly, closing a long position means selling stock and buying ETFs

*I don't necessarily subscribe to this.

**The paper addresses this briefly in the following comment "Another possibility is to consider scenarios in which one of the stocks is expected to out-perform the other over a significant period of time. In this case the co-integration residual should not be stationary. This paper will be principally concerned with mean-reversion, so we don’t consider such scenarios," but this is a convergence model. In the case where the number of chess players goes parabolic upwards for whatever reason, you lose a lot of money without having a divergence model as well.