Robust Returns-Based Style Analysis

Question

Sharpe's Return-Based Style Analysis is an interesting theory but flawed in practice when working with long-short funds or funds that are changing strategies over shorter periods of time due to the limits of linear regression.

I have found a few papers looking into improvements to make the calculations more robust Markov, Muchnik, Krasotkina, Mottl (2006) seems fairly reasonable for instance. However, they commonly only deal with the time-varying beta issue.

I was wondering if there was anyone out there doing work on the limitations of linear regression for style analysis. I particular more robust variance-covariance matrices for the minimization of the objective function.

Answer 1

Whether or not it is flawed in practice depends on dynamic the risk exposures really are. Many factors or indices used for style analysis actually require dynamic trading to maintain - so you could potentially have a fund that trades a lot while still generating a return series that can be be modeled out of sample with static exposures.

One relatively simple approach for what you are trying to do is to use the Lasso (discussed in the paper). This will achieve your goal of reducing factors as they coefficients will be shrunk towards zero. Another more complex option would be to use Bayesian regression with informative priors to estimate factor exposures. For example, you might have different priors on the exposure to SPY of a long/short equity fund vs. a merger arb fund. Kruschke, author of Doing Bayesian Data Analysis, also showed an example of "robust" regression where the errors are assumed to follow a t-distribution. Both of these approaches are pretty straightforward in R.

Finally, if you do you want to explore dynamic exposures you could use a state space model to estimate time-varying parameters. This is a bit more complex to implement, but one of the R packages that is useful here is dlm. The package's author has written a book: Dynamic Linear Models with R. There are also various slides from Yollin floating around online demonstrating how to estimating time-varying beta exposures using dlm. You might want to check out Understanding Hedge Fund Alpha Using Improved Replication Methodologies by Chen & Tindall, which I believe a number of these approaches.

Answer 2

Thanks for the answers and comments above. In particular to Eric Brady, who had me reading a lot of Bayesian papers.

In the end, I think the answer to the question is that on the monthly time-frame robust factor algorithms aren't really necessary. On daily and lower time frames, large spikes in returns due to events (earnings ect.) can really mess with factor loadings and robust methods like Principal Component Pursuit run on the whole universe and then applied to the factors and return streams will give much better results. Bayesian methods are interesting as well but tough to apply.

However, on the longer, lower-frequency time scale that I was interested in above the spikes in returns aren't important enough to mess with the variance-covarience matrix. The real issue is just that the betas need to vary in time in a more robust manner than the standard rolling-window linear regressions from Sharpe. For this the Kalman Filter borrowed from signal analysis appears to be a very good solution.

Answer 3

As I know, style analysis is not a linear regression, it's not trying to minimize the square of the error, it's trying to make a combination of the benchmark portfolios to track your portfolio as close as possible. That is to say, your portfolio can outperform the benchmark like 4 percent, it's trying to make the 4 percent as stable as possible, the variance of the 4 percent alpha be as small as possible; while linear regression is trying to say the sum square of all the outperform and underperform is small as possible.

Style analysis make your portfolio has a parallel outperform or underperform relative to a selection of benchmarks, and from the weights of these benchmarks, you can know your portfolio's style; while linear regression is trying to make the benchmarks curve as close as possible to your portfolio.

Answer 4

I've often thought about the same thing. To try and figure out some concrete (in my opinion) information about a stock or mutual fund, I wrote something in Python to simulate:

• buying stock at some interval
• stock paying dividend at some interval
• adjusting returns for inflation
• subtracting out fees (if a mutual fund or something with an expense ratio)
• do this 100 or many more times for a given time period and see what you get

I've found some gems (so far) using this method. Long term reliably good returns over most any period in time (so far) seems to be a pretty legit way to look at things. I usually will look at thousands of them and pick ones with good, 5, 10, 15 good return results. Plotting these things on a scatter plot is very helpful to see how you'd have fared at any randomly selected time in the past if you make it for some stock you are interested in.