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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.

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You might want to make this question a bit more specific, such as listing what type of analysis you want to perform or the types of questions you want to be able to answer (e.g. what is a robust approach to know how my exposure to x changed over time). – John Sep 18 '13 at 19:34
Sure. I'm interested in understanding how long/short fund monthly returns can be properly characterized using RBSA over large number of possible benchmarks. – rhaskett Sep 18 '13 at 23:40
To clarify on robustness, I'm talking about automatic dropping of unnecessary factors like a robust principle component analysis will reduce its factor space. Also, robustness to large sparse moves in returns is always useful in finance. – rhaskett Sep 18 '13 at 23:43
I do not see why a risk-adjusted return analysis (which Sharpe is) is flawed just because a fund frequently changes strategies. Sharpe calculations have nothing to do with linear regressions. Of course there are improvements that can be made, I am not saying that Sharpe measures are perfect, but it still is industry standard by which you, funds, anyone is measured before people take a closer look. – Matt Wolf Sep 19 '13 at 4:45
Matt, Dr. William Sharpe has done quite a bit of work in his life on mutual fund performance including the Sharpe ratio you are likely referencing and RBSA en.wikipedia.org/wiki/Returns-based_style_analysis. – rhaskett Sep 19 '13 at 22:51

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.

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up vote 1 down vote accepted

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.

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Thanks. In case you haven't found it I would recommend a quick Google for the Zivot and Yollin presentation if you want some examples of working R code of estimating time-varying betas with the Kalman Filter. Here is a currently working link: rinfinance.com/agenda/2012/workshop/Zivot+Yollin.pdf – Eric Brady Oct 21 '14 at 15:17
Thanks again. A LaTeX presentation is nice for reference. I'm biased toward Python/Pandas these days so I ended up using PyKalman which is pretty impressive. – rhaskett Oct 21 '14 at 17:02

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.

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That's not right, OLS does that not necessarily linear regression. – user2763361 Jan 20 '14 at 3:50
@user2763361 Would you please explain your reasoning? – StayFoolish Jan 21 '14 at 18:15
Beacuse you can also minimise perpendicular distance (PCA-equivalent) or absolute distance (median regression) – user2763361 Jan 22 '14 at 1:59

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.

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