The main idea is to compare mean reversion (MR) vs. follow through (FT). For simplicity I define mean reversion as an up day being followed by a down day, and a down day being followed by an up day. Conversely, follow through sees an up day followed by another up day, and a down day followed by a down day.
I took a look at the major US equity indices, SPX (GSPC), NDX and RUT.
For each series we calculate daily log returns for the current period and shift the forward to get the return for the next period. Then we calculate realized volatility (RV) and split the data set into "low volatility" and "high volatility", by looking at median realized vol for the whole series.
Then, for each series, we use bootstrapped samples to simulate a number of trajectories/equity curves for each strategy (MR/FT) under the two classes of RV. Finally we take an average of the total return of each trajectory to get a ballpark idea of how they went.
The data is from the start of 1999 to the present, so roughly 15 years. Each run generates 1000 trajectories with a sample size of roughly 950.
For the low vol case, the results are unfortunately ambiguous. Follow through in a low vol environment seemed to do well for NDX and RUT, but the opposite was the case for SPX.
The TR column is the sum of the series over the whole period for the volatility class (i.e. a simple long only strategy), giving an idea of a directional bias that may be present in the sampling.
In the high vol environment, mean reversion was a clear winner, and consistent over the different underlyings.
The results seem relatively stable across trajectory size/sample size.
I'm not really sure what is going on SPX. My intuition was that FT would do well in low vol environments, but that doesn't seem to be the case, at least not for SPX.
I was actually getting consistent votes for FT in the low vol case, then restarted R to run with a clean environment and started getting the above instead. You can't spell argh without R it seems.
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