# Momentum and Mean Reversion in Different Time Frames

In a recent blog post, I rather glibly stated that the market tends to revert to a mean. A reader called me out about the time frame I was using, which raises a good point. A market can tend toward both mean reversion and momentum over different time frames. Many traders would argue that different markets show different characteristics over specific time frames, and that these characteristics are persistent.

For example, one might argue that the S&P 500 is mean-reverting on a weekly basis but shows momentum characteristics over a yearly basis. Or that soybean futures show momentum on a weekly basis but are mean reverting on a yearly basis.

I decided to take a look at a few markets, to see if they exhibit a certain MR-vs-momentum ‘character’ at different time frames. Here’s how I went about the test:

For each trading day of the time series, I compared the close of the current day vs the close n days ago. I also compared the close n days ago vs the close of n * 2 days ago. If the signs of both changes were either positive or negative, that day would get a ‘1’ for that time frame. If the signs were different, it got a ‘0’.

In the lead image, you can see two periods marked. The short one is n=1 and the long one, n=13. Both of these time periods get a ‘1’, because they show momentum (both legs are up). Had they been falling, they would have also gotten a ‘1’. If both periods weren’t heading in the same direction, they would get a ‘0’.

My data thus looked like this:

Each column indicates a different time period. Each column was averaged to get a value between 0 and 1, and then .5 was subtracted to make it centered around 0. Negative averages mean there’s a tendency to revert to a mean over that time frame, and a positive number means momentum is more prevalent. The further away from zero, the stronger the tendency.

For my time periods, I chose the Fibonacci sequence of 1, 2, 3, 5, 8, 13, 21, 34, 55 and 89 trading days. Why the Fibonacci sequence? Because the ancient Egyptians believed that the Fibonacci sequence was the key to understanding the Cairo stock market.

Not.

Actually, it’s an easy way to get an ever-increasing sequence that incorporates the 1 and 5 day time frames, and also gets close to the monthly and quarterly time frame. Don’t worry, I haven’t gone over to the Technical Analysis dark side.

And now, the big reveal. Let’s look at the S&P 500 index from 1985 to the present.

We see here that the S&P tends to be very mildly mean-reverting on a one-day time frame, just about noise at the 2 and 3 day time frames, and then it starts to show momentum characteristics. The longer the time frame, the more momentum-y it gets. Cool! But how does that compare to other bits of money that move around?

Above we’re looking at \$XAU, or the PHLX Gold/Silver Sector Index. It shows a strong tendency to revert to a mean over just about all time frames.

Compare that to USDJPY:

I’d say that’s pretty uncoordinated. I wouldn’t have guessed a mean reversion in the short time frames, but the values are pretty close to zero throughout.

Then I started wondering what actually makes for significance? I decided to concoct some random-walk price series. I did the same test of MR-vs-momentum on four synthetic random-walk sets of data. Here are the data:

Looks pretty realistic, huh? I made sets of data that were roughly as long as the real data I was testing. Here are the results of the four sets of synthetic data:

I think we can use the max/min values of the four synthetic data runs to give us an idea what can result from mere chance. It makes sense that the larger time periods will show a greater excursion, because they will have had the opportunity to wander further away from the start over time. One run shows a momentum value of .06 for the 89 day period. The SPX value is almost double that. Without busting out some fancy math, that value makes me feel comfortable that the SPX is momentum-driven in the longest time frames.

It’s a little harder to judge visually for the smaller time frames, so take my word for it. At the one-day level, the SPX shows a -.009 value. The largest synthetic run showed a .006 positive value. On an absolute basis, the SPX is 50% greater. Does that mean its slight mean-reversion tendency is larger than what we would see from mere noise? I would think so…but without creating a ton of additional synthetic data and testing for statistical significance, it’s hard to say.

My job is not to do exhaustive mathematical stuff. My job is to get you thinking!

I think this sort of testing is worthwhile for two reasons:

First, markets behave differently. They have different ‘characters’ for want of a better word. It might be foolish to try and develop systems that work equally well across all tradables. What works on currencies should not necessarily be expected to work on equities, etc.

And second, if we know the MR/Mom (“mister mom”?) characteristics of a market, we can design systems around specific trade time frames. Or we can do the reverse: pick a time frame we’d like to trade (intra-day, daily, quarterly etc) and then design a system we know is more likely to work for that.

## One thought on “Momentum and Mean Reversion in Different Time Frames”

1. Mindaugas says: