Paper Review: Trends and Reversion in Financial Markets on Time Scales from Minutes to Decades

A paper by Sara Safari and Christof Schmidhuber

Thanks to a recent podcast episode on Top Traders Unplugged, I came across this paper on the performance of trend following at different speeds, as well as using both daily and intraday data.

Paper investigates the efficacy of trend following strategies at different lookback periods, and makes the claim that the reversion effect at any time horizon can be modeled by a third degree polynomial fit as such:

\(R(t+1) = a + b \cdot \phi(t) + c \cdot \phi(t)^3 + \epsilon\)

In the formula, \phi represents the trend strength, b is the transfer coefficient between the trend strength and next period returns, and c is the transfer coefficient between the cube of trend strength and next day’s returns. They deliberately did not include a second order term in the regression, as this would have removed the sign dependence in the relationship which is important in trading settings.

Unlike many papers where you have to read pages of nonsense to get to the bottom of the results, this paper gives us the results right away:

The subchart on the left shows the empirical expected return for a given signed trend strength value of \phi. There are some key observations in this chart that I agree with, which are not emphasized in the paper:

• It is important to keep an eye on the confidence intervals (assume these are 95% CI) as well as the values for E(r) for each bin

• As the superimposed polynomial indicates, it should be possible to fit a third-degree polynomial through the fits for each bin

• From a trading perspective, the 3rd degree polynomial fit seems useless

In practice, given how wide the confidence intervals around the E(r) estimates are at high values of trend strength \phi, one can either:

1. Cap the trend signal (i.e. \phi_max = +/-2)

2. Fade the trend signal down to zero, as a profit taking mechanism

3. Bet on mean-reversion at high values of \phi

Pretty much every CTA on the street does at least (1). Some of them, typically of the European ilk would likely do (2) as well. CTAs that employ (2) would likely taper their signal at zero, meaning the signal would never bet in the opposite direction of market trend.

Employing (3) as a trading strategy and betting against trend is not something I have seen work. Main problem is that trend strength is so strong very infrequently, therefore statistical significance goes out the window. One thing that goes in favor of such a mean reversion strategy is that the span of the confidence interval does not cross zero, so we can say that the direction of the mean-reversion strategy is likely to be correct.

An important observation from the paper also struck me: “While c has been fairly stable over time, b appears to have vanished over the decades.” As a reminder, c here is the coefficient of the 3rd degree polynomial coefficient, which determines the strength of the mean reversion relationship. I think this is quite important. Perhaps the mean reversion effect is so infrequent that it hasn’t been washed away. Another explanation is that unlike trend alphas, this effect cannot be removed by markets becoming more efficient. In fact, if the markets were perfectly efficient, all new information would be incorporated into the market immediately when it becomes available, and any additional noise would be immediately faded by the participants. In such a setup, strong trends would be less likely, and if such a strong trend exists, we would be able to mean-revert it with ease. I know I am writing a story to fit the narrative, but the mean-reversion aspect is something that deserves a bit more research in my opinion.

The authors then delve into a discussion of lattice gas model of financial markets without much of a derivation or demonstration. I find this part of the paper less useful. They claim more work is coming on this area.

There is clearly an interesting handoff between the time horizons where trend persists (down to roughly 1mo), and where there mean-reversion effects are clear, in the sub-hour timescale. In this area, trend sometimes work and sometimes it does not. Even in the 1 week or so timeframe that we expect trends to somewhat persist, it is not clear to me whether a naive implementation of trend following would survive trading costs. Of course, there are ways to overcome such costs with smarter execution and alpha embellishments, but this is beyond the scope of my review.

I should also say that I am not a fan of the 5th degree polynomial fit extension, I think this is heavily going in the direction of overfitting.

Overall, this is a decent paper which independently verifies some of the stylized facts about trend and mean reversion that practitioners have taken to heart, but it is lacking detailed results and discussion.