Trading strategies are often loosely divided into two categories: trend-following and mean-reverting. They’re designed to exploit the mean-reverting or trending properties of asset prices. These properties are often investigated through time series techniques or Hurst exponent. Reference [1] provided, however, a different perspective and approach for studying the mean-reverting and trending properties of assets. It compared the long-run variances of mean-reverting and trending assets to that of a random-walk process. It stated,

*We have explored using a probabilistic model for investment styles to show that the variance of a financial asset is directly dependent on the probability of moving in the same direction on successive days. The theoretical analysis shows that variance may actually be reduced through reversal strategies – capturing the case that the asset is more likely to move in opposing directions on subsequent days. We have applied a simple model to US stock data, showing that such a regime is indeed prevalent in 97 of the largest stocks and thereby proven that relative to a random walk the variance of these stocks is actually reduced as a result of this frenetic behaviour. Indeed such a result suggests that these stocks are actually more predictable than a random walk due to this artefacct.*

In short, the paper concluded that most large-cap US stocks are mean-reverting, and the mean reversion resulted in a reduction of the variances of the assets. This means that mean-reverting asset prices are more predictable as compared to a random walk. The opposite is true for trending assets: larger variances and less predictability.

It’s refreshing to find a paper that elegantly combined theoretical and empirical research. Our observations are as follows,

- It’s not surprising that most large-cap stocks exhibit mean-reverting behavior, especially in the daily timeframe.
- The paper suggested that mean-reverting
**strategies**have lower variances than trending-following ones, but it did not provide a proof. Intuitively, this could be true, since mean-reverting strategies operate in a more predictable space, hence they have smaller variances. Also, this is consistent with the empirical fact that mean-reverting strategies have higher win rates. - The above claim can be investigated through numerical simulations.
- Trend-following strategies can be designed to exploit the expansion of variances, i.e. capturing the tail risks, by letting the profit run. But note that empirically they have lower win rates.

**References**

[1] L. Middleton, J. Dodd, S. Rijavec, *Trading styles and long-run variance of asset prices*, 2021, arXiv:2109.08242