Long-Run Variances of Trending and Mean-Reverting Assets

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

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