Z-Score Factor: Signals, Limits, and Mean Reversion

The January 2024 issue of Technical Analysis of Stocks & Commodities features Z-score as its fourth article, titled Z-score: How to use it in Trading. We'll take this opportunity to share our own views on building quant factors with Z-score.

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Calculating the Z-Score

The Z-score is calculated as: $$ Z = \frac{X-\mu}{\sigma} $$

Here $Z$ is the z-score, $X$ is typically the price, $\mu$ is the mean, and $\sigma$ is the standard deviation, also known as volatility.

If X follows a normal distribution, then:

1. The probability of abs(z-score) >= 2 is less than 2.3%
2. The probability of abs(z-score) >= 3 is less than 0.13%

This can be used as a kind of reversal signal: once the z-score moves beyond ±2, there is a 97.7% chance it will revert back inside ±2 — in other words, the price should revert toward the mean.

The scipy.stats package offers a zscore function, but it is computed over the full sample you pass in. For factor construction we need a rolling z-score. So we compute it ourselves with pandas rolling:

1234def rolling_zscore(s, win=20):
    ma = s.rolling(window=win).mean()
    std = s.rolling(window=win).std()
    return (s - ma)/std

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## Factor Testing

We won't run the full pipeline in this note. Instead, we'll pick a random ticker, compute its z-score over the last 250 bars, mark z-scores below -2 as buys and above +2 as sells, plot them, and dig deeper into what the chart tells us:

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Almost every buy sits near a local low, and almost every sell sits near a local high. But in a downtrend, even buying the exact low doesn't help much — the bounce is short-lived and shallow, and price often turns down again long before a z-score sell signal appears.

In other words, a z-score above 2 is sufficient but not necessary for a sell; a z-score below -2 is sufficient but not necessary for a buy. It does not forecast the future trend, and 97.7% of the time it emits no trading signal at all, leaving capital underutilized. Therefore, a z-score can make a factor, but not a strategy. As a factor, however, it is excellent because its signals are well-defined and have a very high win rate.

Relationship with Bollinger Bands

If you know Bollinger Bands, you'll see the Z-score uses exactly the same math. The difference is that Bollinger Bands plot upper and lower bands at ±2 standard deviations from the mean, whose absolute levels can swing widely, while the z-score itself lives in a fixed range around (-3, +3), which acts like normalization. That makes it easy to use as a factor in machine learning. For true normalization, just take the cumulative probability of the z-score — it is distributed over [0,1], with a value of 0.977 when the z-score equals 2.

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## Beware of Black Swans Unlike many other factors, the validity of the z-score factor rests on the normality assumption. Only if price fluctuations follow a normal distribution can we claim that a deviation of more than two standard deviations from the mean occurs less than 2.3% of the time. In reality, price moves are not normally distributed (indexes are closer, but fit a generalized hyperbolic distribution better than a normal). The theoretical footing for the Z-score factor — and for Bollinger Bands — is therefore shaky.

More importantly, out at two standard deviations, the event is rare but can be severe when it does happen. That is the black swan effect described by Taleb. In China A-shares, the rule is simple: if a ±2-sigma deviation comes with a price limit lock-up, drop the Z-score signal decisively. In that situation, sentiment is extreme.

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Quiz

If price changes follow no normal — or no known — distribution, how should we capture their statistical properties?

For example: if the Shanghai Composite (沪指) is down 4% today, what is the probability it keeps falling, based on the past 1,000 trading days? How would you answer? Anyone who can answer this correctly can catch washout bottoms — or fade panic sell-offs triggered by surprise events.

Hint: this is the question we use in our course to introduce PDF/CDF concepts.

Source code for this article: [source code](https://blog.quantide.cn/assets/notebooks/zscore.ipynb)