Z-Score

A z-score tells you how extreme a daily move is relative to “normal” moves in the same window.

It converts returns into “how many standard deviations away from the mean”:

$$z_t = \frac{\ell_t - \mu}{\sigma}$$

Where:

• $\ell_t$ is the daily log return,
• $\mu$ is the mean of daily log returns in the window,
• $\sigma$ is the standard deviation of daily log returns in the window.

We compute log returns as:

$$\ell_t = \ln\left(\frac{P_t}{P_{t-1}}\right)$$

Where:

• $P_t$ is the price at time $t$,
• $P_{t-1}$ is the price at the prior time step.

What “2σ” and “3σ” mean

• A -2σ day is a day where $z_t \le -2$ (a big downside move).
• A +2σ day is a day where $z_t \ge 2$ (a big upside move).
• A |3σ| day is a day where $|z_t| \ge 3$ (an extreme move either direction).

If returns were perfectly normal (they aren’t), you’d expect roughly:

• ~4.6% of days beyond |2σ| (two-sided),
• ~0.27% of days beyond |3σ| (two-sided).

How we use z-scores at Gale Finance

On compare pages and scorecards we use z-scores to flag “sigma events”:

• Tail days (down / up). Counts of days below $-2\sigma$ and above $+2\sigma$ on each asset’s own daily log return series.
• |z| > 3σ days (observed vs expected). A quick sanity check: how many extreme days actually happened vs how many a normal distribution would predict for the same number of observations.

Because we show down and up tail-day counts separately, at 2σ the normal “expected” rate is one-sided (~2.3% beyond +2σ or below −2σ).

This is deliberately simple. It’s not a model. Instead, it’s a readable way to see whether an asset’s return distribution is behaving “normal-ish” or doing fat-tail things.

Important caveats

• Window-dependent. Change the window and $\mu$ and $\sigma$ change, so z-scores change too.
• Fat tails make “σ language” optimistic. If an asset has fat tails, “3σ events” will happen more often than a normal baseline.
• Not a forecast. This is descriptive: it summarizes what happened in the chosen window.

If you care about “do these assets crash together?”, pair this with tail dependency (downside co-moves).