Kurtosis

Kurtosis describes how weight is distributed between the center and the tails. In return series, we use excess kurtosis as a quick signal that tails are heavier than normal (even though kurtosis also reflects how “peaked” the center is).

We report excess kurtosis, which is kurtosis minus 3. A normal distribution has kurtosis 3, so it has excess kurtosis 0.

The definition

Excess kurtosis is the fourth standardized moment minus 3:

$$\text{Excess Kurtosis} = \frac{E[(r - \mu)^4]}{\sigma^4} - 3$$

How we calculate kurtosis at Gale Finance

1. Daily log returns. Like our other distribution-shape metrics, we compute kurtosis on daily log returns:

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

2. We report excess kurtosis. If you see a value like 2 or 5, that means the tails are materially fatter than normal.

3. Short-window caution. In one‑year samples, kurtosis can be dominated by a few extreme days. That’s not “wrong” — it’s literally telling you the tails matter — but you shouldn’t treat it like a stable parameter.

What kurtosis does (and doesn’t) tell you

Kurtosis tells you tails are fat, but not whether they’re fat mostly on the downside or upside. For that, combine it with skew and with direct tail metrics like Expected Shortfall.