Kurtosis — Return

Excess kurtosis of daily returns on the Return curve (low-to-low). >0 = fat tails, more extreme moves than normal distribution.

Computed from: Equity curve
Scope: Single report
Range: Any real number
Direction: Lower is better

Basis

Computed on the low-to-low Return Series (money-weighted) — measured day-worst-point to day-worst-point, not close-to-close.

Kurtosis measures how fat the tails are — how prone the daily results are to extreme days, big surprises in either direction. Picture a bell curve: a hump in the middle for ordinary days, thinning out to tails on the far left and right for the rare big days. Fat tails mean those far ends are heavier than a normal bell curve predicts — extreme days happen more often than the calm middle suggests. A high reading means the quiet stretches are deceptive: most days are ordinary, but the rare day is really extreme. It is the natural partner of skewness — skew says which tail is fatter, kurtosis says how fat both tails are.

Key takeaways

What it is — the fatness of the tails of your daily results: how often extreme days happen versus a normal bell curve. (Technically a fourth-power average, but you only need to read whether it's big.)
Read it as — EquityTruth reports excess kurtosis (excess just means "compared to a normal bell curve"): 0 = normal tails, positive = fat tails, negative = thin. Some positive reading is normal for daily returns (markets are fat-tailed) — only an extreme reading, paired with negative skew, is the real warning. A near-zero or negative reading is suspicious, not safe.
Why it matters — fat tails mean the calm is deceptive: a smooth-looking record can still hide a rare, violent day. Read it with skewness — negative skew and high kurtosis is the worst combination, because the fat tail you're most likely to hit is the losing one.
In EquityTruth — measured on each day's return (one value per trading day, not on individual trades), on the Return, TWR, and P&L curves.

How it's calculated

The "tails" of a distribution are its stretched-out far ends — the rare extreme days. Kurtosis asks how heavy those ends are: raise each day's standardized result to the fourth power (so far-out days dominate overwhelmingly, and — unlike skew — the sign is lost, since both tails count), average, then subtract the value a normal distribution would give. What's left is excess kurtosis, centered at zero.

Kurtosis = [ n(n+1) / ((n−1)(n−2)(n−3)) ] · Σ ( (rᵢ − r̄) / σ )⁴  −  3(n−1)² / ((n−2)(n−3))

rᵢ: each day's return
r̄, σ: the mean and standard deviation (typical spread) of those returns
n: the number of days

In this product: This is excess kurtosis, centered at 0 (0 = normal-distribution tails, positive = fatter): the subtracted term removes the value a normal distribution would produce. EquityTruth uses the bias-corrected sample form shown — the n(n+1)/((n−1)(n−2)(n−3)) weighting and the 3(n−1)²/((n−2)(n−3)) adjustment, σ with ddof=1 — so short records aren't systematically off. It's computed on the daily low-to-low returns — one value per trading day, each day's lowest equity to the next day's lowest — the same series as [skewness](/glossary/skewness), [Sharpe](/glossary/sharpe-ratio), Sortino, and [volatility](/glossary/volatility).

On the Return and TWR curves kurtosis is computed on the equity curve's returns. Because it's standardized and unitless, it's comparable across accounts. Return is money-weighted (deposits/withdrawals move it); TWR strips cashflows out for the purest read of the strategy's own tail behavior.

What it tells you

Kurtosis is a fourth moment — even noisier than skewness. Because deviations are raised to the fourth power, a single outlier dominates the estimate more than it does anywhere else, and on a short record the number means very little. Trust a strong reading only on a long record (roughly 250+ days) that has actually seen some extreme days.

Value	Reading	Notes
< 0	Thin tails — too calm	Fewer extreme days than even a normal distribution. Genuinely rare for real returns — usually a short, sheltered, or hand-picked record, not a calm strategy. Treat as a flag, not comfort.
0 – 3	Mildly fat — quiet for markets	Tails a bit heavier than normal. For daily returns this is the calm end of the range.
3 – 10	Fat tails — normal for daily returns	The everyday range for real trading records. Not an alarm by itself — read it with skewness to see which side the extremes fall on.
> 10	Very fat — extreme-prone	A few days dominate the whole distribution. With negative skew, this is the short-vol / blow-up signature — inspect leverage and the worst day.

A fat-tailed (leptokurtic) account: a tall, narrow peak of ordinary days in the center, but small non-zero bars far out on both sides — rare extreme days in either direction. The peak is taller and the tails heavier than a normal bell curve, so excess kurtosis is positive. Kurtosis is symmetric: it counts both tails equally and says nothing about which side — that's skewness.

Worked example

Two accounts post the same average daily return and the same volatility. Account A grinds out a tight band of small, similar days — its results cluster near the center with almost nothing far out, giving low (even negative) excess kurtosis. Account B looks just as calm most of the time, but a few times a year it prints a violent day — a sudden +8% or −8% — and those rare days stretch both tails far from the center, giving high kurtosis. The two have identical headline risk numbers, yet B is the one that can hand you a shock the volatility figure never warned you about. Fat tails are exactly the risk that standard deviation, and the Sharpe ratio built on it, quietly understate — because both assume a normal bell curve, and fat tails mean the real odds of a disaster day are far higher than that curve admits. A 3-Sharpe short-vol book is the classic trap. (You never compute the kurtosis formula by hand; the tool does it — read the bands.)

Pitfalls

Pitfalls & caveats

Even noisier than skewness. Kurtosis is a fourth moment, so one outlier dominates the estimate massively — a single extreme day can set the whole number. It's the most sample-hungry of the shape metrics; trust it only on a long record.
It says nothing about which tail. High kurtosis alone is neutral — the extremes could be wins or losses. You have to read it with skewness: high kurtosis with negative skew is the dangerous one; with positive skew it's a feature, not a bug (big upside surprises).
A low reading is a warning, not comfort. Financial returns are almost always fat-tailed — markets have more crashes and spikes than a bell curve predicts. So an unnaturally smooth, low- or negative-kurtosis daily curve on a real leveraged account usually means the record is too short, hand-picked, or running a martingale that simply hasn't met its bad day yet — the calm before, not a calm strategy.
Look at the day, not just the moment. Kurtosis is a noisy summary statistic; the fastest, most reliable tail check is to ignore the number and look at the worst single day and worst week directly — and at Value at Risk / CVaR, which average the bad tail instead of summarizing it.
Daily, not per-trade. This measures the shape of daily returns, not individual trades. The two can differ, and it's the daily/aggregate tail that empties the account.

Kurtosis and skewness (the pair)

Skewness and kurtosis are read together: skew tells you which tail is fatter, kurtosis tells you how fat both tails are. The combination is what matters. High kurtosis with negative skew is the worst case — a rare extreme day that is both one-sided and severe, and on the loss side: the short-volatility / blow-up signature worn by grid systems, carry trades, option-writing, and the famous failures (LTCM, the February 2018 short-vol funds). High kurtosis with positive skew is desirable — the rare extreme day is a large win, the convex trend-following profile — though it pays for those rare wins with long stretches of small losses. Same tail fatness, opposite meaning; only skew tells them apart. For a steadier empirical cross-check than either raw moment, see tail ratio.

Skewness tells you which tail is fatter (read it alongside kurtosis), Tail Ratio gives a steadier right-vs-left tail read than the raw moments, Volatility measures everyday spread (which fat tails sit beyond), and Value at Risk puts a number on the bad tail kurtosis warns about.

Compare withSkewness