Distribution Fitter

Fit Normal, t-distribution, and empirical distributions to historical returns and see how often real data exceeds Normal tail predictions.

Feb 23, 2026, Eric

The Core Idea

Most risk models — VaR, portfolio optimization, options pricing — assume returns are Normally distributed. They are not. Real returns have fat tails: extreme events happen far more often than the Normal distribution predicts. The t-distribution fits better, and the empirical KDE shows the actual shape. This tool makes the gap between Normal and reality concrete and quantified.

Data

Upload CSVCSV: one column of daily returns (% or decimal)

Distribution Overlay

Gray bars = empirical histogram (density). Red = Normal fit. Amber = t-distribution fit. Blue = KDE.

Sample Size (N)

1,260

Ann. Mean

-3.4%

Ann. Std Dev

19.0%

Skewness

-0.163

Roughly symmetric

Excess Kurtosis

4.771

Leptokurtic — fat tails

t-dist degrees of freedom (ν)

Fat tails

Max Single-Day Loss

-7.07%

Max Single-Day Gain

+6.80%

Left-Tail Analysis — How Often Does Reality Beat the Normal Model?

"Ratio" = actual count ÷ Normal predicted count. >1.0 means more tail events than Normal expects.

Threshold	Actual	Normal predicts	t-dist predicts	Ratio (actual/normal)
<−1σ (-1.21%)	167	199.9	158.6	0.84×
<−2σ (-2.41%)	22	28.7	30.7	0.77×
<−3σ (-3.61%)	8	1.7	7.2	4.70×
<−4σ (-4.80%)	6	0.0	2.1	150.28×

Fat tail summary: Real returns exceeded the Normal −3σ prediction 18 times vs. Normal's predicted 3 — that's 6.0× more frequent than expected. This is the fat tail.

What This Means

Excess kurtosis: 4.77. High kurtosis: the return distribution has significantly fatter tails than Normal predicts. Risk models assuming Normality will systematically underestimate tail losses for this dataset.

t-distribution fit: Fitted ν = 5 degrees of freedom — this is a fat-tailed distribution. Lower ν means heavier tails. For reference, the S&P 500 historically fits ν ≈ 4–6.

Tail count vs. model: Beyond −3σ: 18 actual vs. 3 Normal-predicted. That's 6.0× more frequent — the gap that blows up Normal-based VaR models.