Returns Distribution Analyzer

First four moments (mean, std, skew, kurtosis), goodness-of-fit to standard distributions (Normal, Student-t, Skewed-t, GED), tail statistics (VaR, CVaR at 95% and 99%), and visual diagnostics (QQ plot, density vs. fit). The methodology page documents each.

Kurtosis = E[(X-μ)⁴]/σ⁴. Normal distribution has kurtosis 3. Excess kurtosis = kurtosis − 3, so a Normal has excess kurtosis 0. The analyzer reports excess kurtosis (more interpretable). Most return series have excess kurtosis 3-15: 'fatter than Normal' is the rule, not the exception.

Returns have fat tails — extreme moves are more common than Normal predicts. The Student-t with low degrees of freedom (3-6) reproduces this. The analyzer reports the fitted ν parameter; ν<5 is fat-tailed, ν>30 is essentially Normal.

Value at Risk at 95% says: 'on a typical day, you won't lose more than X.' Specifically, the 5th percentile of the daily-return distribution. CVaR (Conditional VaR) is the expected loss given that the 5% threshold is breached — answering 'when it's bad, how bad is it on average?' CVaR > VaR by definition.

Only as one input. VaR has known weaknesses: it's not subadditive (combining two assets can make VaR worse), it ignores tail beyond the threshold, and it's sample-dependent. For risk management, prefer CVaR or expected shortfall. The methodology page links to the canonical critiques (Acerbi, Tasche).