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Risk

Skewness & Kurtosis

Skewness is the third standardised moment of a return distribution (asymmetry); kurtosis is the fourth (tail heaviness). Together they describe departures from normality.

Also known as: third moment · fourth moment · excess kurtosis

A Gaussian distribution is fully described by its mean and variance. Real return distributions are not Gaussian — they are typically asymmetric (skewed) and have fatter tails (high kurtosis). Reporting skewness and kurtosis explicitly is the lightest-touch way to disclose that.

Formulas

Skewness = E[ ( R − μ )³ ] / σ³
Kurtosis = E[ ( R − μ )⁴ ] / σ⁴
Excess Kurtosis = Kurtosis − 3

The Gaussian distribution has kurtosis = 3 by construction, so "excess kurtosis" (= kurtosis − 3) is the convention in finance: 0 = Gaussian, > 0 = fatter tails, < 0 = thinner tails.

Interpretation

  • ·Skewness < 0: more frequent small gains, occasional large losses (long equity, short volatility).
  • ·Skewness > 0: more frequent small losses, occasional large gains (long volatility, trend-following).
  • ·Excess kurtosis > 3: noticeably fatter tails than Gaussian. Equity returns typically show 3-8.
  • ·Excess kurtosis > 10: extreme tail risk. Single-name leveraged exposure can show this.

How MEDGE Capital uses skewness and kurtosis

Both are reported in every portfolio analysis alongside the headline triple (CAGR, vol, Sharpe). When excess kurtosis > 5 the Cornish-Fisher VaR adjustment is automatically applied to the reported VaR/CVaR numbers, making the tail-honesty visible without the user having to do anything.

See also

VaR (Value at Risk)

VaR (Value at Risk) is the maximum loss not expected to be exceeded with a given confidence level over a given holding period.

CVaR (Conditional Value at Risk)

CVaR (Conditional Value at Risk) is the average loss conditional on the VaR threshold being breached at a given confidence level.

Cornish-Fisher VaR

Cornish-Fisher VaR is the Gaussian VaR adjusted for empirical skewness and kurtosis via the Cornish-Fisher expansion of the standard-normal quantile.