Two numbers describe how a distribution departs from the bell curve, and both are computed from the same deviations you already have.
How are skewness and kurtosis calculated?
Sample skewness is the standardised third moment, with a small-sample correction:
S = [n / ((n−1)(n−2))] × Σ(rᵢ − r̄)³ / s³
On the twenty-quarter series the cubed deviations sum to −14,747.799 and s³ = 809.834535, giving S = −1.064964. Negative skew means the left tail is longer. The losses sit further from the middle than the gains do, which matches a series whose worst quarter is −21.40% and whose best is +15.30%.
Sample excess kurtosis is the standardised fourth moment, corrected and then shifted so that a normal distribution scores zero:
K_E = [n(n+1) / ((n−1)(n−2)(n−3))] × Σ(rᵢ − r̄)⁴ / s⁴ − 3(n−1)² / ((n−2)(n−3))
The fourth powers sum to 501,086.04, s⁴ = 7548.518506, and the answer is K_E = +1.256184, or a non-excess kurtosis of 4.256184 against a normal's 3. Positive excess kurtosis, leptokurtosis, means more of the probability sits in the peak and in the tails, and less in the shoulders.
Both estimators are fragile at n = 20. The uncorrected population moments give skewness −0.983367 and excess kurtosis +0.677675 from the same twenty observations, materially different numbers. The Jarque-Bera statistic built on them is 20/6 × (0.983367² + 0.677675²/4) = 3.606070 against a 5% critical 5.991465 on two degrees of freedom, so at a p-value of 0.1648 this sample cannot reject normality despite a visible left tail. A test with no power is evidence about the sample size, not about the distribution.
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Why does a normal VaR understate risk?
Kurtosis stops being an abstraction the moment it meets a value-at-risk number. Worked example 4 states a two-regime daily model whose moments can be written down exactly: 90% of days drawn from a normal with a 0.70% standard deviation and 10% from a normal with a 2.60% one. That mixture has a standard deviation of 1.056882% a day and an excess kurtosis of 8.5073. A 99% one-day VaR computed the normal way takes 2.3263479 × 1.056882% = 2.458676% of value. Under the true mixture, the probability of a loss worse than that cut is 1.741634%, not 1%. The number promising 2.5 breaches in 250 trading days delivers 4.35.
The genuine 99% cut is 3.332163%, so on a ₹2,00,00,000 position the normal number understates the loss by ₹1,74,697, or 35.5% of itself. At 99.9% the understatement grows to 85.2%. This is the mechanism behind every risk model that behaved well until the week it did not, and a fourth moment anybody can compute measures it in advance.
One honest caveat about the fix. A Cornish-Fisher expansion adjusts the normal quantile using the skewness and kurtosis directly, and on these moments it returns a z of 4.3153 and a VaR of 4.560715%, overshooting the true 3.332163% by more than the normal number undershoots it. It is a two-term approximation, reliable for mild non-normality and unreliable at an excess kurtosis of 8.5. Knowing where a tool's working range ends is the actual skill.