The Skew-Normal Distribution

A plain normal distribution is perfectly symmetric — its left and right tails are mirror images. Real return data often isn't: equity returns tend to have a longer left tail (occasional sharp crashes) than right tail (steadier grinding gains), a shape a symmetric normal simply cannot represent no matter how its mean and standard deviation are tuned. The skew-normal distribution fixes this by adding one extra shape parameter, $\alpha$, on top of the usual location and scale, controlling how much and in which direction the distribution leans.

When $\alpha = 0$, the skew-normal is exactly an ordinary normal distribution — it's a strict generalization, not a different family. As $\alpha$ moves away from zero, probability mass shifts, stretching one tail and compressing the other while keeping the overall shape still recognizably bell-like rather than turning into something exotic. A negative $\alpha$ produces the long-left-tail shape that resembles daily equity return histograms; a positive $\alpha$ stretches the right tail instead, closer to what some commodity or volatility-related series can look like.

In practice, the skew-normal is used as a slightly richer building block than the normal when a return series is clearly asymmetric but a full heavy-tailed model like a skew-t is more machinery than the problem calls for — fitting three parameters (location, scale, skew) instead of two, and recovering the normal exactly as a special case.

As a rough sense of scale: fitting a skew-normal to a daily equity return series with a shape parameter of $\alpha \approx -4$ produces a skewness coefficient around $-0.5$, meaningfully negative but still a recognizable bell curve — compared to $\alpha=0$, which forces that skewness coefficient to exactly zero regardless of what the data actually looks like.