Skewness

Understanding asymmetry in return distributions

What Is Skewness?

Skewness measures the asymmetry of a return distribution:

Right-skewed (positive skew)
Fat right-tail → occasional large gains.
Left-skewed (negative skew)
Fat left-tail → occasional large losses.

In portfolio analytics skewness helps answer:

"Does this strategy make small profits most of the time and rare blow-ups, or the opposite?"

Mathematical Definition

For a series of demeaned returns $r_t$ ( $t=1\dots n$ ) with standard deviation $\sigma$ :

\text{Skewness} = \gamma_1 = \frac{1}{n\sigma^3}\sum_{t=1}^{n} r_t^3

$\gamma_1 > 0$ → right-skew.
$\gamma_1 < 0$ → left-skew.
$\gamma_1 = 0$ → perfect symmetry (Gaussian).

Because the third moment amplifies tails, even a handful of extreme days can swing skewness dramatically.

How Our Backend Calculates Skewness

Inside srv.py → compute_custom_metrics() we capture it in one line:

skewness = port_returns.skew()

port_returns are daily simple returns (not log).
pandas.Series.skew() uses the Fisher-Pearson (bias-adjusted) estimator, matching the formula above.
The scalar is then stored in performance.skewness and surfaced in the results table for every optimisation method (MVO, MinVol, HRP, CLA, …).

Tip: When you generate the distribution plot (histogram) the fat-tail visual directly matches the sign of this skewness value.

Interpreting Skewness in Portfolios

Skewness	Typical Strategy Profile	Risk Narrative
$\gamma_1 > 1$	Trend-followers, lottery-like stocks	Many small losses, rare big wins
$0 < \gamma_1 < 1$	Broad equity indices	Mild right-tail (crash up-days)
$-1 < \gamma_1 < 0$	High-yield credit, carry trades	Mildly left-tailed (bleed)
$\gamma_1 < -1$	Short-vol / options selling	Small steady gains, rare large losses

Combine skewness with kurtosis (tail-fatness) for a fuller shape picture.

Why It Matters

Risk Management – Left-skewed strategies need extra tail-risk controls (stop-loss, hedges).
Product Design – Retail investors often prefer right-skew ("lottery ticket" pay-offs).
Optimization Constraints – Advanced portfolio construction (e.g., Higher-moment MVO) can set skewness targets.

Visual Diagnostics

Our portfolio optimization app uses histograms with Freedman-Diaconis bins to visualize return distributions, which helps in identifying skewness patterns in your portfolio.

These histograms provide an intuitive way to see the asymmetry in return distributions and complement the numeric skewness measure.

Caveats

Issue	Comment
Sample Sensitivity	Few extreme values can dominate $\gamma_1$ . Robust checks (bootstrap) are wise.
Non-stationarity	Rolling skewness (e.g., 1-year window) reveals regime shifts.
Interpretation vs. Utility	Positive skew isn't automatically good if gains are tiny; always look at mean & risk metrics together.

Academic References

Kraus, A., & Litzenberger, R. H. (1976). "Skewness Preference and the Valuation of Risk Assets." Journal of Finance, 31(4), 1085-1100.
Harvey, C. R., & Siddique, A. (2000). "Conditional Skewness in Asset Pricing Tests." Journal of Finance, 55(3), 1263-1295.
Cont, R. (2001). "Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues." Quantitative Finance, 1(2), 223-236.

By measuring skewness alongside Sharpe, Sortino, and β you gain a shape dimension of risk—vital for understanding tail behaviour that variance-based metrics alone can't reveal.