Kurtosis

Understanding the "Tailedness" of Return Distributions

What Is Kurtosis?

Kurtosis quantifies the weight of a distribution's tails relative to the normal bell-curve:

Name	Excess Kurtosis ( $\gamma_2$ )	Shape	Intuition
Leptokurtic	$\gamma_2 > 0$	Peaked centre, fat tails	Higher chance of extreme gains and losses
Mesokurtic	$\gamma_2 = 0$	Normal-like	Gaussian benchmark
Platykurtic	$\gamma_2 < 0$	Flatter centre, thin tails	Extremes rarer than Gaussian

Rule of thumb: A strategy with $\gamma_2 \gg 0$ hides more "black-swan" risk than variance alone suggests.

Mathematical Definition

For de-meaned returns $r_t$ with standard deviation $\sigma$ :

\text{Excess Kurtosis }(\gamma_2) = \frac{1}{n\sigma^4}\sum_{t=1}^{n} r_t^4 - 3

Subtracting 3 converts raw kurtosis to excess kurtosis so that a normal distribution sits at 0.
Fourth power makes the metric hypersensitive to outliers.

Our Backend Implementation

Inside srv.py → compute_custom_metrics:

kurtosis = port_returns.kurt()

pandas.Series.kurt() produces the bias-adjusted Fisher–Pearson estimator of excess kurtosis—exactly the $\gamma_2$ above. Each optimisation method's value is stored in performance.kurtosis and displayed in your results cards.

Interpreting Excess Kurtosis

$\gamma_2$	Typical Strategy Examples	Risk Narrative
> 3 (very fat)	Short-vol (option selling), carry trades, stable-coin yields	Many small profits punctuated by rare, catastrophic draw-downs
≈ 0–1	Broad equity indices, balanced funds	Tail risk comparable to Gaussian assumption
< 0	Certain trend-followers, lottery stocks	Extremes are dampened (rare) but centre flattens – performance may be choppy

Use kurtosis in tandem with skewness and VaR/CVaR to map full tail risk.

Visual Diagnostics

Our portfolio optimization app provides histograms to visualize return distributions, which can help identify the kurtosis of your portfolio returns.

For a more detailed examination, you might consider:

Histogram with log-y axis clarifies tails.
QQ-plot vs. normal line quickly shows tail divergence.
Kurtosis–Time Chart (rolling 1-year windows) reveals if tail risk is creeping up.

Why Investors Should Care

Use-Case	How Kurtosis Helps
Stress testing	Identify strategies where "10-sigma" events aren't so rare.
Risk budgeting	Allocate less capital to highly leptokurtic sleeves unless adequately hedged.
Product disclosure	Flag fat-tailed pay-offs to regulators / clients.

Caveats & Best Practice

Issue	Recommendation
Extreme sample-sensitivity	Winsorise or bootstrap to test stability.
Window length trade-off	Longer windows → stable estimate; shorter windows → regime detection.
Ambiguous sign	Positive kurtosis alone isn't "bad" if accompanied by high right-tail skew—context matters.

Academic References

Taleb, N. N. (2010). The Black Swan: The Impact of the Highly Improbable. Random House.
Cont, R. (2001). "Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues." Quantitative Finance, 1(2), 223–236.
Scott, D. (1992). Multivariate Density Estimation. Wiley – Ch. 4 (Higher-moment estimators).

By exposing kurtosis alongside variance-based and downside-based metrics, our platform equips users to see beyond average volatility, recognising those hidden tail risks where true portfolio disasters—and sometimes spectacular windfalls—originate.