Kurtosis

Rolling excess kurtosis (fourth standardised central moment minus 3) of the last period values. Positive readings flag fat tails (heavy outliers compared to normal); negative readings flag light tails (more concentrated than normal). 0 = Gaussian baseline.

Quick reference

Item	Value
Family	Price Statistics
Input type	f64
Output type	f64
Output range	[-2, ∞)
Default parameters	period required
Warmup period	period
Interpretation	> 3 very heavy tails; < 0 light tails; 0 Gaussian

Formula

mean     = (1/n) · Σ x
m2       = (1/n) · Σ (x - mean)²
m4       = (1/n) · Σ (x - mean)⁴
Kurtosis = m4 / m2² - 3

The unshifted kurtosis m4/m2² equals 3 for the normal distribution; subtracting 3 gives excess kurtosis so that 0 is Gaussian baseline. See crates/wickra-core/src/indicators/kurtosis.rs.

Parameters

Name	Type	Default	Constraint	Description
period	usize	none	>= 4	Rolling window.

Inputs / Outputs

Indicator<Input = f64, Output = f64>. Standard binding shapes.

Warmup

warmup_period() == period.

Edge cases

• Constant input. m2 = 0; Kurtosis undefined → returns 0.0.
• Lower bound. Mathematical floor for excess kurtosis is -2 (achieved by a 2-point bimodal distribution).
• Sample size sensitivity. Kurtosis estimates from small windows are noisy; statistics literature warns against reading kurtosis on n < 50.
• Reset. Clears the rolling window.

Examples

Rust

use wickra::{BatchExt, Indicator, Kurtosis};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut returns: Vec<f64> = (0..100)
        .map(|i| (f64::from(i) * 0.2).sin() * 0.01)
        .collect();
    returns[50] = 0.1;  // outlier
    let mut k = Kurtosis::new(50)?;
    println!("row 80 = {:?}", k.batch(&returns)[80]);  // positive (fat tail)
    Ok(())
}

Python

import numpy as np
import wickra as ta

returns = np.sin(np.linspace(0, 20, 100)) * 0.01
returns[50] = 0.1  # outlier
k = ta.Kurtosis(50)
print(k.batch(returns)[80])

Node

const wickra = require('wickra');
const k = new wickra.Kurtosis(50);
const returns = Array.from({ length: 100 }, (_, i) => Math.sin(i * 0.2) * 0.01);
returns[50] = 0.1;
console.log(k.batch(returns)[80]);

Streaming

use wickra::{Indicator, Kurtosis};

let mut k = Kurtosis::new(252).unwrap();
let return_stream: Vec<f64> = Vec::new(); // your stream of daily returns
for daily_return in return_stream {
    if let Some(v) = k.update(daily_return) {
        if v > 3.0 { /* fat-tail regime — tail-risk premia rising */ }
    }
}

Interpretation

• Excess kurtosis = 0. Returns look Gaussian.
• Excess kurtosis > 3. Very heavy tails — common in financial returns. Risk models that assume normality underestimate tail risk in this regime.
• Excess kurtosis < 0. Light tails — rare but possible in range-bound markets.
• Pair with Skewness. Together describe the shape: skewed + fat-tailed = highly asymmetric tail risk.

Common pitfalls

• Sample-size noise. Kurtosis is the most-noisy of the four moments. Don't read short-window kurtosis as anything more than directional.
• Population vs sample. Wickra uses population (n). Some references use sample formulas with bias corrections; ratios differ slightly.
• Outlier sensitivity. Single large outlier dramatically inflates kurtosis. By design — that's how it detects tail risk.

References

• Standard statistics — documented in any introductory text.
• For finance applications: Cont, Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance, 2001.

See also

• Skewness — third-moment sibling.
• Variance — second-moment.
• StdDev — sqrt of Variance.
• Indicators-Overview — full taxonomy.