Beta

Rolling sensitivity of an asset to a benchmark. Beta of 1 means the two move together one-for-one; > 1 means the asset amplifies the benchmark; < 1 means it dampens. The slope from a rolling OLS regression of asset returns on benchmark returns.

Quick reference

Item	Value
Family	Price Statistics
Input type	(f64, f64) — (asset, benchmark) pair
Output type	f64
Output range	unbounded
Default parameters	period required (> 1)
Warmup period	period
Interpretation	Market-risk sensitivity

Formula

cov_ab = (1/n) · Σ a·b - ā·b̄
var_b  = (1/n) · Σ b² - b̄²
Beta   = cov_ab / var_b

Population formulas (not sample). Each update is O(1). See crates/wickra-core/src/indicators/beta.rs.

Parameters

Name	Type	Default	Constraint	Description
period	usize	none	> 1	Rolling window.

Inputs / Outputs

Indicator<Input = (f64, f64), Output = f64>. Python: Beta(period).batch(asset, benchmark) returns a 1-D np.ndarray with NaN warmup. Node: same.

Warmup

warmup_period() == period.

Edge cases

• Flat benchmark. var(b) = 0; indicator returns 0.0 rather than NaN.
• Negative Beta. Asset moves opposite to benchmark (e.g. gold vs equities sometimes).
• Reset. Clears running sums.

Examples

Rust

use wickra::{Beta, Indicator};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut b = Beta::new(50)?;
    for i in 0..100 {
        let bench = (f64::from(i) * 0.1).sin() * 0.01;
        let asset = bench * 1.2 + 0.0005;  // beta ~1.2
        let _ = b.update((asset, bench));
    }
    Ok(())
}

Python

import numpy as np
import wickra as ta

bench = np.sin(np.linspace(0, 10, 100)) * 0.01
asset = bench * 1.2 + 0.0005
b = ta.Beta(50)
print(b.batch(asset, bench)[-1])  # ~1.2

Node

const wickra = require('wickra');
const b = new wickra.Beta(50);
// feed asset, benchmark

Streaming

use wickra::{Beta, Indicator};

let mut b = Beta::new(252).unwrap();
let return_stream: Vec<(f64, f64)> = Vec::new(); // your stream of periodic returns
for (asset_return, bench_return) in return_stream {
    if let Some(v) = b.update((asset_return, bench_return)) {
        // v is 1-year Beta on daily returns
    }
}

Interpretation

• Beta > 1. Asset amplifies benchmark moves — high-beta growth stocks vs SPX.
• Beta < 1. Asset dampens benchmark — defensive sectors (utilities, staples).
• Beta ≈ 0. Asset moves independently of benchmark — managed futures, market-neutral.
• Beta < 0. Inverse correlation — gold and long-duration bonds vs equities in some regimes.

Common pitfalls

• Period choice. Short windows make Beta noisy; long windows hide regime changes. Industry convention: 60 months for fund reporting, 252 days for active management.
• Benchmark dependence. Beta is benchmark-specific. AAPL has different Beta vs SPX, vs NDX, vs sector ETF.
• Asymmetric Beta. Asset's response to up moves and down moves can differ — single Beta hides this. Look at up-beta / down-beta separately for skew.

References

• William F. Sharpe, Capital Asset Prices, Journal of Finance, 1964 — CAPM and Beta.

See also

• Alpha — CAPM's other half.
• PearsonCorrelation — correlation cousin (Beta = correlation · σ_a / σ_b).
• TreynorRatio — uses Beta in denominator.
• Indicators-Overview — full taxonomy.