Beta and Market-Relative Risk

Install and import#

npm install fintech-algorithms

import { betaAndMarketRelativeRisk } from "fintech-algorithms/foundations/financial-risk-and-performance-statistics/beta-and-market-relative-risk";

Signature#

betaAndMarketRelativeRisk(input)

Divides the sample covariance between a return series and its benchmark by the benchmark's sample variance.

Parameters#

Returns#

D00Output

beta is the covariance over the benchmark variance, marketVariance is the benchmark's sample variance, and covariance is the sample covariance of the two series.

Errors#

• When returns or benchmark is absent, empty, or holds a non-finite number — throws RangeError
• When returns and benchmark differ in length, or hold fewer than two observations — throws RangeError
• When frequency is zero or negative — throws RangeError
• When the benchmark is constant, so its sample variance is zero — throws RangeError
• When confidence is absent, or outside the range zero to one — the engine reads it for every topic from A04 onward, including those that never use it — throws RangeError

Complexity: time O(n log n), space O(n).

Worked example#

verified This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

Input#

{
  "returns": [0.01, -0.02, 0.015, -0.01, 0.03],
  "benchmark": [0.008, -0.01, 0.012, -0.006, 0.02],
  "frequency": 252,
  "target": 0,
  "confidence": 0.8,
  "riskFree": 0.0001,
  "weights": [0.6, 0.4],
  "covarianceMatrix": [
    [0.04, 0.01],
    [0.01, 0.09]
  ]
}

Call#

betaAndMarketRelativeRisk(input)

Returns#

object with 2 fields: beta, marketVariance

{
  "beta": 1.5903307888040712,
  "marketVariance": 0.0001572
}

Diagrams#

How it works#

This page states the contract — how to call it correctly. The article explains the concept: why it works, and where it breaks.

Read the article →

References#

• Expected Shortfall — Basel Framework MAR33
• Mutual Fund Performance — William F. Sharpe
• Measures of Scale — NIST/SEMATECH e-Handbook
• Historical-example decision

The rest of the Financial Risk and Performance Statistics family#

• Volatility and Annualized VolatilityTakes the sample standard deviation of a return series and rescales it to an annual horizon by the square root of the observati…
• Downside Deviation and Target ShortfallMeasures dispersion of returns below a target only, so upside swings do not count as risk.
• Drawdown and Maximum DrawdownCompounds a return series into a wealth path and tracks how far that path sits below its own running peak.
• Loss Distributions and Loss QuantilesFlips a return series into losses, sorts them, and reads off the quantile at the requested confidence by linear interpolation b…
• Value at Risk IntuitionReads historical value at risk straight off the sorted loss series and counts how many observed losses ran past it.
• Expected Shortfall IntuitionAverages the losses that sit at or beyond the value-at-risk threshold, describing the tail rather than only its edge.
• Sharpe, Sortino, and Information Ratio IntuitionComputes three reward-per-unit-of-risk ratios from the same series, each dividing an average by a different notion of risk.
• Active Return and Tracking ErrorSubtracts the benchmark from the portfolio period by period, then reports the average and the annualized volatility of that dif…
• Covariance Matrices, Portfolio Variance, and DiversificationCombines portfolio weights with a covariance matrix into portfolio variance, and isolates the share of it that comes from the o…