Glossary

Beta (Finance) — Definition, Formula, and CAPM Context

Beta measures a portfolio's systematic exposure to a benchmark. Definition, regression formula, worked example, and why beta is benchmark- and window-dependent.

By NakedPnL Research·May 7, 2026·4 min read

TL;DR

Beta is the slope coefficient from a regression of asset excess returns on benchmark excess returns.
It captures only the systematic, benchmark-correlated component of risk.
Beta is benchmark-dependent and window-dependent — neither is canonical.

Definition

In the Capital Asset Pricing Model (CAPM), beta (β) measures the sensitivity of a portfolio's excess return to the excess return of a chosen benchmark. A beta of 1.0 means the portfolio moves in lockstep with the benchmark; 0.5 means it moves half as much; 2.0 means it amplifies benchmark moves by a factor of two. A negative beta indicates an inverse relationship.

Formula

beta = Cov(R_p - R_f, R_m - R_f) / Var(R_m - R_f)

or equivalently, the slope coefficient in the regression:
  (R_p - R_f) = alpha + beta * (R_m - R_f) + epsilon

where:
  R_p = portfolio return
  R_m = benchmark (market) return
  R_f = risk-free rate

Beta is conventionally estimated by ordinary least squares over a fixed lookback window — 60 days for high-frequency analysis, 36 to 60 months for traditional equity research.

Worked example

A daily-return series is regressed against a benchmark over 252 trading days. The covariance of excess portfolio returns with excess benchmark returns is 0.00018; the variance of excess benchmark returns is 0.00012. Beta is 0.00018 / 0.00012 = 1.5. The portfolio is 50% more sensitive than the benchmark on average over this window. Re-estimate beta over a different 252-day window or against a different index, and the value can shift materially — beta is a sample statistic, not a property of the underlying strategy.

Why NakedPnL doesn't display this on profile pages

Beta is only meaningful relative to a specified benchmark and a specified window. Choosing a benchmark on a trader's behalf is editorially fraught — a Polymarket trader's beta against the S&P 500 is essentially noise; a Bitcoin perpetuals trader's beta against gold is similarly uninformative. There is no canonical benchmark for the cross-asset population NakedPnL hosts. Displaying one beta on a profile would force a hidden modelling choice that allocators are better positioned to make themselves. NakedPnL therefore restricts profile-level metrics to TWR, total PnL, and trade count, and exposes the daily-return series via the API for any factor regressions an external user wants to run.

Alpha — the regression intercept from the same CAPM regression
Capital Asset Pricing Model (CAPM)
R-squared — share of variance explained by the regression
Tracking error

Frequently asked questions

What does a beta of zero mean?

Statistically, that the portfolio's returns have no linear relationship with the chosen benchmark over the regression window. It does not mean the portfolio has no risk — it has no benchmark-correlated risk. Idiosyncratic risk can still be substantial.

Can beta be greater than 1 for a fund using no leverage?

Yes. Beta measures co-movement, not gross exposure. A fund concentrated in stocks more cyclical than the benchmark can show beta above 1.0 with no borrowed capital.

How stable is beta over time?

Empirically, not very. Rolling-window beta estimates for individual portfolios drift considerably. This is one reason allocators rerun the regression frequently rather than treating any single beta number as a fixed parameter.

References