How to calculate and interpret beta

How is beta calculated?

Beta measures how strongly an asset’s return moves with the market’s return. It is the systematic risk of the asset, expressed per unit of market risk. The market’s variance is the yardstick, so beta is the asset’s covariance with the market divided by the market’s variance.

$$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2}$$

Covariance equals correlation times both standard deviations, so one σ_m cancels and beta can be written from correlation instead.

$$\text{Cov}(R_i, R_m) = \rho_{i,m}\,\sigma_i\,\sigma_m$$

$$\beta_i = \rho_{i,m}\,\frac{\sigma_i}{\sigma_m}$$

What does a beta mean?

A beta of 1.3 means that, on average, when the market’s return is 1 percentage point above normal, this asset’s return is 1.3 points above normal. The market portfolio has a beta of 1 and the risk-free asset has a beta of 0, because T-bills do not move with the market.

Beta of an assetsensitivity of its return to the market return

• Above 1: aggressive Use it amplifies market moves; required return above the market’s
• Equal to 1: moves with the market Memorize the market portfolio itself
• Between 0 and 1: defensive Use it moves less than the market
• Zero: no market sensitivity Memorize the risk-free asset; capital asset pricing model (CAPM) return equals Rf
• Negative: moves against the market Use it rare, for example some hedges; CAPM return below Rf

How is beta estimated in practice?

Regress the asset’s returns on the market’s returns. The fitted line is the security characteristic line , and its slope is beta. A common choice is 60 monthly returns against a broad index.

$$R_i - R_f = \alpha_i + \beta_i\,(R_m - R_f) + e_i$$

Change the correlation and the two standard deviations and the cloud regenerates. Beta is the slope of the fitted line, not the width of the scatter: a noisy stock can still have a beta of 1.

The estimate depends on three choices: which index stands in for the market, how often returns are measured (daily, weekly or monthly) and how long the period is. Different choices give different betas for the same stock.

How do you find a portfolio’s beta?

A portfolio’s beta is the weighted average of its holdings’ betas, because covariance is additive.

$$\beta_p = \sum_{i=1}^{n} w_i\,\beta_i$$

Go deeper Adjusted beta, and why betas drift toward 1

Betas estimated from history tend to move toward 1 over time, as companies diversify and mature. Many data providers publish an adjusted beta that blends the raw estimate with 1.

$$\beta_{adj} = \tfrac{2}{3}\,\beta_{raw} + \tfrac{1}{3}$$

For a raw beta of 1.8 the adjusted figure is (2 × 1.8 + 1) ÷ 3 = 1.53.

Adjusted beta

TI BA II Plus HP 12C

1. 1.8×2+1=÷3= 1.5333

1. 1.8ENTER2×1+3÷ 1.5333

ENTER enter (on the HP, puts the number on the stack)

Because beta is a regression slope, it comes with a standard error, as in the regression module of Quantitative Methods. A company’s beta also depends on how much debt it carries, a point that returns in Corporate Issuers when betas are adjusted for leverage.

Exam tip Covariance over variance, or correlation times the ratio of standard deviations. If the question gives ρ and two σ values, use the second form; do not square anything.