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Work out beta and explain what it means

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part II · PM.4.c

The idea

Beta measures an asset's sensitivity to market movements: the covariance of its return with the market's, divided by the market's variance. Equivalently it is the correlation with the market times the ratio of the two standard deviations. A beta of one means the asset has moved with the market on average, above one means amplified moves and below one means damped ones. A portfolio's beta is the weighted average of its holdings' betas, which is what makes beta the practical tool for managing market exposure at the portfolio level.

Formulas

Beta

β = Cov ÷ market variance

Cov
the covariance of the asset's return with the market's
market variance
the variance of the market's return

How much of the market's move the asset takes. The market has a beta of one by construction, and a beta above one means the asset amplifies the market rather than that it is more volatile in any absolute sense.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does a beta of 1.4 imply?

  1. The asset returned 40% more than the market last yearWhy not: Beta is an expected sensitivity across many periods, not a single period's realised return.
  2. The asset has historically moved about 1.4 times as much as the market — correct
  3. The asset has 40% more total volatility than the marketWhy not: Total volatility also depends on the specific risk, which beta deliberately excludes.

Beta describes sensitivity to market moves, not realised outperformance and not total volatility, which also includes the asset's specific risk.

A harder one

An asset has a standard deviation of 30%, the market's is 20%, and their correlation is 0.4. What is the asset's beta?

  1. 0.60 — correct
  2. 0.27Why not: Inverts the volatility ratio, multiplying the correlation by 20/30 instead of 30/20.
  3. 1.20Why not: Divides the covariance by the market's standard deviation rather than its variance, which leaves the result a factor of 0.20 too large.

Beta is 0.4 x (0.30 / 0.20) = 0.60. Despite being half again as volatile as the market, its weak correlation makes its systematic exposure modest.

The hardest tier

A manager wants to reduce a portfolio's beta from 1.1 to 0.8 without selling any holdings. What does the additivity of beta allow?

  1. Nothing — beta can only be changed by selling securitiesWhy not: Because portfolio beta is a weighted average, an overlay position changes it without touching the underlying holdings.
  2. Reducing beta by increasing the number of holdingsWhy not: More names diversifies away specific risk. Systematic exposure is unchanged by how many securities carry it.
  3. Adding a short index futures overlay — correct

Portfolio beta is the weighted average of its components' betas, so a negative-beta overlay offsets part of the existing exposure. This is the standard mechanism for adjusting market exposure quickly.

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