Work out beta and explain what it means

CFA® Level I · Portfolio Construction · 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.

Beta from correlation and risk

β = ρ × σshare ÷ σmarket

ρ: the correlation between the share and the market
σshare: the standard deviation of the share's return
σmarket: the standard deviation of the market's return

The same beta as covariance over market variance. It shows that a share with a high correlation or a high volatility ratio has a high beta.

Beta of a portfolio

portfolio beta = w₁ × β₁ + w₂ × β₂ + …

w₁, w₂: the weight of each holding
β₁, β₂: the beta of each holding

Beta is a weighted average, so it adds up simply. A portfolio with a beta of 1.2 is expected to move 12% when the market moves 10%.

A practice question

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

What does a beta of 1.4 imply?

The asset earned a return 40% above the market's last yearWhy not: Beta is an expected sensitivity across many periods, not a single year's outperformance.
Its return tends to move about 1.4 times as much as the market's - correct
The asset's total volatility is 40% higher than the market'sWhy not: Total volatility also depends on specific risk, which beta does not capture.

Beta measures how strongly an asset's return reacts to moves in the whole market. Expected asset move = Beta x Market move. With a beta of 1.4, the asset tends to move about 1.4 times as much as the market. Example: if the market rises 5%, the asset is expected to rise 1.4 x 5% = 7%. If the market falls 5%, it is expected to fall 7%. Beta is a typical tendency over many periods, not last year's result. It also ignores risk specific to the asset, so it is not total volatility.

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?

0.60 - correct
0.27Why not: Inverts the volatility ratio, multiplying the correlation by 20/30 instead of 30/20.
1.50Why not: Ignores the correlation and uses only the volatility ratio 30 / 20. That would be the beta only if the correlation were 1.

Beta shows how much the asset moves with the market. Beta = Correlation x (Asset standard deviation / Market standard deviation). Standard deviation measures how widely returns swing. Correlation measures how closely the asset follows the market. Beta = 0.4 x (30% / 20%) = 0.4 x 1.5 = 0.60. Check with the other form, Beta = Covariance / Market variance: covariance = 0.4 x 0.30 x 0.20 = 0.024, market variance = 0.20 x 0.20 = 0.04, and 0.024 / 0.04 = 0.60. The asset swings a lot, but its weak link to the market keeps beta 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?

Adding more holdings to the portfolio to spread the riskWhy not: More names diversifies away specific risk. Systematic exposure, which beta measures, is largely unchanged.
Swapping some holdings for lower-beta securitiesWhy not: That works, but it requires selling holdings, which the manager wants to avoid.
Adding a short position in index futures as an overlay - correct

Beta shows how much a portfolio moves with the market. Portfolio beta is the weighted average of the betas of what it holds: Portfolio beta = Sum of (weight x beta). A short position has a negative weight. So selling index futures short (a contract that gains when the market falls) cuts beta without selling any holdings. Example: shorting an amount equal to 30% of the portfolio, using an index with beta 1, gives 1.1 - 0.3 x 1.0 = 0.8. Holding more securities does not help, because it only removes company-specific risk.

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