Explain what beta means and derive its value for one asset or for a whole portfolio.

FRM® Part I · Foundations of Risk Management · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) · FRM1.FR.5.f

The idea

Beta measures an asset's sensitivity to the market. It is the covariance of the asset's return with the market's return, divided by the variance of the market return: Beta = Cov(Ri, Rm) ÷ Var(Rm). Equivalently, beta is the correlation times the ratio of volatilities: Beta = ρ × σi ÷ σm. It is the slope of a regression of the asset's returns on the market's. The market has a beta of 1 and the risk-free asset 0, and a beta above 1 means the asset amplifies market moves. Portfolio beta is the value-weighted average of the asset betas, with cash entering at zero, so weights must be on the whole portfolio including any cash or borrowing. Beta also splits an asset's variance in two. In the market-model regression, whose residual is uncorrelated with the market, the systematic part is β² × σm² and the rest is idiosyncratic. The share of variance that the market explains is the regression R² = β² × σm² ÷ σi². Both are model-based, not identities. Beta is not correlation, so do not confuse the two, and do not divide covariance by the market's standard deviation.

Formulas

Beta from correlation

β = ρ × σi ÷ σm

ρ: the correlation between the asset's return and the market's (or the factor's)
σi: the volatility of the asset's return
σm: the volatility of the market (or factor) return

Beta is not correlation: it also scales by the volatility ratio. An efficient portfolio has ρ = 1, so its beta is σp ÷ σm.

Portfolio beta

βp = w1 × β1 + w2 × β2 + … + wn × βn

w1, w2, wn: each holding's weight in the whole portfolio, cash included
β1, β2, βn: each holding's beta, zero for cash

Weights must be on the whole portfolio including cash or borrowing, so cash pulls the beta down.

Share of variance explained by the market

R² = β² × σm² ÷ σi²

β: the asset's beta
σm: the market's volatility
σi: the asset's total volatility

Holds in the market-model regression whose residual is uncorrelated with the market. R² is the square of the correlation, not the correlation.

A practice question

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

A portfolio holds 75% in a market index fund (beta 1.0) and 25% in Treasury bills. Its beta is:

1.00Why not: Ignores the Treasury bill holding, as if the whole portfolio tracked the market.
0.25Why not: Swaps the weights, using the T-bill weight as the market exposure.
0.50Why not: Takes a simple average of the betas 1.0 and 0.0 instead of the value-weighted average.
0.75 - correct

Portfolio beta = 0.75 x 1.0 + 0.25 x 0.0 = 0.75. Cash and risk-free assets have a beta of zero.

A harder one

A stock has a volatility of 30% and a correlation of 0.70 with the market, whose volatility is 20%. Its beta is closest to:

1.05 - correct
1.50Why not: Uses the volatility ratio alone and ignores the 0.70 correlation.
0.47Why not: Inverts the volatility ratio, putting the market in the numerator.
0.21Why not: Multiplies correlation by the stock volatility without dividing by the market volatility.

Beta = rho x sigma_i / sigma_m = 0.70 x 30% / 20% = 1.05.

The hardest tier

A stock has a beta of 1.2 and a volatility of 25%. The market volatility is 16%. What proportion of the stock's variance is explained by the market?

77%Why not: Reports the correlation, 1.2 x 16% / 25%, instead of its square.
41%Why not: Gives the idiosyncratic share of variance rather than the systematic share.
59% - correct
92%Why not: Squares beta but not the volatilities, so the units no longer match variance.

Systematic variance = beta² x sigma_m² = 1.44 x 0.0256 = 0.0369; total variance = 0.0625. R² = 0.590. The correlation is 0.768, which is not the same thing as R².

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