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Explain models that generate returns, the market model among them

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part II · PM.8.b

The idea

A return-generating model estimates an asset's expected return from one or more factors. The simplest is the market model, a single-index model: Ri = alpha i + beta i x Rm + ei. The beta term is the systematic part of the return, the alpha is the intercept, and the residual captures firm-specific news with an expected value of zero. Because residuals of different firms are assumed uncorrelated, total variance splits into beta squared times market variance plus residual variance, and the covariance between two assets reduces to beta i times beta j times market variance. Beta itself can be estimated as the correlation times the asset's standard deviation divided by the market's. Multifactor models add further factors: macroeconomic models use surprises in variables such as GDP growth or inflation, while fundamental models use company characteristics, as in the Fama-French model's size and value factors alongside the market.

A practice question

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

In the market model, Ri = alpha i + beta i x Rm + ei, which term captures the systematic component of the return?

  1. beta i x Rm — correct
  2. eiWhy not: The residual is the firm-specific part of the return, which is expected to be zero and diversifiable.
  3. alpha iWhy not: The intercept is the return not explained by the market; it does not vary with market moves.

Only the beta term moves with the market return, so it is the systematic, non-diversifiable part of the asset's return.

A harder one

Under the market model, a stock has an alpha of 0.5% and a beta of 1.2. If the market return is 8%, what is the stock's predicted return?

  1. 9.6%Why not: Applies beta to the market return but leaves off the 0.5% intercept.
  2. 8.5%Why not: Adds alpha to the market return without scaling the market by beta.
  3. 10.1% — correct

Predicted return = 0.5% + 1.2 x 8% = 0.5% + 9.6% = 10.1%. The residual has an expected value of zero.

The hardest tier

Under the market model, a stock has a beta of 0.8 and a standard deviation of 25%; the market's standard deviation is 20%. What proportion of the stock's total variance is systematic?

  1. 64.0%Why not: This is beta x market SD / stock SD, i.e. the correlation of 0.64. The systematic share is its square.
  2. 41.0% — correct
  3. 59.0%Why not: This is the share of variance that is firm-specific, the complement of the answer.

Systematic variance = 0.8^2 x 0.20^2 = 0.0256. Total variance = 0.25^2 = 0.0625. Share = 0.0256 / 0.0625 = 40.96%, about 41.0%.

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