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Characterise multifactor models and set them side by side with the CAPM, pointing out similarities and differences.

FRM® Part II · Risk Management and Investment Management · Factor Theory · FRM2.IM.1.d

The idea

A multifactor model builds expected return from several priced factors. Let βk be the asset's exposure to factor k and λk the risk premium of that factor. The expected return, written ER, is the risk-free rate plus the sum over factors of each exposure times its premium: ER = rf + Σ βk × λk. Factors may be macroeconomic (growth, inflation), statistical, or characteristic-based such as size, value and momentum. The CAPM is the special case with one factor, the market. Both approaches are linear, pay only for systematic exposure and leave idiosyncratic risk unpriced. The CAPM is an equilibrium result for mean-variance investors and names its factor. The arbitrage pricing theory rests on no-arbitrage with well-diversified portfolios and does not say which factors to use. Exposures can be negative and then subtract from expected return. Alpha under the CAPM can disappear once omitted factors are added.

Formulas

Multifactor expected return

E(R) = Rf + β1 × λ1 + β2 × λ2 + … + βk × λk

Rf
the risk-free rate
β1, β2, βk
the asset's beta to each factor
λ1, λ2, λk
each factor's risk premium, the expected return of its pure factor portfolio minus Rf

Linear in exposures and pays only for systematic risk; the CAPM is the one-factor case. Keep the sign of a negative exposure, which subtracts from the expected return.

A practice question

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

Which statement best describes a multifactor model of expected returns?

  1. Expected return depends on a single exposure to the market, multiplied by the market risk premium and added to the risk-free rateWhy not: This describes the one-factor CAPM, not a multifactor model.
  2. Expected excess return is each factor exposure times that factor's premium, summed over factors - correct
  3. Expected return is the average of the factors' past returns, whatever the asset's exposuresWhy not: Ignores asset-specific exposures, which determine how much of each premium the asset earns.
  4. Total volatility is the sum of the standalone volatilities of the factorsWhy not: Mixes up risk aggregation with return determination; volatilities do not simply add.

E(Ri) = rf + sum of beta_ik x lambda_k. The CAPM is the one-factor special case.

A harder one

Which statement correctly distinguishes the arbitrage pricing theory (APT) from the CAPM?

  1. APT rests on no-arbitrage and leaves the factors open; CAPM rests on equilibrium and names the market - correct
  2. The CAPM requires no market proxy, while the APT needs the market portfolio to be observable and mean-variance efficientWhy not: Reversed: the CAPM depends on the market portfolio, which the APT does not need.
  3. The APT predicts a reward for bearing idiosyncratic risk in a well-diversified portfolio, whereas the CAPM does notWhy not: Neither theory rewards idiosyncratic risk; it is diversified away.
  4. The APT is always a one-factor model, whereas the CAPM allows several factorsWhy not: The APT allows several factors; the CAPM is a single-factor model.

The APT derives a linear factor relation from absence of arbitrage among well-diversified portfolios, leaving factor choice open. The CAPM is an equilibrium model with the market as the one factor.

The hardest tier

Risk-free rate 2%. Portfolio P is 60% asset X and 40% asset Y. Asset X has a market exposure of 1.2 and a value exposure of 0.5. Asset Y has a market exposure of 0.7 and a value exposure of -0.2. The market premium is 5% and the value premium is 3%. The expected return on P is closest to:

  1. 12.40%Why not: Adds the asset exposures instead of weighting them.
  2. 6.74%Why not: Swaps the weights, using 40% for X and 60% for Y.
  3. 7.66% - correct
  4. 5.66%Why not: Leaves out the risk-free rate.

Market exposure 0.6 x 1.2 + 0.4 x 0.7 = 1.00; value exposure 0.6 x 0.5 + 0.4 x (-0.2) = 0.22. Expected return = 2 + 1.00 x 5 + 0.22 x 3 = 7.66%.

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