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Explain multi-factor models and arbitrage pricing theory as used for equities

CFA® Level I · Equity Investments · The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models · EQ.18.b

The idea

Arbitrage pricing theory generalises the CAPM: expected return is the risk-free rate plus the sum of each factor sensitivity multiplied by that factor's risk premium, E(R) = Rf + b1 x RP1 + b2 x RP2 and so on. It assumes returns follow a factor model and that arbitrage removes mispricing, but it does not say which factors matter. Multi-factor models fill that gap in practice. Macroeconomic models use surprises in variables such as GDP growth, inflation and interest rates. Fundamental models use company characteristics such as size, value and momentum; statistical models extract factors from return data. They are used to estimate required returns, explain performance and control a portfolio's exposure to particular risks.

A practice question

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

Arbitrage pricing theory derives expected returns from:

  1. mean-variance efficiency of the marketWhy not: That is the foundation of the CAPM, which APT does not require.
  2. the absence of arbitrage opportunities — correct
  3. the firm's historical dividend growthWhy not: Dividend growth belongs to discounted cash flow models, not to APT.

APT assumes that returns follow a factor model and that well-diversified portfolios cannot earn riskless profits, which pins down factor risk premiums.

A harder one

Under APT, the risk-free rate is 3%. A stock has a sensitivity of 1.2 to factor 1 (risk premium 4%) and 0.5 to factor 2 (risk premium 2%). Its expected return is:

  1. 8.8% — correct
  2. 5.8%Why not: Adds up the factor risk premiums correctly but omits the risk-free rate.
  3. 13.2%Why not: Multiplies the sum of the sensitivities (1.7) by the sum of the premiums (6%) and adds the risk-free rate, instead of pairing each sensitivity with its own premium.

Expected return = 3% + 1.2 × 4% + 0.5 × 2% = 3% + 4.8% + 1.0% = 8.8%.

The hardest tier

A single-factor APT applies with a risk-free rate of 4% and a factor risk premium of 4%. Well-diversified portfolio A has a factor sensitivity of 1.5 and an expected return of 11%. An investor should:

  1. buy A and short an equal-risk replica — correct
  2. short A and buy an equal-risk replicaWhy not: Reverses the trade: A's 11% exceeds the 10% required for its risk, so A is underpriced.
  3. take no position, as A is fairly pricedWhy not: Miscalculates the APT return; 4% + 1.5 × 4% = 10%, so A offers 1% more than its risk warrants.

APT expected return for A = 4% + 1.5 × 4% = 10%. A offers 11%, so buying A and shorting a portfolio with the same sensitivity locks in about 1% risk-free.

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