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

CFA® Level I · Equities · 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.

Formulas

Arbitrage pricing theory

E(R) = R_f + b₁ × RP₁ + b₂ × RP₂ + …

E(R)
the expected return
R_f
the return with no factor risk
b₁
the sensitivity to factor 1
RP₁
the reward per unit of factor 1 risk
b₂
the sensitivity to factor 2
RP₂
the reward per unit of factor 2 risk

Each factor has its own sensitivity and its own reward. CAPM is the one-factor case.

A practice question

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

Multi-factor models are important in equity investing mainly because they:

  1. eliminate all firm-specific risk directlyWhy not: Diversification, not the choice of model, removes firm-specific risk.
  2. require no estimates of factor sensitivitiesWhy not: Every factor model needs sensitivities (betas) to each factor to be estimated.
  3. capture several sources of systematic risk - correct

A single market beta may miss systematic risks such as size, value or macroeconomic factors; multi-factor models explain returns and risk exposures more fully.

A harder one

Which statement about arbitrage pricing theory is most accurate?

  1. It uses the market portfolio as its only factorWhy not: That is the CAPM; APT allows several factors and does not need the market portfolio.
  2. It needs more restrictive assumptions than CAPMWhy not: APT is generally regarded as needing fewer assumptions, such as no specific investor preferences.
  3. It does not specify the number or identity of factors - correct

APT sets out the structure of expected returns but leaves the analyst to identify and estimate the relevant factors.

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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