Questions: Arbitrage Pricing Theory (APT) and Factor Models

Question 1

According to APT, why must every unit of systematic (factor) risk be compensated by a positive expected return premium in equilibrium?

• A. Because investors are risk-averse and demand compensation for all risk, systematic and idiosyncratic alike
• B. Because if uncompensated systematic risk existed, investors could construct a zero-investment, zero-risk portfolio with positive expected return — an arbitrage opportunity that cannot persist
• C. Because the Capital Asset Pricing Model establishes that beta determines expected returns, and APT extends this result to multiple betas
• D. Because empirical regressions consistently show that factor loadings predict future returns, confirming the theoretical relationship

Answer: (B)

APT's foundation is the no-arbitrage condition, not utility theory or risk aversion assumptions. The argument is: if a well-diversified portfolio had zero factor exposure (no systematic risk) but positive expected return, it would require no investment and bear no risk while earning positive profit — a free lunch. Rational investors would demand arbitrarily large positions, which is impossible in equilibrium. Therefore, any systematic risk exposure must be exactly compensated. This derivation requires no assumptions about investor preferences beyond the desire for more return.

Question 2

A mutual fund shows positive alpha (α > 0) when evaluated using the Fama-French three-factor model. What does this most likely indicate?

• A. The fund has more market risk than the model accounts for, so the alpha is actually a mismeasured beta
• B. The fund earns return in excess of what its exposure to market, size, and value factors predicts — suggesting either genuine skill or a missing risk factor
• C. The fund is risk-free and earns exactly the risk-free rate plus a premium for its factor exposures
• D. The fund has negative exposure to one of the three factors, which artificially inflates its alpha calculation

Answer: (B)

Alpha in a factor model measures return unexplained by the model's risk factors. Positive alpha means the fund earns more than its systematic exposures (market, size, value) warrant. This can mean genuine manager skill (true alpha) or that the three-factor model is incomplete and the fund actually earns a premium for exposure to a fourth risk factor the model omits. Distinguishing these interpretations requires additional analysis. The APT framework treats alpha as the benchmark: active management is only valuable if it produces true alpha after controlling for factor exposures.

Question 3

APT specifies a precise, theoretically derived list of macroeconomic factors (such as inflation, GDP growth, and interest rates) that is expected to be used to price assets correctly.

• T. True
• F. False

Answer: False

This is a key limitation of APT relative to CAPM. APT's theory is agnostic about which factors to use — it says that whatever systematic risk factors drive returns must be compensated, but it does not identify what those factors are. Factor identification is purely empirical, which means practitioners must search for co-movements in returns that command a premium. This flexibility allows APT to fit real-world complexity but also opens the door to data mining: factors can be added post-hoc until almost any return pattern is 'explained.'

Question 4

CAPM can be understood as a special case of APT in which there is only one systematic risk factor — the market portfolio return minus the risk-free rate.

• T. True
• F. False

Answer: True

APT's general form E[rᵢ] = rₓ + β₁λ₁ + β₂λ₂ + … + βₖλₖ reduces to CAPM's E[rᵢ] = rₓ + βᵢ(E[rₘ] − rₓ) when there is exactly one factor (the market premium) and one loading (the market beta). Both models share the no-arbitrage foundation, and CAPM's single-factor structure can be derived from APT with one additional assumption: that all investors hold the same mean-variance efficient portfolio. APT does not make CAPM obsolete — it generalizes it.

Question 5

What is the logical foundation of APT, and why does the theory not require assumptions about the shape of investor utility functions?

Model answer: APT is founded entirely on the no-arbitrage condition: if a risk-free, zero-investment portfolio with positive expected return could be constructed, rational investors of any preference type would demand infinite amounts of it, which is impossible. Therefore such opportunities cannot exist in equilibrium, and expected returns must be proportional to factor exposures. No specific form of utility (quadratic, power, log) is needed because the argument applies to any investor who prefers more wealth to less — a much weaker assumption than CAPM's specific mean-variance preferences.

CAPM requires investors to have quadratic utility or normally distributed returns to derive the mean-variance frontier from which the market portfolio emerges. APT bypasses this entirely by using no-arbitrage, which is a market-level condition rather than an individual-level one. This makes APT's logic more general and its assumptions more plausible, at the cost of leaving factor identification to empirical work rather than theory.