Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT) is a multi-factor asset pricing framework: if there are no persistent arbitrage opportunities, then an asset’s expected return is determined by its exposures to a set of systematic risk factors.

A standard representation

A common starting point is a factor model for returns:

[ R_i = a_i + \beta_i’ f + \varepsilon_i, ]

where (f) is a vector of factor shocks, (\beta_i) are factor loadings (exposures), and (\varepsilon_i) is idiosyncratic risk.

Under no-arbitrage conditions, the cross section of expected returns satisfies a linear restriction:

[ E(R_i) = R_f + \beta_{i1}\lambda_1 + \beta_{i2}\lambda_2 + \cdots + \beta_{ik}\lambda_k, ]

where (R_f) is the risk-free rate and (\lambda_j) is the risk premium per unit exposure to factor (j).

Intuition (why no-arbitrage implies linear pricing)

If two portfolios have the same factor exposures, they should have the same expected return. Otherwise, it is possible to construct a long-short position that:

has (approximately) zero net factor exposure,
requires little net investment,
but earns a systematic excess return.

Arbitrage pressure would then push prices (and expected returns) back toward the APT restriction.

APT vs CAPM

CAPM is a one-factor special case (market beta).
APT allows multiple factors and does not require identifying a single “market portfolio,” but it also does not uniquely specify which factors matter.

Practical use (what can go wrong)

Empirical APT work depends on:

choosing factors (macro variables, statistical components, or style factors),
estimating betas and risk premia,
assuming the factor structure is stable over time.

If factors are misspecified or unstable, the model’s pricing implications can fail.

Knowledge Check

### APT is built around which core pricing restriction? - [x] No-arbitrage implies expected returns are linear in factor exposures - [ ] Equilibrium prices always equal marginal costs - [ ] Inflation is always monetary - [ ] Markets always clear instantly > **Explanation:** Under no-arbitrage, assets with the same systematic risk exposures should offer the same expected return, implying a linear factor-pricing form. ### In the APT expected-return equation, what do the \(\beta_{ij}\) terms represent? - [x] Factor loadings (exposures) of asset \(i\) to factor \(j\) - [ ] The inflation rate - [ ] The risk-free rate - [ ] Firm accounting profit margins > **Explanation:** Betas measure how sensitive the asset’s return is to each systematic factor. ### How does APT differ from CAPM in its basic structure? - [x] APT allows multiple factors; CAPM is a one-factor model (market beta) - [ ] APT assumes risk neutrality; CAPM assumes risk aversion - [ ] APT requires a balanced government budget; CAPM does not - [ ] APT applies only to bonds; CAPM applies only to stocks > **Explanation:** CAPM is a special case with one systematic factor, while APT is a more flexible multi-factor no-arbitrage framework.