How APT Constrains Factor-Model Alphas and Exposures

Summary

The document presents a return regression with a market factor and a zero-cost, market-neutral long-short factor, then asks which intercept and factor loadings are consistent with arbitrage pricing theory. It specifies assumptions about factor means, variances, and the residual variance, and reports a proposed answer: alphas should be zero for most assets, while the factor loadings can vary freely.

The author questions why this follows from APT, noting that the theory connects expected returns to factor exposures but does not make the restriction immediately clear from the regression alone. No derivation or answer is included, so the text frames a conceptual issue rather than resolving it. The key distinction for study is between restrictions on expected returns or pricing errors and restrictions on an asset’s sensitivity to priced factors; the document leaves the role of the market-neutral factor’s zero expected return for further explanation.

Key ideas

• The regression models excess returns using market and market-neutral long-short factors.
• The posed APT claim restricts intercepts for most assets while leaving factor loadings unrestricted.
• APT relates expected returns to exposure to priced sources of risk, which differs from directly constraining factor loadings.
• The document asks why the stated alpha restriction follows and does not supply a derivation.

Tags

# APT assumptions


# APT assumptions












Suppose an investor identifies an asset with return $R_i$ whose excess return, when regressed on the market portfolio with return $R_M$ and a (zero-cost) long-short portfolio with return $R_{LS}$ gives:

$$ R_i - R_f = \alpha_i + \beta_i (R_M - R_f) + \nu_i R_{LS} + \epsilon_i $$

The long-short portfolio is market neutral that is $R_{LS}$ is uncorrelated with the market. Further $E[R_{LS}] = 0$ and $\text{Var}[R_{LS}] = \sigma^2_{LS}$, $E[R_M] = \mu_M$, $\text{Var}[R_M] = \sigma^2_M$, and $\text{Var}[\epsilon_i] = \sigma^2_\epsilon$.

What conditions on $\alpha_i$, $\beta_i$, $\nu_i$ are required for the model to be consistent with the APT (arbitrage pricing theory)? The answer I was given is $\alpha_i = 0$ for most $i$, $\beta_i$ and $\nu_i$ unrestricted. However, I really don't see why. For me APT says just that if we have returns are modeled with factors then the expectation of the return satisfies some form.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.