Multifactor Models, the APT and Active Risk Decomposition

Learning objectives

You can:

1. State the three assumptions of arbitrage pricing theory (returns follow a k-factor process, markets are sufficiently competitive and populated to hold many well-diversified portfolios, and no arbitrage is available among them) and explain why the theory delivers a pricing relation without a market portfolio, a utility function or a normal distribution.
2. Solve for the factor risk premiums implied by well-diversified portfolios with known exposures, identify a portfolio that does not lie on the resulting plane, and construct the arbitrage that closes it: a replicating combination with matched exposures, zero net investment and a positive expected return, plus the price change that ends the opportunity.
3. Decompose a realised return under a macroeconomic factor model into its expected component, its factor-surprise component and its firm-specific component, and explain why the factors must be surprises rather than levels.
4. Estimate a fundamental factor model by cross-sectional regression on a panel of standardised attributes, read off each name's fitted and idiosyncratic return, and verify the two orthogonality conditions the fit must satisfy.
5. Split active return into factor and selection components that reconcile exactly, and split active risk into factor and idiosyncratic parts using a factor covariance matrix, reporting each factor's share and cross-checking the model's tracking error against the realised one.
6. Compute both information ratios a factor model supports, the naive ratio of active return to active risk and the appraisal ratio of selection return to specific risk, and read a manager whose first ratio is excellent and whose second is zero.
7. Build a pure factor portfolio by solving a linear system for weights carrying unit exposure to one factor and zero to the others, use its zero-investment version to hedge an unwanted exposure, and find the risk-minimising hedge, which is generally not a full hedge.
8. Read a statistical factor model honestly: compute the variance share of the leading principal components on a small correlation matrix, and state what is lost when factors are extracted rather than named.
9. Apply the multiple-testing arithmetic to a factor claim: expected false discoveries, the Bonferroni critical value, the expected maximum t-statistic of N pure-noise trials, and the post-publication haircut.
10. (Productivity objective: R10.) Reproduce every estimation above in Python, name the two library defaults that change the answer, and use an assistant to build and cross-check factor code while keeping the by-hand estimation that the gate rewards.

The duality, stated once (R10). The gated skill is the estimation done by hand: a two-equation solve for factor premiums, a cross-sectional regression whose residuals sum to zero, a quadratic form that turns exposures into tracking error. The productivity payoff is running the same thing across a thousand funds in twenty lines. The risk that sits between them is specific to this material and worth naming now. An assistant will fit any factor model you ask it to fit, on any data, and report exposures and an alpha that look exactly like the output of a correct model, because the arithmetic almost always succeeds. What fails silently is the specification: a factor that does not belong, a benchmark whose own exposures were never measured, a sample chosen after the result was known. There is no error message for a well-fitted wrong model, and the reconciliation identities below are the only cheap defence.

Prerequisites & connections

Builds on. M3.02 supplies the single-factor machinery used here without re-derivation: expected return and variance, covariance, beta as covariance-with-the-market over market variance, the security market line, and the honest scorecard on where CAPM fails empirically. That scorecard is the motivation for everything that follows, and it is not repeated. QM1.02 supplies the regression discipline: ordinary least squares, standard errors, the t-statistic, R-squared and the diagnostics, all of which are applied here rather than taught. QM1.04 supplies the inference half, and in particular the multiple-testing correction that the factor-zoo section turns on published factor claims. M9.02 supplies the factor menu (value, quality, momentum, size, low volatility) with the evidence and the caveat for each, and the argument that a bottom-up investor should know her implicit tilts. AA1.01 supplies the covariance matrix and the estimation-error problem, which reappears here as the reason factor models exist at all. AA1.03 supplies the Brinson decomposition, the information ratio in its portfolio sense, and the reconciliation habit.