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CFA® Level II · Portfolio Construction · free lesson
Using Multifactor Models
Multifactor models explain returns and risk with a handful of common drivers, and they sit behind most institutional risk systems, performance reports and factor products. This module covers arbitrage pricing theory and the arbitrage argument behind it, the three families of factor models, and how practitioners use them to attribute returns and risk and to build portfolios. Level II item sets combine an APT calculation with an active risk decomposition, so the emphasis is on getting premiums, active exposures and variance arithmetic right.
What you will learn
1. Explain the assumptions and pricing equation of arbitrage pricing theory, calculate expected returns from factor sensitivities and premiums, and identify and exploit an arbitrage opportunity.
- APT assumes a factor model, enough assets to diversify specific risk, and no persistent arbitrage among well-diversified portfolios.
- Expected return = risk-free rate + sum of (factor sensitivity x factor risk premium).
- A factor risk premium is the expected return on the pure factor portfolio minus the risk-free rate.
- Arbitrage: buy the portfolio with the higher expected return and short a replicating mix with identical sensitivities; net investment and factor exposure are zero.
- APT is more general than the CAPM but does not identify the factors or their premiums.
2. Compare macroeconomic, fundamental and statistical factor models, including how each defines factors and estimates sensitivities.
- Macroeconomic models: factors are surprises (actual minus expected); sensitivities come from time-series regressions; the intercept is the expected return.
- Fundamental models: sensitivities are standardized attributes known in advance; factor returns come from cross-sectional regressions.
- Standardized sensitivity = (company value - average) / standard deviation; industry factors use 0/1 dummies.
- Statistical models extract factors from return data; good fit, poor interpretability.
- Fama-French style factors are returns on long-short attribute portfolios, used with time-series regressions.
3. Use factor models to attribute portfolio returns and risk to factor exposures and security selection, and to build tracking, pure factor and tilted portfolios.
- Factor contribution to active return = active exposure (portfolio minus benchmark sensitivity) x factor return.
- Security selection is the part of active return the factors do not explain.
- Risk attribution tests whether the sources of risk match the manager's claimed skill.
- Tracking portfolios copy benchmark factor exposures; pure factor portfolios isolate one factor.
- Shorting a pure factor portfolio hedges that factor without changing other exposures.
4. Decompose active risk into active factor risk and active specific risk, and interpret tracking risk and the information ratio.
- Active risk (tracking risk) = standard deviation of portfolio minus benchmark returns.
- Active risk squared = active factor risk + active specific risk; add variances, never standard deviations.
- Active factor risk comes from style and industry tilts; active specific risk from individual stock bets.
- Information ratio = mean active return / active risk; it measures consistency of value added.
- A self-described stock picker should show a high share of active specific risk.
Module at a glance
- APT: expected return = risk-free rate + sum of factor sensitivities x factor risk premiums.
- A factor premium is the pure factor portfolio's expected return minus the risk-free rate.
- Arbitrage: buy the portfolio with the higher return, short a mix with identical sensitivities; zero investment, zero factor risk.
- Macro models use surprises and time-series regressions; fundamental models use standardized attributes and cross-sectional regressions; statistical models let the data pick the factors.
- Factor contribution to active return = (portfolio - benchmark sensitivity) x factor return; the residual is security selection.
- Active risk squared = active factor risk + active specific risk; shares are computed on variances.
- Information ratio = mean active return / active risk; check that a manager's risk sources match its stated skill.
This free module has 4 lessons, 14 practice questions with full explanations.