CFA Level II Exam · Using Multifactor Models

Types of Multifactor Models for CFA Level II

Updated 7 October 2026 · Fact-checked

A multifactor model explains an asset's return using several risk factors. Macroeconomic models use surprises in economic variables, fundamental models use company attributes such as size or book-to-market, and statistical models extract factors from return data. To solve questions, identify the model type, then apply the sensitivities.

Understand Types of Multifactor Models

A single-factor model, like the CAPM, says an asset's return depends on one thing: the market. A multifactor model says returns depend on several systematic factors. Each asset has a factor sensitivity (also called a factor beta or loading) that tells you how much its return moves when a factor moves by one unit. Whatever is not explained by the factors is the asset-specific (idiosyncratic) return, shown as the error term ε.

There are three types, and the exam tests whether you can tell them apart.

A macroeconomic factor model uses surprises in macroeconomic variables as the factors. Examples are surprises in GDP growth, inflation, or interest rates. The key word is surprise: the actual value minus the expected value. The expected return a_i already includes the expected part of the economy. Only the unexpected part moves returns. The factor surprise has a mean of zero. The sensitivities are found by time-series regression of asset returns on the surprises.

A fundamental factor model uses attributes of the company itself as factors. Examples are book-to-market ratio, market capitalization, earnings yield, leverage, and industry membership. The sensitivities are the standardized attributes (for example, a standardized book-to-market score), and the factor returns are estimated by cross-sectional regression across many stocks at one point in time. Fundamental models are often used by practitioners for portfolio construction and risk control.

A statistical factor model uses statistical methods, mainly factor analysis or principal components analysis, to find factors that best explain the covariances or variances of returns in the data. The factors are not labelled in advance. They are portfolios of assets, and their economic meaning can be hard to interpret. Principal components models explain as much total variance as possible with a small number of components.

The well-known models fit this frame. The Fama-French three-factor model uses market excess return, size (SMB) and value (HML). Carhart adds momentum (WML or UMD). Fama-French and Carhart are multifactor models whose factors are returns on long-short portfolios formed on company attributes. They resemble fundamental models but are not described as a separate type. The Fama-French models are often used to explain expected return.

Key formulas to remember

Macroeconomic factor model: R_i = a_i + b_i1 × F1 + b_i2 × F2 + … + b_ik × Fk + ε_i; F are surprises in macro variables (actual − expected). a_i is the expected return. b are sensitivities. ε_i is the asset-specific return.
Factor surprise: Surprise = Actual value − Expected value; Only the surprise enters the model. The expected part is already in a_i.
Fundamental factor model: R_i = a_i + b_i1 × F1 + … + b_ik × Fk + ε_i; Same form, but the sensitivities b_i are the standardized attributes of the company (for example standardized book-to-market) and the factor returns F are estimated by cross-sectional regression.
Fama-French three-factor model: R_i − R_f = a_i + b_mkt × (R_m − R_f) + b_SMB × SMB + b_HML × HML + ε_i; SMB is small minus big. HML is high minus low book-to-market. The factors are returns on long-short portfolios formed on company attributes.
Carhart four-factor model: Fama-French three factors + b_WML × WML; WML is winners minus losers, the momentum factor.
APT / factor risk premium expected return: E(R_i) = R_f + b_i1 × λ1 + … + b_ik × λk; λ is the risk premium of each factor and b are the sensitivities. Use it when the question gives the risk-free rate and factor risk premiums. Do not confuse it with the macro model above, where a_i is the expected return and F are surprises.

How to solve Types of Multifactor Models questions

Most questions ask you to classify a model, interpret a coefficient, or compute a return. Work through the vignette in this order.

1Read the exhibit title and find what the factors are. Economic variables or surprises point to a macroeconomic model. Company attributes point to a fundamental model. Unlabelled components point to a statistical model.
2Check whether inputs are actual values or surprises. If the question gives actual and expected values, compute surprise = actual − expected first.
3Identify the intercept. In a macro model it is the expected return when all surprises are zero.
4Multiply each sensitivity by its factor value, keeping signs. Add the products to the intercept.
5Add the asset-specific term ε only if the question gives it. Otherwise state the result as the return explained by the factors.
6Check the answer: unexpected good news with a positive beta must raise return above the expected level.
7If the question is conceptual, match the statement to the model's strengths and weaknesses: interpretability, estimation method, and how factors are obtained.

Quickest way: Three-second model classifier

When to use it: Use it on any question that asks which type of model is described or which is best for a stated purpose.

Factors are economic surprises: macroeconomic model, time-series regression.
Factors are company attributes: fundamental model, cross-sectional regression.
Factors are unnamed and found from return data: statistical model, factor analysis or principal components.
For return calculations, convert to surprises first, then compute intercept + Σ(beta × surprise).

Common mistakes in Types of Multifactor Models

Plugging actual GDP growth or inflation into the model instead of the surprise.
The vignette lists actual and forecast figures, and the actual figure looks like the input.
Fix: Subtract the expected value first. Only the difference enters a macro model. The expected part is already in the intercept.
Calling a model with factors like book-to-market and size a macroeconomic model.
Students think any systematic factor is macroeconomic.
Fix: Ask whether the factor is a company attribute. If yes, it is fundamental.
Saying statistical factors are easy to interpret.
Students remember that statistical models fit the data well.
Fix: Statistical factors are found from the data and often have no clear economic meaning. This is their main weakness.
Using cross-sectional regression for macro models and time-series regression for fundamental models.
Both are described as regressions on factors.
Fix: Macro models use time-series regression of one asset's returns on surprises. Fundamental models use cross-sectional regression across assets at one date to estimate factor returns.
Forgetting the sign of a negative sensitivity.
Students multiply magnitudes and ignore the minus sign.
Fix: Write the sign next to each beta. A positive surprise with a negative beta lowers return.

Worked examples

Example 1

An analyst uses a macroeconomic factor model for Stock X: R = 8% + 1.2 × (GDP surprise) − 0.8 × (inflation surprise) + ε. Expected GDP growth was 2.0% and actual was 2.5%. Expected inflation was 3.0% and actual was 3.5%. Q1: What is the surprise in each factor? Q2: What is the return explained by the factors, ignoring ε? Q3: What does the 8% represent?

Show the solution

Q1: GDP surprise = 2.5% − 2.0% = +0.5%. Inflation surprise = 3.5% − 3.0% = +0.5%.
Q2: GDP contribution = 1.2 × 0.5% = +0.6%.
Inflation contribution = −0.8 × 0.5% = −0.4%.
Return = 8% + 0.6% − 0.4% = 8.2%.
Q3: With zero surprises both factors contribute nothing, so 8% is the expected return.

Answer: Q1: both surprises are +0.5%. Q2: 8.2%. Q3: the expected return of Stock X.

Example 2

A portfolio manager reviews three models. Model A has factors that are standardized book-to-market, market capitalization and leverage of each company, with factor returns estimated across 500 stocks each month. Model B has factors that are unexpected changes in industrial production and credit spreads. Model C uses principal components extracted from 10 years of returns. Q1: Classify each model. Q2: Which model is likely hardest to interpret economically? Q3: How are the sensitivities in Model B estimated?

Show the solution

Q1: Model A uses company attributes, so it is fundamental. Model B uses surprises in economic variables, so it is macroeconomic. Model C extracts components from return data, so it is statistical.
Q2: Model C, since principal components are portfolios found from the data and often lack clear economic meaning.
Q3: Macroeconomic sensitivities come from a time-series regression of asset returns on the factor surprises.

Answer: Q1: A fundamental, B macroeconomic, C statistical. Q2: Model C. Q3: time-series regression of asset returns on the surprises.

Exam tips

Look for the word surprise. If the vignette gives actual and expected values, expect a calculation step before the model is applied.
Classification questions usually hinge on what the factors are. Read the factor list before reading anything else.
Know one strength and one weakness for each type: macro is intuitive but surprises are hard to measure; fundamental is easy to interpret; statistical fits well but has unclear factors.
For Fama-French and Carhart, memorize the factor names (market, SMB, HML, WML) and what each long-short portfolio is.
There is no penalty for wrong answers, so answer every question even if you must guess between two options.

Types of Multifactor Models in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Types of Multifactor Models: frequently asked questions

What is the difference between macroeconomic and fundamental factor models?

Macroeconomic models use surprises in economic variables such as GDP or inflation as factors. Fundamental models use company attributes such as size, book-to-market or leverage. Macro sensitivities come from time-series regression, while fundamental factor returns come from cross-sectional regression.

What is a factor surprise?

A factor surprise is the actual value of a macro variable minus its expected value. It has an expected value of zero. Only the surprise changes an asset's return relative to its expected return.

What is a factor sensitivity?

It is the asset's return change for a one-unit move in a factor, also called a factor beta or loading. A positive sensitivity means return rises when the factor rises. A negative one means return falls.

Where do Fama-French and Carhart fit among the types?

They are multifactor models whose factors are returns on long-short portfolios formed on company attributes, such as size and book-to-market. They resemble fundamental models but are not described as a separate type. Carhart adds a momentum factor to the Fama-French three factors.