FRM Exam Part II · Factor Theory

Factor Theory and Asset Pricing Foundations for FRM Part II

Updated 11 October 2026 · Fact-checked

Factor theory says an asset's expected return comes from its exposure to systematic risk factors, not from its own idiosyncratic risk. CAPM uses one factor, the market. APT and multifactor models use several. To solve questions, find each factor beta, multiply by its risk premium, add the risk-free rate.

Understand Factor Theory and Asset Pricing Foundations

Start with a simple idea. Some risks can be diversified away. Others hit almost every asset at once, such as a recession, a rate shock or a market sell-off. These are systematic risks. Investors only get paid for bearing risk they cannot diversify. That is the core of factor theory.

A risk factor is a source of systematic return variation that investors demand compensation for. The reward for bearing it is the factor risk premium. An asset's beta (or factor loading) measures how strongly its return moves with a factor. Higher exposure to a priced factor means higher expected return.

The CAPM is the one-factor case. The only priced factor is the market portfolio. Expected excess return equals beta times the market risk premium. Idiosyncratic risk earns nothing, because it can be diversified. The CAPM relies on assumptions: investors hold mean-variance efficient portfolios, share the same beliefs, and can borrow and lend at the risk-free rate.

The APT relaxes this. It assumes returns follow a linear factor model and that there are enough assets to diversify away idiosyncratic risk. If expected returns did not line up with factor exposures, arbitrageurs could earn riskless profit, so prices adjust. APT does not name the factors. You must choose them. They can be macroeconomic (growth, inflation, rates, credit spreads) or style factors (value, size, momentum).

Multifactor models such as Fama-French add factors beyond the market. Expected excess return is the sum of each beta times its factor premium. In practice, these models help explain returns, decompose portfolio risk and measure alpha. Alpha is the return left over after paying for factor exposure. A manager who only loads on known factors has not shown skill.

Key formulas to remember

CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf]; βi = Cov(Ri, Rm) ÷ Var(Rm). Only market (systematic) risk is priced.
Multifactor / APT expected return: E(Ri) = Rf + βi1 × λ1 + βi2 × λ2 + … + βik × λk; λ is the risk premium for each factor. Betas are exposures to each factor.
Factor return model: Ri = αi + βi1 F1 + … + βik Fk + εi; εi is idiosyncratic and assumed uncorrelated with the factors.
Alpha: α = actual return − expected return from factor model; Measured in excess of Rf. Positive alpha means return above what factor exposure explains.
Portfolio beta: βp = Σ wi × βi; Weights are portfolio weights. Betas add linearly.
Systematic variance: Var(Ri) = β² × Var(F) + Var(ε) (single factor); The first term is systematic risk. The second is diversifiable.

How to solve Factor Theory and Asset Pricing Foundations questions

Use this order for almost any factor theory or CAPM/APT question.

1Identify the model asked for: CAPM (one factor), APT or a multifactor model.
2List the inputs: risk-free rate, betas or loadings, factor premiums or market return.
3Check whether premiums are given as excess returns. If the market return is given, subtract Rf first.
4Multiply each beta by its factor premium and sum the results.
5Add the risk-free rate to get expected return.
6If alpha is asked, subtract the model expected return from the actual return, both on the same basis.
7For risk questions, split variance into systematic (β² × factor variance) and idiosyncratic parts.
8Interpret the result: is the asset priced fairly, and which risk is being compensated?

Quickest way: Premium-times-beta shortcut

When to use it: Use it for numeric expected return and alpha questions when time is short.

Write Rf first.
Compute beta × premium for each factor and add them mentally.
Convert any market return into a premium by subtracting Rf.
Compare with the actual return to get alpha.
Rule out options that treat idiosyncratic risk as rewarded.

Common mistakes in Factor Theory and Asset Pricing Foundations

Using the market return instead of the market risk premium in CAPM.
The question gives E(Rm) and students multiply it by beta directly.
Fix: Always compute E(Rm) − Rf first, then multiply by beta and add Rf.
Saying total volatility is priced under CAPM.
Students link higher risk with higher return.
Fix: Only beta, the non-diversifiable part, earns a premium. Idiosyncratic risk does not.
Claiming APT names the factors.
APT is confused with Fama-French.
Fix: APT only says returns are linear in some factors. The user must pick and estimate them.
Treating a positive alpha as skill without checking factors.
Alpha is measured against only one factor.
Fix: Add relevant factors. Alpha shrinks if the manager simply loaded on value, size or momentum.
Averaging betas with the wrong weights.
Equal weights are assumed when position sizes differ.
Fix: Use market-value weights summing to 1, including cash at beta 0 or any short at negative weight.

Worked examples

Example 1

The risk-free rate is 4%, the expected market return is 10% and a stock has beta 1.3. What is the CAPM expected return? If the stock is expected to return 11.5%, what is its alpha?

Show the solution

Market risk premium = 10% − 4% = 6%.
Risk premium of the stock = 1.3 × 6% = 7.8%.
Expected return = 4% + 7.8% = 11.8%.
Alpha = 11.5% − 11.8% = −0.3%.

Answer: CAPM expected return is 11.8%. Alpha is −0.3%, so the stock is slightly overpriced against CAPM.

Example 2

A two-factor APT model has Rf = 3%, factor 1 premium 5%, factor 2 premium 2%. A fund has loadings of 0.8 on factor 1 and 1.5 on factor 2. Its realised return is 10.5%. Find expected return and alpha.

Show the solution

Factor 1 contribution = 0.8 × 5% = 4.0%.
Factor 2 contribution = 1.5 × 2% = 3.0%.
Expected return = 3% + 4.0% + 3.0% = 10.0%.
Alpha = 10.5% − 10.0% = 0.5%.

Answer: Expected return is 10.0% and alpha is +0.5%.

Exam tips

Read whether the question gives a market return or a premium. This is the most common trap.
Know the CAPM versus APT contrast: CAPM has one factor and strong investor assumptions, APT has multiple unspecified factors and relies on no-arbitrage.
Expect interpretation questions: which risk is priced, and why diversifiable risk earns nothing.
When alpha appears, check whether it is measured against one factor or several.
Keep units consistent, excess returns with excess returns.

Practice questions from Factor Theory

Factor Theory and Asset Pricing Foundations in other exams

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

CA FinalArbitrage Pricing Theory and Factor Models

Factor Theory and Asset Pricing Foundations: frequently asked questions

What is the difference between CAPM and APT?

CAPM is a one-factor model where the market portfolio is the only priced risk. APT allows several factors and relies on no-arbitrage rather than investor equilibrium assumptions. APT does not say what the factors are.

What is a risk factor in asset pricing?

It is a source of systematic return variation that investors want paying for. Examples are market, interest rate, inflation, credit spread, value and momentum. Exposure to it is measured by beta.

How do multifactor models explain expected returns?

They sum each factor exposure times its risk premium and add the risk-free rate. Assets with higher loadings on priced factors have higher expected returns. Whatever is left over is alpha.

Is idiosyncratic risk rewarded?

No, not in CAPM or APT. It can be diversified away in a large portfolio, so investors do not need compensation for it.