FRM Exam Part I · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

APT vs CAPM: Key Differences for FRM Part I

Updated 11 October 2026 · Fact-checked

CAPM says expected return depends on one factor: beta to the market portfolio. APT says expected return depends on exposures to several systematic factors, derived from no-arbitrage in well-diversified portfolios. CAPM needs strong investor assumptions and the market portfolio. APT needs fewer assumptions but does not name its factors.

Understand APT versus CAPM

Both models answer one question: why do some assets earn higher expected returns than others? Both say only systematic risk is rewarded. Firm-specific risk can be diversified away, so the market does not pay you for bearing it.

The CAPM gets there through investor behaviour. All investors are mean-variance optimisers with the same beliefs, so they all hold the market portfolio combined with the risk-free asset. Risk is then measured by one number, beta to the market. The result is E(Ri) = Rf + βi × [E(Rm) − Rf].

The APT gets there through arbitrage. Returns follow a factor model: Ri = E(Ri) + βi1F1 + βi2F2 + ... + ei, where the factors F have zero mean and ei is firm-specific noise. In a large market, a well-diversified portfolio has almost no firm-specific risk. If its expected return did not match its factor exposures, you could build a risk-free profit. So no-arbitrage forces E(Ri) = Rf + βi1λ1 + βi2λ2 + ..., where λ is the risk premium per unit of each factor.

The key contrasts are these. CAPM has one factor, the market. APT allows K factors and does not say what they are. CAPM requires the market portfolio to be mean-variance efficient and observable in principle. APT does not require the market portfolio at all. CAPM is a statement about all assets in equilibrium. APT holds approximately for most assets, and a few may be mispriced. APT also uses weaker assumptions about investors: they only prefer more wealth to less and arbitrage opportunities do not persist.

CAPM is a special case of APT: if the single factor is the market portfolio's excess return, the APT equation collapses to the CAPM. That is why a multifactor model with one factor equals the CAPM.

On testability, the CAPM is hard to test because the true market portfolio includes every asset and is unobservable (Roll's critique). The APT avoids that but has its own problem: theory does not tell you which factors to use or how many, so tests depend on your choice.

Key formulas to remember

CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf]; One factor. βi = Cov(Ri, Rm) ÷ Var(Rm). Market risk premium is E(Rm) − Rf.
APT return-generating process: Ri = E(Ri) + βi1F1 + βi2F2 + ... + βiKFK + ei; Factors F have zero expected value. ei is firm-specific risk, uncorrelated with the factors.
APT pricing equation: E(Ri) = Rf + βi1λ1 + βi2λ2 + ... + βiKλK; λk is the risk premium per unit of exposure to factor k, a factor portfolio's expected return minus Rf.
CAPM as special case of APT: K = 1, F1 = Rm − E(Rm), λ1 = E(Rm) − Rf; The APT equation then becomes the CAPM.
Alpha under a factor model: αi = E(Ri) − [Rf + Σ βik λk]; Non-zero alpha means mispricing relative to the model.

How to solve APT versus CAPM questions

Use this method for any APT versus CAPM question, whether it is conceptual or numerical.

1Identify what is being asked: a comparison of assumptions, a statement about factors, or a required return calculation.
2For a conceptual question, sort each statement into one of the two models. CAPM: one factor, market portfolio, investor equilibrium. APT: many factors, no-arbitrage, well-diversified portfolios.
3For a numerical question, list the inputs: Rf, each factor's risk premium (λ or expected factor return minus Rf), and each beta.
4If the model is CAPM, compute Rf + β × market risk premium. If APT, compute Rf + Σ(βk × λk).
5Check the units. Use the same basis (annual or monthly) for all inputs and express the answer as a percentage.
6If asked about mispricing, compare the model return to the quoted expected return. Higher quoted return means underpriced (positive alpha).
7Re-read the options for common traps such as 'APT identifies the factors' or 'CAPM allows multiple betas'.

Quickest way: Four-question sort for APT vs CAPM

When to use it: Use it on conceptual multiple-choice questions where each option makes a claim about one model.

Ask: how many factors? One means CAPM, several means APT.
Ask: is the market portfolio needed? Yes means CAPM, no means APT.
Ask: what drives the result? Investor equilibrium means CAPM, no-arbitrage means APT.
Ask: does the theory name the factors? CAPM names one. APT does not name any.
For calculations, multiply each beta by its premium, add them, then add Rf.

Common mistakes in APT versus CAPM

Saying APT tells you which factors to use.
Students see a named model such as Fama-French and assume APT supplies it.
Fix: APT gives the structure but not the factors. You choose them (macro, statistical or fundamental).
Believing APT needs the market portfolio.
The market portfolio is central in CAPM, so it is assumed to be central everywhere.
Fix: APT needs no market portfolio. The market can be one of the factors, but it is not required.
Treating APT as holding exactly for every asset.
Students mix up the CAPM, which holds for all assets in equilibrium, with APT.
Fix: APT holds exactly for well-diversified portfolios. Individual assets can deviate slightly.
Adding betas instead of multiplying each beta by its own premium.
Rushing through the multifactor formula.
Fix: Compute βk × λk for each factor first, then sum, then add Rf.
Subtracting Rf from a premium that is already a premium.
Questions vary between giving factor returns and factor premiums.
Fix: Read the label. If a factor 'return' is given, subtract Rf. If a 'premium' is given, use it directly.
Saying the CAPM is a special case of APT without the condition.
Memorising the slogan.
Fix: It holds only when there is a single factor, the market's excess return.

Worked examples

Example 1

A stock has beta 1.2 to the market. The risk-free rate is 3% and the expected market return is 9%. Under the CAPM, what is its required return? If an APT model with one factor equal to the market's excess return gives the same premium, what is the APT-implied return?

Show the solution

Market risk premium = 9% − 3% = 6%.
CAPM: E(R) = 3% + 1.2 × 6% = 3% + 7.2% = 10.2%.
One-factor APT with λ1 = 6% and β = 1.2: E(R) = 3% + 1.2 × 6% = 10.2%.
The results match because the CAPM is the one-factor special case of APT.

Answer: 10.2% under both models.

Example 2

Rf = 2%. A stock has factor betas of 0.8 to factor 1 (premium 5%), 1.5 to factor 2 (premium 2%) and −0.5 to factor 3 (premium 4%). Its expected return in the market is 6.5%. Using APT, what is its alpha and is it under- or overpriced?

Show the solution

Factor 1 contribution: 0.8 × 5% = 4.0%.
Factor 2 contribution: 1.5 × 2% = 3.0%.
Factor 3 contribution: −0.5 × 4% = −2.0%.
Sum of contributions = 4.0% + 3.0% − 2.0% = 5.0%.
Model return = 2% + 5.0% = 7.0%.
Alpha = 6.5% − 7.0% = −0.5%.
Expected return is below the model return, so the stock is overpriced.

Answer: Alpha is −0.5%; the stock is overpriced.

Exam tips

Expect statement-style questions: 'Which is true of APT but not CAPM?' Use the four-question sort.
Remember the pairing: CAPM needs the market portfolio and investor assumptions, APT needs no-arbitrage and diversification.
On calculations, write each βk × λk on its own line. Negative betas are a common trap.
Know the weakness of each: CAPM is hard to test because the market portfolio is unobservable, APT does not specify its factors.
A financial calculator adds little here. Use a simple sum and check signs.

Practice questions from The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

APT versus CAPM: frequently asked questions

What is the main difference between APT and CAPM?

CAPM uses a single factor, beta to the market portfolio, and relies on investor equilibrium. APT allows several factors and relies on no-arbitrage in well-diversified portfolios. APT does not require the market portfolio.

Is CAPM a special case of APT?

Yes, under a condition. If APT has one factor and that factor is the market's excess return, its pricing equation becomes the CAPM. With more factors, or a different single factor, it does not.

Which is better, APT or CAPM?

Neither is better in every respect. APT is more flexible and needs weaker assumptions, but it does not tell you the factors. CAPM is simple and gives a clear answer, but depends on strong assumptions and an unobservable market portfolio.

What are the limitations of arbitrage pricing theory?

It does not identify the factors or their number, so results depend on your choice. Its pricing holds exactly only for well-diversified portfolios in a large market. Estimating factor betas and premiums also involves error.