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

Arbitrage Pricing Theory (APT) for FRM Part I

Updated 11 October 2026 · Fact-checked

Arbitrage pricing theory says a security's expected return is a linear function of its exposures (betas) to a few systematic factors. If prices break this line, investors can build zero-cost, zero-risk portfolios with positive profit. To solve questions, compute E(R) = Rf + Σ βk × λk and compare it with the stated return.

Understand Arbitrage Pricing Theory (APT)

Start with the idea of factors. Many things move stock returns together: GDP growth, inflation, interest rates, oil prices. A multifactor model writes a return as an expected part, plus betas times factor surprises, plus a firm-specific shock.

The APT asks a simple question: what must be true so that no one can earn a riskless profit? Take a well-diversified portfolio. Its firm-specific risk has averaged away, so only factor risk remains. Two well-diversified portfolios with identical factor betas must then have identical expected returns. If they do not, you buy the one with the higher return, short the other, and lock in a gain with no net investment and no risk. That is an arbitrage. Buying and selling pressure removes it.

This no-arbitrage argument forces expected return to be linear in the betas: E(Rp) = Rf + β1 λ1 + β2 λ2 + ..., where each λ is the factor risk premium, the extra expected return for one unit of exposure to that factor. In the single-factor case, all well-diversified portfolios must plot on one straight line of expected return against beta.

The APT's main assumptions are: returns follow a factor structure, there are enough securities to diversify away idiosyncratic risk, and markets have no persistent arbitrage. Unlike CAPM, it does not need mean-variance investors or a market portfolio. It also does not name the factors. You must choose or estimate them.

The pricing result holds exactly for well-diversified portfolios. For single stocks it holds only approximately, because a few can deviate without being arbitraged away.

Key formulas to remember

Multifactor return model: Ri = E(Ri) + βi1 F1 + βi2 F2 + ... + βik Fk + εi; F are factor surprises with mean zero. ε is firm-specific risk, uncorrelated with the factors.
APT expected return (K factors): E(Ri) = Rf + βi1 λ1 + βi2 λ2 + ... + βiK λK; λk is the risk premium on factor k, often the expected return of a factor portfolio with beta 1 on it minus Rf.
Single-factor APT: E(Rp) = Rf + βp λ; Applies to well-diversified portfolios. All must lie on one line.
Portfolio beta: βp = Σ wi βi; Weights sum to 1. Use the same for each factor separately.
Arbitrage profit for a mispriced portfolio: Profit = E(R) of the long portfolio − E(R) of the short replicating portfolio (identical betas), per unit invested; Go long the portfolio with the higher return and short a mix of assets with the same betas on every factor. Multiply the return difference by the amount invested. The profit is approximately riskless only when both sides are well diversified.
Alpha: α = actual expected return − APT required return; Non-zero alpha for a well-diversified portfolio signals arbitrage.

How to solve Arbitrage Pricing Theory (APT) questions

Use this routine for any APT question, whether it asks for a required return, a mispricing or an arbitrage.

1Identify the factors and the risk-free rate given in the question.
2Write down the factor risk premiums (λ) and the betas of the portfolio or stock.
3If the portfolio is a mix of assets, compute each factor beta as the weighted average of the asset betas.
4Compute the required return: Rf + Σ βk λk.
5Compare it with the stated expected return. Equal means fairly priced. Higher means underpriced, lower means overpriced.
6For an arbitrage, go long the underpriced portfolio and short a combination of assets with identical betas but the correct return, with weights summing to the same investment.
7Check that net investment is zero and net factor exposure is zero. The profit is the return difference times the amount.
8State the answer with units and sign, as a percentage or in currency.

Quickest way: Required return, then gap

When to use it: Use when you see a multiple-choice question asking for expected return, mispricing or the direction of an arbitrage trade.

Compute Rf + Σ β λ in one line on your calculator.
Subtract it from the quoted return. The gap is the alpha.
Positive gap: buy. Negative gap: sell short.
For a two-portfolio arbitrage with the same beta, the profit per unit is simply the return difference.
Eliminate options that ignore the risk-free rate or use a total premium instead of the factor premium.

Common mistakes in Arbitrage Pricing Theory (APT)

Forgetting to add the risk-free rate when λ is a premium over Rf.
Students copy the CAPM habit of reading the premium as the whole return.
Fix: Read the wording. If λ is a risk premium, add Rf. If it is the total expected factor-portfolio return, subtract Rf first.
Treating APT as requiring the market portfolio or mean-variance investors.
It is confused with CAPM.
Fix: Remember APT needs only a factor structure, diversification and no arbitrage. A single-factor APT with the market as the factor resembles CAPM, but CAPM rests on mean-variance assumptions and the market portfolio.
Claiming APT holds exactly for every individual stock.
The conclusion is stated loosely in notes.
Fix: The exact result is for well-diversified portfolios. Single stocks may deviate because idiosyncratic risk is not priced away by arbitrage.
Using factor levels instead of factor betas or surprises.
Problems list GDP growth numbers next to betas.
Fix: Betas multiply unexpected factor movements. Expected return uses betas times risk premiums.
Calling a trade an arbitrage when it has net factor exposure.
The short leg has a different beta from the long leg.
Fix: Match every factor beta between the long and short sides and keep net investment at zero.
Averaging betas without weighting.
Rushing under time pressure.
Fix: Multiply each beta by its portfolio weight before summing. Do this separately for each factor.

Worked examples

Example 1

A two-factor APT has Rf = 3%, λ1 = 5% for factor 1 and λ2 = 2% for factor 2. A well-diversified portfolio has β1 = 1.2 and β2 = 0.5. Its quoted expected return is 9.5%. What is the required return, and is the portfolio under- or overpriced?

Show the solution

Required return = Rf + β1 λ1 + β2 λ2.
= 3% + 1.2 × 5% + 0.5 × 2%.
= 3% + 6% + 1% = 10%.
Quoted return is 9.5%, which is below the required 10%.
Alpha = 9.5% − 10% = −0.5%.

Answer: The required return is 10%. The portfolio is overpriced (alpha −0.5%), so it offers too little return for its risk.

Example 2

Well-diversified portfolio A has beta 1.0 to a single factor and expected return 8%. Portfolio B has beta 1.0 and expected return 11%. Rf = 4%. You invest $1,000,000 long in B and short in A. What is the expected arbitrage profit and the net factor exposure?

Show the solution

Both portfolios have the same beta, so factor risks should be rewarded equally.
Rf = 4% is not needed here, because the profit comes only from the return gap between two portfolios with identical beta. APT would require both to earn 4% + 1.0λ.
Long $1,000,000 in B and short $1,000,000 in A gives net investment of zero.
Net beta = 1.0 × 1,000,000 − 1.0 × 1,000,000 = 0, so there is no factor risk.
Both are well diversified, so firm-specific risk is negligible.
Expected profit = ($1,000,000 × 11%) − ($1,000,000 × 8%) = $110,000 − $80,000 = $30,000.

Answer: The expected profit is $30,000 with zero net investment and zero net factor exposure. It is an expected profit, approximately riskless only because both portfolios are well diversified, so a small residual risk remains. This arbitrage pushes B's price up and A's price down until the returns converge.

Exam tips

Expect numerical questions where you compute Rf + Σ βλ. Practise doing this in under a minute.
Know the contrast with CAPM: APT is multi-factor, needs no market portfolio, and is derived from no-arbitrage.
Questions often say the portfolio is well diversified. That phrase signals that only factor risk matters.
In arbitrage questions, check that betas match on both sides before choosing the long and short legs.
Read whether λ is a premium over Rf or a total return. Option distractors often exploit this.

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

Arbitrage Pricing Theory (APT) in other exams

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

Arbitrage Pricing Theory (APT): frequently asked questions

What is the difference between APT and CAPM?

CAPM uses one factor, the market, and relies on investor behaviour and the market portfolio. APT allows several factors and rests on no-arbitrage for well-diversified portfolios. APT does not say which factors to use.

What are the assumptions of APT?

Returns follow a factor model, there are enough assets to diversify away firm-specific risk, and markets allow no lasting arbitrage. It does not need normally distributed returns or mean-variance investors.

Why must the APT relationship be linear?

If expected returns were not linear in betas, you could combine portfolios to create one with the same exposures as another but a different return. That is an arbitrage, so prices adjust until the relationship is linear.

How do I find an arbitrage opportunity in an APT question?

Compute the required return from the APT formula and compare it with the quoted return. Then buy the portfolio with the excess return and short a combination with matching factor betas, keeping net investment at zero.