Strategic Financial Management · Asset Pricing Theories

Arbitrage Pricing Theory (APT) Formula and Numericals

Updated 11 October 2026 · Fact-checked

Arbitrage Pricing Theory says a security's expected return equals the risk-free rate plus the sum of each factor's sensitivity (beta) multiplied by that factor's risk premium. To solve a problem, list the factor betas and premiums, add the products to the risk-free rate, then compare with the stated return to spot mispricing.

Understand Arbitrage Pricing Theory (APT)

CAPM explains return with one factor: the market. Stephen Ross's Arbitrage Pricing Theory (APT) says a security's return can depend on several macroeconomic factors, such as inflation, GDP growth, interest rates or oil prices.

Each security has a sensitivity (factor beta) to each factor. A beta of 1.5 to inflation means the return moves by 1.5 percentage points for each 1 percentage point of unexpected change in that factor. Each factor also has a risk premium: the extra return investors demand for bearing that factor's risk.

The core idea is the law of one price. Two assets with the same factor exposures must offer the same expected return. If they do not, investors buy the underpriced one, sell the overpriced one, and make a risk-free profit with no net investment. That is arbitrage. Buying pressure raises the price of the cheap asset and selling pressure lowers the price of the dear one, so returns converge to the APT value.

The main assumptions are:
- Returns follow a linear factor model.
- Markets are perfectly competitive and frictionless, with no transaction costs.
- The portfolio is well diversified, so specific (unsystematic) risk is eliminated.
- Investors prefer more wealth to less and will exploit arbitrage.
- The number of securities is much larger than the number of factors.

APT does not name the factors or their number. This is its strength (flexibility) and weakness (hard to apply). CAPM needs the market portfolio and a single beta. APT needs no market portfolio and no assumption about investor utility or mean-variance preferences.

Key rules to remember

APT expected return (multi-factor): E(Rᵢ) = Rf + β₁ × RP₁ + β₂ × RP₂ + … + βₙ × RPₙ; Rf is the risk-free rate. RPₖ is the risk premium of factor k, that is, the expected return on a portfolio with beta 1 to factor k and 0 to others, minus Rf. Use the premium directly if given.
APT using factor expected returns: E(Rᵢ) = Rf + Σ βₖ × (E(Fₖ) − Rf); Use when the question gives the expected return of each factor portfolio, not the premium. Subtract Rf from each factor return first.
Actual return under factor model: Rᵢ = E(Rᵢ) + β₁ × (F₁ − E(F₁)) + β₂ × (F₂ − E(F₂)) + … + ε; Surprises in factors move the actual return away from expected. ε is the firm-specific term, which is assumed to be zero for a well-diversified portfolio.
Portfolio beta and return: βₚ,ₖ = Σ wᵢ × βᵢ,ₖ ; E(Rₚ) = Σ wᵢ × E(Rᵢ); Weights wᵢ sum to 1. Portfolio factor betas are weighted averages of security betas.
Single-factor APT: E(Rᵢ) = Rf + βᵢ × (E(Rm) − Rf); With one factor being the market, APT looks like CAPM. This is a special case.
Arbitrage test: Mispricing = Given expected return − APT expected return; Positive means underpriced: buy. Negative means overpriced: sell or short.

How to solve Arbitrage Pricing Theory (APT) questions

Use this order for any APT question, whether it asks for expected return, mispricing or an arbitrage strategy.

1Write down the risk-free rate, every factor beta of the security, and every factor risk premium.
2If factor returns are given instead of premiums, convert each to a premium by subtracting the risk-free rate.
3Multiply each beta by its matching premium. Keep the factors in the same order so you do not mismatch.
4Add all the products to the risk-free rate to get the APT required (equilibrium) return.
5If asked about a portfolio, first find the weighted-average beta for each factor, then apply the formula once. Or compute each security's return and take the weighted average. Both give the same answer.
6Compare the APT return with the market's expected return for the security. If expected return is higher, the security is underpriced: buy. If lower, it is overpriced: sell.
7For an arbitrage strategy, build a zero-investment portfolio: long the underpriced asset, short the overpriced one, with matching factor exposures. State the profit.
8Write the conclusion in one line, with the decision and the reason.

Quickest way: Beta × premium sum-up

When to use it: Use for the routine 'find expected return and decide whether to buy' question, where betas and premiums are given directly.

Set the risk-free rate as your starting total.
Go down the factor list. For each, multiply beta by premium and add to the total. Do it in your head or in the margin.
Check that each premium is in percentage points, not already net of Rf twice.
Compare the total with the given return. Write 'Underpriced: buy' or 'Overpriced: sell'.
Add the unit (%) to your answer.

Common mistakes in Arbitrage Pricing Theory (APT)

Using the factor's expected return as the premium without subtracting the risk-free rate.
Students copy the CAPM habit of using Rm, and the question wording 'return on factor' is ambiguous.
Fix: Read whether the figure is a premium or a total return. If it is a total return on a factor portfolio, subtract Rf first.
Forgetting to add the risk-free rate at the end.
The sum of beta × premium looks like a complete answer.
Fix: Write the formula first, with Rf at the front, and tick it off after the final addition.
Matching a beta to the wrong factor.
Factors are listed in a different order in the table and in the question text.
Fix: Label each product by factor name, such as 'Inflation: 1.2 × 2% = 2.4%'.
Saying APT requires the market portfolio or names the factors.
Students mix up CAPM and APT theory.
Fix: Remember that APT does not specify the factors or their number, and does not need the market portfolio. It relies on no-arbitrage.
Taking a negative or positive mispricing and recommending the wrong action.
The subtraction direction is reversed.
Fix: Always compute Expected (market) − Required (APT). Positive means buy, negative means sell.
Averaging factor betas across factors for a portfolio.
Students want a single portfolio beta, as in CAPM.
Fix: Average across securities for each factor separately. Keep one portfolio beta per factor.

Worked examples

Example 1

The risk-free rate is 6%. A stock of Bharat Textiles Ltd has these factor sensitivities and premiums: inflation beta 1.5 with premium 2%; GDP growth beta 0.8 with premium 5%; interest rate beta −0.5 with premium 3%. The market expects a return of 11% on the stock. Using APT, find the required return and advise whether to buy or sell.

Show the solution

Inflation: 1.5 × 2% = 3.0%.
GDP growth: 0.8 × 5% = 4.0%.
Interest rate: −0.5 × 3% = −1.5%.
Sum of factor premiums = 3.0 + 4.0 − 1.5 = 5.5%.
Required return = 6% + 5.5% = 11.5%.
Market expected return is 11%, which is below 11.5%. Mispricing = 11 − 11.5 = −0.5%.

Answer: APT required return is 11.5%. The stock offers only 11%, so it is slightly overpriced. Sell or avoid it.

Example 2

The risk-free rate is 7%. Two factors affect returns. Factor 1 portfolio has an expected return of 12% and Factor 2 portfolio has an expected return of 10%. A portfolio is invested 60% in Stock A and 40% in Stock B. Stock A has betas of 1.2 (Factor 1) and 0.5 (Factor 2). Stock B has betas of 0.6 and 1.5. Find the portfolio's APT expected return.

Show the solution

Factor 1 premium = 12% − 7% = 5%. Factor 2 premium = 10% − 7% = 3%.
Portfolio beta on Factor 1 = 0.6 × 1.2 + 0.4 × 0.6 = 0.72 + 0.24 = 0.96.
Portfolio beta on Factor 2 = 0.6 × 0.5 + 0.4 × 1.5 = 0.30 + 0.60 = 0.90.
Factor 1 contribution = 0.96 × 5% = 4.8%.
Factor 2 contribution = 0.90 × 3% = 2.7%.
Expected return = 7% + 4.8% + 2.7% = 14.5%.
Check using stocks: A = 7 + 1.2×5 + 0.5×3 = 7 + 6 + 1.5 = 14.5%. B = 7 + 0.6×5 + 1.5×3 = 7 + 3 + 4.5 = 14.5%. Weighted = 14.5%.

Answer: The portfolio's APT expected return is 14.5%.

Exam tips

Write the formula with the factor names before substituting. It earns method marks even if arithmetic slips.
In theory questions on CAPM versus APT, give a short comparison: number of factors (one versus many), need for the market portfolio (yes versus no), basis (equilibrium versus no-arbitrage), and factors specified (market beta versus not specified).
In MCQs, watch for the trap of a premium given as a total factor return. Check for the words 'excess' or 'premium'.
State the decision in words: buy, sell or hold, with the reason. SFM answers need a clear recommendation.
If a question gives several securities, find the APT return for each, rank by mispricing, and recommend the most underpriced.

Practice questions from Asset Pricing Theories

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 CAPM and APT?

CAPM uses one factor, the market, and one beta, and rests on mean-variance investors and the market portfolio. APT allows several factors and relies on the no-arbitrage argument. APT does not say which factors matter, so you must be given them.

What are the assumptions of APT?

Returns follow a linear multi-factor model, markets are competitive with no transaction costs, and portfolios are well diversified so firm-specific risk disappears. Investors prefer more wealth and will exploit any arbitrage. There are many more securities than factors.

What is the APT formula with an example?

E(R) = Rf + β₁RP₁ + β₂RP₂ + … For Rf of 6%, betas of 1.0 and 0.5, and premiums of 4% and 2%, the expected return is 6 + 4 + 1 = 11%.

How do I know if a security is underpriced under APT?

Compute the APT required return and compare it with the return the market expects. If the expected return is higher than the required return, the security is underpriced and you buy. If lower, it is overpriced.