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Advanced Financial Management · Portfolio Management

Arbitrage Pricing Theory and Factor Models (CA Final AFM)

Updated 5 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 beta times that factor's risk premium. Sharpe's single index model uses one factor, the market. To solve, list betas and premiums, compute the required return, compare it with the expected return, then buy underpriced and sell overpriced securities.

Understand Arbitrage Pricing Theory and Factor Models

A factor model explains a security's return using a few common influences, such as the market index, GDP growth or inflation. Whatever the factors do not explain is a security-specific (unsystematic) part. This is far simpler than estimating the covariance of every pair of securities.

Sharpe's single index model uses one factor, the market return. A security's return is Ri = αi + βi Rm + ei. Here α is the return that does not depend on the market, β is sensitivity to the market, and e is the random firm-specific error with an expected value of zero. The market part is systematic risk. The e part is unsystematic risk and falls as you diversify.

Arbitrage Pricing Theory (APT) extends this to several factors. It says expected return = risk-free rate + Σ (beta to each factor × risk premium of that factor). The risk premium is the extra return investors expect for bearing one unit of that factor's risk. Unlike CAPM, APT does not name the factors. You are told them in the question.

Arbitrage is a risk-free profit with no net investment. If a security's expected return is above its APT return, it is underpriced. Buyers push its price up until the gap closes. If it is below, it is overpriced, and selling pushes the price down. So in APT, prices are pulled to the fair level by arbitrageurs, not by a single market-wide equilibrium.

CAPM versus APT: CAPM has one factor (market beta) and rests on mean-variance investors. APT allows many factors and needs only that diversified portfolios leave no arbitrage. Both give a linear link between return and beta.

Key rules to remember

Single index model
Ri = αi + βi Rm + ei
Taking expected values: E(Ri) = αi + βi E(Rm), because E(ei) = 0.
Security variance
σi² = βi² σm² + σei²
First term is systematic risk, second is unsystematic risk.
Covariance between two securities
Cov(i, j) = βi × βj × σm²
Holds in the single index model because errors are assumed uncorrelated.
Portfolio alpha and beta
αp = Σ wi αi ; βp = Σ wi βi
Weights wi are market-value proportions and sum to 1.
Portfolio variance (single index)
σp² = βp² σm² + Σ wi² σei²
Unsystematic part shrinks as the number of securities rises.
APT expected return
E(R) = Rf + β1 λ1 + β2 λ2 + … + βn λn
λ is the risk premium of each factor, that is, the expected return of the factor over Rf.
Arbitrage rule
Expected return > APT return: underpriced, buy. Expected return < APT return: overpriced, sell.
Fair price today = (expected price + dividend) ÷ (1 + APT return).

How to solve Arbitrage Pricing Theory and Factor Models questions

Use this order for any numerical on the single index model or APT.

  1. 1Identify the model. One market index with alpha, beta and σe means single index. Several factors with risk premiums means APT.
  2. 2Write down all inputs in one place: Rf, market or factor premiums, betas, alphas, standard deviations and weights. Keep percentages consistent.
  3. 3For single index, find each security's expected return as α + β × E(Rm). For APT, find the required return as Rf + Σ β × premium.
  4. 4For a portfolio, take weighted averages of alpha, beta and returns. Never average standard deviations.
  5. 5For risk, compute systematic variance as βp² σm² and unsystematic variance as Σ w² σe². Add them, then take the square root for standard deviation.
  6. 6For arbitrage questions, compare the expected return given in the question with the APT return. State clearly which security is underpriced or overpriced.
  7. 7Give the action: buy the underpriced security and sell or short the overpriced one, and say the price will move until returns equal the APT return.
  8. 8Write a one-line conclusion with the numbers, as the examiner expects interpretation.

Quickest way: Table-and-compare shortcut

When to use it: Use when the question has several securities or factors and time is short.

  1. Draw a small table with one row per security and columns for betas, APT return and given return.
  2. Compute the APT return row by row: Rf plus each beta times its premium.
  3. Subtract APT return from the given return. A positive gap means buy, a negative gap means sell.
  4. For portfolio beta and alpha, use weights directly on the table values.
  5. For portfolio variance, compute βp² σm² first, then add Σ w² σe² in a single line.

Common mistakes in Arbitrage Pricing Theory and Factor Models

  • Adding the risk-free rate inside each factor premium or subtracting it twice

    Some questions give the factor's expected return, others give its premium over Rf.

    Fix: Read the wording. If the figure is a premium, use it as is. If it is a factor's total expected return, subtract Rf first.

  • Taking a weighted average of standard deviations for portfolio risk

    It looks like the same step as averaging returns.

    Fix: Average only alpha, beta and returns. Compute portfolio risk from variances using σp² = βp² σm² + Σ w² σe².

  • Forgetting to square the weights on the unsystematic term

    Students copy the βp formula pattern.

    Fix: Residual risks are uncorrelated, so each contributes w² σe². Square the weight and the σe.

  • Treating alpha as zero in the expected return

    CAPM has no alpha, so students carry that habit over.

    Fix: In the single index model, E(Ri) = α + β E(Rm). Include alpha whenever it is given.

  • Naming the wrong action in arbitrage

    Confusion over which direction the gap runs.

    Fix: Expected return above APT return means price too low, so buy. Below means price too high, so sell.

  • Saying APT tells you which factors to use

    Students mix up the theory with its inputs.

    Fix: APT does not specify factors or their number. The question or the analyst supplies them.

Worked examples

Example 1

Under the single index model, the market's expected return is 12% and its standard deviation is 20%. Security A has α = 1%, β = 1.2, σe = 10%. Security B has α = 0.5%, β = 0.8, σe = 15%. You hold 60% in A and 40% in B. Find the portfolio's expected return, beta and standard deviation.

Show the solution
  1. Portfolio alpha = 0.6 × 1 + 0.4 × 0.5 = 0.6 + 0.2 = 0.8%.
  2. Portfolio beta = 0.6 × 1.2 + 0.4 × 0.8 = 0.72 + 0.32 = 1.04.
  3. Expected return = 0.8 + 1.04 × 12 = 0.8 + 12.48 = 13.28%.
  4. Systematic variance = 1.04² × 20² = 1.0816 × 400 = 432.64.
  5. Unsystematic variance = 0.6² × 10² + 0.4² × 15² = 0.36 × 100 + 0.16 × 225 = 36 + 36 = 72.
  6. Total variance = 432.64 + 72 = 504.64 (in %²).
  7. Standard deviation = √504.64 ≈ 22.46%.

Answer: Expected return 13.28%, portfolio beta 1.04, standard deviation about 22.46%. Most of the risk is systematic, so adding more securities would only trim the 72 of unsystematic variance.

Example 2

The risk-free rate is 7%. Two factors drive returns: GDP growth with a risk premium of 5%, and inflation with a risk premium of 2%. Stock X has a beta of 1.2 to GDP growth and 0.5 to inflation. X trades at ₹200. You expect it to be ₹220 after a year, with a dividend of ₹14. Is X fairly priced? What should you do, and what is the fair price?

Show the solution
  1. APT return = 7 + 1.2 × 5 + 0.5 × 2 = 7 + 6 + 1 = 14%.
  2. Expected return = (220 + 14 − 200) ÷ 200 = 34 ÷ 200 = 17%.
  3. Expected return 17% is above the APT return 14%, so X is underpriced.
  4. Fair price = (220 + 14) ÷ 1.14 = 234 ÷ 1.14 ≈ ₹205.26.
  5. Action: buy X, funded if needed by selling an overpriced security. Buying pressure will lift the price towards ₹205.26 until the return falls to 14%.

Answer: X is underpriced. APT return is 14% against an expected 17%. Buy X. Its fair price is about ₹205.26.

Exam tips

  • Show the APT formula with numbers substituted. Marks are given for the method even if arithmetic slips.
  • In arbitrage questions, always write the final line: underpriced or overpriced, the action, and the fair price if asked.
  • For the single index model, check whether variance or standard deviation is asked, and square or root accordingly.
  • A theory question on CAPM versus APT is common. Prepare points on number of factors, assumptions, and whether factors are specified.
  • Keep percentages in the same units through the sum. Convert to decimals only at the end if needed.

Practice questions from Portfolio Management

Arbitrage Pricing Theory and Factor Models 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 and Factor Models: frequently asked questions

What is the difference between CAPM and APT?

CAPM uses one factor, the market beta, and assumes investors choose by mean and variance. APT allows several factors and only needs that no arbitrage exists. APT does not name its factors, while CAPM names the market.

Why does the single index model reduce workload?

It relates every security to one index, so you need only alpha, beta and residual risk for each security plus the market's figures. Covariance between two securities becomes βi × βj × σm², so you avoid estimating each pair directly.

How do I spot an arbitrage opportunity in APT?

Compute the APT required return from the betas and premiums. Compare it with the expected return in the question. If expected is higher, the security is underpriced. If lower, it is overpriced.

Is the factor risk premium the same as the factor return?

No. The risk premium is the factor's extra return over the risk-free rate. Check the wording, and subtract Rf if the question gives a total expected return for the factor.