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

Well-Diversified Portfolios and Risk Premiums in APT

Updated 11 October 2026 · Fact-checked

A well-diversified portfolio holds so many assets that firm-specific (nonsystematic) risk averages out to almost zero. Only factor (systematic) risk remains, so only factor betas earn a premium. Expected return = risk-free rate + Σ (beta × factor risk premium). Solve by listing betas, premiums, and adding.

Understand Well-Diversified Portfolios and Risk Premiums

Every stock's return has two parts. One part moves with broad economic factors such as the market, inflation or interest rates. This is systematic risk. The other part is specific to the firm: a product recall, a lawsuit, a new contract. This is firm-specific (nonsystematic) risk.

Firm-specific shocks are not related across firms. When you hold many stocks, some shocks are good and some are bad, and they offset. If you hold n equally weighted stocks, each with firm-specific standard deviation σ(e), the firm-specific standard deviation of the portfolio is σ(e) ÷ √n. It shrinks toward zero as n grows. Factor risk does not shrink, because every stock responds to the same factors.

A well-diversified portfolio is one where each weight is small enough that firm-specific risk is negligible. Its return is driven only by its factor betas. In a single-factor model, its variance is essentially β² × σ²(F).

This is the core of Arbitrage Pricing Theory (APT). Investors can remove firm-specific risk for free, so the market does not pay for bearing it. If a well-diversified portfolio offered an expected return out of line with its betas, you could build a zero-investment, zero-risk position that earned a sure profit. Buying and selling would push prices back until the line holds.

The result is a linear relationship. With one factor, expected return is E(r) = rf + β × RP, where RP is the factor risk premium. With several factors, you add one term per factor. Each factor risk premium is the expected return on a portfolio with beta 1 on that factor and beta 0 on all others, minus rf. The one-factor case with the market as the factor looks like the CAPM security market line.

Key formulas to remember

Single-factor return model: r = E(r) + β × F + e; F is the surprise in the factor (mean zero). e is firm-specific noise with mean zero.
Firm-specific risk of an equally weighted portfolio: σ²(e_P) = σ²(e) ÷ n, so σ(e_P) = σ(e) ÷ √n; Assumes firm-specific shocks are uncorrelated and all stocks have the same σ(e). Falls toward zero as n rises.
Total variance of a portfolio (one factor): σ²_P = β_P² × σ²_F + σ²(e_P); For a well-diversified portfolio, σ²(e_P) ≈ 0, so σ_P ≈ |β_P| × σ_F.
Portfolio beta: β_P = Σ wᵢ βᵢ; Weights sum to 1. Beta is a weighted average.
One-factor APT expected return: E(r_P) = rf + β_P × [E(r_F) − rf]; Holds for well-diversified portfolios. Applies to individual assets only approximately.
Multifactor APT expected return: E(r_P) = rf + β₁ × RP₁ + β₂ × RP₂ + … + β_k × RP_k; Each RP is the premium of a factor portfolio with beta 1 on that factor and 0 on the others.
Alpha (mispricing): α = E(r) − [rf + Σ βⱼ RPⱼ]; Non-zero alpha on a well-diversified portfolio signals an arbitrage opportunity.

How to solve Well-Diversified Portfolios and Risk Premiums questions

Use this order for any question on well-diversified portfolios and APT risk premiums.

1Read what is given: factor betas, factor risk premiums or factor portfolio returns, the risk-free rate, and any stated firm-specific risk.
2Decide if the portfolio is well diversified. If yes, ignore firm-specific risk for pricing and for total risk.
3If you are given factor portfolio returns, convert them to premiums: RP = E(r_factor portfolio) − rf.
4Compute portfolio betas as weighted averages if the portfolio is a mix of assets.
5Compute expected return: rf + Σ (beta × RP). Keep all numbers in the same units, such as percent.
6If asked about risk, use σ_P = |β| × σ_F for a well-diversified portfolio. Add firm-specific variance only if the portfolio is not well diversified.
7If asked about arbitrage, compare the stated expected return with the APT value. The difference is alpha. Long the underpriced one, short the overpriced one.
8Check that the answer is sensible: higher beta should mean higher expected return when the premium is positive.

Quickest way: Plug-in method for APT expected return

When to use it: Use when the question gives betas and premiums and asks for an expected return or a mispricing.

Write rf first, then add beta × premium for each factor in turn.
Convert percentages carefully. Write 0.8 × 6% = 4.8%.
For a mix of two portfolios, take the weighted average of their expected returns. Do not recompute from scratch.
For firm-specific risk of n equal holdings, divide σ(e) by √n. Use a calculator square-root key.
Compare with the quoted return. Only the sign of the difference decides which side to trade.

Common mistakes in Well-Diversified Portfolios and Risk Premiums

Adding firm-specific risk to the expected return calculation.
Students think more total risk should earn more return.
Fix: Only factor betas are priced. Firm-specific risk can be diversified away, so it carries no premium.
Using the factor portfolio's total return as the premium.
The question gives a return, and students plug it straight in.
Fix: Subtract rf first. The risk premium is the excess return over the risk-free rate.
Dividing firm-specific variance by √n instead of standard deviation by √n.
Mixing up variance and standard deviation.
Fix: Variance falls as 1 ÷ n. Standard deviation falls as 1 ÷ √n.
Saying diversification removes all risk.
Overstating the diversification benefit.
Fix: It removes only firm-specific risk. Factor (systematic) risk stays, and it is what earns the premium.
Assuming the APT holds exactly for every individual stock.
The pricing line is shown for portfolios and applied loosely to single assets.
Fix: The arbitrage argument is strict for well-diversified portfolios. For single assets, mispricing can persist because of firm-specific risk.
Forgetting that the market factor's premium is not the only one in a multifactor model.
Habit from CAPM, where one premium is used.
Fix: Add a beta × premium term for every factor listed, even if its premium is small or negative.

Worked examples

Example 1

A well-diversified portfolio has beta 1.2 on factor 1 and 0.5 on factor 2. The risk-free rate is 3%. Factor 1 risk premium is 5% and factor 2 risk premium is 4%. Find the APT expected return.

Show the solution

Formula: E(r) = rf + β₁ × RP₁ + β₂ × RP₂.
Factor 1 contribution: 1.2 × 5% = 6.0%.
Factor 2 contribution: 0.5 × 4% = 2.0%.
Add rf: 3% + 6.0% + 2.0% = 11.0%.

Answer: 11.0%

Example 2

A fund holds 25 stocks in equal weights. Each stock has firm-specific standard deviation of 20%, and firm-specific shocks are uncorrelated. The portfolio beta is 1.0 on a factor with standard deviation 15%. Find the portfolio's total standard deviation.

Show the solution

Firm-specific variance of one stock: 0.20² = 0.04.
Portfolio firm-specific variance: 0.04 ÷ 25 = 0.0016. (Standard deviation = 20% ÷ √25 = 4%.)
Factor variance: β² × σ²_F = 1.0² × 0.15² = 0.0225.
Total variance: 0.0225 + 0.0016 = 0.0241.
Total standard deviation: √0.0241 ≈ 0.1552, about 15.5%.

Answer: About 15.5% (versus 15% from factor risk alone)

Exam tips

Questions often hide the risk premium inside a factor portfolio return. Subtract rf before using it.
If the question says the portfolio is well diversified, you can drop firm-specific variance. If it does not, check whether to include it.
Be ready to explain in words why firm-specific risk earns no premium. The arbitrage argument is a common conceptual choice.
In arbitrage questions, compute alpha first. Positive alpha means buy that portfolio and fund it by shorting a matching-beta portfolio.
On the 100-question paper, these are short calculations. Do them in under two minutes and move on.

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

Well-Diversified Portfolios and Risk Premiums: frequently asked questions

What is a well-diversified portfolio in APT?

It is a portfolio with many assets, each with a small weight, so firm-specific risk is negligible. Its return is driven almost entirely by factor exposures. APT pricing applies to it directly.

Why is firm-specific risk not priced?

Investors can remove it at almost no cost by diversifying. If it carried a premium, investors would capture it by holding it in a diversified way, and prices would adjust. Only risk that cannot be diversified earns a premium.

How do I calculate a factor risk premium?

Take the expected return on a factor portfolio, which has beta 1 on that factor and 0 on others, and subtract the risk-free rate. In the market-only case, this is the market risk premium used in CAPM.

How is the APT line different from the security market line?

With one factor that is the market portfolio, they look the same: rf plus beta times the premium. APT allows several factors and relies on no-arbitrage, while CAPM relies on investors holding mean-variance efficient portfolios.