Financial Management and Business Data Analytics · Risk and Return

Arbitrage Pricing Theory, CAPM Difference and Sharpe Ratio

Updated 10 October 2026 · Fact-checked

Arbitrage Pricing Theory (APT) 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 it, list the factors, multiply each beta by its premium, add them, then add the risk-free rate.

Understand Arbitrage Pricing Theory and Other Models

CAPM explains a share's expected return using one factor: its beta against the market. In practice, returns move with more than the market. Inflation, interest rates, industrial output and exchange rates also push prices up or down. Arbitrage Pricing Theory (APT) accepts this and uses several factors.

The idea behind APT is simple. If two assets carry the same risk but offer different expected returns, traders will buy the cheap one and sell the dear one to make a riskless profit. This is arbitrage. Their trading pushes prices until the profit disappears. So in equilibrium, expected return must line up with the asset's exposure to the common risk factors.

In APT, each factor has a sensitivity (factor beta): how much the asset's return changes for a one-unit move in that factor. Each factor also has a risk premium: the extra return investors expect above the risk-free rate for bearing that factor's risk. APT does not name the factors. You are given them in the question.

APT and CAPM differ mainly in structure. CAPM is a single-factor model and needs a market portfolio. APT is multi-factor, rests on the no-arbitrage argument and does not need the market portfolio. CAPM is a special case of APT with one factor, the market.

CAPM and APT predict return. Performance measures judge how well a portfolio rewarded you for the risk taken. The Sharpe ratio divides excess return by total risk (standard deviation). The Treynor ratio divides excess return by systematic risk (beta). Jensen's alpha is the actual return minus the CAPM-required return.

Key rules to remember

APT expected return: E(R) = Rf + β1 × RP1 + β2 × RP2 + … + βn × RPn; Rf is the risk-free rate. Each RP is that factor's risk premium, not the factor's total return, unless the question says so.
CAPM (single-factor comparison): E(R) = Rf + β × (Rm − Rf); Use when only market beta is given.
Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; σp is the standard deviation of portfolio returns. Measures reward per unit of total risk.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Measures reward per unit of systematic risk. Suited to well-diversified portfolios.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Positive alpha means the portfolio beat its CAPM-required return.

How to solve Arbitrage Pricing Theory and Other Models questions

Use this order for any question on APT, CAPM comparison or performance ratios.

1Read the question and note what is asked: required return, a comparison, or a performance ratio.
2Write down the given data with labels: Rf, each factor beta, each factor premium, Rp, σ, β, Rm.
3Check whether the factor data is a risk premium or a total factor return. If it is a total return, subtract Rf first.
4For APT, multiply each beta by its premium, add all products, then add Rf.
5For Sharpe or Treynor, compute excess return (Rp − Rf) first, then divide by σ or β respectively.
6Compare with the benchmark or other portfolios. The higher ratio is better, and say so in one sentence.
7Write the answer in percentage with the unit and a short conclusion.

Quickest way: Three-line APT and ratio shortcut

When to use it: Use in MCQs and when time is short in a 14-mark question.

Convert every percentage to a decimal or keep all in percent consistently.
For APT, add the products β × premium mentally, then add Rf last.
For ranking portfolios, compute only the ratio asked. Do not compute alpha unless requested.
Sanity check: if all betas are positive and premiums positive, the answer must exceed Rf.

Common mistakes in Arbitrage Pricing Theory and Other Models

Forgetting to add the risk-free rate at the end of the APT formula.
Students focus on the factor products and treat their sum as the answer.
Fix: The sum of β × premium is only the risk premium. Always add Rf to get expected return.
Using total factor returns instead of premiums.
The question gives a factor's expected return, and students multiply it directly by beta.
Fix: Check whether the data is a premium. If it is a total return, deduct Rf to get the premium.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio.
The two ratios look alike and students mix the denominators.
Fix: Remember: Sharpe uses Standard deviation (S for S), Treynor uses beta (T for the systematic 'tilt').
Saying APT names the factors.
Students assume it lists inflation and GDP as fixed factors.
Fix: State that APT does not specify the factors or their number. They must be identified or given.
Concluding that a higher return means better performance.
Risk is ignored while comparing portfolios.
Fix: Compare risk-adjusted ratios. A portfolio with a lower return can still have a higher Sharpe ratio.

Worked examples

Example 1

The risk-free rate is 6%. A share has sensitivities of 1.2 to inflation risk, 0.8 to industrial output and 0.5 to interest rate changes. The risk premiums are 3%, 4% and 2% respectively. Find the expected return using APT.

Show the solution

Inflation: 1.2 × 3% = 3.6%
Industrial output: 0.8 × 4% = 3.2%
Interest rate: 0.5 × 2% = 1.0%
Total factor premium = 3.6% + 3.2% + 1.0% = 7.8%
Expected return = 6% + 7.8% = 13.8%

Answer: Expected return as per APT is 13.8%.

Example 2

Portfolio A earns 16% with a standard deviation of 12% and a beta of 1.2. Portfolio B earns 14% with a standard deviation of 8% and a beta of 0.8. The risk-free rate is 6% and the market return is 13%. Compute the Sharpe ratio, Treynor ratio and Jensen's alpha for each, and state which is better.

Show the solution

Excess return A = 16% − 6% = 10%. Excess return B = 14% − 6% = 8%.
Sharpe A = 10 ÷ 12 = 0.833. Sharpe B = 8 ÷ 8 = 1.000.
Treynor A = 10 ÷ 1.2 = 8.33. Treynor B = 8 ÷ 0.8 = 10.00.
Market premium = 13% − 6% = 7%.
Required return A = 6% + 1.2 × 7% = 14.4%. Alpha A = 16% − 14.4% = 1.6%.
Required return B = 6% + 0.8 × 7% = 11.6%. Alpha B = 14% − 11.6% = 2.4%.
B is higher on all three measures.

Answer: Sharpe: A 0.833, B 1.000. Treynor: A 8.33, B 10.00. Alpha: A 1.6%, B 2.4%. Portfolio B gives better risk-adjusted performance.

Exam tips

In APT numericals, the whole mark scheme is the list of beta × premium lines. Show each line separately to earn step marks.
For the theory question on difference between CAPM and APT, write four points: number of factors, basis, market portfolio need, and specification of factors.
In MCQs, check the denominator first: standard deviation means Sharpe, beta means Treynor.
Always add a one-line interpretation after ratios, such as which portfolio gives more reward per unit of risk.
Keep all rates in the same form, either percent or decimal, through the sum.

Practice questions from Risk and Return

Arbitrage Pricing Theory and Other 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 Other Models: frequently asked questions

What is the difference between CAPM and APT?

CAPM uses one factor, the market, and depends on the market portfolio. APT uses several factors and rests on the no-arbitrage argument. CAPM is a special case of APT with one factor.

Does APT tell us which factors to use?

No. APT does not name the factors or fix their number. In exam questions, the factors, betas and premiums are given to you.

When should I use Sharpe ratio and when Treynor ratio?

Use Sharpe when total risk matters, for example for a whole portfolio. Use Treynor when the portfolio is well diversified and only systematic risk matters.

Can the Sharpe ratio be negative?

Yes. If the portfolio return is below the risk-free rate, excess return is negative and so is the ratio. It means the portfolio did worse than a risk-free investment.