If investors are paid only for the risk they cannot diversify away, then the return that investors require from a share should depend on one thing: how strongly the share moves with the whole market. That idea, worked out into a formula, is the capital asset pricing model.
It is the most widely used way to estimate the cost of equity of a listed company, to judge whether a share is cheap or dear, and to set the required return for a project. It is a model with many simplifications, and an alternative, arbitrage pricing theory, allows more than one source of systematic risk.
This topic explains beta and how it is calculated, the capital asset pricing model and the security market line, beta of a portfolio, alpha and what it says about mispricing, the assumptions, advantages and limitations of the model, and arbitrage pricing theory, with worked examples. The next topic covers the cost of debt.
The capital asset pricing model was used to find the cost of equity in November Diet, 2025, Question 3(a), together with the asset beta and the re-geared beta (7 marks in all). Read systematic and unsystematic risk and the efficient frontier first.
In 30 seconds
The capital asset pricing model (CAPM): expected return = Rf + β(Rm − Rf), where Rf is the risk-free rate, Rm the return on the market and (Rm − Rf) the market risk premium. Beta measures systematic risk: β = covariance of the share with the market ÷ variance of the market = ρσsσm ÷ σm2; the market has β = 1 and the risk-free asset β = 0. A portfolio’s beta is the weighted average of its betas.
On the security market line, a share whose forecast return is above the required return is undervalued (positive alpha: buy); below, overvalued (negative alpha: sell). The cost of equity = the CAPM return. Arbitrage pricing theory (APT) uses several risk factors: R = Rf + β1(premium 1) + β2(premium 2) + … . The model’s inputs are hard to estimate and its assumptions are simplifications.
The capital asset pricing model
The capital asset pricing model (CAPM) is an equilibrium model that predicts the expected return on a security from the return on the market, the security’s beta and the risk-free rate. It assumes that investors hold diversified portfolios and are therefore concerned only with systematic risk. The formula:
R = Rf + β (Rm − Rf)
Here Rf is the risk-free rate, the return on an investment with no risk (in practice the yield on a government security of matching term; the real risk-free rate is the yield less inflation); Rm is the expected return on the market portfolio; and Rm − Rf is the market risk premium, the extra return that investors require for holding the whole market instead of the risk-free asset.
Example: Rf = 7%, Rm = 15% (premium 8%) and β = 1.3: the required return is 7 + 1.3 × 8 = 17.4%. A share with β above 1 amplifies market movements and requires a higher return than the market; one with β below 1 requires less. For a listed company the required return on its shares is the cost of equity, which can be set beside the estimate from the dividend valuation model.
Beta
Beta is the measure of a security’s systematic risk relative to the market: β = covariance(share, market) ÷ variance(market) = ρsmσsσm ÷ σm2. If a share has σ = 30%, its correlation with the market is 0.6 and the market’s σ is 20%, then β = 0.6 × 30 × 20 ÷ 400 = 0.9. The beta of the market portfolio is 1 and of the risk-free asset 0.
The beta of a portfolio is the weighted average of the betas of its holdings: shares with weights 40%, 35% and 25% and betas 1.2, 0.8 and 1.5 give a portfolio beta of 0.48 + 0.28 + 0.375 = 1.135, and a required return of 7 + 1.135 × 8 = 16.08%.
The security market line and alpha
The security market line (SML) is the graph of the CAPM: required return against beta. It starts at Rf when β = 0 and has a slope equal to the market risk premium. (The capital market line uses total risk and holds only efficient portfolios; the SML uses beta and holds every correctly priced security.) A security’s forecast return can be compared with its required return: the difference is its alpha.
If the forecast return is above the line (positive alpha), the security gives more than the required return for its risk and is undervalued: buy. If it is below (negative alpha), it is overvalued: sell. If it is on the line (alpha zero) it is correctly priced: hold.
Example: Rf = 7%, Rm = 15% and the variance of the market is 16 (%2). Four shares:
| Share | Covariance with market | Beta | Required return % | Forecast return % | Alpha % | Action |
|---|---|---|---|---|---|---|
| A | 20 | 1.25 | 17.0 | 16.0 | −1.0 | Sell (overvalued) |
| B | 12 | 0.75 | 13.0 | 13.0 | 0.0 | Hold |
| C | 16 | 1.00 | 15.0 | 15.8 | +0.8 | Buy (undervalued) |
| D | 28 | 1.75 | 21.0 | 21.4 | +0.4 | Buy (undervalued) |
Alpha is temporary: if the market agrees the share is mispriced, its price will move until the forecast return equals the required return. For decisions on portfolios under portfolio theory choose the highest return for the lowest risk; under CAPM, compare the forecast with the required return and prefer the highest positive alpha.
Assumptions, advantages and limitations
- Assumptions: investors can borrow and lend at the risk-free rate (which makes the capital market line straight); the market is efficient and investors share the same expectations; they are risk averse and diversified; one time period; no transaction costs; and taxes do not distort choices.
- Advantages: it gives a measurable relationship between risk and return; it concentrates on the systematic risk that matters; and it provides a way to estimate the cost of equity (and, with the debt, the cost of capital) and the required return on projects with different risk.
- Limitations: the risk-free rate, the beta and the market return are hard to estimate and differ with the data and period used (a beta measured on past returns may not hold in future); a single factor may not explain returns; the assumptions are unrealistic (borrowing at the risk-free rate, no costs); and in markets with few traded shares or low liquidity, betas are less reliable.
Arbitrage pricing theory
The CAPM is a single-factor model: the only source of systematic risk is the market. Arbitrage pricing theory (APT) suggests there are several macroeconomic and industry factors that affect returns, such as inflation, interest rates, industrial output and the price of oil, and each factor has its own beta (the sensitivity of the share’s return to the factor) and a risk premium: R = Rf + β1(premium1) + β2(premium2) + β3(premium3) + ….
Example: Rf = 6%; inflation premium 3% with β = 0.8; industrial production premium 5% with β = 1.2; oil price premium 2% with β = −0.5. The required return is 6 + 0.8 × 3 + 1.2 × 5 + (−0.5) × 2 = 6 + 2.4 + 6.0 − 1.0 = 13.4%. If a security’s price implies a different return from the model it is mispriced, and arbitrageurs will buy or sell until it is in line. The difficulty with APT is that the theory does not say which factors matter or what their premiums are; they must be estimated from data.
The ideas in more depth
Why beta is a covariance ratio
Beta asks how much of the market’s movement shows up in the share. The covariance measures the joint movement; dividing by the market variance converts it into the change in the share’s return for each 1% change in the market. A beta of 1.25 means that if the market return rises by 1% the share is expected to rise by 1.25%, and fall by the same proportion if the market falls.
Using the model in practice
In Nigeria the risk-free rate is usually taken from the yield on Federal Government of Nigeria bonds or treasury bills of suitable term, the market return from the long-run return on the stock market index, and beta from the regression of the share’s returns on the market’s. Each choice affects the answer, so a question will give the figures, and an analyst should show the sensitivity of the cost of equity to them.
A share that trades rarely gives a poor beta. An unlisted company has no beta of its own, which is why the betas of similar listed companies are adjusted and used (the next topics).
Sources and references
- No accounting standard governs the model. The capital asset pricing model derives from the work of Sharpe, Lintner and Mossin; arbitrage pricing theory from Ross. The CAPM formula and the asset and equity beta formulae are on the formula sheet of the paper.
These points are stated as at 5 November 2026.
Worked examples
Example 1: the required return. Rf 7%; Rm 15%; β 1.3. Required: the cost of equity.
- Premium.
15 − 7 = 8%. - Cost.
7 + 1.3 × 8 = 17.4%.
Example 2: beta from covariance. Covariance with the market 20; market variance 16. Required: the beta.
- Beta.
20 ÷ 16 = 1.25.
Example 3: alpha. Required return 17.0%; forecast 16.0%. Required: the alpha and the advice.
- Alpha.
16.0 − 17.0 = −1.0%. - Advice. Overvalued: sell.
How to answer in the exam
- Calculate beta first if only covariance or correlation is given.
- Apply the CAPM with the risk premium shown and state what the result is used for.
- For alpha, compare forecast with required return and give the action.
- Evaluate the model: advantages, limitations and the assumptions that fail in practice.
Common mistakes in the exam
- Using Rm instead of the premium Rm − Rf multiplied by beta.
- Dividing covariance by the market standard deviation instead of the variance.
- Using total risk (standard deviation) in place of beta.
- Reversing the buy and sell advice for alpha.
- Forgetting that a portfolio beta is a weighted average.
- Describing APT as a single-factor model.
Quick revision
- CAPM: R = Rf + β(Rm − Rf); β = covariance ÷ market variance.
- Market beta 1; risk-free beta 0; portfolio beta = weighted average.
- Above the SML: undervalued (buy); below: overvalued (sell).
- APT: several factors, each with a beta and a premium.
Practice questions
These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.
Recall check: multiple choice
Question 1. The capital asset pricing model states that the expected return is:
- Rm + βRf
- β × Rm
- Rf + β(Rm − Rf)
- Rf − β
Show the answer
C. The risk-free rate plus beta times the market risk premium.
Question 2. The beta of the market portfolio is:
- 0
- 1
- 2
- The risk-free rate
Show the answer
B. It moves exactly with itself.
Question 3. Beta equals:
- Covariance with the market ÷ variance of the market
- Covariance ÷ market standard deviation
- Variance of the share
- Rm − Rf
Show the answer
A. The share’s sensitivity to the market.
Question 4. The beta of the risk-free asset is:
- 1
- 2
- The market return
- Zero
Show the answer
D. It has no systematic risk.
Question 5. The security market line shows:
- Return against total risk
- Required return against beta
- Price against earnings
- Profit against sales
Show the answer
B. It is the graph of the CAPM.
Question 6. A share plotting above the security market line is:
- Undervalued, with a positive alpha
- Overvalued
- Correctly priced
- Risk free
Show the answer
A. Its forecast return exceeds the required return.
Question 7. The beta of a portfolio is:
- The highest beta
- The lowest beta
- The weighted average of the betas
- Always 1
Show the answer
C. Weighted by value.
Question 8. Arbitrage pricing theory differs from the CAPM in using:
- No risk-free rate
- One factor
- Total risk
- Several factors
Show the answer
D. Each with its own beta and premium.
Question 9. A limitation of the CAPM is that:
- It ignores risk
- It needs no inputs
- Its inputs are hard to estimate and its assumptions are unrealistic
- It is only for bonds
Show the answer
C. The beta and market return vary with the data used.
Question 10. The market risk premium is:
- Rm − Rf
- Rf
- Rm
- Rf − Rm
Show the answer
A. The extra return for holding the market.
Section B style: written questions
Question 11 (10 marks). (a) State the CAPM formula and define each term. (4 marks) (b) State THREE advantages and TWO limitations. (5 marks) (c) State how APT differs. (1 mark)
Question 12 (15 marks). The risk-free rate is 8% and the market return 16%. (a) Calculate the cost of equity of a share with beta 0.9. (4 marks) (b) A portfolio is half in a share with beta 0.8 and half in one with beta 1.4; calculate its beta and its required return. (6 marks) (c) State what the cost of equity is used for. (5 marks)
A further written question
Question 13 (20 marks). The risk-free rate is 6%, the market return 14% and the market variance 16 (%2). Three shares have these covariances with the market and forecast returns: P 24 and 18.6%; Q 16 and 14.0%; R 12 and 12.9%.
(a) Calculate each beta. (6 marks) (b) Calculate each required return. (6 marks) (c) Calculate each alpha and state the advice. (6 marks) (d) State what alpha means. (2 marks)
Model answers
Show the model answer to Question 11
(a) R = Rf + β(Rm − Rf), where Rf is the risk-free rate, Rm the market return, (Rm − Rf) the market risk premium and β the share’s systematic risk relative to the market (4).
(b) Advantages: measurable risk and return link; uses systematic risk; estimates the cost of equity and project returns (1 each, any three). Limitations: inputs hard to estimate; unrealistic assumptions (1 each, any two).
(c) APT uses several factors, not only the market (1).
Show the model answer to Question 12
(a) 8 + 0.9 × (16 − 8) = 8 + 7.2 = 15.2% (4).
(b)
- Beta 0.5 × 0.8 + 0.5 × 1.4 = 1.1 (3)
- required return 8 + 1.1 × 8 = 16.8% (3).
(c) As the discount rate for the company’s equity cash flows, as the equity element in the weighted average cost of capital, and as the basis of the return required on projects of similar risk (5).
Show the model answer to Question 13
(a)
- P 24 ÷ 16 = 1.5
- Q 16 ÷ 16 = 1.0
- R 12 ÷ 16 = 0.75 (2 each).
(b)
- P 6 + 1.5 × 8 = 18.0%
- Q 6 + 1.0 × 8 = 14.0%
- R 6 + 0.75 × 8 = 12.0% (2 each).
(c)
- P 18.6 − 18.0 = +0.6, undervalued: buy
- Q 14.0 − 14.0 = 0, correctly priced: hold
- R 12.9 − 12.0 = +0.9, undervalued: buy (2 each).
(d) The difference between the forecast and the required return, a sign of mispricing that should disappear as the market adjusts (2).
Pathfinder on The capital asset pricing model and arbitrage pricing theory
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Part (a) (7 marks) asks for the asset beta, the re-geared equity beta, the cost of equity using the capital asset pricing model and the project cost of capital; part (b) (13 marks) asks why asset betas differ and about proxy companies.
Revise first: The capital asset pricing model
Source: ICAN Pathfinder, Financial Management. Diets covered: Every diet in the Pathfinder, November Diet, 2025 and May Diet, 2026, was reviewed here. The diets listed above test this topic directly..
Spotted a mistake? Tell us and we will check it.
