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IAI Actuarial Core Principles · Economic Modelling

Capital Asset Pricing Model (CAPM): Formula, Beta and Risk Premium

The **Capital Asset Pricing Model** says an asset's expected return equals the risk-free rate plus beta times the market risk premium: E(Rᵢ) = r_f + βᵢ[E(R_M) − r_f]. To solve questions, find beta as Cov(Rᵢ, R_M) ÷ Var(R_M), then substitute. Only systematic risk earns a premium.

What this chapter covers

This chapter explains how investors choose portfolios and how risk is priced in a market. It starts with mean-variance portfolio theory, where you judge a portfolio by its expected return and variance. From there you build the efficient frontier, then add a risk-free asset to reach the Capital Market Line and the Security Market Line.

The central result is that only systematic risk, measured by beta, is rewarded with extra expected return. Diversification removes specific risk, so the market does not pay for it. The last part of the chapter looks at where the model fails, how it is tested, and what extensions exist, such as multi-factor models.

In CM2 this chapter links to measures of investment risk, asset valuations and the wider economic modelling material. Beta and required return feed into discount rates and valuation. The ideas of utility, risk aversion and diversification from earlier chapters are used directly here. Expect both calculation and explanation questions.

CAPM is a core idea in CM2 and it supports many later topics, so time spent here pays back across the paper. Questions mix short calculations (portfolio variance, beta, required return) with written parts that ask you to state assumptions, interpret results or criticise the model. The calculations are mechanical once you know the formulas, so they are reliable marks. The written parts reward clear structure. Paper B can also use the same ideas in a computer-based task, such as computing beta from return data. Learn both the working and the reasoning.

Capital Asset Pricing Model (CAPM): topics in the order to study them

  1. 1Mean-Variance Portfolio Theory and Efficient FrontierEverything else rests on expected return, variance, covariance and diversification, so start here.
  2. 2CAPM Assumptions and the Capital Market LineOnce you have the frontier, adding a risk-free asset gives the CML and the market portfolio, and you need the assumptions that make this work.
  3. 3Security Market Line, Beta and Required ReturnThis turns the CML result into a pricing rule for individual assets, which is the part most often calculated.
  4. 4Limitations, Tests and Extensions of CAPMCriticism only makes sense after you know the model and its assumptions, so leave it for last.

How to prepare Capital Asset Pricing Model (CAPM)

Treat this chapter as one chain of ideas, from portfolio choice to pricing. Practise the algebra by hand first, then confirm it with data.

  1. Revise the basics: expected return, variance and covariance of two assets, and the formula for portfolio variance with weights w₁ and w₂.
  2. Draw the efficient frontier and explain in words why some portfolios are dominated. Do this from memory until it is easy.
  3. Write the assumptions of CAPM as a short list. For each one, note what would break if it failed.
  4. Derive or at least state the CML and SML, and be clear on the difference: the CML uses total risk (standard deviation) and applies to efficient portfolios, while the SML uses beta and applies to any asset.
  5. Practise calculations: beta from covariance and variance, required return from the SML, and whether an asset plots above or below the line.
  6. Learn the main criticisms and tests, with one sentence on each extension, then write a short answer under timed conditions.
  7. If you have access to the computer-based practice, calculate beta from a return series by regression and interpret the slope and intercept.

Common mistakes in Capital Asset Pricing Model (CAPM)

  • Mixing up the CML and the SML.

    Fix: Remember the axis: the CML plots against standard deviation and covers efficient portfolios only; the SML plots against beta and covers all assets.

  • Using the market return instead of the market risk premium in the SML.

    Fix: Write the premium E(R_M) − r_f as its own line before substituting.

  • Forgetting the covariance term in portfolio variance.

    Fix: Always write all three terms for two assets and check the sign of the correlation.

  • Squaring weights incorrectly or letting weights not sum to 1.

    Fix: Check that weights sum to 1 first, then compute w² terms separately.

  • Saying beta measures total risk.

    Fix: State that beta measures sensitivity to market movements, which is systematic risk only.

  • Giving a list of limitations with no explanation.

    Fix: Pair each criticism with the assumption it attacks and say what the consequence is for the model.

Last-day revision: Capital Asset Pricing Model (CAPM)

  • Portfolio expected return: E(Rₚ) = Σ wᵢE(Rᵢ), with weights summing to 1.
  • Two-asset variance: σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁, R₂).
  • Correlation: ρ = Cov(R₁, R₂) ÷ (σ₁σ₂), and it lies between −1 and +1.
  • The efficient frontier holds portfolios with the highest return for a given risk.
  • The CML: E(Rₚ) = r_f + [E(R_M) − r_f] ÷ σ_M × σₚ, for efficient portfolios.
  • The SML: E(Rᵢ) = r_f + βᵢ[E(R_M) − r_f], for any asset or portfolio.
  • Beta: βᵢ = Cov(Rᵢ, R_M) ÷ Var(R_M). The market has beta 1 and the risk-free asset has beta 0.
  • Only systematic risk is priced. Specific risk is removed by diversification.
  • An asset above the SML is underpriced (positive alpha). One below is overpriced.
  • The market risk premium is E(R_M) − r_f.
  • Key assumptions: mean-variance investors, common expectations, one period, borrowing and lending at r_f, no taxes or transaction costs.
  • Main limits: unrealistic assumptions, unobservable market portfolio, and weak empirical support for beta alone.

Capital Asset Pricing Model (CAPM) practice questions

Capital Asset Pricing Model (CAPM) in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Capital Asset Pricing Model (CAPM): frequently asked questions

Is CAPM tested in CM2 Paper A or Paper B?

Expect it in the written Paper A, where you may be asked to calculate and to explain. Paper B is a computer-based exam, so be ready to apply the ideas to data, such as estimating beta. Check the latest syllabus and past papers for the current pattern.

What is the difference between beta and standard deviation?

Standard deviation measures total risk, both systematic and specific. Beta measures only the sensitivity of an asset's return to the market's return. CAPM prices beta, not standard deviation.

Which formulas must I memorise for CAPM?

Learn the portfolio variance formula, the beta formula, the SML and the CML. If you can also explain each one in a sentence, you are well covered for both calculation and written questions.

Can an asset have a negative beta?

Yes. Its returns tend to move opposite to the market. The SML then gives an expected return below the risk-free rate, because the asset reduces portfolio risk.