FRM Exam Part I · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

CAPM Limitations and Extensions for FRM Part I

Updated 11 October 2026 · Fact-checked

CAPM says expected return = Rf + β(E[Rm] − Rf), with beta as the only priced risk. Empirically, low-beta stocks earn more than predicted and the market portfolio cannot be observed. Extensions such as zero-beta CAPM and multifactor models relax the assumptions. You solve questions by identifying which assumption fails, then applying the right formula.

Understand CAPM Limitations and Extensions

The CAPM rests on strong assumptions: investors hold mean-variance efficient portfolios, share the same beliefs, borrow and lend at one risk-free rate, and face no taxes or frictions. Under these, only systematic risk (beta) earns a premium. Total risk splits into systematic risk and idiosyncratic risk, which diversification removes.

The model has known empirical problems. The estimated security market line is often flatter than predicted: low-beta assets earn higher returns than CAPM implies and high-beta assets earn less. Size, value and momentum effects show returns that beta does not explain. Betas also change over time, so estimates are noisy.

Roll's critique is a testing problem. The true market portfolio contains every risky asset, including human capital and real estate, and cannot be observed. Any test uses a proxy such as a stock index. If the proxy is mean-variance efficient, the SML fits exactly by construction. If it is not, CAPM may appear to fail even when it is true. So CAPM is not cleanly testable.

Extensions relax the assumptions. Zero-beta CAPM (Black) drops the risk-free asset: the SML intercept becomes the expected return of a portfolio with zero beta to the market, usually above the risk-free rate. Multifactor models and the APT let several factors earn premiums. The APT needs no market portfolio, but assumes returns follow a factor structure and that arbitrage is ruled out in well-diversified portfolios. CAPM is a special case with one factor, the market.

For total risk decomposition, a single-factor regression gives σ² = β²σm² + σe². The first term is systematic variance and the second is idiosyncratic variance. R² is the systematic share of total variance.

Key formulas to remember

CAPM: E[Ri] = Rf + βi × (E[Rm] − Rf); Beta is the only priced risk. Beta = Cov(Ri, Rm) ÷ σm².
Zero-beta CAPM: E[Ri] = E[Rz] + βi × (E[Rm] − E[Rz]); Rz is the return on a portfolio with zero beta to the market. Used when there is no risk-free borrowing and lending at one rate.
Multifactor model: E[Ri] = Rf + βi1·λ1 + βi2·λ2 + … + βik·λk; λ is the risk premium of each factor. βik is the exposure to factor k.
Total variance decomposition: σi² = βi² × σm² + σe²; Systematic variance plus idiosyncratic variance. Assumes the residual is uncorrelated with the market.
R-squared link: R² = βi²σm² ÷ σi² = ρ² (single-factor case); Share of total variance that is systematic. Idiosyncratic share = 1 − R².
Alpha: αi = Ri − [Rf + βi(Rm − Rf)]; Excess return over the CAPM-required return. Zero under CAPM in equilibrium.

How to solve CAPM Limitations and Extensions questions

Most questions test either a conceptual weakness of CAPM or a calculation using an extended model. Use this order.

1Read the question and decide: concept (which assumption fails) or calculation (which model to apply).
2For concept questions, match the issue to the assumption: unobservable market means Roll's critique; no riskless asset means zero-beta CAPM; extra priced factors means multifactor or APT.
3For calculations, write down the model formula before inserting numbers.
4Convert all percentages to decimals and check that betas and premiums match the right factors.
5For zero-beta, use E[Rz] as the intercept, not Rf, and the market premium over E[Rz].
6For variance decomposition, compute systematic variance as β²σm², then idiosyncratic variance as total minus systematic.
7Take square roots only at the end when a standard deviation is requested.
8Check reasonableness: idiosyncratic variance cannot be negative and R² must lie between 0 and 1.

Quickest way: Match the clue to the fix

When to use it: Use for conceptual multiple-choice questions where you have about a minute.

Unobservable market portfolio or testability: Roll's critique.
No risk-free asset or restricted borrowing: zero-beta CAPM.
Returns explained by size, value, momentum: multifactor model.
Needs no market portfolio, arbitrage argument: APT.
Variance split: β²σm² is systematic, the rest is idiosyncratic.
Eliminate options that claim CAPM is proven false or that beta explains all returns.

Common mistakes in CAPM Limitations and Extensions

Saying Roll's critique proves CAPM is false.
The critique sounds like a rejection of the model.
Fix: It says CAPM cannot be tested without the true market portfolio. It is a testability issue, not a disproof.
Using Rf as the intercept in zero-beta CAPM.
Students memorise the standard SML and plug in directly.
Fix: Replace Rf with E[Rz] in both places in the formula.
Assuming the zero-beta portfolio must have a return equal to Rf.
Zero beta sounds riskless.
Fix: A zero-beta portfolio can still have idiosyncratic risk. Its expected return is typically above Rf when borrowing is restricted.
Treating APT and CAPM as identical.
Both give linear risk-return relations.
Fix: CAPM needs a mean-variance efficient market portfolio and one factor. APT allows several factors and relies on no-arbitrage with well-diversified portfolios.
Using beta instead of β² in the variance decomposition.
Mixing the return formula with the variance formula.
Fix: Variance scales with the square of beta: systematic variance = β²σm².
Taking the idiosyncratic standard deviation as total minus systematic standard deviation.
Subtracting standard deviations feels natural.
Fix: Subtract variances, then take the square root.

Worked examples

Example 1

A stock has beta 1.2. The expected market return is 9%, and the expected return on a zero-beta portfolio is 3%. The risk-free rate is 2%. Using zero-beta CAPM, what is the stock's expected return?

Show the solution

Formula: E[Ri] = E[Rz] + β × (E[Rm] − E[Rz]).
Substitute: 3% + 1.2 × (9% − 3%).
Market premium over zero-beta: 6%.
1.2 × 6% = 7.2%.
3% + 7.2% = 10.2%. The risk-free rate is not used.

Answer: 10.2%

Example 2

A stock has beta 1.5 and total volatility 30%. Market volatility is 16%. What is the idiosyncratic volatility, to one decimal place, and what is R²?

Show the solution

Total variance = 0.30² = 0.09.
Systematic variance = 1.5² × 0.16² = 2.25 × 0.0256 = 0.0576.
Idiosyncratic variance = 0.09 − 0.0576 = 0.0324.
Idiosyncratic volatility = √0.0324 = 0.18, or 18.0%.
R² = 0.0576 ÷ 0.09 = 0.64.

Answer: Idiosyncratic volatility is 18.0% and R² is 0.64 (64%).

Exam tips

Know the exact link: Roll's critique concerns the unobservable market portfolio, not the logic of the model.
In zero-beta questions, check which rate sits in the intercept before calculating.
For variance splits, use variances throughout and take the square root last.
Expect conceptual comparisons of CAPM and APT: number of factors, role of the market portfolio and the basis (equilibrium versus arbitrage).
A calculator helps with squares and square roots. Keep four decimals in intermediate steps.

Practice questions from Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

CAPM Limitations and Extensions in other exams

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

Actuarial Core Principles (IAI)Limitations, Tests and Extensions of CAPM

CAPM Limitations and Extensions: frequently asked questions

What are the main limitations of CAPM for FRM Part I?

The assumptions are unrealistic, such as one risk-free rate and homogeneous expectations. The market portfolio cannot be observed, per Roll's critique. Empirically, beta does not fully explain returns, since size, value and momentum matter.

What is zero-beta CAPM in simple terms?

It is CAPM without a risk-free asset. The intercept of the SML is the expected return on a portfolio uncorrelated with the market. That return is usually above the risk-free rate.

What is Roll's critique of CAPM?

The true market portfolio includes all assets and cannot be measured. Tests use proxies, so results depend on whether the proxy is efficient. CAPM is therefore hard to test conclusively.

What is the difference between CAPM and APT?

CAPM is an equilibrium model with one factor, the market, and relies on mean-variance efficiency. APT is a no-arbitrage model with several factors and does not need the market portfolio. CAPM can be seen as a special case of a factor model.