Economic Modelling · Capital Asset Pricing Model (CAPM)

CAPM Assumptions and the Capital Market Line Explained

Updated 11 October 2026 · Fact-checked

CAPM assumes investors are rational mean-variance optimisers with the same beliefs, one period, and a common risk-free rate. Everyone then holds the risk-free asset plus the market portfolio (two-fund separation). The Capital Market Line plots expected return against standard deviation for these efficient portfolios: E(R_P) = r_f + [(E(R_M) − r_f) ÷ σ_M] × σ_P.

Understand CAPM Assumptions and the Capital Market Line

The Capital Asset Pricing Model (CAPM) starts from mean-variance theory. Each investor cares only about the expected return and variance of a portfolio over one period. More return is preferred, and less variance is preferred. Investors are risk averse.

CAPM then adds assumptions so that all investors reach the same answer. The usual set is:

All investors have the same one-period horizon.
All investors have the same (homogeneous) beliefs about expected returns, variances and covariances.
Investors are rational, risk averse and use mean-variance criteria.
Markets are perfect: no taxes, no transaction costs, assets are infinitely divisible, and investors are price takers.
All investors can lend and borrow unlimited amounts at the same risk-free rate.
All assets are marketable, and short selling is allowed in the standard form.

A risk-free asset has a known return and zero variance, and so zero covariance with every risky asset. Mix it with a risky portfolio P with weight w in P. Then the expected return is (1 − w) r_f + w E(R_P), and the standard deviation is w σ_P (for w ≥ 0). Both are linear in w, so the mix lies on a straight line. Borrowing at r_f means w > 1, which extends the line beyond P.

With a risk-free asset, the best straight line from the risk-free point is the one with the steepest slope. It touches the risky-asset efficient frontier at a single point, the tangency portfolio. Every investor holds this same risky portfolio, whatever their risk aversion. This is the two-fund separation theorem: every efficient portfolio is a mix of the risk-free asset and the tangency portfolio. Risk aversion only decides the mix, not the risky holdings.

If all investors hold the same risky portfolio and markets clear, that portfolio must be the market portfolio: every risky asset in proportion to its market value. The line from r_f through the market portfolio M is the Capital Market Line (CML). Its slope, (E(R_M) − r_f) ÷ σ_M, is the market price of risk per unit of standard deviation. The CML applies only to efficient portfolios. The Security Market Line (SML) is different: it uses beta, not standard deviation, and applies to every asset and portfolio.

Key rules to remember

Portfolio of risk-free asset and risky portfolio P: E(R) = (1 − w) r_f + w E(R_P); σ = |w| σ_P; w is the proportion in P. w > 1 means borrowing at r_f to invest more in P.
Capital Market Line: E(R_P) = r_f + [(E(R_M) − r_f) ÷ σ_M] × σ_P; Applies to efficient portfolios only. Intercept is r_f; slope is the market price of risk.
Slope of the CML (Sharpe ratio of the market): (E(R_M) − r_f) ÷ σ_M; Extra expected return per unit of standard deviation.
Market portfolio weights: w_i = market value of asset i ÷ total market value of all risky assets; All investors hold the risky assets in these proportions under CAPM.
Standard deviation needed for a target return: σ_P = [E(R_P) − r_f] ÷ slope; Rearranged CML. Use it when the question gives a target return.

How to solve CAPM Assumptions and the Capital Market Line questions

Use this method for assumption, theory and CML calculation questions.

1Read what is asked: assumptions, a theory explanation, or a calculation on the CML.
2For theory, list the relevant assumptions and link each to its consequence, for example homogeneous beliefs lead to one tangency portfolio.
3For a calculation, write down r_f, E(R_M) and σ_M, and check that all are in the same units (per cent or decimals).
4Compute the CML slope: (E(R_M) − r_f) ÷ σ_M.
5Write the CML equation and substitute the given σ_P or the target return.
6Interpret the weight w in the market portfolio: w < 1 is lending, w > 1 is borrowing. Check the sign.
7State the result with units and one sentence of interpretation, and state any assumption you used.

Quickest way: Weight-first shortcut for CML questions

When to use it: Use it when the question gives a target return or target risk and asks for the split between the risk-free asset and the market portfolio.

Find w directly. For a target return: w = (target − r_f) ÷ (E(R_M) − r_f).
For a target risk: w = σ_target ÷ σ_M.
Then 1 − w is the risk-free weight. Negative means borrowing.
Get the missing quantity: σ = w σ_M, or E(R) = r_f + w (E(R_M) − r_f).

Common mistakes in CAPM Assumptions and the Capital Market Line

Applying the CML to individual securities or inefficient portfolios.
The CML and the SML look similar and both involve r_f and the market.
Fix: Remember: the CML uses σ and holds for efficient portfolios only. Individual assets lie below it. Use the SML with beta for any asset.
Using beta on the x-axis of the CML.
Confusion with the SML diagram.
Fix: CML axes are standard deviation (x) and expected return (y). SML axes are beta (x) and expected return (y).
Saying risk-averse investors hold different risky portfolios.
Students link risk aversion to asset choice.
Fix: Under two-fund separation, all hold the same tangency portfolio. Risk aversion changes only the split with the risk-free asset.
Forgetting that borrowing means w > 1 and the risk-free weight is negative.
Weights are assumed to lie between 0 and 1.
Fix: Always compute w and check it. If target return exceeds E(R_M), you must borrow.
Listing assumptions without explaining why they matter.
Memorising a list for recall.
Fix: Attach a consequence to each: common risk-free rate gives one line, homogeneous beliefs give one tangency portfolio, no taxes or costs let all hold M.
Mixing per cent and decimals, for example using 8 and 0.12 together.
Data are given in different forms.
Fix: Convert everything to one form before computing the slope.

Worked examples

Example 1

Under CAPM, r_f = 6%, E(R_M) = 12% and σ_M = 15%. (a) Write the CML. (b) An efficient portfolio has standard deviation 20%. Find its expected return and the weights in the market portfolio and the risk-free asset.

Show the solution

Slope = (12% − 6%) ÷ 15% = 0.4.
CML: E(R_P) = 6% + 0.4 σ_P.
For σ_P = 20%: E(R_P) = 6% + 0.4 × 20% = 14%.
Weight in M: w = σ_P ÷ σ_M = 20 ÷ 15 = 4/3 ≈ 1.3333.
Risk-free weight = 1 − 4/3 = −1/3, so borrowing.
Check: (−1/3) × 6% + (4/3) × 12% = −2% + 16% = 14%. Correct.

Answer: CML: E(R_P) = 6% + 0.4 σ_P. The portfolio has expected return 14%. It holds 133.33% in the market portfolio and borrows 33.33% at the risk-free rate.

Example 2

With r_f = 5%, E(R_M) = 11% and σ_M = 12%, an investor wants an expected return of 8%. Find the weights and the standard deviation of the portfolio, and explain why another investor with a lower risk tolerance holds the same risky assets.

Show the solution

Market risk premium = 11% − 5% = 6%.
w = (8% − 5%) ÷ 6% = 0.5.
Risk-free weight = 1 − 0.5 = 0.5.
σ_P = w σ_M = 0.5 × 12% = 6%.
Check with the CML: slope = 6% ÷ 12% = 0.5; E(R) = 5% + 0.5 × 6% = 8%. Correct.
By two-fund separation, every investor holds the market portfolio as the only risky holding. A more risk-averse investor just puts a smaller w in M and more in the risk-free asset.

Answer: Hold 50% in the market portfolio and 50% in the risk-free asset. The standard deviation is 6%. All investors share the same risky portfolio; only w differs.

Exam tips

For assumption questions, give 5 or 6 assumptions and say what each one delivers. Marks usually go for the link, not the list.
In a CML sketch, label the axes (σ and expected return), the intercept r_f, the point M and the slope. Show that the line continues beyond M for borrowing.
Compare CML and SML in one line each when asked: CML is for efficient portfolios and uses σ; SML is for all assets and uses beta.
In computer-based papers, compute the slope in a separate cell or variable, then reuse it. Show the formula in your answer.
State when the assumptions fail, such as differing borrowing and lending rates, which breaks the single CML.

Practice questions from Capital Asset Pricing Model (CAPM)

CAPM Assumptions and the Capital Market Line in other exams

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

CAPM Assumptions and the Capital Market Line: frequently asked questions

What is the two-fund separation theorem?

It says that, with a risk-free asset available, every investor's optimal portfolio is a mix of the risk-free asset and one risky portfolio, the tangency portfolio. The investor's risk aversion decides the mix only. In CAPM, the tangency portfolio is the market portfolio.

How do you derive the Capital Market Line?

Combine the risk-free asset with a risky portfolio P: E(R) = r_f + w (E(R_P) − r_f) and σ = w σ_P. Eliminate w to get a straight line in (σ, E) space. Choose P to give the steepest slope, which is the tangency portfolio M, and you get the CML.

What is the difference between the CML and the SML?

The CML plots expected return against standard deviation and applies to efficient portfolios, mixes of the risk-free asset and M. The SML plots expected return against beta and applies to every asset and portfolio. Only efficient portfolios lie on both.

Why must the tangency portfolio be the market portfolio?

All investors have the same beliefs and so the same tangency portfolio. Markets must clear, so total demand for each risky asset equals its supply. The only portfolio that does this is the one holding assets in proportion to market value.