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

Capital Market Line and Two-Fund Separation Explained

Updated 11 October 2026 · Fact-checked

The capital market line (CML) is the capital allocation line that joins the risk-free asset to the tangency portfolio, the risky portfolio with the highest Sharpe ratio. Its formula is E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp. Two-fund separation says every investor holds just the risk-free asset and the tangency portfolio.

Understand Capital Market Line and Two-Fund Separation

Start with one risky portfolio and one risk-free asset. If you put weight w in the risky portfolio and 1 − w in the risk-free asset, both expected return and risk move in a straight line. Expected return is Rf + w × (E(Rr) − Rf). Standard deviation is w × σr, because the risk-free asset has zero volatility and zero covariance with anything. Plot these combinations and you get a straight line from the risk-free rate. This line is the capital allocation line (CAL). Its slope is the Sharpe ratio of the risky portfolio.

Every risky portfolio on the Markowitz efficient frontier gives a different CAL. A steeper line is better, because for any level of risk it offers more return. So you want the CAL with the highest slope. That line just touches the efficient frontier at one point. This point is the tangency portfolio. It is the risky portfolio with the maximum Sharpe ratio.

The CAL drawn through the tangency portfolio is the capital market line (CML). Points between Rf and the tangency portfolio mean you lend at the risk-free rate (0 < w < 1). Points beyond the tangency portfolio mean you borrow at the risk-free rate and invest more than 100% in the tangency portfolio (w > 1). This assumes you can borrow at Rf.

The two-fund separation theorem follows. The choice has two separate parts. First, find the tangency portfolio. This is the same for all investors who share the same inputs (expected returns, volatilities and correlations). Second, decide how much to hold in it versus the risk-free asset. This depends on your risk aversion. Investors differ only in the mix, not in the risky portfolio.

In CAPM, all investors have the same beliefs, so the tangency portfolio is the market portfolio. Then the CML is E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp. Do not confuse it with the security market line (SML). The CML plots return against total risk (σ) and applies only to efficient portfolios. The SML plots return against beta and applies to any asset or portfolio.

Key formulas to remember

Two-asset combination (risk-free plus risky): E(Rp) = w × E(Rr) + (1 − w) × Rf, and σp = w × σr; w is the weight in the risky portfolio. w > 1 means borrowing at Rf. Assumes w ≥ 0 for the risky portfolio.
Capital allocation line (CAL): E(Rp) = Rf + [(E(Rr) − Rf) ÷ σr] × σp; Slope is the Sharpe ratio of the risky portfolio r.
Capital market line (CML): E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp; Uses the market (tangency) portfolio. Slope is the market price of risk per unit of σ.
Sharpe ratio: SR = (E(Rp) − Rf) ÷ σp; Equal to the CML slope for any portfolio on the CML.
Weight for a target risk: w = σtarget ÷ σm; Weight in the tangency portfolio needed to reach a chosen total volatility.
Tangency portfolio (definition): Maximise [E(Rp) − Rf] ÷ σp over risky portfolios; The point where the CAL touches the efficient frontier.

How to solve Capital Market Line and Two-Fund Separation questions

Use this method for any CML, CAL or separation question. Work with total risk (σ), not beta, unless the question asks for the SML.

1Identify the inputs: Rf, the risky or market portfolio's expected return and σ, and what you must find (return, σ, weight or Sharpe ratio).
2Compute the slope: (E(Rm) − Rf) ÷ σm. This is the Sharpe ratio of the tangency portfolio.
3If a weight is given, find the portfolio σ as w × σm and the return as Rf + w × (E(Rm) − Rf).
4If a target return or σ is given, solve for w. For σ, w = σtarget ÷ σm. For return, w = (target − Rf) ÷ (E(Rm) − Rf).
5Interpret w: below 1 means lending at Rf, above 1 means borrowing at Rf, and the money in the risk-free asset is 1 − w.
6Check that the portfolio lies on the line. Its Sharpe ratio must equal the slope.
7For concept questions, state that the risky mix is the same for all investors and only w differs.

Quickest way: Slope-and-scale shortcut

When to use it: Use it when the question gives Rf, E(Rm) and σm and asks for the expected return, risk or weight of a mixed portfolio.

Compute the market risk premium: E(Rm) − Rf.
Compute w from the target. For a given σ, w = σp ÷ σm. For a given return, w = (target − Rf) ÷ premium.
Expected return = Rf + w × premium. σp = w × σm.
Sanity check: with w = 1 you must get the market portfolio. With w = 0 you get Rf.
On a calculator, store the slope in memory and reuse it for each option.

Common mistakes in Capital Market Line and Two-Fund Separation

Confusing the CML with the SML.
Both lines start at Rf and slope upward, and both involve the market premium.
Fix: CML: x-axis is σ, applies to efficient portfolios only. SML: x-axis is beta, applies to all assets. Look at the axis label first.
Using the weighted average of volatilities when a risk-free asset is mixed in, then adding the risk-free asset's risk.
Students apply the general two-asset variance formula with a correlation term.
Fix: The risk-free asset has σ = 0 and zero covariance, so σp = w × σr exactly.
Treating the tangency portfolio as the minimum-variance portfolio.
Both sit on the efficient frontier, so they get blurred together.
Fix: The tangency portfolio maximises the Sharpe ratio. The minimum-variance portfolio has the lowest σ. They are different portfolios on the frontier for realistic values of Rf.
Thinking more risk-averse investors hold a different risky portfolio.
Intuition says risk preferences should change the whole portfolio.
Fix: Separation theorem: risk aversion only sets the weight w. The risky portfolio is the same for everyone who shares the same inputs.
Forgetting that w > 1 means borrowing, and putting a positive weight in the risk-free asset.
Students assume weights must lie between 0 and 1.
Fix: Risk-free weight is 1 − w. If w = 1.5, the risk-free weight is −0.5, meaning borrowing at Rf.
Applying the CML to an individual stock.
The formula looks like it prices any asset.
Fix: A single stock lies below the CML because it holds diversifiable risk. Use the SML for individual securities.

Worked examples

Example 1

The risk-free rate is 3%. The market portfolio has expected return 9% and standard deviation 15%. An investor wants a portfolio with a standard deviation of 12%, using the CML. What is its expected return, and what is the weight in the risk-free asset?

Show the solution

Slope = (9% − 3%) ÷ 15% = 0.40.
Weight in market: w = 12% ÷ 15% = 0.80.
Expected return = 3% + 0.40 × 12% = 3% + 4.8% = 7.8%.
Check: 0.8 × 9% + 0.2 × 3% = 7.2% + 0.6% = 7.8%.
Risk-free weight = 1 − 0.80 = 0.20.

Answer: Expected return is 7.8%, with 20% in the risk-free asset and 80% in the market portfolio.

Example 2

Rf = 2%. The tangency portfolio has expected return 8% and standard deviation 10%. An investor with $1,000,000 wants an expected return of 11%. Using the CML, how much must be borrowed at the risk-free rate, and what is the portfolio's standard deviation?

Show the solution

Market premium = 8% − 2% = 6%.
Weight in tangency: w = (11% − 2%) ÷ 6% = 9% ÷ 6% = 1.5.
Invested in tangency portfolio = 1.5 × $1,000,000 = $1,500,000.
Borrowing = $1,500,000 − $1,000,000 = $500,000 at 2%.
Standard deviation = 1.5 × 10% = 15%.
Check Sharpe ratio: (11% − 2%) ÷ 15% = 0.60, equal to the slope 6% ÷ 10% = 0.60.

Answer: Borrow $500,000 at the risk-free rate. The portfolio has a standard deviation of 15%.

Exam tips

Read the axis. If the question uses σ, the answer is on the CML. If it uses beta, it is the SML.
Expect questions that test the separation theorem in words: the risky portfolio is identical for all investors, and only the risk-free weight differs.
Always compute the slope first. It is the Sharpe ratio of the tangency portfolio and is reused in every option.
Watch for w > 1. A required return above the market's means leveraged investing, so the risk-free weight is negative.
An asset or portfolio plotted below the CML is inefficient or holds diversifiable risk. Know how to state that.

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

Capital Market Line and Two-Fund Separation in other exams

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

Capital Market Line and Two-Fund Separation: frequently asked questions

What is the difference between the capital market line and the security market line?

The CML plots expected return against total risk (standard deviation) and covers only efficient portfolios of the risk-free asset and the market portfolio. The SML plots expected return against beta and prices any asset or portfolio. Efficient portfolios lie on both lines; individual stocks lie only on the SML.

What is the tangency portfolio?

It is the risky portfolio with the highest Sharpe ratio. Graphically, it is where a line from the risk-free rate just touches the efficient frontier. In CAPM, with homogeneous beliefs, it is the market portfolio.

What does the two-fund separation theorem say?

Every investor can build their optimal portfolio from two funds: the risk-free asset and the tangency portfolio. Investors who share the same inputs hold the same risky portfolio. Their risk aversion only decides how much goes to each fund.

Is the slope of the CML the Sharpe ratio?

Yes. The slope is (E(Rm) − Rf) ÷ σm, which is the Sharpe ratio of the market portfolio. Any portfolio on the CML has this same Sharpe ratio.