Economic Modelling · Mean-variance portfolio theory

Risk-Free Asset and Capital Market Line: Finding the Tangency Portfolio

Updated 11 October 2026 · Fact-checked

Adding a risk-free asset lets investors combine it with one risky portfolio, the tangency portfolio. All efficient portfolios then lie on a straight line, the capital market line, from the risk-free rate through the tangency point. To solve questions, find the tangency portfolio, then mix it with cash to suit the target risk.

Understand Risk-Free Asset and Capital Market Line

Start with the risky-only case. With only risky assets, the efficient frontier is a curve in a graph of standard deviation (x-axis) against expected return (y-axis). Each investor picks a point on it according to their risk appetite.

Now add a risk-free asset. It has return Rf, zero variance and zero covariance with every risky asset. Mix a fraction w in a risky portfolio P with 1 − w in the risk-free asset. The expected return is w·E(P) + (1 − w)·Rf. The standard deviation is w·σP, because the risk-free part adds no variance and no covariance. Both are linear in w, so the mixes lie on a straight line.

You want the steepest line from the point (0, Rf) that still touches the risky frontier. Steepest means the highest slope (E(P) − Rf) ÷ σP, the Sharpe ratio. The point where this line touches the frontier is the tangency portfolio, T. Every investor who uses only mean and variance holds the same risky mix T.

This is the separation theorem. The decision splits in two. First, find T. This depends only on expected returns, variances and covariances, not on the investor. Second, decide how much to put in T and how much in the risk-free asset. This depends on risk aversion. A cautious investor holds some cash. An aggressive investor may borrow at Rf and hold more than 100% in T. Borrowing is allowed only if the question assumes borrowing and lending at the same risk-free rate.

The straight line from (0, Rf) through T is the capital market line (CML). Under CAPM, T is the market portfolio M. The CML then says: expected return = Rf + [(E(M) − Rf) ÷ σM] × σ. The slope is the market price of risk per unit of standard deviation.

Key rules to remember

Mix of risk-free asset and risky portfolio P: E(R) = w·E(P) + (1 − w)·Rf and σ = w·σP (for w ≥ 0); w is the fraction in P. If w > 1 the investor borrows at Rf. Use |w|·σP if w could be negative, but the CML normally uses w ≥ 0.
Sharpe ratio: S = (E(P) − Rf) ÷ σP; The tangency portfolio has the highest Sharpe ratio of all risky portfolios.
Capital market line: E(R) = Rf + [(E(M) − Rf) ÷ σM] × σ; Applies to efficient portfolios only. Under CAPM, the tangency portfolio is the market portfolio M.
Tangency portfolio, two risky assets (weight in asset 1): w1 = [μ1′·σ2² − μ2′·Cov] ÷ [μ1′·σ2² + μ2′·σ1² − (μ1′ + μ2′)·Cov], where μi′ = E(Ri) − Rf and Cov = Cov(R1, R2); w2 = 1 − w1. This gives the tangency weights when the risky weights sum to 1.
General tangency weights: w ∝ Σ⁻¹(μ − Rf·1); Σ is the covariance matrix, μ the vector of expected returns. Scale w so the weights sum to 1.

How to solve Risk-Free Asset and Capital Market Line questions

Use this order for any question on the risk-free asset, tangency portfolio or CML. State your assumptions first: single period, mean-variance investors, same borrowing and lending rate.

1Write down Rf, the expected returns, standard deviations and the covariance or correlation. Convert correlation to covariance using Cov = ρ·σ1·σ2.
2Compute excess returns μi′ = E(Ri) − Rf for each risky asset.
3Find the tangency portfolio. For two assets use the w1 formula. For more assets solve Σ·z = μ′ and set w = z ÷ Σzi.
4Compute E(T) and σT from the weights. Use σT² = w1²σ1² + w2²σ2² + 2w1w2Cov.
5Find the CML slope (E(T) − Rf) ÷ σT and write the line E(R) = Rf + slope × σ.
6For a given investor, set w = target σ ÷ σT, or solve target return = Rf + w(E(T) − Rf). Put 1 − w in the risk-free asset.
7Check: the weights sum to 1, w > 1 means borrowing, and the final point lies on the CML. State the answer with units.

Quickest way: Excess return over risk, then scale

When to use it: Use for MCQs and for any question that gives the tangency portfolio or the market portfolio and asks for a mix.

If T or M is given, skip finding weights. Go straight to the slope (E(T) − Rf) ÷ σT.
For a target standard deviation s, expected return = Rf + slope × s. Weight in T is s ÷ σT.
For a target return r, weight in T is (r − Rf) ÷ (E(T) − Rf).
Weight in the risk-free asset is 1 minus that. A negative value means borrowing.
For two risky assets, compute the numerator and denominator of the w1 formula separately and check that w1 is between the signs you expect.

Common mistakes in Risk-Free Asset and Capital Market Line

Using E(R) instead of the excess return E(R) − Rf when finding the tangency portfolio.
The no-risk-free formulas for the frontier use returns directly, so students carry that habit over.
Fix: Subtract Rf from every expected return first. Then use the excess returns in the formula.
Adding the risk-free asset's variance or covariance when computing portfolio risk.
Students apply the general two-asset variance formula without remembering that Rf has zero variance and covariance.
Fix: Write σ = w·σP. Only the risky part contributes to risk.
Thinking each investor has a different risky portfolio.
The separation theorem is misread as saying the risk-free weight is the same for everyone.
Fix: Everyone holds the same risky mix T. Only the split between T and the risk-free asset changes with risk aversion.
Treating the CML as the line for every asset or inefficient portfolio.
The CML looks like the security market line.
Fix: The CML plots expected return against standard deviation and holds for efficient portfolios only. The SML plots against beta and holds for any asset under CAPM.
Forgetting that weights above 100% in T mean borrowing.
Students expect weights to lie between 0 and 1.
Fix: If the target return is above E(T), you need w > 1 and a negative risk-free weight. State that it is borrowing at Rf.
Not scaling the weights to sum to 1 when using the matrix method.
The solution z of Σz = μ′ is only proportional to the weights.
Fix: Divide each zi by the sum of the zi.

Worked examples

Example 1

Two risky assets have E(R1) = 12%, E(R2) = 8%, σ1 = 20%, σ2 = 10% and covariance 0.0 between them. The risk-free rate is 4%. Find the tangency portfolio and the expected return and standard deviation of T.

Show the solution

Excess returns: μ1′ = 12% − 4% = 0.08 and μ2′ = 8% − 4% = 0.04.
Variances: σ1² = 0.04 and σ2² = 0.01. Cov = 0.
Use w1 = [μ1′σ2² − μ2′Cov] ÷ [μ1′σ2² + μ2′σ1² − (μ1′ + μ2′)Cov].
Numerator = 0.08 × 0.01 = 0.0008.
Denominator = 0.0008 + 0.04 × 0.04 = 0.0008 + 0.0016 = 0.0024.
w1 = 0.0008 ÷ 0.0024 = 1/3 and w2 = 2/3.
E(T) = (1/3)(12%) + (2/3)(8%) = 4% + 5.3333% = 9.3333%.
σT² = (1/9)(0.04) + (4/9)(0.01) = 0.004444 + 0.004444 = 0.008889. σT = 0.09428, or 9.43%.

Answer: T holds 1/3 in asset 1 and 2/3 in asset 2. E(T) = 9.33% and σT = 9.43%.

Example 2

Using the tangency portfolio from the previous example (E(T) = 9.3333%, σT = 9.4281%, Rf = 4%), an investor wants a portfolio with standard deviation 5%. Find the weights, the expected return and the CML equation. Then find the weight in T needed for a 12% return.

Show the solution

Slope = (9.3333% − 4%) ÷ 9.4281% = 5.3333 ÷ 9.4281 = 0.5657.
CML: E(R) = 4% + 0.5657 × σ.
For σ = 5%: weight in T = 5 ÷ 9.4281 = 0.5303. Weight in the risk-free asset = 0.4697.
Expected return = 4% + 0.5657 × 5% = 6.83%.
For a 12% return: w = (12 − 4) ÷ (9.3333 − 4) = 8 ÷ 5.3333 = 1.5.
Risk-free weight = 1 − 1.5 = −0.5, so the investor borrows 50% of their wealth at 4% and invests 150% in T.
Check: σ = 1.5 × 9.4281% = 14.14%. Return = 4% + 0.5657 × 14.14% = 12.0%.

Answer: For σ = 5%, hold 53.0% in T and 47.0% in the risk-free asset, giving 6.83%. The CML is E(R) = 4% + 0.5657σ. A 12% return needs 150% in T and −50% in the risk-free asset, with σ = 14.14%.

Exam tips

Write the assumptions: single period, mean-variance preferences, same borrowing and lending rate, and agreement on expected returns and covariances. These often earn marks in written parts.
Explain the separation theorem in two stages: find T without reference to preferences, then mix T with cash according to risk aversion.
Show the line algebra for the mix. State that σ = w·σP because the risk-free asset has zero variance and covariance.
In MCQs, check the sign of the risk-free weight. A negative value means borrowing.
Link the CML to CAPM: T becomes the market portfolio M. In the computer paper, set up Σ⁻¹(μ − Rf·1) clearly and normalise the weights.

Practice questions from Mean-variance portfolio theory

Risk-Free Asset and 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.

Risk-Free Asset and Capital Market Line: frequently asked questions

What is the separation theorem in portfolio theory?

It says the investment decision splits into two parts. First, every investor finds the same tangency portfolio of risky assets. Second, each investor decides how much to hold in it versus the risk-free asset, based on risk aversion.

How do I find the tangency portfolio?

Subtract Rf from each expected return. Then choose risky weights that maximise the Sharpe ratio. For two assets use the closed formula. For more assets solve Σz = μ′ and scale z so the weights sum to 1.

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

The CML plots expected return against standard deviation and applies to efficient portfolios. The security market line plots expected return against beta and applies to any asset or portfolio under CAPM.

Can an investor hold more than 100% in the tangency portfolio?

Yes, if the model assumes unlimited borrowing at the risk-free rate. The investor borrows and invests the proceeds in T. This moves them along the CML above T, with higher return and higher risk.