CFA Level I Exam · Portfolio Risk and Return: Part II

Capital Allocation Line and Risk-Free Asset Combinations

Updated 7 October 2026 · Fact-checked

The capital allocation line (CAL) shows the risk-return combinations from mixing a risk-free asset with one risky portfolio. Expected return = Rf + w(E(Rp) − Rf). Standard deviation = w × σp. Plot return against risk and you get a straight line whose slope is the risky portfolio's Sharpe ratio.

Understand Capital Market Theory and the Capital Allocation Line

Start with two assets. One is a risk-free asset with return Rf and zero standard deviation. The other is a risky portfolio P with expected return E(Rp) and standard deviation σp. You choose a weight w in the risky portfolio and 1 − w in the risk-free asset.

Because the risk-free asset has zero variance and zero covariance with anything, the portfolio's standard deviation is simply w × σp. Risk scales in direct proportion to w. Expected return is a weighted average, so it is also linear in w. Two linear relationships give a straight line in expected return and standard deviation space. That line is the capital allocation line (CAL).

The intercept is Rf and the slope is (E(Rp) − Rf) ÷ σp. This slope is the Sharpe ratio of the risky portfolio. It is the extra expected return you earn per unit of risk. Moving along the line changes how much risk you take, but not the reward per unit of risk.

If 0 < w < 1, you are lending: part of your money sits in the risk-free asset. If w = 1, you hold only the risky portfolio. If w > 1, you are borrowing at the risk-free rate and investing more than 100% of your capital in the risky portfolio. This is leverage. Weight in the risk-free asset is then negative. The line extends beyond P, and both return and risk rise.

A CAL can be drawn through any risky portfolio. The CAL with the steepest slope is best, because it offers more return for every level of risk. For a given set of expectations, that steepest CAL is the line through the tangency portfolio on the efficient frontier. It is called the capital market line (CML) only when all investors share the same (homogeneous) expectations and can borrow and lend at Rf. In that case the tangency portfolio is the market portfolio. So the CML is a special CAL. The CAL is general. The CML uses the market portfolio.

Key formulas to remember

Expected return of the combined portfolio: E(Rc) = w × E(Rp) + (1 − w) × Rf = Rf + w × [E(Rp) − Rf]; w is the weight in the risky portfolio. w > 1 means borrowing at Rf.
Standard deviation of the combined portfolio: σc = w × σp; Holds because the risk-free asset has zero standard deviation and zero covariance with the risky portfolio. Use w as a positive number here.
CAL equation: E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc; Slope is the Sharpe ratio of the risky portfolio. Intercept is Rf.
Weight needed for a target risk: w = σtarget ÷ σp; Use this to find leverage or lending for a given standard deviation.
Weight needed for a target return: w = [E(Rtarget) − Rf] ÷ [E(Rp) − Rf]; If w > 1, borrow; the risk-free weight (1 − w) is negative.
Sharpe ratio (CAL slope): Sharpe = [E(Rp) − Rf] ÷ σp; Same for every point on the CAL.
Capital market line: E(Rc) = Rf + [(E(RM) − Rf) ÷ σM] × σc; A CAL where the risky portfolio is the market portfolio M.

How to solve Capital Market Theory and the Capital Allocation Line questions

Use this routine for any question on mixing a risk-free asset with a risky portfolio.

1Write down Rf, E(Rp) and σp. Convert percentages to decimals only if your arithmetic needs it.
2Identify the weight w in the risky portfolio. If given a weight in the risk-free asset, use w = 1 − wf. If the investor borrows, w is above 1.
3Compute expected return with E(Rc) = Rf + w × (E(Rp) − Rf).
4Compute risk with σc = w × σp.
5If the question gives a target risk or return, solve for w first using the target formulas, then compute the other measure.
6Check the slope: (E(Rp) − Rf) ÷ σp. For comparing CALs, the higher slope is better.
7Sense-check: lending gives risk below σp and return below E(Rp). Borrowing gives both above. Then eliminate options that break this.

Quickest way: Scale the excess return and the risk by w

When to use it: Any numerical question with a given weight, target risk or target return on a single CAL.

Compute excess return of the risky portfolio: E(Rp) − Rf.
Multiply excess return by w and add Rf. That is the expected return.
Multiply σp by w. That is the standard deviation.
For a target σ, set w = σ ÷ σp. Then apply the two steps above.
Eliminate options: with w < 1, return must lie between Rf and E(Rp). With w > 1, it must exceed E(Rp).

Common mistakes in Capital Market Theory and the Capital Allocation Line

Using σc = w × σp with w as the risk-free weight.
The question states the risk-free weight and students plug it in directly.
Fix: Always convert to the risky weight first: w = 1 − wf. Standard deviation scales with the risky weight only.
Calculating portfolio standard deviation as a weighted average including the risk-free asset's term or with a correlation.
Students apply the two-asset variance formula by habit.
Fix: The risk-free asset has zero variance and zero covariance, so the formula collapses to σc = w × σp.
Treating borrowing as a negative weight in the risky portfolio.
Confusion over which asset carries the negative sign.
Fix: If w > 1, borrow; the risk-free weight (1 − w) is negative and the risky weight exceeds 1. Example: borrowing 30% of capital gives w = 1.3 and wf = −0.3.
Confusing the CAL with the CML.
Both are straight lines starting at Rf.
Fix: The CAL goes through any chosen risky portfolio. The CML goes through the market portfolio only and is the best-case CAL under common assumptions.
Forgetting that the slope is the same at every point on the line.
Students think leveraging improves reward per unit of risk.
Fix: Leverage raises both return and risk proportionally. The Sharpe ratio stays unchanged on the same CAL.
Using the risky portfolio's return instead of its excess return when scaling.
Mixing up w × E(Rp) with w × (E(Rp) − Rf).
Fix: Use Rf + w × (E(Rp) − Rf), or the full version w × E(Rp) + (1 − w) × Rf. Both give the same value.

Worked examples

Example 1

The risk-free rate is 3%. A risky portfolio has expected return 11% and standard deviation 16%. An investor puts 75% of funds in the risky portfolio and 25% in the risk-free asset. What is the expected return of the combined portfolio? (A) 6%, (B) 9%, (C) 11%.

Show the solution

w = 0.75 in the risky portfolio.
Expected return = 3% + 0.75 × (11% − 3%) = 3% + 6% = 9%.
Check: the investor is lending, so the return must lie between 3% and 11%. This range check eliminates only option C, which equals the risky portfolio's return and would need w = 1. Option A (6%) is also inside the range, so the check does not remove it.
Option A is removed by direct computation: 0.75 × 8% = 6% leaves out the risk-free rate, so it is below the correct 9% and is wrong.

Answer: B: expected return 9%. (The standard deviation is 0.75 × 16% = 12%.)

Example 2

The risk-free rate is 2%. A risky portfolio has expected return 10% and standard deviation 20%. An investor wants a combined portfolio with standard deviation 30%, borrowing at the risk-free rate. What is the expected return? (A) 10%, (B) 12%, (C) 14%.

Show the solution

Find w = σtarget ÷ σp = 30% ÷ 20% = 1.5.
The risk-free weight is 1 − 1.5 = −0.5, so the investor borrows 50% of capital.
Expected return = 2% + 1.5 × (10% − 2%) = 2% + 12% = 14%.
Check: Sharpe ratio = 8% ÷ 20% = 0.4. At σ = 30%, return = 2% + 0.4 × 30% = 14%. Same result.

Answer: C: expected return 14%.

Exam tips

Numerical options run from smallest to largest. Compute w first, then return and risk, and match the value asked for.
Use the sense-check: lending keeps both measures below the risky portfolio's, borrowing pushes both above. This can eliminate two options in seconds.
Watch the wording: 'borrows 40% of his own capital' means w = 1.4. 'Invests 40% in the risk-free asset' means w = 0.6.
Conceptual items often ask what changes along a CAL. Answer: return and risk change, the Sharpe ratio does not.
If asked which CAL is better, pick the higher slope. No calculator keystrokes are needed beyond basic arithmetic.

Practice questions from Portfolio Risk and Return: Part II

Capital Market Theory and the Capital Allocation Line 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 Theory and the Capital Allocation Line: frequently asked questions

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

The CAL describes combinations of the risk-free asset with any risky portfolio. The CML is the CAL drawn through the market portfolio. It is the CAL with the highest slope when investors share the same expectations and can borrow and lend at the risk-free rate.

How do you calculate the standard deviation of a risk-free and risky portfolio?

Multiply the risky portfolio's standard deviation by its weight: σc = w × σp. The risk-free asset adds no variance and has zero covariance with the risky portfolio. So no correlation term is needed.

What does the slope of the CAL represent?

The slope is the Sharpe ratio of the risky portfolio: (E(Rp) − Rf) ÷ σp. It tells you the extra expected return per unit of standard deviation. A steeper CAL offers more return for the same risk.

What happens to the CAL if the investor can only lend, not borrow?

The line stops at the risky portfolio P, because w cannot exceed 1. Only points from Rf up to P are available. If borrowing is allowed at a higher rate than Rf, the line beyond P has a lower slope.