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

Capital Allocation Line and Optimal Portfolio Explained

Updated 7 October 2026 · Fact-checked

The capital allocation line (CAL) shows the risk-return combinations you get by mixing a risk-free asset with one risky portfolio. Its slope is the Sharpe ratio. The optimal risky portfolio has the highest Sharpe ratio. Your optimal mix on that line depends on your risk aversion.

Understand Capital Allocation Line and Optimal Portfolio

Start with two building blocks. 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 can split your money between them. Put 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 mix is simple. Expected return is a straight weighted average. Risk is just w × σp. Plot every possible w and you get a straight line from Rf through P. This is the capital allocation line (CAL).

The slope of the CAL is (E(Rp) − Rf) ÷ σp. This is the Sharpe ratio: extra return per unit of total risk. If w > 1, you borrow at the risk-free rate and invest more than your wealth in P. The line then continues beyond P with the same slope. This assumes you can borrow at Rf, which the exam usually states.

Many risky portfolios exist, so many CALs exist. You want the steepest one. The risky portfolio that gives the steepest CAL, the highest Sharpe ratio, is the optimal risky portfolio. It is where the CAL touches the efficient frontier of risky assets. Every investor holds this same risky portfolio.

Where you sit on that line depends on your utility. A more risk-averse investor holds more risk-free asset (w small). A less risk-averse investor holds more risky portfolio (w large, maybe above 1). The optimal portfolio is the point where your highest indifference curve is tangent to the CAL. When the CAL is built from the market portfolio of all risky assets, it is called the capital market line (CML). So the CML is a special CAL, and a CAL can use any risky portfolio.

Key formulas to remember

Expected return of the mix: E(Rc) = Rf + w × [E(Rp) − Rf]; w is the weight in the risky portfolio. Same as w × E(Rp) + (1 − w) × Rf.
Standard deviation of the mix: σc = w × σp; Works because the risk-free asset has zero standard deviation and zero correlation with P.
Capital allocation line: E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc; Straight line. Intercept is Rf, slope is the Sharpe ratio.
Sharpe ratio: Sharpe = (E(Rp) − Rf) ÷ σp; Uses total risk (standard deviation). Higher is better.
Weight for a target risk: w = σtarget ÷ σp; Use to find how much to hold in P for a chosen standard deviation.
Optimal risky portfolio: Choose the portfolio with the highest Sharpe ratio; It is the tangency point of the CAL with the efficient frontier of risky assets.
Utility of a portfolio: U = E(R) − 0.5 × A × σ²; A is the risk-aversion coefficient. Higher A means more risk averse. The investor picks the point on the CAL with the highest U.

How to solve Capital Allocation Line and Optimal Portfolio questions

Use this method for any CAL, Sharpe ratio or optimal portfolio question. Write returns as decimals to avoid slips.

1Identify Rf and the risky portfolio's E(Rp) and σp. Check whether there are several risky portfolios.
2Compute the Sharpe ratio (E(Rp) − Rf) ÷ σp for each risky portfolio. The highest one is the optimal risky portfolio.
3Decide the weight w in the risky portfolio. Use the given w, or w = σtarget ÷ σp if a risk level is given, or solve for w from a target return.
4Compute the mix's return: Rf + w × (E(Rp) − Rf).
5Compute the mix's risk: w × σp. Remember w > 1 means borrowing at Rf.
6If utility is given, compute U = E(R) − 0.5 × A × σ² for each choice and pick the highest.
7Check the answer: it must lie on the CAL, and the Sharpe ratio of the mix must equal that of P.

Quickest way: Sharpe-first elimination

When to use it: Use when a question asks which portfolio is best, or which line is steepest, and you are short on time.

Compute the Sharpe ratio for each risky portfolio and ignore everything else.
Pick the highest. Eliminate the other two options.
If a mix is asked, find w from the target risk and apply the straight-line formula.
Sanity check: with w < 1 the return lies between Rf and E(Rp). With w > 1 it lies above E(Rp).

Common mistakes in Capital Allocation Line and Optimal Portfolio

Using the wrong denominator in the Sharpe ratio, such as dividing by E(Rp) or by variance.
Students mix up the Sharpe ratio with a simple return-per-unit ratio, or forget σ means standard deviation.
Fix: Always write (E(Rp) − Rf) ÷ σp. The numerator is excess return and the denominator is standard deviation.
Forgetting to subtract the risk-free rate in the numerator.
The word return is read as total return.
Fix: The CAL slope measures reward above the risk-free rate. Subtract Rf before dividing.
Treating the CAL and the CML as identical.
Both are straight lines starting at Rf, so they look the same.
Fix: The CML uses the market portfolio as the risky asset. A CAL can use any risky portfolio. The CML is the CAL built on the market portfolio.
Choosing the optimal risky portfolio by picking the highest return or lowest risk.
Students look at one dimension only.
Fix: Compare Sharpe ratios. The optimal risky portfolio is the same for all investors, whatever their risk aversion.
Saying risk-averse investors hold a different risky portfolio from others.
Confusing the choice of risky portfolio with the choice of mix.
Fix: Everyone holds the same optimal risky portfolio. Risk aversion only changes w, the split with the risk-free asset.
Computing the mix's risk as a weighted average of σp and 0 with a correlation term, or adding weights incorrectly when borrowing.
Students apply the two-risky-asset formula unnecessarily.
Fix: With a risk-free asset, σc = w × σp. For borrowing, w is above 1 and the risk-free weight is negative.

Worked examples

Example 1

The risk-free rate is 3%. Portfolio P has expected return 11% and standard deviation 16%. An investor puts 60% of wealth in P and 40% in the risk-free asset. What is the expected return of the investor's portfolio? Options: (A) 7.8%; (B) 9.6%; (C) 11.0%.

Show the solution

Rf = 0.03, E(Rp) = 0.11, σp = 0.16, w = 0.60.
Expected return = 0.03 + 0.60 × (0.11 − 0.03) = 0.03 + 0.60 × 0.08 = 0.03 + 0.048 = 0.078.
Check with the weighted average: 0.6 × 0.11 + 0.4 × 0.03 = 0.066 + 0.012 = 0.078.
For reference, the standard deviation of the mix is 0.60 × 0.16 = 0.096. Option B is this risk figure, not a return.
Option C is the expected return of P alone. It ignores the 40% held in the risk-free asset.

Answer: (A) 7.8%. The investor's standard deviation is 9.6%.

Example 2

The risk-free rate is 2%. Portfolio X has expected return 8% and standard deviation 12%. Portfolio Y has expected return 10% and standard deviation 20%. An investor chooses the better risky portfolio and wants a total standard deviation of 15%. What expected return can the investor earn? Options: (A) 8.0%; (B) 9.5%; (C) 10.0%.

Show the solution

Sharpe ratio of X = (0.08 − 0.02) ÷ 0.12 = 0.06 ÷ 0.12 = 0.50.
Sharpe ratio of Y = (0.10 − 0.02) ÷ 0.20 = 0.08 ÷ 0.20 = 0.40.
X has the higher Sharpe ratio, so X is the optimal risky portfolio.
Weight in X = 0.15 ÷ 0.12 = 1.25. The investor borrows 25% of wealth at the risk-free rate.
Expected return = 0.02 + 1.25 × (0.08 − 0.02) = 0.02 + 0.075 = 0.095.
Cross-check using the CAL: 0.02 + 0.50 × 0.15 = 0.095.

Answer: (B) 9.5%. Using X with 125% invested earns 9.5% at 15% risk. Y at 15% risk earns only 2% + 0.40 × 15% = 8.0%, which is below X's 9.5%.

Exam tips

When asked for the best risky portfolio, compute Sharpe ratios first. Do not look at return or risk alone.
If a question says the investor can borrow at the risk-free rate, expect w above 1 and a return above E(Rp).
Know the difference between the CAL and the CML. The CML uses the market portfolio only.
Questions on utility often give A and ask which mix is preferred. Compute U = E(R) − 0.5 × A × σ² for each option and pick the highest.
Convert percentages to decimals before squaring σ in a utility calculation. A common slip is using 16 instead of 0.16.

Practice questions from Portfolio Risk and Return: Part I

Capital Allocation Line and Optimal Portfolio in other exams

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

Capital Allocation Line and Optimal Portfolio: frequently asked questions

What is the capital allocation line formula in CFA Level I?

E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc. It is a straight line with intercept Rf and slope equal to the Sharpe ratio of the risky portfolio. It shows every return and risk pair you can reach by mixing the risk-free asset with that portfolio.

How do I calculate the Sharpe ratio?

Subtract the risk-free rate from the portfolio's expected return, then divide by the portfolio's standard deviation. For example, (11% − 3%) ÷ 16% = 0.50. A higher ratio means more excess return per unit of risk.

What is the optimal risky portfolio?

It is the risky portfolio with the highest Sharpe ratio. It gives the steepest CAL and sits where that line touches the efficient frontier. All investors hold it, and they differ only in how much they combine with the risk-free asset.

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

The CAL uses any risky portfolio combined with the risk-free asset. The CML is the CAL where the risky portfolio is the market portfolio. So every CML is a CAL, but not every CAL is a CML.