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CFA Level I Exam · The Return and Risk of a Financial Portfolio

Capital Allocation Line and Optimal Portfolio for CFA Level 1

Updated 7 October 2026 · Fact-checked

The capital allocation line (CAL) shows the risk-return combinations you get by mixing the risk-free asset with one risky portfolio. Its slope is the Sharpe ratio. The tangency portfolio is the risky portfolio with the highest Sharpe ratio. To solve questions, find the weight in the risky portfolio, then compute return and risk.

Understand Capital Allocation Line and Optimal Portfolio

Start with two building blocks: a risk-free asset (return Rf, standard deviation 0) and a risky portfolio P (expected return E(Rp), standard deviation σp). You can split your money between them. Put weight w in P and 1 − w in the risk-free asset.

Because the risk-free asset has zero variance and zero covariance with P, the mix is simple. Expected return is a weighted average. Risk is just w × σp. Both move in a straight line as w changes, so every mix sits on a straight line in return-risk space. That line is the capital allocation line (CAL).

The slope of the CAL is the extra return you earn per unit of risk: (E(Rp) − Rf) ÷ σp. This is the Sharpe ratio. A higher Sharpe ratio means a steeper CAL and a better deal for every investor, whatever their risk appetite. If w is above 1, you borrow at the risk-free rate to buy more of P. This is a leveraged position, and it extends the line past P. This assumes you can borrow at Rf, which the basic model does.

You could draw a CAL through any risky portfolio. The best one is the line that touches the efficient frontier of risky assets at a single point, the tangency portfolio. It has the highest Sharpe ratio of all risky portfolios, so its CAL lies above all others. Every investor, whatever their risk aversion, holds the same risky portfolio (the tangency portfolio) and differs only in how much they put in the risk-free asset.

The optimal portfolio for a given investor is where their highest reachable indifference curve touches the CAL. A more risk-averse investor sits lower on the line (more risk-free asset). A less risk-averse investor sits higher, possibly with leverage. When all investors share the same expectations and the tangency portfolio is the market portfolio, the line is called the capital market line (CML). The CML is a special CAL. The CAL can use any risky portfolio. The CML uses the market portfolio.

Key formulas to remember

Expected return of the mix
E(Rc) = w × E(Rp) + (1 − w) × Rf
w is the weight in the risky portfolio. w > 1 means borrowing at Rf.
Standard deviation of the mix
σc = w × σp
Works because the risk-free asset has zero standard deviation and zero covariance with P. Use w as a positive number here.
Capital allocation line
E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc
Intercept is Rf. Slope is the Sharpe ratio of P.
Sharpe ratio
Sharpe = (E(Rp) − Rf) ÷ σp
Excess return per unit of total risk. Higher is better. Do not use beta here.
Tangency portfolio
Tangency portfolio = risky portfolio with the maximum Sharpe ratio
It is where the CAL touches the efficient frontier of risky assets.
Weight for a target risk
w = σtarget ÷ σp
Use this when a question gives the desired portfolio standard deviation.

How to solve Capital Allocation Line and Optimal Portfolio questions

Use this method for any question on the CAL, Sharpe ratio or optimal mix of the risk-free asset and a risky portfolio.

  1. 1Write down Rf, E(Rp) and σp. Check whether returns and risk are in the same units (percent or decimals).
  2. 2Work out the weight w in the risky portfolio. If the question gives a budget split, use it. If it gives a target risk, use w = σtarget ÷ σp. If it gives a target return, solve w from the return formula.
  3. 3Compute expected return: w × E(Rp) + (1 − w) × Rf. If w > 1, the risk-free weight is negative (borrowing).
  4. 4Compute standard deviation: w × σp.
  5. 5Compute the Sharpe ratio (E(Rp) − Rf) ÷ σp. Remember every point on the CAL has the same Sharpe ratio as P.
  6. 6To compare portfolios, pick the higher Sharpe ratio. The best risky portfolio is the tangency portfolio.
  7. 7Check the answer: the mix must lie on the straight line, and with w < 1 its return and risk must sit between Rf and P.

Quickest way: Slope-and-line shortcut

When to use it: Use when the question asks for the return at a given risk level, or asks which portfolio is better.

  1. Compute the Sharpe ratio once: (E(Rp) − Rf) ÷ σp.
  2. For a target risk σc, the return is Rf + Sharpe × σc. No weights needed.
  3. To compare two risky portfolios, compare Sharpe ratios only. Ignore raw returns.
  4. Check the answer option order: options run from smallest to largest, so a quick estimate usually removes two choices.
  5. On a TI BA II Plus, using the first worked example (Rf = 3%, E(Rp) = 11%, σp = 16%), key 11 − 3 = 8, then ÷ 16 = 0.5 using the normal keys. Store the result with STO 1 and recall it with RCL 1 if you need it again.

Common mistakes in Capital Allocation Line and Optimal Portfolio

  • Using beta or standard deviation of the market in the Sharpe ratio instead of the portfolio's own standard deviation.

    Sharpe, Treynor and CAPM formulas look alike under time pressure.

    Fix: Sharpe divides by the portfolio's total standard deviation. Treynor divides by beta.

  • Forgetting to subtract Rf in the numerator of the Sharpe ratio.

    Students treat it like a return-per-risk ratio.

    Fix: Always write 'excess return' first: E(Rp) − Rf.

  • Adding the risk-free weight into the standard deviation or using w² × σp instead of w × σp.

    Mixing up the variance rule with the standard deviation rule.

    Fix: With the risk-free asset, standard deviation scales linearly: σc = w × σp. Variance would be w² × σp².

  • Treating CAL and CML as the same thing.

    Both are straight lines starting at Rf.

    Fix: The CAL can pass through any risky portfolio. The CML is the CAL through the market portfolio, which is the tangency portfolio under common expectations.

  • Ignoring borrowing when w is above 1, then putting a positive weight on the risk-free asset.

    Students assume weights must be between 0 and 1.

    Fix: If w = 1.5, the risk-free weight is −0.5. The investor borrows at Rf.

  • Choosing the portfolio with the highest return as the best one.

    Return is easy to see, risk is easy to ignore.

    Fix: Compare Sharpe ratios. A higher return with much higher risk can have a lower Sharpe ratio.

Worked examples

Example 1

The risk-free rate is 3%. Risky portfolio P has an expected return of 11% and a standard deviation of 16%. An investor puts 75% of her money in P and 25% in the risk-free asset. What are the expected return and standard deviation of her portfolio?

Show the solution
  1. w = 0.75, Rf = 3%, E(Rp) = 11%, σp = 16%.
  2. Expected return = 0.75 × 11% + 0.25 × 3% = 8.25% + 0.75% = 9.00%.
  3. Standard deviation = 0.75 × 16% = 12.00%.
  4. Check with the CAL: Sharpe = (11 − 3) ÷ 16 = 0.5. Return = 3% + 0.5 × 12% = 9.00%. It matches.

Answer: Expected return 9.00% and standard deviation 12.00%.

Example 2

The risk-free rate is 2%. Portfolio X has an expected return of 10% and a standard deviation of 20%. Portfolio Y has an expected return of 8% and a standard deviation of 10%. The investor can borrow at the risk-free rate. She wants a portfolio with a standard deviation of 15% using the better of X and Y combined with the risk-free asset. What is the expected return? Options: A) 9.0%, B) 11.0%, C) 12.0%.

Show the solution
  1. Sharpe of X = (10 − 2) ÷ 20 = 0.40.
  2. Sharpe of Y = (8 − 2) ÷ 10 = 0.60. Y is better, so use Y.
  3. Weight in Y = 15 ÷ 10 = 1.5. The investor borrows 50% at the risk-free rate.
  4. Expected return = 1.5 × 8% + (−0.5) × 2% = 12% − 1% = 11%.
  5. Check with the CAL: 2% + 0.60 × 15% = 2% + 9% = 11%.

Answer: B) 11.0%. Use Y because its Sharpe ratio (0.60) is higher than X (0.40), and leverage it to 15% risk.

Exam tips

  • Questions often hide the answer in the Sharpe ratio. Compute it first, then use the CAL as a straight line.
  • Watch for leverage: a target risk above the risky portfolio's standard deviation means w > 1 and borrowing at Rf.
  • Know the conceptual points: the tangency portfolio has the highest Sharpe ratio, and all investors hold the same risky portfolio under common expectations.
  • Know the CAL versus CML difference in one sentence. Expect a three-option question testing it.
  • With no penalty for wrong answers, never leave a question blank. Eliminate options that break the straight-line logic, such as a mix with higher risk than P but w < 1.

Practice questions from The Return and Risk of a Financial Portfolio

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. The intercept is the risk-free rate and the slope is the Sharpe ratio of the risky portfolio. It gives the expected return for any level of portfolio risk on that line.

How do you 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% = 0.5. A higher ratio means more excess return per unit of risk.

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

The CAL combines the risk-free asset with any risky portfolio. The CML is the CAL that uses the market portfolio as the risky portfolio. The CML is therefore one specific CAL, and it uses the tangency portfolio when all investors share the same expectations.

What is the tangency portfolio?

It is the risky portfolio on the efficient frontier with the highest Sharpe ratio. The CAL drawn through it just touches the frontier and lies above every other CAL. Investors then decide how much to hold in it versus the risk-free asset.