Optimal Portfolio Selection Using Utility and the Capital Allocation Line

Updated for the 2026-2027 CFA® Level I curriculum.

An investor's optimal portfolio depends on two things: the best available combination of risky assets and how much risk that investor is willing to accept. CFA Level I tests this as a two-step decision. First, find the optimal risky portfolio. Second, combine it with the risk-free asset based on the investor's utility function.

This note walks through both steps using the capital allocation line and the standard utility formula.

Key Takeaways About Optimal Portfolio Selection Using Utility and the Capital Allocation Line

• The optimal risky portfolio is found first and does not depend on any single investor's risk preference.

• The optimal risky portfolio is the tangency point between the CAL and the efficient frontier, where the Sharpe ratio is highest.

• The complete portfolio combines the risk-free asset and the optimal risky portfolio in proportions set by the investor's utility function.

• Risk aversion determines how much weight goes to the risky portfolio, not which risky portfolio is chosen.

• A CAL with a steeper slope offers a better risk-return tradeoff and is preferred by all investors, regardless of risk aversion.

• Lending (holding some risk-free asset) and borrowing (using leverage) are both positions along the same CAL, just on opposite sides of the risky portfolio's weight.

What You Need to Know for CFA Level I

• Know the difference between the optimal risky portfolio (one answer for all investors) and the complete portfolio (different for each investor).

• Be able to write and apply the CAL equation to find expected return or standard deos or solve for the optimal weight.

• Understand that the optimal risky portfolio is the tangency portfolio, identified by thviation at any point on the line.

• Be able to apply the utility formula to compare portfolie highest Sharpe ratio on the efficient frontier.

• Recognize that a higher risk-aversion coefficient shifts the investor toward the risk-free asset, not toward a different risky portfolio.

• Know how to interpret  as a borrowing position and as a lending position.

Efficient Portfolios and the Capital Allocation Line

An efficient portfolio offers the highest expected return for a given level of risk. Plotting all efficient portfolios of risky assets produces the efficient frontier. Adding a risk-free asset changes the problem. Instead of choosing among curved combinations of risky assets, an investor can draw a straight line from the risk-free rate to any risky portfolio on the frontier.

This straight line is the capital allocation line. Its equation is:

Notation:

• = expected return of the complete portfolio

• = risk-free rate

• = expected return of the chosen risky portfolio

• = standard deviation of the chosen risky portfolio

• = standard deviation of the complete portfolio

• = the slope of the CAL, also the Sharpe ratio of the risky portfolio

Every risky portfolio on the efficient frontier has its own CAL. The steeper the slope, the better the tradeoff between risk and return. An investor should choose the risky portfolio with the steepest possible CAL. That single CAL becomes the relevant opportunity set for every investor, no matter their personal risk tolerance.

Investor Utility and Risk Aversion

Once the CAL is fixed, the investor still has to decide where on that line to sit. This decision depends on utility, a way to score portfolios based on expected return and risk.

Notation:

• = utility value (higher is preferred)

• = expected return of the portfolio

• = risk-aversion coefficient

• = variance of the portfolio's returns

A higher means the investor loses more utility for each unit of variance. A risk-averse investor with penalizes risk more heavily than an investor with . This formula lets candidates rank portfolios or solve directly for the allocation that produces the highest utility along the CAL.

Finding the Optimal Risky Portfolio

The optimal risky portfolio is the point on the efficient frontier where the CAL is tangent to the frontier. This tangency point maximizes the Sharpe ratio, so it offers the best possible reward per unit of risk. It is the same for every investor because it does not depend on individual risk aversion. Only the combination of this portfolio with the risk-free asset changes from investor to investor.

The graph shows the CAL running from the risk-free rate through the tangency portfolio. Indifference curves represent constant utility for a given investor. A more risk-averse investor has steeper, more curved indifference curves. The optimal complete portfolio sits where the highest attainable indifference curve just touches the CAL.

Selecting the Complete Portfolio

The complete portfolio is the actual mix an investor holds: part risk-free asset, part optimal risky portfolio. The weight placed in the risky portfolio, , is found by maximizing utility:

Notation:

• = proportion of the complete portfolio invested in the optimal risky portfolio

• = risk premium of the optimal risky portfolio

• = risk-aversion coefficient

• = variance of the optimal risky portfolio

If y is less than 1, the investor lends, holding some risk-free asset alongside the risky portfolio.

If y is greater than 1, the investor borrows at the risk-free rate to invest more than 100% of the portfolio in the risky asset. Both positions sit on the same CAL. Only the split between lending and borrowing changes.

Worked Example

Priya is evaluating her retirement account. She has already identified her optimal risky portfolio: expected return of 12%, standard deviation of 20%. The risk-free rate is 3%. Priya's risk-aversion coefficient is .

Step 1: Find the risk premium

Step 2: Find the variance of the risky portfolio

Step 3: Solve for

Step 4: Find the complete portfolio's expected return

Step 5: Find the complete portfolio's standard deviation

Priya should place about 56% of her portfolio in the optimal risky portfolio and 44% in the risk-free asset. This split reflects her specific risk aversion, not a change in which risky portfolio she holds. A less risk-averse investor with the same risky portfolio and risk-free rate would hold a larger share in the risky asset.

Common Exam Traps

Confusing the optimal risky portfolio with the complete portfolio

The optimal risky portfolio is the same for every investor. The complete portfolio changes based on each investor's risk aversion. A question that asks for "the investor's optimal portfolio" usually wants the complete portfolio, not just the tangency portfolio.

Using the CML when the question describes a general risky portfolio

The capital market line applies specifically when the risky portfolio is the market portfolio under CAPM assumptions. If the question gives a risky portfolio without stating it is the market portfolio, use the CAL, not the CML.

Ignoring the investor's risk aversion

Two investors can share the same optimal risky portfolio and still hold very different complete portfolios. Skipping the risk-aversion coefficient when solving for is a common calculation error.

Misreading borrowing and lending positions

A weight above 1 in the risky portfolio means the investor is borrowing at the risk-free rate, not investing more than 100% of their own capital. A weight below 1 means lending, or holding some risk-free asset.

Practice Question

An investor has a risk-aversion coefficient of 3. The optimal risky portfolio has an expected return of 14% and a standard deviation of 25%. The risk-free rate is 4%. What proportion of the investor's complete portfolio should be allocated to the optimal risky portfolio?

1. 13.3%

2. 53.3%

3. 26.7%

FAQs About Optimal Portfolio Selection Using Utility and the Capital Allocation Line