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

Portfolio Selection and Optimal Portfolio for CFA Level 1

Updated 7 October 2026 · Fact-checked

The optimal portfolio is the point where an investor's highest reachable indifference curve just touches the efficient frontier (or the capital allocation line when a risk-free asset exists). Risk-averse investors with steeper curves pick lower-risk portfolios. Return and risk are measured as expected return and standard deviation.

Understand Portfolio Selection and Optimal Portfolio

Every investor wants high return and low risk. The efficient frontier shows the best return available for each level of risk using risky assets. It does not tell you which point to pick. That choice depends on the investor.

An indifference curve joins all combinations of risk (standard deviation) and expected return that give an investor the same utility. For a risk-averse investor, the curve slopes upward, because extra risk must be paid for with extra return. Curves higher and to the left give more utility. Two curves of the same investor never cross.

The optimal portfolio is where the investor's highest attainable indifference curve is tangent to the efficient frontier. A point on a higher curve is unreachable. A point on a lower curve is worse than the tangent point. So tangency is the best feasible choice.

Risk aversion sets the slope. A more risk-averse investor has steeper indifference curves. The tangent point then sits further left on the frontier, at lower risk and lower expected return. A less risk-averse investor has flatter curves and picks a point further right.

When a risk-free asset exists, the opportunity set becomes the capital allocation line (CAL) drawn from the risk-free rate through the optimal risky portfolio. With the market portfolio as the risky portfolio, it is the capital market line (CML). All investors hold the same risky portfolio, the tangency portfolio with the highest Sharpe ratio. They differ only in how much they put in it versus the risk-free asset. Risk-averse investors lend (hold some risk-free asset). Less risk-averse investors may borrow at the risk-free rate and hold more than 100% in the risky portfolio.

Key formulas to remember

Utility of a portfolio (mean-variance): U = E(R) − 0.5 × A × σ²; A is the risk-aversion coefficient. Higher A means more risk-averse. Use σ² as a decimal, e.g. 20% gives 0.04.
Capital allocation line: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp; Slope is the Sharpe ratio of the risky portfolio. On the CML the risky portfolio is the market portfolio.
Two-asset mix with risk-free asset: E(Rp) = w × E(Rrisky) + (1 − w) × Rf; σp = w × σrisky; The risk-free asset has zero standard deviation, so its covariance with any risky asset is zero (correlation is not defined). w above 1 means borrowing.
Optimal weight in risky portfolio: w* = [E(Rrisky) − Rf] ÷ (A × σ²risky); Maximizes U on the CAL. Higher A gives smaller w*.
Tangency rule: Optimal point: indifference curve slope = frontier (or CAL) slope; Tangent to the highest feasible indifference curve.

How to solve Portfolio Selection and Optimal Portfolio questions

Use this method for any question on choosing the optimal portfolio, whether it is conceptual or numerical.

1Identify the opportunity set: efficient frontier only, or CAL/CML because a risk-free asset is available.
2Identify the investor's preferences: risk aversion, indifference curve steepness, or coefficient A.
3If the question is conceptual, apply tangency: the highest reachable indifference curve touches the opportunity set at one point.
4Link risk aversion to position: steeper curves mean a point further left or more in the risk-free asset.
5If numerical with a risk-free asset, compute the risky portfolio's excess return and variance, then use w* = (E(R) − Rf) ÷ (A × σ²).
6Compute the portfolio return and standard deviation using the weights: return is the weighted average, risk is w × σrisky.
7Check that the answer is sensible: w above 1 means borrowing, below 1 means lending, and risk cannot exceed what the weights imply.
8Compare utilities if asked which portfolio is best: pick the highest U.

Quickest way: Tangency and weight shortcut

When to use it: Use when you have 90 seconds and either a conceptual choice or a single optimal-weight calculation.

Conceptual: more risk-averse means steeper curves, lower risk, further left, and more risk-free asset.
Remember that with a risk-free asset every investor holds the same risky portfolio, the tangency portfolio.
Numerical: compute w* = (E(R) − Rf) ÷ (A × σ²) in decimals.
Then portfolio risk = w* × σ and return = Rf + w* × (E(R) − Rf).
Eliminate options that put risk above w × σ or give return outside the mix.

Common mistakes in Portfolio Selection and Optimal Portfolio

Saying the optimal portfolio is the one with the highest return on the frontier.
Students ignore risk preferences and treat the frontier as a ranking.
Fix: Every frontier point is efficient. The optimal point is chosen by tangency with the investor's indifference curve.
Thinking more risk-averse investors hold a different risky portfolio when a risk-free asset exists.
Confusing the choice of risky portfolio with the choice of mix.
Fix: The risky portfolio is the same for all investors. Only the weight in it versus the risk-free asset changes.
Drawing indifference curves of one investor that cross, or reading higher curves as lower utility.
Mixing up direction of preference.
Fix: Utility rises up and to the left. Curves of one investor never cross.
Using percentages instead of decimals for σ² in the utility or weight formula.
Plugging 20 instead of 0.20 into σ².
Fix: Convert to decimals first: 20% gives σ² = 0.04.
Assuming a steeper indifference curve means less risk aversion.
Steepness is confused with the slope of the frontier.
Fix: Steeper curves mean the investor demands a lot more return per unit of risk, so greater risk aversion.

Worked examples

Example 1

A risky portfolio has expected return 12% and standard deviation 20%. The risk-free rate is 4%. An investor has risk-aversion coefficient A = 4 and uses U = E(R) − 0.5 × A × σ². What proportion should be invested in the risky portfolio? A. 0.50 B. 1.00 C. 2.00

Show the solution

Excess return = 12% − 4% = 8% = 0.08.
σ² = 0.20² = 0.04.
w* = 0.08 ÷ (4 × 0.04) = 0.08 ÷ 0.16 = 0.50.
So 50% goes in the risky portfolio and 50% in the risk-free asset.

Answer: A. 0.50

Example 2

Using the same data, what is the standard deviation of the optimal complete portfolio? A. 10% B. 20% C. 30%

Show the solution

Weight in risky portfolio w* = 0.50.
Standard deviation = w* × σrisky = 0.5 × 20% = 10%.
Check: the risk-free asset adds no risk, so portfolio risk is only the risky share of 20%.
Cross-check with return and utility: expected return = 0.5 × 12% + 0.5 × 4% = 8%, and U = 0.08 − 0.5 × 4 × 0.01 = 0.08 − 0.02 = 0.06.

Answer: A. 10%

Exam tips

Questions are three-option MCQs, so read the stem for what is held constant: the risky portfolio, the risk-free rate, or the investor.
For conceptual items, remember the sentence: tangency of the highest reachable indifference curve with the opportunity set.
If w* is above 1, the investor borrows at the risk-free rate. Do not discard it as impossible.
Check units: convert percentages to decimals before using σ² in formulas.
A wrong option often applies risk to the full portfolio instead of w × σ. Eliminate it first.

Practice questions from Portfolio Risk and Return: Part II

Portfolio Selection 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.

Portfolio Selection and Optimal Portfolio: frequently asked questions

What is the optimal portfolio in CFA Level I?

It is the portfolio on the efficient frontier, or on the CAL/CML when a risk-free asset exists, that touches the investor's highest attainable indifference curve. It is the best feasible combination of risk and return for that investor.

How does risk aversion affect the optimal portfolio?

A more risk-averse investor has steeper indifference curves. The tangent point is at lower risk and lower expected return. With a risk-free asset, that means a larger share in the risk-free asset.

Why do all investors hold the same risky portfolio on the CML?

This holds when all investors share the same expectations (homogeneous expectations) and can borrow and lend at the same risk-free rate. Then the tangency portfolio has the highest Sharpe ratio, so the CAL through it dominates every other CAL. Investors only decide how much to lend or borrow at the risk-free rate.

Do indifference curves ever cross?

No, not for one investor with consistent preferences. Crossing would mean two portfolios could give both the same and different utility. Different investors have different curves.