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

Diversification and the Efficient Frontier Explained

Q: How does correlation reduce portfolio risk?

When two assets are not perfectly correlated, their gains and losses do not happen together, so some movements cancel out. The lower the correlation, the more cancellation and the lower the portfolio standard deviation for the same weights. Expected return stays the weighted average.

Q: What is the minimum-variance portfolio?

It is the mix of risky assets with the lowest possible portfolio variance. For two assets, you find its weights with w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2Cov12). On the frontier it is the leftmost point.

Q: What is the difference between the minimum-variance frontier and the efficient frontier?

The minimum-variance frontier shows the lowest risk for every level of expected return. The efficient frontier is only its upper part, starting at the global minimum-variance portfolio. Portfolios on the lower branch offer less return for the same risk, so they are inefficient.

Q: Can diversification remove all risk?

Only in the special case of two assets with a correlation of −1 and the right weights. In practice, correlations between risky assets are above −1. Diversification then reduces risk but cannot remove it.

Updated 7 October 2026 · Fact-checked

Diversification lowers portfolio risk when assets are not perfectly correlated, because their returns do not move in lockstep. The lower the correlation, the larger the benefit. The minimum-variance frontier shows the lowest risk at each return level. The efficient frontier is the part of it from the global minimum-variance portfolio upward.

Understand Diversification and the Efficient Frontier

Portfolio risk is not the average of the risk of its assets. It depends on how the assets move together. If two assets rise and fall at different times, a loss in one is partly offset by a gain in the other. That offset is the diversification benefit.

Correlation measures this co-movement and ranges from −1 to +1. At ρ = +1 there is no benefit: portfolio standard deviation is the weighted average of the two standard deviations. At any ρ below +1, portfolio standard deviation is lower than that weighted average. At ρ = −1 the benefit is greatest, and a particular mix can have zero risk. Expected return is always the weighted average, so diversification cuts risk without cutting expected return.

Plot expected return against standard deviation for every mix of two assets. You get a curve. The lower the correlation, the more it bends to the left. With more than two assets, the minimum-variance frontier is the set of risky portfolios with the lowest variance for each level of expected return. Its leftmost point is the global minimum-variance portfolio.

Below the global minimum-variance portfolio, the frontier curves back down. Those portfolios give lower return for more risk than a portfolio directly above them on the curve, so no rational investor holds them. The part of the frontier from the global minimum-variance portfolio upward is the efficient frontier. Each of its portfolios offers the highest expected return for its level of risk.

The efficient frontier does not tell you which portfolio to hold. That depends on risk aversion. A risk-averse investor picks the point where the highest attainable indifference curve just touches the frontier. When a risk-free asset is added, the choice changes. You then study the capital allocation line.

Key formulas to remember

Two-asset portfolio expected return: E(Rp) = w1 × E(R1) + w2 × E(R2); Weights sum to 1. It is always a weighted average, whatever the correlation.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Standard deviation is the square root of this. The last term is 2 w1 w2 Cov12.
Covariance from correlation: Cov12 = ρ12 × σ1 × σ2; Use it to move between the covariance and correlation forms.
Perfect positive correlation (ρ = +1): σp = w1σ1 + w2σ2; No diversification benefit. Risk is the weighted average of the two standard deviations.
Perfect negative correlation (ρ = −1): σp = |w1σ1 − w2σ2|; Risk can reach zero. This happens at w1 = σ2 ÷ (σ1 + σ2).
Minimum-variance weight, two assets: w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2 Cov12); w2 = 1 − w1; Gives the global minimum-variance portfolio of two risky assets.
Efficient frontier rule: Efficient frontier = minimum-variance frontier from the global minimum-variance portfolio upward; Portfolios below the global minimum-variance point are inefficient.

How to solve Diversification and the Efficient Frontier questions

Use this method for any question on correlation, diversification, the minimum-variance frontier or the efficient set.

1Identify what is asked: portfolio risk, a minimum-variance weight, or a conceptual point about the frontier.
2Write down the weights, standard deviations and correlation. If you are given covariance, convert with Cov = ρσ1σ2, or the reverse.
3Set bounds first. Portfolio standard deviation lies between the ρ = −1 value and the weighted average (ρ = +1). Expected return is always the weighted average.
4For risk, compute variance with σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2, then take the square root. Keep variances in decimals (0.04, not 4%).
5For the minimum-variance portfolio, use w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2Cov12). Check that the two weights sum to 1.
6For concept questions, recall that lower correlation shifts the curve left, the global minimum-variance portfolio is the leftmost point, and only the part above it is efficient.
7Compare your answer with the bounds. If it falls outside them, recheck your arithmetic.

Quickest way: Bounds and elimination

When to use it: Use it when options are numbers close together or when the question is conceptual. You often do not need the full calculation.

Compute the weighted-average standard deviation. That is the ceiling when ρ < +1, so any option above it is wrong.
If ρ is positive but well below 1, expect a result a little under the ceiling. If ρ is 0 or negative, expect a clearly lower result.
If one option equals the weighted average and ρ < 1, eliminate it.
Calculate only if two options remain. Use the calculator memory to keep intermediate values: compute the three variance terms, add them, then press √.
For frontier concepts, remember: efficient means above the global minimum-variance portfolio, and lower correlation means more benefit.

Common mistakes in Diversification and the Efficient Frontier

Taking portfolio standard deviation as the weighted average of the asset standard deviations when ρ is below 1.
Expected return works that way, so students assume risk does too.
Fix: Only at ρ = +1 is risk a weighted average. For any lower ρ, risk is below it.
Forgetting to square the weights, or forgetting the factor of 2 on the covariance term.
Students rush the formula from memory.
Fix: Write the three terms separately: w1²σ1², w2²σ2² and 2w1w2Cov12. Then add them.
Using standard deviations in percent inside the variance formula, or forgetting to take the square root at the end.
Units get mixed: 20% becomes 20 in one place and 0.20 in another.
Fix: Convert to decimals first. The sum is variance. Take √ to get standard deviation, which the options usually show.
Calling the whole minimum-variance frontier the efficient frontier.
The two terms sound alike.
Fix: The efficient frontier is only the part from the global minimum-variance portfolio upward. The lower branch is inefficient.
Believing that diversification reduces expected return, or that a lower correlation changes expected return.
Students link lower risk with lower return.
Fix: Expected return is the weighted average of asset returns. Correlation affects only risk.
Assuming the efficient frontier identifies the single best portfolio for every investor.
Students overlook investor preferences.
Fix: The frontier is the set of candidates. The optimal portfolio depends on risk aversion, at the tangency with an indifference curve.

Worked examples

Example 1

Asset 1 has a standard deviation of 20% and Asset 2 has 10%. Their correlation is 0.20. A portfolio holds 50% in each. Which is closest to the portfolio standard deviation? A) 12.0% B) 15.0% C) 16.2%

Show the solution

Bound: the weighted average is 0.5 × 20% + 0.5 × 10% = 15%. With ρ = 0.20 < 1, the answer must be below 15%. That eliminates B and C.
Check by calculation. w1²σ1² = 0.25 × 0.04 = 0.0100.
w2²σ2² = 0.25 × 0.01 = 0.0025.
2w1w2ρσ1σ2 = 2 × 0.5 × 0.5 × 0.20 × 0.20 × 0.10 = 0.5 × 0.004 = 0.0020.
Variance = 0.0100 + 0.0025 + 0.0020 = 0.0145. √0.0145 ≈ 0.1204, or 12.0%.

Answer: A) 12.0%

Example 2

Using the same assets (σ1 = 20%, σ2 = 10%, ρ = 0.20), what weight in Asset 1 gives the global minimum-variance portfolio? A) 14.3% B) 25.0% C) 50.0%

Show the solution

Covariance: Cov12 = 0.20 × 0.20 × 0.10 = 0.004.
Variances: σ1² = 0.04 and σ2² = 0.01.
Numerator: σ2² − Cov12 = 0.01 − 0.004 = 0.006.
Denominator: σ1² + σ2² − 2Cov12 = 0.04 + 0.01 − 0.008 = 0.042.
w1 = 0.006 ÷ 0.042 = 0.1429, so w2 = 0.8571.
Check: the portfolio variance is about 0.00914, or a standard deviation of about 9.56%. That is below even the lower-risk asset's 10%, which is expected because the low-risk asset gets most of the weight and the correlation is low.

Answer: A) 14.3%

Exam tips

For risk questions, compute the weighted-average standard deviation first. It removes options at once, because with ρ < 1 the true answer is lower.
Watch the units. Variance is in decimals squared, and the options usually show standard deviation in percent.
Know the three correlation cases cold: +1 means no benefit, 0 gives partial benefit, and −1 gives the maximum benefit with a possible zero-risk mix.
For frontier questions, use exact words. The global minimum-variance portfolio is the leftmost point, and the efficient frontier is the part above it.
Read what the question asks. Expected return of the portfolio never depends on correlation, and many distractors rely on that confusion.

Practice questions from The Return and Risk of a Financial Portfolio

Diversification and the Efficient Frontier in other exams

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

Diversification and the Efficient Frontier: frequently asked questions

How does correlation reduce portfolio risk?

When two assets are not perfectly correlated, their gains and losses do not happen together, so some movements cancel out. The lower the correlation, the more cancellation and the lower the portfolio standard deviation for the same weights. Expected return stays the weighted average.

What is the minimum-variance portfolio?

It is the mix of risky assets with the lowest possible portfolio variance. For two assets, you find its weights with w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2Cov12). On the frontier it is the leftmost point.

What is the difference between the minimum-variance frontier and the efficient frontier?

The minimum-variance frontier shows the lowest risk for every level of expected return. The efficient frontier is only its upper part, starting at the global minimum-variance portfolio. Portfolios on the lower branch offer less return for the same risk, so they are inefficient.

Can diversification remove all risk?

Only in the special case of two assets with a correlation of −1 and the right weights. In practice, correlations between risky assets are above −1. Diversification then reduces risk but cannot remove it.