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

Diversification and the Efficient Frontier Explained

Updated 7 October 2026 · Fact-checked

Diversification combines assets that are not perfectly correlated, so portfolio risk falls below the weighted average of individual risks. The efficient frontier is the set of portfolios giving the highest expected return for each level of risk. To solve questions, use the two-asset variance formula, check the correlation, and remember the minimum-variance portfolio is the frontier's leftmost point.

Understand Diversification and Efficient Frontier

Diversification means holding several assets instead of one. A portfolio's expected return is the weighted average of the asset returns. Its risk is usually lower than the weighted average of the asset risks. The gap comes from correlation. When one asset falls, another may rise or fall less, so the ups and downs partly cancel.

The lower the correlation, the bigger the benefit. With a correlation of +1, there is no benefit: portfolio standard deviation equals the weighted average of the standard deviations. With any correlation below +1, portfolio risk is lower than that average. With a correlation of −1, risk can be reduced to zero at the right weights.

Total risk splits into two parts. Systematic risk (market risk) affects all assets and cannot be diversified away. Nonsystematic risk (unsystematic, firm-specific or idiosyncratic risk) is unique to a company or sector, and diversification reduces it. As you add more assets, nonsystematic risk shrinks and the portfolio's risk approaches systematic risk. The marginal benefit of each extra asset gets smaller.

Now plot every possible portfolio of risky assets with risk (standard deviation) on the x-axis and expected return on the y-axis. This is the opportunity set (investment opportunity set). The global minimum-variance portfolio is the leftmost point, the portfolio with the lowest risk of all. The efficient frontier is the part of the opportunity set from that point upward. These portfolios give the highest expected return for a given risk, or the lowest risk for a given return. Portfolios below the minimum-variance point are dominated, because a portfolio with the same risk and a higher return exists above them.

The frontier bows to the left because of diversification. The lower the correlation between assets, the more the curve bows. Different investors choose different points on it, depending on their risk tolerance.

Key formulas to remember

Covariance from correlation
Cov(1,2) = ρ12 × σ1 × σ2
Use this when the question gives correlation and standard deviations, so you can plug covariance into the variance or weights formula.
Two-asset portfolio variance
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2
Portfolio standard deviation is the square root. Weights are squared in the first two terms. Do not forget the final square root.
Perfect positive correlation (ρ = +1)
σp = w1σ1 + w2σ2
The only case where portfolio risk equals the weighted average of the standard deviations. There is no diversification benefit.
Perfect negative correlation (ρ = −1)
σp = |w1σ1 − w2σ2|
Risk can be reduced to zero by choosing w1 = σ2 ÷ (σ1 + σ2).
Two-asset minimum-variance weight
w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2 Cov12), and w2 = 1 − w1
With ρ = 0 this simplifies to w1 = σ2² ÷ (σ1² + σ2²). The lower-risk asset gets the larger weight.
Risk decomposition
Total risk = Systematic risk + Nonsystematic risk
Diversification removes nonsystematic risk only. Investors are not rewarded for bearing risk that can be diversified away.
Efficient frontier definition
Efficient portfolios: maximum expected return for a given σ (or minimum σ for a given return)
It starts at the global minimum-variance portfolio and runs upward and to the right.

How to solve Diversification and Efficient Frontier questions

Use this sequence for any question on diversification, minimum variance or the efficient frontier.

  1. 1Identify what is asked: portfolio risk, a weight, a risk type, or which portfolios are efficient.
  2. 2Write down weights, standard deviations and the correlation or covariance. Convert percentages to decimals.
  3. 3For portfolio risk, compute each variance term, add them, then take the square root.
  4. 4For the minimum-variance weight, use the weight formula. Check that w1 + w2 = 1 and the lower-risk asset has the larger weight when ρ is low.
  5. 5Sense-check: with ρ < +1, portfolio σ must be below the weighted average of the σs. With ρ = +1 it equals that average.
  6. 6For frontier questions, ask whether a portfolio is above the minimum-variance point and has the highest return for its risk. Anything below it is not efficient.
  7. 7For risk-type questions, ask whether the risk affects the whole market (systematic) or one firm (nonsystematic). Only the latter diversifies away.

Quickest way: Bounds check to eliminate options

When to use it: Use this on numerical portfolio-risk questions where you have about 90 seconds and the options are listed smallest to largest.

  1. Compute the weighted average of the standard deviations. This is the upper bound, reached only when ρ = +1.
  2. If ρ < +1, eliminate any option at or above that bound.
  3. If ρ = 0 or negative, expect the answer well below the bound and often below the lower-risk asset's σ at the right weights.
  4. Compute the full variance only if two options remain. On the BA II Plus enter the variance, then press 2nd √. On the HP 12C enter the variance, then press g √x.
  5. For the minimum-variance weight, use w1 = σ2² ÷ (σ1² + σ2²) when ρ = 0, and give the larger weight to the lower-risk asset.

Common mistakes in Diversification and Efficient Frontier

  • Averaging standard deviations to get portfolio risk

    Expected return is a weighted average, so students assume risk is too.

    Fix: Use the variance formula with the correlation term. The weighted average of σs is correct only when ρ = +1.

  • Saying diversification can eliminate all risk

    Students remember that risk falls as assets are added.

    Fix: Diversification removes nonsystematic risk only. Systematic risk remains, apart from the special case of ρ = −1 between two assets.

  • Treating the whole opportunity set as the efficient frontier

    Both are drawn as curves on the same graph.

    Fix: The frontier is only the upper part, from the global minimum-variance portfolio upward. Portfolios below it are dominated.

  • Forgetting to square the weights or to take the square root

    Time pressure and several terms in the formula.

    Fix: Write each term separately. The answer to the formula is variance, so take √ for standard deviation before choosing an option.

  • Mixing up covariance and correlation

    Both measure co-movement, but only correlation is limited to −1 to +1.

    Fix: Covariance = ρσ1σ2. If a question gives a covariance, use it directly in the formula and do not multiply by ρ again.

  • Expecting a higher expected return from adding diversifiable risk

    Students link all risk to return.

    Fix: Investors are rewarded for systematic risk only. Nonsystematic risk can be diversified away, so it earns no extra return.

Worked examples

Example 1

Asset 1 has a standard deviation of 20% and Asset 2 has 10%. A portfolio holds 60% in Asset 1 and 40% in Asset 2. The correlation is +0.5. The portfolio standard deviation is closest to: A) 12.0%, B) 14.4%, C) 16.0%.

Show the solution
  1. Square the weighted terms: w1²σ1² = 0.36 × 0.04 = 0.0144.
  2. w2²σ2² = 0.16 × 0.01 = 0.0016.
  3. Covariance term: 2 × 0.6 × 0.4 × 0.5 × 0.20 × 0.10 = 0.0048.
  4. Variance = 0.0144 + 0.0016 + 0.0048 = 0.0208.
  5. Standard deviation = √0.0208 = 0.1442, or 14.4%.
  6. Check: the weighted average is 0.6 × 20% + 0.4 × 10% = 16%. Since ρ < 1, the answer must be below 16%, which rules out C.

Answer: B) 14.4%

Example 2

Asset 1 has a standard deviation of 20% and Asset 2 has 10%. Their correlation is zero. What weight in Asset 1 gives the minimum-variance portfolio? A) 20%, B) 50%, C) 80%.

Show the solution
  1. With ρ = 0, covariance is zero, so w1 = σ2² ÷ (σ1² + σ2²).
  2. σ2² = 0.01 and σ1² = 0.04, so w1 = 0.01 ÷ 0.05 = 0.20.
  3. So w2 = 0.80.
  4. Check: portfolio variance = 0.04 × 0.04 + 0.64 × 0.01 = 0.0016 + 0.0064 = 0.0080, so σ = 8.94%. This is below even the 10% of the safer asset.
  5. The riskier asset gets the smaller weight, as expected.

Answer: A) 20% in Asset 1 (and 80% in Asset 2)

Exam tips

  • When a question gives a low or negative correlation, expect the answer to be below the weighted average of the standard deviations. Use this to eliminate options fast.
  • Read carefully whether the question asks for variance or standard deviation. Wrong-unit options are common distractors.
  • Risk-type questions often hinge on one word. Market-wide factors such as interest rates or recessions are systematic. Strikes, lawsuits or management errors at one firm are nonsystematic.
  • For frontier questions, compare portfolios on both axes. A portfolio is inefficient if another has higher return at the same or lower risk.
  • Keep the options in order. Numerical options run from smallest to largest, so after computing, match your answer carefully to A, B or C, and guess if time runs out because there is no penalty.

Practice questions from Portfolio Risk and Return: Part I

Diversification and 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 Efficient Frontier: frequently asked questions

What is the efficient frontier in CFA Level I?

It is the set of risky-asset portfolios that offer the highest expected return for each level of risk. It starts at the global minimum-variance portfolio and slopes upward. Portfolios below it are inefficient.

How does correlation affect diversification benefits?

The lower the correlation between assets, the greater the reduction in portfolio risk. At +1 there is no benefit. Below +1 portfolio risk is lower than the weighted average of the asset risks, and at −1 it can fall to zero.

What is the minimum-variance portfolio?

It is the portfolio with the lowest possible standard deviation from the available risky assets. In the two-asset case you find its weights from the variances and the covariance. It is the leftmost point of the opportunity set and the start of the efficient frontier.

What is the difference between systematic and unsystematic risk?

Systematic risk affects the whole market and cannot be diversified away. Unsystematic risk is specific to a company or industry and can be reduced by holding many assets. Only systematic risk is rewarded with higher expected return.