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

Portfolio Expected Return and Risk for CFA Level I

Updated 7 October 2026 · Fact-checked

Portfolio expected return is the weighted average of each asset's expected return. Portfolio variance is not a weighted average: it adds each asset's weighted variance plus twice each weighted covariance. Compute the return, build the variance from the weights, standard deviations and correlations, then take the square root for standard deviation.

Understand Portfolio Expected Return and Risk

A portfolio is a set of assets with weights that add up to 1. The weight of an asset is its market value divided by the total portfolio value. Weights can be negative for short positions, and then the other weights add up to more than 1.

Expected return is easy. The portfolio's expected return is the sum of weight × expected return for each asset. Return is linear, so a 60/40 mix earns 60% of one return plus 40% of the other.

Risk is different. Variance is not linear, because assets move together. The extra term is covariance, which measures how two returns move together. Covariance equals correlation × standard deviation of A × standard deviation of B. Correlation always lies between -1 and +1.

This is why diversification works. If correlation is below +1, the portfolio standard deviation is less than the weighted average of the individual standard deviations. At correlation +1 there is no reduction. The lower the correlation, the larger the benefit. At -1 you can build a portfolio with zero risk using the right weights.

With three or more assets you use a covariance matrix. The diagonal holds each asset's variance. Off-diagonal cells hold covariances, and the matrix is symmetric. Portfolio variance is the sum of weight × weight × covariance over every cell of the matrix.

Key formulas to remember

Portfolio expected return
E(Rp) = Σ wᵢ × E(Rᵢ)
Weights sum to 1. Use decimals or percentages consistently.
Covariance from correlation
Cov(A,B) = ρ(A,B) × σA × σB
Rearrange to get ρ = Cov ÷ (σA × σB). Covariance of an asset with itself is its variance.
Two-asset portfolio variance
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2
Take the square root for standard deviation. Do not square-root before adding.
General portfolio variance
σp² = Σᵢ Σⱼ wᵢ wⱼ Cov(i,j)
Sum over every cell of the covariance matrix. Off-diagonal pairs appear twice, so each pair carries a factor of 2.
Number of distinct covariances
n(n − 1) ÷ 2
For n assets. Three assets need three covariances, four need six.
Perfect positive correlation (ρ = +1)
σp = w1σ1 + w2σ2
Only in this case is portfolio standard deviation the weighted average of the standard deviations (long positions).
Perfect negative correlation (ρ = −1)
σp = |w1σ1 − w2σ2|
Risk can fall to zero if w1σ1 = w2σ2.

How to solve Portfolio Expected Return and Risk questions

Use this method for any question on expected return or risk of a multi-asset portfolio.

  1. 1Write down the weights and check they sum to 1. If market values are given, divide each by the total.
  2. 2Compute expected return as Σ weight × expected return. Keep percentages as decimals if you will square them later.
  3. 3If the question gives correlations, convert each to a covariance: ρ × σ × σ. If it gives a covariance matrix, use it directly.
  4. 4Compute the variance terms: weight² × variance for each asset.
  5. 5Compute the covariance terms: 2 × weight × weight × covariance for each pair. Count each pair once, with the factor of 2.
  6. 6Add all terms to get portfolio variance. Take the square root if the question asks for standard deviation.
  7. 7Sense-check: for long positions, the answer should be below the weighted average of the standard deviations unless correlation is +1 (or all assets have identical risk and perfect correlation). It should also be above zero unless correlation is -1 and the weights offset exactly.

Quickest way: Bounds check and elimination

When to use it: Use when you have about 90 seconds and the options are far apart. It often removes two options without full calculation.

  1. Compute the weighted average of the standard deviations. For long positions, this is the upper limit of portfolio risk, reached only when correlation is +1.
  2. When all correlations are below +1, any option at or above that figure is wrong. This usually removes one option.
  3. Run the full formula only if the two remaining options are close.
  4. For the square root, key the variance and press √ on the TI BA II Plus. On the HP 12C press g then √x.
  5. For expected return, use the memory keys or simply add the weighted terms. Do not retype intermediate answers rounded to few digits.

Common mistakes in Portfolio Expected Return and Risk

  • Taking the weighted average of standard deviations as portfolio risk

    Expected return works that way, so students assume risk does too.

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

  • Forgetting the factor of 2 on the covariance term

    The pair w1w2Cov appears twice in the matrix, once in each off-diagonal cell, but the two-asset formula is often remembered without the 2.

    Fix: Always write 2 × w1 × w2 × Cov. In a matrix, either use both cells or use one cell with a factor of 2.

  • Squaring the weight but not the standard deviation, or the reverse

    Mixed-up units under time pressure.

    Fix: Each variance term is w² × σ². Covariance terms use w × w × σ × σ × ρ. Write the term out before you key it.

  • Giving variance when asked for standard deviation

    The calculation ends one step early.

    Fix: Check the last line of the question. Standard deviation needs the square root. Options such as 0.0189 and 13.7% can both appear.

  • Mixing percentages and decimals

    Variance in percent squared is hard to read, for example 20% squared is 400, not 4.

    Fix: Convert to decimals first: 20% = 0.20, so variance = 0.04. Convert back at the end.

  • Using correlation where a covariance is needed, or the reverse

    Both measure co-movement and the names look alike.

    Fix: Correlation is unit-free and between -1 and +1. Covariance can be any size. If the value is outside -1 to +1 it must be a covariance.

Worked examples

Example 1

A portfolio holds 60% in Asset X and 40% in Asset Y. X has an expected return of 10% and a standard deviation of 20%. Y has an expected return of 6% and a standard deviation of 10%. The correlation between them is 0.3. What is the portfolio standard deviation? A. 11.0%, B. 13.7%, C. 16.0%

Show the solution
  1. Weights are 0.6 and 0.4, which sum to 1.
  2. Expected return check: 0.6 × 10% + 0.4 × 6% = 6% + 2.4% = 8.4%. Not asked, but useful for context.
  3. Variance of X term: 0.6² × 0.20² = 0.36 × 0.04 = 0.0144.
  4. Variance of Y term: 0.4² × 0.10² = 0.16 × 0.01 = 0.0016.
  5. Covariance term: 2 × 0.6 × 0.4 × 0.3 × 0.20 × 0.10 = 0.48 × 0.3 × 0.02 = 0.00288.
  6. Portfolio variance = 0.0144 + 0.0016 + 0.00288 = 0.01888.
  7. Standard deviation = √0.01888 = 0.1374, or 13.7%.
  8. Elimination check: the weighted average of standard deviations is 0.6 × 20% + 0.4 × 10% = 16.0%. Since ρ < 1, the answer must be lower, so C is out.

Answer: B. 13.7%

Example 2

A portfolio has weights of 50% in Asset 1, 30% in Asset 2 and 20% in Asset 3. Variances are 0.0400, 0.0225 and 0.0100. Covariances are Cov(1,2) = 0.0060, Cov(1,3) = 0.0020 and Cov(2,3) = 0.0030. What is the portfolio standard deviation? A. 8.0%, B. 12.2%, C. 16.5%

Show the solution
  1. Variance terms: 0.5² × 0.0400 = 0.0100; 0.3² × 0.0225 = 0.002025; 0.2² × 0.0100 = 0.0004.
  2. Sum of variance terms = 0.0100 + 0.002025 + 0.0004 = 0.012425.
  3. Covariance terms: 2 × 0.5 × 0.3 × 0.0060 = 0.0018; 2 × 0.5 × 0.2 × 0.0020 = 0.0004; 2 × 0.3 × 0.2 × 0.0030 = 0.00036.
  4. Sum of covariance terms = 0.0018 + 0.0004 + 0.00036 = 0.00256.
  5. Portfolio variance = 0.012425 + 0.00256 = 0.014985.
  6. Standard deviation = √0.014985 = 0.1224, or about 12.2%.
  7. Trap check: the standard deviations are 20%, 15% and 10%. Their weighted average is 0.5 × 20% + 0.3 × 15% + 0.2 × 10% = 16.5%, which is option C. That is the answer only if all correlations were +1.

Answer: B. 12.2%

Exam tips

  • Options are listed smallest to largest. The weighted average of standard deviations is often the largest option and is a planted trap.
  • Questions may give a correlation matrix instead of covariances. Convert each correlation to covariance with ρ × σ × σ before using the variance formula.
  • Conceptual questions ask what happens to risk when correlation falls. The answer is that portfolio risk falls, with expected return unchanged.
  • Know the special cases: ρ = +1 gives the weighted average, ρ = -1 can give zero risk, ρ = 0 leaves only the variance terms.
  • Keep four or five decimals in variance. Rounding early can push you between two close options.

Practice questions from Portfolio Risk and Return: Part I

Portfolio Expected Return and Risk in other exams

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

Portfolio Expected Return and Risk: frequently asked questions

What is the portfolio variance formula for two assets?

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2. The last term is twice the weighted covariance. Take the square root to get standard deviation.

How do I calculate the expected return of a portfolio?

Multiply each asset's weight by its expected return and add the results. The weights must sum to 1. This is a plain weighted average, and correlation does not affect it.

How does correlation affect portfolio risk?

Lower correlation means lower portfolio risk for the same weights and standard deviations. At +1 there is no diversification benefit. At -1 risk can be reduced to zero with the right weights.

How do I find portfolio risk with three assets and a covariance matrix?

Multiply each weight pair by its matrix cell and add everything. The diagonal gives w² × variance. Each off-diagonal pair is counted twice, so use 2 × wᵢ × wⱼ × covariance once per pair.