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

Portfolio Expected Return and Variance Explained

Updated 6 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 also needs covariance: for two assets, w1²σ1² + w2²σ2² + 2w1w2Cov(1,2). Take the square root of variance to get portfolio standard deviation.

Understand Expected Return and Variance of a Portfolio

A portfolio is a set of assets held in given proportions. The weight of an asset is its market value divided by the total portfolio value. Weights add up to 1, or 100%. A short position has a negative weight.

Expected return is simple. Each asset contributes its own expected return times its weight. The portfolio's expected return is the sum of these contributions. Correlation does not matter here.

Risk is different. Variance measures how widely portfolio returns spread around the mean. Portfolio variance depends on three things: how much you hold of each asset, how volatile each asset is, and how the assets move together. That last part is covariance. If two assets do not move in lockstep, their ups and downs partly cancel. This is the source of diversification.

This is why portfolio standard deviation is usually less than the weighted average of the individual standard deviations. The two are equal only when correlation is +1. Lower correlation gives more risk reduction.

Remember the order of work: find the weights, find the expected return, find the variance, then take the square root if the question asks for standard deviation.

Key formulas to remember

Portfolio expected return
E(Rp) = Σ wi × E(Ri)
Weights must sum to 1. Short positions use negative weights.
Two-asset portfolio variance (with covariance)
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 Cov(R1, R2)
Cov is the covariance of the two assets' returns.
Covariance from correlation
Cov(R1, R2) = ρ12 × σ1 × σ2
Use this when the question gives correlation instead of covariance.
Two-asset portfolio variance (with correlation)
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 σ1 σ2 ρ12
Same formula as above, with covariance replaced.
Portfolio standard deviation
σp = √σp²
Convert variance back to standard deviation at the end.
Perfect positive correlation case
σp = w1σ1 + w2σ2 when ρ12 = +1 (weights non-negative)
No diversification benefit. Risk is the weighted average.
General n-asset variance
σp² = Σi Σj wi wj Cov(Ri, Rj)
Includes i = j terms, where Cov(Ri, Ri) = σi².

How to solve Expected Return and Variance of a Portfolio questions

Use this order for any portfolio return or risk question. It keeps units and terms straight.

  1. 1Write down the weights. If given amounts, divide each by the total. Check they sum to 1.
  2. 2Compute expected return as the weighted sum of expected returns.
  3. 3Check whether you are given covariance or correlation. If correlation, convert: Cov = ρ × σ1 × σ2.
  4. 4Check whether you are given standard deviations or variances. Square standard deviations to get variances.
  5. 5Compute each part of the variance: w1²σ1², w2²σ2², and 2w1w2Cov.
  6. 6Add the three parts to get variance.
  7. 7Take the square root if the question asks for standard deviation.
  8. 8Sanity check: with ρ below 1, σp must be below the weighted average of the two standard deviations.

Quickest way: Work in decimals and compare with the +1 bound

When to use it: Use this when you have about 90 seconds and the options are far enough apart to estimate.

  1. Compute the weighted average of standard deviations. This is the upper bound for non-negative weights.
  2. Any option above this bound cannot be correct when ρ is below 1. Eliminate it.
  3. Calculate the exact variance using decimals, for example 0.40 and 0.15, not percentages.
  4. On the BA II Plus, use the memory keys or chain the calculation. Take the square root with the √x key.
  5. Match your result to the option. Numerical options run from smallest to largest, so check you picked the right position.

Common mistakes in Expected Return and Variance of a Portfolio

  • Using a weighted average of standard deviations as portfolio risk.

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

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

  • Forgetting the factor of 2 on the covariance term.

    The covariance appears twice, once as Cov(1,2) and once as Cov(2,1), and students drop one.

    Fix: Always write 2 × w1 × w2 × Cov before you substitute numbers.

  • Not squaring the weights in the variance terms.

    Students copy the return formula and replace the return with variance.

    Fix: Variance terms are w², not w. Write w1²σ1² explicitly.

  • Mixing up variance and standard deviation.

    Questions give σ but the formula needs σ². Or the answer needs σ but you stop at variance.

    Fix: Square the standard deviations on the way in. Take the square root on the way out.

  • Using covariance as if it were correlation, or the reverse.

    Both measure co-movement. Correlation is bounded between -1 and +1, while covariance is not.

    Fix: If the number is between -1 and +1, it is likely correlation. Convert with Cov = ρσ1σ2 before using the variance formula.

  • Weights that do not sum to 1 or ignoring a short position.

    Students use raw amounts or forget the sign.

    Fix: Divide by total portfolio value. Enter short positions as negative weights and check the sum.

Worked examples

Example 1

A portfolio holds 60% in Asset A and 40% in Asset B. Asset A has an expected return of 10% and standard deviation of 20%. Asset B has an expected return of 5% and standard deviation of 10%. The correlation between them is 0.25. What is the portfolio's standard deviation? (A) 12.6%, (B) 13.6%, (C) 16.0%.

Show the solution
  1. Covariance = 0.25 × 0.20 × 0.10 = 0.005.
  2. w1²σ1² = 0.36 × 0.04 = 0.0144.
  3. w2²σ2² = 0.16 × 0.01 = 0.0016.
  4. 2w1w2Cov = 2 × 0.60 × 0.40 × 0.005 = 0.0024.
  5. Variance = 0.0144 + 0.0016 + 0.0024 = 0.0184.
  6. Standard deviation = √0.0184 = 0.1356, or about 13.6%.
  7. Check: the weighted average of standard deviations is 0.6 × 20% + 0.4 × 10% = 16%. Option C equals this bound, which is only correct when ρ = +1, so it is wrong here. Option A (12.6%) is what you get if you leave out the covariance term: √(0.0144 + 0.0016) = √0.016 = 12.6%. That leaves option B.

Answer: Portfolio standard deviation is about 13.6%, which is option B.

Example 2

An investor holds €60,000 in Fund X and €40,000 in Fund Y. The variance of Fund X returns is 0.0225 and the variance of Fund Y returns is 0.0100. The covariance between the funds is -0.0030. What is the portfolio variance? (A) 0.0064, (B) 0.0083, (C) 0.0111.

Show the solution
  1. Total value = €1,00,000, so wX = 0.60 and wY = 0.40.
  2. wX²σX² = 0.36 × 0.0225 = 0.0081.
  3. wY²σY² = 0.16 × 0.0100 = 0.0016.
  4. 2wXwYCov = 2 × 0.60 × 0.40 × (-0.0030) = 0.48 × (-0.0030) = -0.00144.
  5. Variance = 0.0081 + 0.0016 - 0.00144 = 0.0097 - 0.00144 = 0.00826.
  6. Round to four decimals: 0.00826 is about 0.0083, which matches option B.
  7. Option C (0.0111) is what you get if you add the covariance term instead of subtracting it: 0.0097 + 0.00144 = 0.01114, or about 0.0111. This is the sign trap. Option A (0.0064) is too low and does not match the correct calculation.

Answer: Portfolio variance is 0.00826, or about 0.0083 (standard deviation about 9.1%). The answer is option B.

Exam tips

  • Expect questions that give correlation and standard deviations. Convert to covariance first, then apply the variance formula.
  • Use the +1 correlation bound to eliminate options fast. Portfolio risk above the weighted average of standard deviations is impossible with non-negative weights and ρ < 1.
  • Work in decimals, not percentages, to avoid scale errors. 20% is 0.20, and its variance is 0.04.
  • Read what is asked: variance, standard deviation, or expected return. Examiners place the intermediate value as a wrong option.
  • Know the conceptual points: lower correlation gives more diversification benefit, and expected return does not depend on correlation.

Practice questions from The Return and Risk of a Financial Portfolio

Expected Return and Variance of a Portfolio in other exams

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

Expected Return and Variance of a Portfolio: frequently asked questions

What is the portfolio expected return formula in CFA Level I?

It is E(Rp) = Σ wi × E(Ri). You multiply each asset's weight by its expected return and add the results. Weights must sum to 1.

How do I calculate portfolio variance for two assets?

Use σp² = w1²σ1² + w2²σ2² + 2w1w2Cov(1,2). If you are given correlation, replace Cov with ρ12σ1σ2. Take the square root for standard deviation.

Why is portfolio risk not a weighted average of asset risks?

Because assets do not move perfectly together. The covariance term captures how their returns interact. Only when correlation is +1 does portfolio standard deviation equal the weighted average.

Can portfolio variance be lower than the variance of every asset in it?

Yes, it can, when correlation is low or negative and the weights are well chosen. This is the benefit of diversification.