FRM Exam Part II · Portfolio Construction

Mean-Variance Optimization and Portfolio Construction Basics

Updated 11 October 2026 · Fact-checked

Mean-variance optimization picks portfolio weights that maximize expected return for a given variance, or minimize variance for a given return. Inputs are expected returns, volatilities and correlations. Solve by writing the objective, adding constraints, tracing the efficient frontier, and then testing how sensitive the weights are to input errors.

Understand Portfolio Construction Basics and Mean-Variance Optimization

Portfolio construction turns views on assets into weights. Mean-variance optimization (MVO) is the standard framework. You give it three inputs: a vector of expected returns, a vector of volatilities and a correlation (or covariance) matrix. It returns the weights that give the best trade-off between return and risk.

The objective can be written two ways. You can minimize variance for a target return, or maximize expected return for a target variance. A third form maximizes utility: E(Rp) − (λ ÷ 2) × σp², where λ is risk aversion. All three trace the same set of portfolios when constraints are the same.

The set of portfolios with the lowest variance for each level of return is the efficient frontier. The portfolio at its left tip is the global minimum variance portfolio. Only the part above it is efficient. Portfolios below it give less return for the same risk. Constraints such as full investment (weights sum to 1), no short selling (weights ≥ 0) or position caps move the frontier inward. A constrained frontier can never lie above the unconstrained one.

MVO has a well-known weakness: it is an error maximizer. It overweights assets with overestimated returns, underestimated volatility or underestimated correlation. Small changes in expected returns can produce large changes in weights. Expected returns are far harder to estimate than covariances, so they cause most of the instability. Typical fixes are constraints, shrinkage of the covariance matrix, resampling, and Black-Litterman style priors.

MVO also assumes that variance captures risk. It ignores skewness and fat tails, and it treats upside and downside the same. Risk parity, by contrast, ignores expected returns and sets weights so each asset contributes equal risk. It is less sensitive to return estimates but depends on volatility and correlation inputs, and it often needs leverage to reach a return target.

Key formulas to remember

Portfolio expected return: E(Rp) = Σ wᵢ × E(Rᵢ) = w′μ; Weights wᵢ usually sum to 1 under the full-investment constraint.
Portfolio variance: σp² = w′Σw = Σᵢ Σⱼ wᵢ wⱼ σᵢⱼ; Σ is the covariance matrix; σᵢⱼ = ρᵢⱼ × σᵢ × σⱼ.
Two-asset variance: σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ σ₁ σ₂; Lower correlation gives lower portfolio variance for the same weights.
Global minimum variance weight (two assets): w₁ = (σ₂² − ρσ₁σ₂) ÷ (σ₁² + σ₂² − 2ρσ₁σ₂); w₂ = 1 − w₁. Valid with full investment and shorting allowed.
Mean-variance utility: U = E(Rp) − (λ ÷ 2) × σp²; Higher λ means more risk aversion and a lower-risk optimal portfolio.
Sharpe ratio: SR = (E(Rp) − Rf) ÷ σp; The tangency portfolio maximizes this ratio when a risk-free asset exists.
Risk parity condition: wᵢ × (Σw)ᵢ ÷ σp = same for all assets i; Each asset contributes equal risk. With zero correlations, weights are proportional to 1 ÷ σᵢ.

How to solve Portfolio Construction Basics and Mean-Variance Optimization questions

Use this sequence for any MVO or portfolio construction question.

1Identify the objective: minimize variance, maximize return, maximize Sharpe ratio or maximize utility.
2List the inputs given: expected returns, volatilities, correlations or covariances, and the risk-free rate if any.
3List the constraints: sum of weights, short-sale limits, caps, turnover or tracking error limits.
4Compute portfolio return and variance for the candidate weights, using the variance formula with the correlation term.
5Locate the portfolio on the frontier: compare with the minimum variance portfolio, the tangency portfolio, or a target return.
6If the question is about inputs, ask which one changed and which assets gain weight. Higher expected return or lower risk raises weight. Lower correlation raises weight.
7Interpret: state how constraints reduce the frontier and how estimation error makes weights unstable.
8Check the answer: weights sum to 1, variance is positive, and volatility is the square root of variance.

Quickest way: Direction-first elimination

When to use it: Use for conceptual or sensitivity MCQs where full calculation is not needed.

Decide the direction of each change: higher return, lower volatility or lower correlation increases an asset's weight.
Remember that constraints can only lower or keep the optimal objective value, never raise it.
Remember that return-estimation error matters more than covariance error.
For risk parity versus MVO, ask whether expected returns are used. MVO uses them; risk parity does not.
For numeric questions, compute variance with the correlation term first, then take the square root last.

Common mistakes in Portfolio Construction Basics and Mean-Variance Optimization

Forgetting the correlation term or using 2ρσ₁σ₂ without the weights
Students rush and treat variance as additive.
Fix: Write all three terms every time: w₁²σ₁², w₂²σ₂² and 2w₁w₂ρσ₁σ₂.
Reporting variance as if it were volatility
The last step of taking the square root is skipped.
Fix: Check whether the question asks for σp or σp². Take √ before answering volatility.
Saying constraints can improve the frontier
Students confuse practical safety with mathematical optimality.
Fix: Constraints shrink the feasible set, so the constrained frontier is at or below the unconstrained one. They may still improve out-of-sample stability.
Blaming covariance errors as the main source of instability
Covariance matrices look more complex.
Fix: Remember that optimal weights are far more sensitive to errors in expected returns than in covariances.
Treating risk parity as the same as minimum variance
Both ignore expected returns.
Fix: Minimum variance minimizes total risk and often concentrates in low-volatility assets. Risk parity equalizes risk contributions and holds all assets.
Thinking the whole frontier curve is efficient
The textbook curve is drawn as a bullet.
Fix: Only the part from the minimum variance portfolio upward is efficient.

Worked examples

Example 1

Asset A has expected return 8% and volatility 12%. Asset B has expected return 4% and volatility 5%. Correlation is 0.20. A portfolio holds 40% A and 60% B. Find expected return and volatility.

Show the solution

Expected return = 0.4 × 8% + 0.6 × 4% = 3.2% + 2.4% = 5.6%.
w₁²σ₁² = 0.16 × 0.0144 = 0.002304.
w₂²σ₂² = 0.36 × 0.0025 = 0.0009.
2w₁w₂ρσ₁σ₂ = 2 × 0.4 × 0.6 × 0.2 × 0.12 × 0.05 = 0.48 × 0.2 × 0.006 = 0.000576.
Variance = 0.002304 + 0.0009 + 0.000576 = 0.00378.
Volatility = √0.00378 ≈ 0.06148, or about 6.15%.

Answer: Expected return 5.6%; volatility about 6.15%.

Example 2

Using the same two assets with correlation 0.20 and shorting allowed, find the weight in A of the global minimum variance portfolio.

Show the solution

σ₁² = 0.0144, σ₂² = 0.0025, σ₁σ₂ = 0.006.
ρσ₁σ₂ = 0.2 × 0.006 = 0.0012.
Numerator = σ₂² − ρσ₁σ₂ = 0.0025 − 0.0012 = 0.0013.
Denominator = 0.0144 + 0.0025 − 2 × 0.0012 = 0.0169 − 0.0024 = 0.0145.
w₁ = 0.0013 ÷ 0.0145 ≈ 0.0897.
w₂ = 1 − 0.0897 ≈ 0.9103.

Answer: About 9.0% in A and 91.0% in B. The minimum variance portfolio is heavily weighted to the low-volatility asset.

Exam tips

Expect questions asking which input change most affects optimal weights. The usual answer is expected returns.
Know that constraints lower the frontier but can reduce estimation-error damage.
Be ready to contrast MVO, minimum variance and risk parity by inputs used: returns, volatilities and correlations.
In numeric questions, check the answer options for variance versus volatility traps.
Remember that MVO uses variance, so it does not capture skewness or tail risk.

Practice questions from Portfolio Construction

Portfolio Construction Basics and Mean-Variance Optimization in other exams

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

CFA Level IIIMean-Variance Optimization and Its Limitations

Portfolio Construction Basics and Mean-Variance Optimization: frequently asked questions

What is the difference between mean-variance optimization and risk parity?

MVO uses expected returns, volatilities and correlations to choose weights on the return-risk trade-off. Risk parity uses only risk inputs and sets weights so each asset contributes equal risk. Risk parity is less exposed to return-forecast error but often needs leverage.

Why is mean-variance optimization sensitive to estimation error?

The optimizer treats inputs as exact and pushes weight into assets that look best, including those with overestimated returns. Small input changes can swing weights sharply, especially when assets are highly correlated. Expected returns are the main source of this problem.

How do constraints change the efficient frontier?

Constraints such as no shorting or position caps reduce the feasible set. The constrained frontier lies at or below the unconstrained one. Weights are usually more diversified and more stable.

What is the global minimum variance portfolio?

It is the portfolio with the lowest possible variance among all feasible portfolios. It sits at the left tip of the frontier and does not depend on expected returns.