Economic Modelling · Capital Asset Pricing Model (CAPM)

Mean-Variance Portfolio Theory and the Efficient Frontier

Updated 11 October 2026 · Fact-checked

Mean-variance portfolio theory judges a portfolio only by its expected return and its variance. You compute both from asset weights, expected returns, variances and covariances. The efficient frontier is the set of portfolios with the highest expected return for each level of risk. A risk-averse investor picks the point that suits their risk aversion.

Understand Mean-Variance Portfolio Theory and Efficient Frontier

Mean-variance theory, due to Markowitz, starts with one idea: an investor cares about two numbers for a portfolio. These are the expected return (the mean) and the risk, measured by the variance or standard deviation of return. More return is preferred. Less variance is preferred.

The key insight is that portfolio risk is not the average of the risks of its assets. It depends on how the assets move together, measured by covariance or correlation. If two assets are not perfectly positively correlated, combining them gives a portfolio standard deviation lower than the weighted average of their standard deviations. This is diversification.

Risk splits into two parts. Unsystematic (specific) risk belongs to one company or sector and can be diversified away by holding many assets. Systematic (market) risk affects all assets and cannot be removed by diversification. This is why investors are rewarded for bearing systematic risk only.

The efficient frontier plots expected return against standard deviation for all feasible portfolios. A portfolio is efficient if no other portfolio has higher expected return with the same or lower risk, or lower risk with the same or higher expected return. The efficient frontier is the upper edge of the feasible set, starting from the minimum variance portfolio and going upward.

Which efficient portfolio you hold depends on your risk aversion. An investor with a utility function increasing in return and decreasing in variance has indifference curves in the return-risk plane. The optimal portfolio is where the highest reachable indifference curve touches the efficient frontier. A more risk-averse investor sits lower on the frontier.

The theory rests on assumptions: investors are rational and risk averse, they care only about mean and variance (which holds exactly if returns are normal or utility is quadratic), they use a single period, markets are frictionless with no transaction costs or taxes, assets are infinitely divisible, and all investors share the same views on returns, variances and covariances.

Key rules to remember

Portfolio expected return: E(Rp) = Σ wᵢ E(Rᵢ); Weights wᵢ sum to 1. Short sales give negative weights.
Two-asset portfolio variance: Var(Rp) = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂); Cov(R₁,R₂) = ρ₁₂σ₁σ₂.
General portfolio variance: Var(Rp) = ΣᵢΣⱼ wᵢwⱼ Cov(Rᵢ,Rⱼ); The sum includes i = j terms, which are the variances.
Correlation: ρ₁₂ = Cov(R₁,R₂) ÷ (σ₁σ₂); Always between -1 and +1.
Minimum variance weight (two assets): w₁ = (σ₂² − σ₁₂) ÷ (σ₁² + σ₂² − 2σ₁₂); σ₁₂ is the covariance. w₂ = 1 − w₁. Weight on asset 1 is found by setting d Var/d w₁ = 0.
Quadratic utility approach (mean-variance utility): U = E(R) − (λ ÷ 2) Var(R); One common form. λ > 0 is the risk aversion coefficient. Larger λ means more risk averse.
Equally weighted n-asset diversification: Var(Rp) = σ̄²÷n + ((n−1)÷n) × average covariance; As n grows, variance tends to the average covariance: the systematic part.

How to solve Mean-Variance Portfolio Theory and Efficient Frontier questions

Use this order for any numerical or descriptive question on portfolios and the efficient frontier.

1Write down the weights, expected returns, variances or standard deviations, and correlations or covariances. Check that weights sum to 1.
2Convert correlations to covariances using Cov = ρσ₁σ₂. Convert standard deviations to variances by squaring.
3Compute the portfolio expected return as the weighted sum of expected returns.
4Compute the portfolio variance using the two-asset formula or the double sum. Do not forget the covariance term, and keep the factor 2.
5Take the square root only at the end if the question asks for standard deviation.
6For minimum variance or optimal weights, differentiate variance with respect to a weight (using w₂ = 1 − w₁) and set to zero, or use the standard formula.
7For descriptive parts, link the result to diversification, systematic and unsystematic risk, and the efficient frontier, and state the assumptions you rely on.

Quickest way: Two-asset variance in under a minute

When to use it: Multiple-choice questions or the first part of a written question asking for portfolio mean and variance of two assets.

Compute the mean: w₁μ₁ + w₂μ₂.
Compute covariance once: ρ × σ₁ × σ₂.
Compute three terms: w₁²σ₁², w₂²σ₂², 2w₁w₂Cov. Add them.
Sanity check: the result must lie between the variance with ρ = −1 and the variance with ρ = +1. With ρ = +1, σp = w₁σ₁ + w₂σ₂.
Square root for standard deviation only if asked.

Common mistakes in Mean-Variance Portfolio Theory 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 covariance term. The weighted average of standard deviations is correct only when ρ = +1.
Leaving out the factor 2 on the covariance term.
The double sum counts Cov(1,2) and Cov(2,1) separately, and students forget this.
Fix: Always write 2w₁w₂Cov(R₁,R₂) in the two-asset formula.
Mixing up covariance and correlation, or variance and standard deviation.
Questions give data in different forms.
Fix: Convert everything first: variance = σ², covariance = ρσ₁σ₂. Then substitute.
Saying diversification removes all risk.
Overstating the benefit of holding many assets.
Fix: Diversification removes unsystematic risk only. Systematic risk remains, equal in the limit to the average covariance in the equal-weight case.
Treating the whole feasible set as the efficient frontier.
The frontier and the feasible region look similar on a diagram.
Fix: Only the upper part, from the minimum variance portfolio upward, is efficient. Portfolios below it are dominated.
Choosing the optimal portfolio without reference to risk aversion.
Students stop once the frontier is drawn.
Fix: Say that the optimum is where an indifference curve is tangent to the frontier. More risk-averse investors have steeper curves and choose lower-risk points.

Worked examples

Example 1

Asset A has expected return 10% and standard deviation 20%. Asset B has expected return 6% and standard deviation 10%. The correlation is 0.25. A portfolio holds 40% in A and 60% in B. Find the portfolio expected return and standard deviation.

Show the solution

Expected return = 0.4 × 10% + 0.6 × 6% = 4% + 3.6% = 7.6%.
Covariance = 0.25 × 0.20 × 0.10 = 0.005.
w₁²σ₁² = 0.16 × 0.04 = 0.0064.
w₂²σ₂² = 0.36 × 0.01 = 0.0036.
2w₁w₂Cov = 2 × 0.4 × 0.6 × 0.005 = 0.0024.
Variance = 0.0064 + 0.0036 + 0.0024 = 0.0124.
Standard deviation = √0.0124 = 0.11136, about 11.14%.

Answer: Expected return 7.6%; variance 0.0124; standard deviation about 11.14%.

Example 2

Two assets have variances 0.04 and 0.09 and covariance 0.012. Find the weights of the minimum variance portfolio and its variance, assuming short sales are allowed.

Show the solution

Let w₁ be the weight in asset 1 (variance 0.04) and w₂ = 1 − w₁.
Use w₁ = (σ₂² − σ₁₂) ÷ (σ₁² + σ₂² − 2σ₁₂).
Numerator = 0.09 − 0.012 = 0.078.
Denominator = 0.04 + 0.09 − 0.024 = 0.106.
w₁ = 0.078 ÷ 0.106 = 0.7358, so w₂ = 0.2642.
Variance = w₁²(0.04) + w₂²(0.09) + 2w₁w₂(0.012).
w₁² = 0.54144, giving 0.021658.
w₂² = 0.06980, giving 0.006282.
2w₁w₂ = 2 × 0.7358 × 0.2642 = 0.38879, giving 0.004666.
Variance = 0.021658 + 0.006282 + 0.004666 = 0.032606.

Answer: Weights about 73.6% in asset 1 and 26.4% in asset 2; minimum variance about 0.0326 (standard deviation about 18.1%).

Exam tips

Show the formula first, then substitute. Markers give method marks even if arithmetic slips.
State assumptions explicitly when asked about the theory: single period, rational risk-averse investors, mean and variance only, no frictions, common beliefs.
For descriptive parts, distinguish systematic from unsystematic risk and say which one diversification removes.
Sketch the frontier with axes labelled: standard deviation on the horizontal, expected return on the vertical. Mark the minimum variance point.
In Paper B or R questions, state the covariance matrix and weights vector clearly, and compute variance as wᵀΣw.

Practice questions from Capital Asset Pricing Model (CAPM)

Mean-Variance Portfolio Theory 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.

Mean-Variance Portfolio Theory and Efficient Frontier: frequently asked questions

What is the difference between systematic and unsystematic risk?

Systematic risk affects the whole market, such as interest rate or economic shocks, and cannot be diversified away. Unsystematic risk is specific to one asset or firm and can be reduced by holding many assets. Investors are rewarded only for bearing systematic risk.

How do you calculate portfolio variance for two assets?

Use w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂). Convert correlation to covariance with ρσ₁σ₂ if needed. Take the square root for standard deviation.

What are the main assumptions of Markowitz portfolio theory?

Investors are rational and risk averse, and they judge portfolios by mean and variance of one-period return. Markets have no taxes or transaction costs, assets are divisible, and all investors share the same beliefs about returns and covariances.

Why does the efficient frontier curve?

When correlation is below +1, combining assets lowers risk relative to the weighted average. This makes the set of risk-return combinations bend towards the return axis rather than lying on a straight line.

How does risk aversion affect the choice of portfolio?

A more risk-averse investor needs more extra return to accept extra variance. Their indifference curves are steeper, so they touch the frontier at a point with lower risk and lower expected return.