Efficient frontier

The efficient frontier contains all portfolios that maximise expected return for each level of total risk.

ByHoang TruongUpdated July 3, 2026

FrameworkModern portfolio theory

See it move

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Plotting every feasible combination of risky assets against risk on the horizontal axis and expected return on the vertical axis produces a cloud of portfolios. The efficient frontier is the upper-left boundary of that cloud: at every risk level, no other feasible portfolio offers a higher return. Rational investors choose only from portfolios lying on this frontier.

Where it fits

TopicRisk, Return & the CAPMAdvancedSubjectCorporate FinanceAdvanced

The formula

LaTeX

E(R_p) = \sum_{i} w_i \cdot E(R_i)

Variables

$E (R_{p})$: expected return on the portfolio
$w_{i}$: portfolio weight of asset i (all weights sum to 1)
$E (R_{i})$: expected return on asset i

Holds for any number of assets. Together with the variance formula, this maps every feasible portfolio; the efficient frontier is the upper boundary of that feasible set.

LaTeX

\sigma^2_p = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2

Variables

$σ_{p}^{2}$: portfolio variance
$w_{1}, w_{2}$: portfolio weights of assets 1 and 2
$σ_{1}, σ_{2}$: standard deviations of returns on assets 1 and 2
$ρ_{12}$: correlation coefficient between assets 1 and 2 (ranges from −1 to +1)

When ρ₁₂ = 1 no variance reduction occurs; when ρ₁₂ = −1 a zero-variance combination is achievable. Tracing all (w₁, w₂) combinations generates the feasible set from which the efficient frontier is identified.

Check yourself

PracticeCORE

Investor Andrés holds a risky portfolio that lies strictly inside the feasible set of mean-variance portfolios but below the efficient frontier. Which statement best describes this portfolio?

Select an answer to check your understanding.