Efficient Frontier
The set of portfolios offering the highest return for each level of risk.
The efficient frontier is a curve in risk-return space representing portfolios that are 'mean-variance optimal'. Each point on the frontier offers the maximum expected return for its level of volatility (risk). Portfolios below the frontier are suboptimal because you could achieve higher return for the same risk or lower risk for the same return.
Formula
Where
=Portfolio weights vector
=Expected portfolio return
=Portfolio volatility
=Target volatility constraint
Variables
| \mathbf{w} | Asset weight vector to optimize |
| \boldsymbol{\mu} | Vector of expected returns (annualized) |
| \mathbf{\Sigma} | Covariance matrix (annualized, positive definite) |
| t | Target portfolio return swept along the frontier |
| L,\,U | Per-asset weight box (long-only when L = 0) |
Assumptions
- Exact convex QP (primal active-set), not a grid-search approximation
- Long-only box constraints (0 ≤ wᵢ ≤ maxWeight) on the public endpoint
- Covariance matrix must be positive definite (no ridge regularization)
- GMV closed form Σ⁻¹1/(1ᵀΣ⁻¹1) and tangency Σ⁻¹(μ−rf·1)/(1ᵀΣ⁻¹(μ−rf·1)) recover the unconstrained solutions when the box does not bind
- Expected returns estimated from the historical mean — the usual mean-variance input sensitivity applies
vs. Industry Tools
Bloomberg PORT — Also uses a quadratic-programming solver for exact mean-variance optimization
Excel Solver — GRG Nonlinear / Simplex; equivalent optimum on the same convex QP
Related Terms
Sharpe Ratio
Risk-adjusted return: excess return divided by volatility.
Minimum Variance Portfolio
The portfolio with the lowest possible volatility.
Maximum Sharpe Portfolio
The portfolio with the highest risk-adjusted return.
Covariance Matrix
Captures how asset returns move together — the foundation of diversification.
Monte Carlo Simulation
Generating thousands of possible future scenarios through random sampling.