The curve of portfolios that have the highest expected return for each level of volatility, from a fixed set of assets. It starts at the minimum variance portfolio. Harry Markowitz described it in 1952.
The efficient frontier is a curve on a chart of Volatility against return. Each point on the curve is a mix of the same set of assets that has the lowest possible volatility for its level of return, at or above the return of the Minimum Variance Portfolio. Every other mix of those assets in that range lies to the right of the curve: it has more volatility for the same return. A mix below that return is never on the curve, because another mix has the same volatility and a higher return.
Harry Markowitz introduced the idea in 1952 in Modern Portfolio Theory, and it earned him a Nobel Prize in 1990. The curve exists because assets do not move in step. When two assets have a low correlation, a mix of them can have less volatility than either asset alone. The left end of the curve is the Minimum Variance Portfolio.
The curve depends on its inputs. It needs an estimate of each asset's expected return and of the covariance between every pair of assets. In practice, both come from past prices. Small changes in the return estimates can move the curve a lot, and research by Chopra and Ziemba (1993) and DeMiguel, Garlappi and Uppal (2009) shows that a frontier built from past means often fails to hold in the future. For this reason, practitioners often shrink the covariance estimate, for example with the Ledoit-Wolf method.
A curve drawn from past prices shows how mixes of the same assets behaved over that window. It is not a forecast. A point on the curve is not a recommendation, because the curve ignores the investor's goals, horizon, taxes and costs.
Formula
For each target return r: minimise wᵀΣw subject to wᵀμ = r, Σwᵢ = 1, wᵢ ≥ 0 (long only)