The efficient frontier

用通俗的话解释

Every mix of the tickers is a point on a risk–return chart. The upper-left edge of all those points is the efficient frontier: you cannot get more return without more risk, or less risk without less return. Portfolios below the curve are inefficient; something on the curve beats them.

计算方法

For a range of target returns, find the weights with the lowest volatility that reach each target (weights add up to 1; long only unless shorts are allowed). The study draws it per unit of capital; the long-and-short frontier lets each dollar be used once, long or short.

For each target m:  minimise wᵀΣw  subject to  wᵀμ = m,  Σwᵢ = 1  (and wᵢ ≥ 0 if long only)
Capital market line (with cash at r_f): μ = r_f + Sharpe_max × σ

如何使用

• In a study, find where each model's dot lies: on the curve means efficient for these inputs.
• In the live analysis, the dot "your portfolio" against the frontier shows how far today's positions are from an efficient mix of the same tickers.
• Pick the point on the curve that matches the risk you can live with; the models are different ways of picking that point (or of distrusting the curve).

得出结论

• The left tip is the minimum-variance portfolio; the right end is the single best-returning ticker.
• The point where a line from the risk-free rate just touches the curve is the maximum-Sharpe (tangency) portfolio.

注意事项

• The frontier is drawn from estimates; the true one is unknown and the estimated one is too optimistic (the "error maximisation" problem, see Robust mean-variance).
• A portfolio exactly on the estimated frontier is often a concentrated, fragile bet.

一点历史

Markowitz described the efficient set in 1952. James Tobin (1958) added cash and showed that every investor should hold the same tangency portfolio, mixed with cash to taste; Sharpe built the capital asset pricing model on it in 1964.