Portfolio construction
The efficient frontier
The curve of the best possible portfolios: for every level of risk, the highest expected return the tickers allow.
The idea in plain words
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.
How it is calculated
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 × σHow to use it
- 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).
Drawing conclusions
- 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.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). The long-only frontier runs from about 6.8% volatility and 2% return (mostly EURUSD) to 18% and 8% (all US500). Around 12% volatility it already reaches 7.5%: almost the full return of US500 for two thirds of its risk, by mixing in gold.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- 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.
A little history
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.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.