Portfolio construction

Maximum Sharpe ratio (tangency portfolio)

The mix with the highest return per unit of risk: the point where a line from cash just touches the efficient frontier.

2 min read · Try it in the portfolio optimiser ↗

The idea in plain words

If you can hold cash or borrow, the best risky portfolio does not depend on how much risk you want: you pick the one with the best Sharpe ratio and then add cash (less risk) or leverage (more). That single portfolio is the tangency portfolio.

How it is calculated

Maximise the Sharpe ratio over the allowed weights. The optimiser sweeps the mean-variance curve to find the best region, then polishes the ratio directly.

maximise  (wᵀμ − r_f) ÷ √(wᵀΣw)
subject to  Σwᵢ = 1,  0 ≤ wᵢ ≤ w_max
Unconstrained: w ∝ Σ⁻¹(μ − r_f)

How to use it

As the "best mix" when you trust the expected returns; then decide separately how much to invest.
Combine it with a maximum weight so that it cannot become a one-ticker portfolio.
Compare its OOS Sharpe with its in-sample Sharpe before using it.

Drawing conclusions

The study marks the best-Sharpe model with a bigger dot on the frontier chart.
If its weights look extreme, the expected returns are probably too far apart to be believed.

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). Maximum Sharpe: about 40% US500, 5% GER40, 55% gold. Return 7.4%, volatility 11.7%, Sharpe 0.63, the best of the unleveraged models (Markowitz with λ = 3 comes close). Gold carries half of the risk because it brings return that does not move with the indices.

The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.

Watch out for

Shares mean-variance's weakness: errors in the expected returns are magnified.
With the risk-free rate set high, few mixes beat cash and the result can become very concentrated.

A little history

James Tobin's separation theorem (1958) and Sharpe's capital asset pricing model (1964) made the tangency portfolio the centre of portfolio theory: under their assumptions it is the market portfolio itself.

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.