Maximum Sharpe ratio (tangency portfolio)

用通俗的话解释

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.

计算方法

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)

如何使用

• 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.

得出结论

• 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.

注意事项

• 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.

一点历史

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.