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.

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Идея простыми словами

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.

Как считается

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)

Как использовать

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.

Выводы

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.

Пример

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.

Входные данные примера условны; веса и цифры — это то, что для них выдаёт наш оптимизатор.

На что обратить внимание

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.

Немного истории

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.

Только образовательный материал. Ни одна модель портфеля не знает будущего: все они работают на оценках из прошлого, и ничто здесь не является инвестиционной рекомендацией.