Sharpe Ratio Maximization (Tangency Portfolio)

Tangency Portfolio

The tangency portfolio is the risky portfolio that maximizes the Sharpe ratio:

$SR = \frac{\mu_p - r_f}{\sigma_p}$

It is the point on the efficient frontier where a line from the risk-free rate is tangent to the frontier.

Analytical Solution for Two Assets

To find the tangency weights, we solve the system of equations z = Σ⁻¹(μ − r_f · 1) and normalize:

$z_i = (\Sigma^{-1} \boldsymbol{e})_i, \quad w_i = \frac{z_i}{\sum_j z_j}$

For two assets with covariance matrix Σ = [[σ₁², ρσ₁σ₂], [ρσ₁σ₂, σ₂²]]:

$z_1 = \frac{\sigma_2^2 e_1 - \rho \sigma_1 \sigma_2 e_2}{\det(\Sigma)}, \quad z_2 = \frac{\sigma_1^2 e_2 - \rho \sigma_1 \sigma_2 e_1}{\det(\Sigma)}$

where eᵢ = μᵢ − r_f and det(Σ) = σ₁²σ₂² − (ρσ₁σ₂)².

Your Task

Implement tangency_weights_2(mu1, mu2, s1, s2, corr, rf) that returns a tuple (w1, w2) of the tangency portfolio weights, each rounded to 4 decimal places.