Analytical Markowitz Tangency Portfolio Weights

Scenario

Consider an investment universe of $N$ risky assets with expected return vector $\boldsymbol{\mu} \in \mathbb{R}^N$ and positive-definite covariance matrix $\boldsymbol{\Sigma} \in \mathbb{R}^{N \times N}$, alongside a risk-free asset with rate $R_f$.

Your Task

1. Formulate the Sharpe ratio maximization problem over portfolio weight vector $\mathbf{w} \in \mathbb{R}^N$.
2. Convert the non-linear fractional objective into an equivalent quadratic program and solve analytically using the Lagrangian multiplier method.
3. Derive the closed-form normalized tangency portfolio weight vector $\mathbf{w}^*$.
4. Given $R_f = 0.03$, $\boldsymbol{\mu} = [0.08, 0.12]^T$, and $\boldsymbol{\Sigma} = \begin{bmatrix} 0.04 & 0.01 \\ 0.01 & 0.09 \end{bmatrix}$, compute $\mathbf{w}^*$ and its annualized Sharpe ratio.

Constraints

• Weights sum to 1: $\mathbf{w}^T \mathbf{1} = 1$.
• Short-selling permitted.

Examples When two assets are uncorrelated, weights are proportional to excess return divided by individual variance.