Markowitz Mean-Variance Optimizer

01 // SELECT ASSET UNIVERSE

02 // CUSTOM ASSET DATA / PARAMETERS

RISK-FREE RATE ($r_f$):

MONTE CARLO SAMPLES:

CUSTOM TICKER LIST (COMMA SEPARATED):

03 // UNIVERSE EXPECTED RETURNS & VOLATILITY

TICKER	E(R) %	VOL ($\sigma$) %	SHARPE

EFFICIENT FRONTIER & CAPITAL ALLOCATION LINE

Individual Assets (●) | Min Variance (▲) | Max Sharpe Tangency (★)

MAX SHARPE: --

04 // OPTIMAL PORTFOLIO WEIGHT ALLOCATIONS ($w_i$)

ASSET	MINIMUM VARIANCE WEIGHT	TANGENCY (MAX SHARPE) WEIGHT	ALLOCATION BARS

05 // QUADRATIC PROGRAMMING & THE AXIOM THESIS

Quadratic Optimization Formulation

The classical Markowitz mean-variance optimization problem is structured as a constrained Quadratic Program (QP):

$$\min_{w} \quad \frac{1}{2} w^T \Sigma w \quad \text{subject to} \quad w^T \mathbf{1} = 1, \quad w^T \mu = r_p, \quad w \ge 0$$

Constructing the Lagrangian $\mathcal{L}(w, \lambda, \gamma, \eta)$ with multipliers for portfolio budget, target return, and no-shorting constraints:

$$\mathcal{L}(w, \lambda, \gamma, \eta) = \frac{1}{2} w^T \Sigma w - \lambda (w^T \mathbf{1} - 1) - \gamma (w^T \mu - r_p) - \eta^T w$$

The Axiom Thesis: Threshold Surface Realizability

"The Efficient Frontier is not merely a curve of optimal risk-return pairs; it is a threshold surface in high-dimensional state space—governed by the exact same mathematics as the graph realizability boundary."

As target return $r_p$ varies, the KKT slackness condition $\eta_i w_i = 0$ acts as a phase transition threshold. At critical Lagrange multipliers $(\lambda^*, \gamma^*)$, inactive assets cross from $w_i = 0$ (boundary) to $w_i > 0$ (interior), creating piecewise analytic phase shifts across the risk-return manifold.