The Mean-Variance Portfolio

The Lagrangian is $\mathcal L=\tfrac12 w^\top\Sigma w-\lambda(w^\top\mu-m)-\gamma(w^\top\one-1)$, and stationarity in $w$ reads $\Sigma w=\lambda\mu+\gamma\one$, so $w=\Sigma^{-1}(\lambda\mu+\gamma\one)$, using that $\Sigma\succ 0$ is invertible. Substituting into the two constraints gives the linear system

using the symmetry $\mu^\top\Sigma^{-1}\one=\one^\top\Sigma^{-1}\mu=A$. The determinant of this two-by-two system is $BC-A^2=D$, which is positive by the Cauchy-Schwarz inequality in the inner product $\langle x,y \rangle_{\Sigma^{-1}}=x^\top\Sigma^{-1}y$, since $A^2=\langle\one,\mu\rangle_{\Sigma^{-1}}^2\le\langle\one, \one\rangle\langle\mu,\mu\rangle=CB$ with equality only if $\mu$ and $\one$ are proportional, excluded when returns are not all equal. Solving gives $\lambda=(Cm-A)/D$ and $\gamma=(B-Am)/D$. The variance is $w^{\ast\top}\Sigma w^\ast=w^{\ast\top}(\lambda\mu+\gamma\one)=\lambda m+\gamma$, and substituting the multipliers yields $\sigma^2(m)=(Cm^2-2Am+B)/D$. Strict convexity makes this stationary point the unique global minimiser.

Minimising $\sigma^2(m)$ over $m$ sets its derivative $2Cm-2A=0$ at $m=A/C$, where $\sigma^2=(C(A/C)^2-2A(A/ C)+B)/D=(B-A^2/C)/D=(BC-A^2)/(CD)=1/C$. Substituting $m=A/C$ into Equation (2) gives $\lambda=0$ and $\gamma=1/C$, so $w_{\min}=\Sigma^{-1}\one/C$, whose return is $w_{\min}^\top\mu=A/C$.

On the budget set $w^\top\one=1$ the excess return is $w^\top\mu-r=w^\top(\mu-r\one)$, so the Sharpe ratio equals $S(w)=w^\top(\mu-r\one)/\sqrt{w^\top\Sigma w}$, which is invariant under positive scaling $w\mapsto tw$, $t>0$. By Cauchy-Schwarz in the $\Sigma$ inner product, $S(w)\le\sqrt{(\mu-r\one)^\top \Sigma^{-1}(\mu-r\one)}=\sqrt{B-2rA+r^2C}$ for all $w\neq 0$, with equality iff $w$ is a positive multiple of $w^\ast=\Sigma^{-1}(\mu-r\one)$, so the global directional maximiser is the ray $\R_+w^\ast$. To realise this ray on the budget set we need $t>0$ with $(tw^\ast)^\top\one=1$, that is $t=1/(\one^\top\Sigma^{-1}(\mu -r\one))=1/(A-rC)$, which exists precisely because $r<A/C$ makes $A-rC>0$. Thus the constrained maximiser coincides with the global maximiser, giving the stated tangency portfolio with attained Sharpe ratio $\sqrt{B-2rA+r^2C}>0$ (positive since $\Sigma^{-1}\succ 0$ and $\mu\neq r\one$). Equivalently the Lagrange condition $\mu-r\one=2\kappa\Sigma w$ on $w^\top\Sigma w=1$ gives $w\propto\Sigma^{-1}(\mu-r\one)$, and substituting it back yields objective value $w^\top(\mu-r\one)=2\kappa$, so the maximiser is the branch with $\kappa>0$ and the minimiser the $\kappa<0$ branch. The line from the risk-free point through $w_T$ is tangent to the efficient frontier, hence the name.