= Sign of the normalized tangency portfolio
{title2=$w=z/(\mathbf1^Tz),\quad z=\Sigma^{-1}(\mu-r\mathbf1)$}
Assume a positive definite <covariance matrix> and nonzero excess-mean vector $d=\mu-r\mathbf1$. The <Cauchy-Schwarz inequality> gives $|d^Tx|\leq\sqrt{d^T\Sigma^{-1}d}\sqrt{x^T\Sigma x}$, with the positive-premium efficient direction $x=cz$, $c>0$. If $s=\mathbf1^Tz\ne0$, its normalized risky portfolio has <Sharpe ratio> $\operatorname{sgn}(s)\sqrt{d^T\Sigma^{-1}d}$. Thus negative normalization reverses the efficient orientation. If $s=0$, the efficient risky direction is self-funded and cannot be normalized to a unit risky budget.
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