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The risk-free weight is $1-1^{t}w$, so a zero risk-free weight requires $1^{t}w=1$. Define $H=1^t{\Sigma }^{-1}d$. If $H\ne 0$, this condition gives $\lambda =1/H$ andThese are the requested tangent-portfolio mean and variance under the usual condition $H>0$. Then $\lambda >0$, the normalized portfolio lies on the upper efficient ray, and its Sharpe ratio is $\sqrt{A}$.

There is a real qualification if the data are allowed to be arbitrary. If $H=0$, the efficient risky direction has zero total risky weight, and no finite multiple makes its risky weights sum to one. If $H<0$, the formal normalized portfolio above has negative excess mean and lies on the lower, dominated branch; it is not an efficient tangent portfolio as described in the question. The sign of the normalized tangency portfolio must therefore be checked, rather than inferred from positive definiteness of $\Sigma$ alone.

All three individual risky assets now have negative excess expected returns. The covariance matrix and the risky portfolio opportunity set remain unchanged, but the risk-free point moves above all three individual means. Solving the new tangency equations givesThe sign of the normalized tangency portfolio now matters. Normalizing to a unit risky budget givesand its Sharpe ratio is $-\sqrt{4555/77}$. This point touches the lower risky frontier, rather than being an efficient positive-premium market portfolio.

The positive-premium efficient direction is still $x=cz$, $c>0$. It shorts the first two assets, holds a hedging long position in the third, and has negative total risky investment. Its risk-free investment exceeds its initial wealth: the net proceeds of short selling are lent. Equivalently it takes a negative position in the normalized tangency portfolio. ThusNegative excess means therefore do not eliminate efficient risky investments when unrestricted short selling is allowed. The second diagram shows both orientations; it would be incorrect to interpret the negative normalized ratio as a negative slope for the upper efficient ray.