= Solution
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 gives
$$
z=\Sigma^{-1}(-15,-8,-1)^T=\frac1{77}(-283,-41,18)^T,\qquad s=\mathbf1^Tz=-\frac{306}{77}.
$$
The <sign of the normalized tangency portfolio> now matters. Normalizing to a unit risky budget gives
$$
w_M=\frac1{306}(283,41,-18)^T,\qquad m_M=\frac{3095}{306}<25,
$$
and 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. Thus
$$
\boxed{\text{upper efficient slope}=+\sqrt{\frac{4555}{77}},\quad \text{normalized tangency Sharpe ratio}=-\sqrt{\frac{4555}{77}}.}
$$
Negative 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.
Back to article page