Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 1 / g / 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 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.