Gaussian one-fund theorem

Let $\xi \sim N(b,\Sigma )$, where $b\ne 0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+{\theta }^T\xi$, every unique optimal portfolio has the formIndeed, the component orthogonal to ${\Sigma }^{-1}b$ in the $\Sigma$ inner product contributes independent mean-zero Gaussian risk without changing the mean. Removing it cannot reduce expected concave utility. A negative coefficient is dominated by the corresponding positive coefficient, which has the same variance and a larger mean.