= Gaussian one-fund theorem
{c}
Let $\xi\sim N(b,\Sigma)$, where $b\ne0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+\theta^T\xi$, every unique optimal portfolio has the form
$$
\theta^*=\lambda\Sigma^{-1}b,
\qquad \lambda\geq0.
$$
Indeed, 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.
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