= Gaussian two-fund theorem
{c}
For nondegenerate Gaussian terminal prices, a finite maximizer of a differentiable increasing concave non-affine utility, subject to a linear budget, lies in the span of $V^{-1}s$ and $V^{-1}\mu$. Gaussian integration by parts gives
$$
\mathbb E[S U'(\pi^TS)]
=\mu\,\mathbb EU'(\pi^TS)+V\pi\,\mathbb EU''(\pi^TS).
$$
The budget first-order condition makes this a multiple of $s$. For a nonzero portfolio, non-affineness and the full Gaussian support imply $\mathbb EU''<0$, so rearrangement gives the span assertion. Affine utility is a genuine exception: when means are proportional to prices, every budget-feasible portfolio is optimal, including portfolios outside that span.
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