Gaussian two-fund theorem

Let terminal asset prices $S$ have mean $\mu$ and positive definite covariance matrix $V$, with a Gaussian distribution. Consider a finite optimizer of $\mathbb{E}[U({\pi }^{T}S)]$ subject to the linear budget $s^T\pi =w$. Assume $U\in C^2(\mathbb{R})$ is increasing, concave and non-affine, and its derivatives satisfy the integrability needed for differentiation and Gaussian integration by parts. Then every optimizer lies in the linear span of $V^{-1}s$ and $V^{-1}\mu$.

The first-order condition and Gaussian identity giveFor $\pi \ne 0$, the terminal wealth has full Gaussian support. Continuity and non-affine concavity make $U^{\prime \prime }<0$ on some interval, so $\mathbb{E}{U}^{\prime \prime }({\pi }^{T}S)<0$. Rearranging proves the two-fund representation. The zero portfolio belongs to the same span automatically.

The regularity conditions justify the displayed derivative proof. Affine utility is an exception to its conclusion: if means are proportional to initial prices, every budget-feasible portfolio can be optimal, including portfolios outside this span.