Exponential-utility portfolio with nonnegative cash (${\theta }^{\ast }=M[\mu -\max (1+r,\eta )S_0]$)

For Gaussian terminal prices with positive definite covariance $V$, set $M=(\gamma V{)}^{-1}$ and $\eta =(S_0^{T}M\mu -w_0)/(S_0^{T}M{S}_0)$, assuming $S_0\ne 0$. The normal moment-generating function turns expected exponential utility into a strictly concave mean-minus-variance objective. Its free optimum has cash $S_0^{T}M{S}_0(1+r-\eta )$. If this is positive, keep the free optimum; otherwise the cash constraint binds and the multiplier becomes $\eta$. Thus zero optimal cash is equivalent to $\eta \ge 1+r$, including equality.