In a one-period market, there are $n$ assets whose prices at time $t$ are given by $S_{t}=\left(S_{t}^{1}, \ldots, S_{t}^{n}\right)^{T}, t=0,1$. The prices $S_{1}$ of the assets at time 1 have a $N(\mu, V)$ distribution, with non-singular covariance $V$, and the prices $S_{0}$ at time 0 are known constants. In addition, there is a bank account giving interest $r$, so that one unit of cash invested at time 0 will be worth $(1+r)$ units of cash at time 1 .

An agent with initial wealth $w_{0}$ chooses a portfolio $\theta=\left(\theta^{1}, \ldots, \theta^{n}\right)$ of the assets to hold, leaving him with $x=w_{0}-\theta \cdot S_{0}$ in the bank account. His objective is to maximize his expected utility

$$E\left(-\exp \left[-\gamma\left\{x(1+r)+\theta \cdot S_{1}\right\}\right]\right) \quad(\gamma>0)$$

Find his optimal portfolio in each of the following three situations:

(i) $\theta$ is unrestricted;

(ii) no investment in the bank account is allowed: $x=0$;

(iii) the initial holdings $x$ of cash must be non-negative.

For the third problem, show that the optimal initial holdings of cash will be zero if and only if

$$\frac{S_{0} \cdot(\gamma V)^{-1} \mu-w_{0}}{S_{0} \cdot(\gamma V)^{-1} S_{0}} \geqslant 1+r$$