Consider a single-period asset price model $\left(\bar{S}_{0}, \bar{S}_{1}\right)$ in $\mathbb{R}^{d+1}$ where, for $n=0,1$,

$$\bar{S}_{n}=\left(S_{n}^{0}, S_{n}\right)=\left(S_{n}^{0}, S_{n}^{1}, \ldots, S_{n}^{d}\right)$$

with $S_{0}$ a non-random vector in $\mathbb{R}^{d}$ and

$$S_{0}^{0}=1, \quad S_{1}^{0}=1+r, \quad S_{1} \sim N(\mu, V) .$$

Assume that $V$ is invertible. An investor has initial wealth $w_{0}$ which is invested in the market at time 0 , to hold $\theta^{0}$ units of the riskless asset $S^{0}$ and $\theta^{i}$ units of risky asset $i$, for $i=1, \ldots, d$.

(a) Show that in order to minimize the variance of the wealth $\bar{\theta} \cdot \bar{S}_{1}$ held at time 1 , subject to the constraint

$$\mathbb{E}\left(\bar{\theta} \cdot \bar{S}_{1}\right)=w_{1}$$

with $w_{1}$ given, the investor should choose a portfolio of the form

$$\theta=\lambda \theta_{m}, \quad \theta_{m}=V^{-1}\left(\mu-(1+r) S_{0}\right)$$

where $\lambda$ is to be determined.

(b) Show that the same portfolio is optimal for a utility-maximizing investor with CARA utility function

$$U(x)=-\exp \{-\gamma x\}$$

for a unique choice of $\gamma$, also to be determined.