In a one-period market, there are $n$ risky assets whose returns at time 1 are given by a column vector $R=\left(R^{1}, \ldots, R^{n}\right)^{\prime}$. The return vector $R$ has a multivariate Gaussian distribution with expectation $\mu$ and non-singular covariance matrix $V$. In addition, there is a bank account giving interest $r>0$, so that one unit of cash invested at time 0 in the bank account will be worth $R_{f}=1+r$ units of cash at time 1 .

An agent with the initial wealth $w$ invests $x=\left(x_{1}, \ldots, x_{n}\right)^{\prime}$ in risky assets and keeps the remainder $x_{0}=w-x \cdot \mathbf{1}$ in the bank account. The return on the agent's portfolio is

$$Z:=x \cdot R+(w-x \cdot \mathbf{1}) R_{f}$$

The agent's utility function is $u(Z)=-\exp (-\gamma Z)$, where $\gamma>0$ is a parameter. His objective is to maximize $\mathbb{E}(u(Z))$.

(i) Find the agent's optimal portfolio and its expected return.

[Hint. Relate $\mathbb{E}(u(Z))$ to $\mathbb{E}(Z)$ and $\operatorname{Var}(Z) .]$

(ii) Under which conditions does the optimal portfolio that you found in (i) require borrowing from the bank account?

(iii) Find the optimal portfolio if it is required that all of the agent's wealth be invested in risky assets.