(i) What does it mean to say that $U$ is a utility function? What is a utility function with constant absolute risk aversion (CARA)?

Let $S_{t} \equiv\left(S_{t}^{1}, \ldots, S_{t}^{d}\right)^{T}$ denote the prices at time $t=0,1$ of $d$ risky assets, and suppose that there is also a riskless zeroth asset, whose price at time 0 is 1 , and whose price at time 1 is $1+r$. Suppose that $S_{1}$ has a multivariate Gaussian distribution, with mean $\mu_{1}$ and non-singular covariance $V$. An agent chooses at time 0 a portfolio $\theta=\left(\theta^{1}, \ldots, \theta^{d}\right)^{T}$ of holdings of the $d$ risky assets, at total cost $\theta \cdot S_{0}$, and at time 1 realises his gain $X=\theta \cdot\left(S_{1}-(1+r) S_{0}\right)$. Given that he wishes the mean of $X$ to be equal to $m$, find the smallest value that the variance $v$ of $X$ can be. What is the portfolio that achieves this smallest variance? Hence sketch the region in the $(v, m)$ plane of pairs $(v, m)$ that can be achieved by some choice of $\theta$, and indicate the mean-variance efficient frontier.

(ii) Suppose that the agent has a CARA utility with coefficient $\gamma$ of absolute risk aversion. What portfolio will he choose in order to maximise $E U(X)$ ? What then is the mean of $X$ ?

Regulation requires that the agent's choice of portfolio $\theta$ has to satisfy the valueat-risk (VaR) constraint

$$m \geqslant-L+a \sqrt{v},$$

where $L>0$ and $a>0$ are determined by the regulatory authority. Show that this constraint has no effect on the agent's decision if $\kappa \equiv \sqrt{\mu \cdot V^{-1} \mu} \geqslant a$. If $\kappa<a$, will this constraint necessarily affect the agent's choice of portfolio?