An investor must decide how to invest his initial wealth $w_{0}$ in $n$ assets for the coming year. At the end of the year, one unit of asset $i$ will be worth $X_{i}, i=1, \ldots, n$, where $X=\left(X_{1}, \ldots, X_{n}\right)^{T}$ has a multivariate normal distribution with mean $\mu$ and non-singular covariance matrix $V$. At the beginning of the year, one unit of asset $i$ costs $p_{i}$. In addition, he may invest in a riskless bank account; an initial investment of 1 in the bank account will have grown to $1+r>1$ at the end of the year.

(a) The investor chooses to hold $\theta_{i}$ units of asset $i$, with the remaining $\varphi=w_{0}-\theta \cdot p$ in the bank account. His objective is to minimise the variance of his wealth $w_{1}=\varphi(1+r)+\theta \cdot X$ at the end of the year, subject to a required mean value $m$ for $w_{1}$. Derive the optimal portfolio $\theta^{*}$, and the minimised variance.

(b) Describe the set $A \subseteq \mathbb{R}^{2}$ of achievable pairs $\left(\mathbb{E}\left[w_{1}\right], \operatorname{var}\left(w_{1}\right)\right)$ of mean and variance of the terminal wealth. Explain what is meant by the mean-variance efficient frontier as you do so.

(c) Suppose that the investor requires expected mean wealth at time 1 to be $m$. He wishes to minimise the expected shortfall $\mathbb{E}\left[\left(w_{1}-(1+r) w_{0}\right)^{-}\right]$subject to this requirement. Show that he will choose a portfolio corresponding to a point on the mean-variance efficient frontier.