A single-period market contains $d$ risky assets, $S^{1}, S^{2}, \ldots, S^{d}$, initially worth $\left(S_{0}^{1}, S_{0}^{2}, \ldots, S_{0}^{d}\right)$, and at time 1 worth random amounts $\left(S_{1}^{1}, S_{1}^{2}, \ldots, S_{1}^{d}\right)$ whose first two moments are given by

$$\mu=E S_{1}, \quad V=\operatorname{cov}\left(S_{1}\right) \equiv E\left[\left(S_{1}-E S_{1}\right)\left(S_{1}-E S_{1}\right)^{T}\right]$$

An agent with given initial wealth $w_{0}$ is considering how to invest in the available assets, and has asked for your advice. Develop the theory of the mean-variance efficient frontier far enough to exhibit explicitly the minimum-variance portfolio achieving a required mean return, assuming that $V$ is non-singular. How does your analysis change if a riskless asset $S^{0}$ is added to the market? Under what (sufficient) conditions would an agent maximising expected utility actually choose a portfolio on the mean-variance efficient frontier?