A market consists of two risky assets and no risk$-$free asset. Let $R_1$ and $R_2$ denote the return on each of the risky assets. Using market data the following have been estimated: $E[R_1]=0\mathrm{.}10,E[R_2]=0\mathrm{.}15,{}_1^2=$Var$(R_1)={0\mathrm{.}1}^2,{}_2^2=$Var$(R_2)={0\mathrm{.}2}^2$ and ${}_{1,2}=-\frac{1}{2}$ where ${}_{1,2}$ denotes the correlation coefficient for $R_1$ and $R_2.$

$($i$)$ Given that an investor is targeting a total expected return of $=0\mathrm{.}125$ on a portfolio, what is the minimum variance that can be achieved?

$($ii$)$ Determine the global minimum variance portfolio and the expected return and variance of return on this portfolio.

$($iii$)$ Using your answers to parts $($i$)$ and $($ii$)$ make a rough sketch of the minimumvariance set in $-{}^2$ space. You should indicate the efficient frontier and the global minimum variance portfolio.

$($iv$)$ Without further calculation, explain how you would expect the variance of return calculated in $($ii$)$ to change if the two risky assets were independent.