Suppose stocks $($denoted by subscript S$)$ offer an expected rate of return of $10\%$ with a standard deviation of $20\%,$ and gold $($denoted by subscript G$)$ offers an expected return of $5\%$ with a standard deviation of $25\%.$

a$.$ Create the investment opportunity set of stocks and gold if the correlation coefficients are $-1,-0\mathrm{.}5,0,0\mathrm{.}5,$ and $1$ respectively. Attach the graph to your spreadsheet.

Hint: copy the formulas to other correlation coefficients and graph the curves.

b$.$ Briefly comment on how the correlation of both assets determines the risk of the risky portfolio consisting of stocks and gold.

c$.$ Suppose that the correlation coefficient is $-0\mathrm{.}5.$ Along the corresponding investment opportunity set, identify the portfolio of risky assets where the weight in stock of the portfolio is $0\mathrm{.}5($and thus the weight in bond is also $0\mathrm{.}5).$ Also, identify another portfolio that would be dominated by this portfolio based on the mean$-$variance criterion.

d$.$ Continuing from $($c$),$ if the investor can buy a Treasury Bill that pays $2\%$ in returns with certainty, what is the Sharpe ratio of this risky portfolio?

e$.$ Continuing from $($c$)$ and $($d$),$ is the portfolio from part $($c$)$ an $$optimal$$ one, or can you find a better portfolio? Give an example of such a portfolio along the investment opportunity set