$11.$ What is the formula for the correlation coefficient $()?$ What does a correlation coefficient of $-1$ indicate about the relationship between two variables? What does a correlation coefficient of $0$ indicate about the relationship between two variables? What does a correlation coefficient of $1$ indicate about the relationship between two variables?$12.$ What must be true about the standard deviation of a portfolio that has a mix of two assets when there is a negative correlation between the two assets?$13.$ What is the expected return of a risky portfolio that has $2$ risky assets? What is the standard deviation of a risky portfolio that has $2$ risky assets?$14.$ How is the investment opportunity set derived?$15.$ What is the minimum$-$variance portfolio? Explain why an investor would never choose a portfolio along the downward sloping portion of the investment opportunity set $($below the minimum$-$variance portfolio$)\mathrm{.}16.$ Draw the investment opportunity set for the special case where the two risky assets are perfectly correlated $(=1\mathrm{.}0).$ In this special case would diversification reduce risk?$17.$ Draw the investment opportunity set for the special case where the two risky assets are perfectly negatively correlated $(=-1\mathrm{.}0).$ In this special case would diversification reduce risk? What would be the level of risk at the minimum$-$variance portfolio in this special case?$18.$ Explain why the minimum$-$variance portfolio cannot be the optimal portfolio of risky assets. How does an investor determine which point along the investor opportunity set they should choose to find the optimal portfolio?$19.$ Set up the optimization problem to find the optimal risky portfolio. What is the formula to find the weight of each risky asset in the optimal risky portfolio?