$2.$ Consider two assets, S and B$.$ The expected return on S is $12\mathrm{.}2\%$ while the expected return on B is $5\mathrm{.}8\%.$ The standard deviation of Asset S returns is $20\%,$ while the standard deviation of Asset B returns is $10\%.$ The correlation between Asset S and Asset B returns is $0\mathrm{.}05.$

$($a$)$ What is the expected return on a portfolio with $20\%$ of invested funds in S and $80\%$ of invested funds in B$?$

$($b$)$ What is the standard deviation of returns for a portfolio with $20\%$ of invested funds in S and $80\%$ of invested funds in B$?$

$($c$)$ Compare your answers obtained in part $($b$)$ to the expected return and standard deviation of a portfolio will all funds invested in B$,$ and summarize the principle demonstrated by the comparison.

$($d$)$ Assume that the correlation of $0\mathrm{.}05$ was based on historical data. However, there is reason to think that this correlation will be closer to $0\mathrm{.}50$ going forward. How is this relevant to the selection of portfolio weights?