The Vanguard Group is one of the largest investment firms in the world with approximately $$7$ trillion in assets in $2021.$

Vanguard offers investment funds in stocks, bonds, annuities, and many other forms. Vanguard estimates the average

correlation between the returns offered by stocks and bonds to be $-0\mathrm{.}32.+$ Suppose that a particular stock index fund

offered by Vanguard has an expected return of $4\mathrm{.}73\%$ with a standard deviation of $18\mathrm{.}74\%,$ and an investment$-$grade bond

fund offered by Vanguard has an expected return of $5\mathrm{.}36\%$ with a standard deviation of $2\mathrm{.}35\%.$

$($a$)$ Based on the information provided, what is the covariance between this stock index fund and the investment$-$grade

bond fund? $($Keep the means and standard deviations as percents when finding the covariance. Round your answer to

four decimal places.$)$

$($b$)$ What is the expected return $($as a percent$)$ for a portfolio composed of $60\%$ invested in the stock index fund and $40\%$

invested in the investment$-$grade bond fund?

$\%$

What is the standard deviation $($as a percent$)$ for a portfolio composed of $60\%$ invested in the stock index fund and

$40\%$ invested in the investment$-$grade bond fund? $($Round your answer to four decimal places.$)$

$\%$

$($c$)$ What is the expected return $($as a percent$)$ for a portfolio composed of $40\%$ invested in the stock index fund and $60\%$

invested in the investment$-$grade bond fund?

$\%$

What is the standard deviation $($as a percent$)$ for a portfolio composed of $40\%$ invested in the stock index fund and

$60\%$ invested in the investment$-$grade bond fund? $($Round your answer to four decimal places.$)$

$\%$

$($d$)$ Which of the portfolios in part $($b$)$ or $($c$)$ has the highest expected return?

Invest $40\%$ in the stock index fund and $60\%$ in the investment$-$grade bond fund.

Invest $60\%$ in the stock index fund and $40\%$ in the investment$-$grade bond fund.

Which portfolio has the lowest standard deviation?

Invest $40\%$ in the stock index fund and $60\%$ in the investment$-$grade bond fund.

Invest $60\%$ in the stock index fund and $40\%$ in the investment$-$grade bond fund.

$($e$)$ Which portfolio would should be preferred?

Invest $40\%$ in the stock index fund and $60\%$ in the investment$-$grade bond fund.

Invest $60\%$ in the stock index fund and $40\%$ in the investment$-$grade bond fund.

Is the portfolio you chose guaranteed to perform better than the other portfolio in any given year?

Yes

No