Investment Portfolio of Index Fund and Core Bonds Fund. J$.$P$.$ Morgan Asset Management publishes information about financial investments. Between $2002$ and $2011$ the expected return for the S&P $500$ was $5\mathrm{.}04\%$ with a standard deviation of $19\mathrm{.}45\%$ and the expected return over that same period for a core bonds fund was $5\mathrm{.}78\%$ with a standard deviation of $2\mathrm{.}13\%($J$.$P$.$ Morgan Asset Management, Guide to the Markets$).$ The publication also reported that the correlation between the S&P $500$ and core bonds is $-.32.$ You are considering portfolio investments that are composed of an S&P $500$ index fund and a core bonds fund.

a$.$ Using the information provided, determine the covariance between the S&P $500$ and core bonds.

b$.$ Construct a portfolio that is $50\%$ invested in an S&P $500$ index fund and $50\%$ in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio?

c$.$ Construct a portfolio that is $20\%$ invested in an S&P $500$ index fund and $80\%$ invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio?

d$.$ Construct a portfolio that is $80\%$ invested in an S&P $500$ index fund and $20\%$ invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio?

e$.$ Which of the portfolios in parts $($b$),($c$),$ and $($d$)$ has the largest expected return? Which has the smallest standard deviation? Which of these portfolios is the best investment alternative?

f$.$ Discuss the advantages and disadvantages of investing in the three portfolios in parts $($b$),($c$),$ and $($d$).$ Would you prefer investing all your money in the S&P $500$ index, the core bonds fund, or one of the three portfolios? Why?