Consider the following scenario analysis: Rate of Return Scen$\%$ario Probability Stocks Bonds Recession $0\mathrm{.}2-5\%14\%$ Normal economy $0\mathrm{.}615\%8\%$ Boom $0\mathrm{.}225\%4\%$

$($i$)$ Calculate the expected return and standard deviation of each investment

$($ii$)$ What is the rate of return in each scenario of a portfolio that is $60\%$ invested in the stock and $40\%$ in the bond?

$($iii$)$ What are the expected return and standard deviation of the portfolio?

$($iv$)$ Would you prefer to invest in the portfolio, in stocks only, or in bonds only?

$($v$)$ Calculate the correlation coefficient for the bond and stock returns

b$)$ Suppose an investor has $$50,000$ to invest in a portfolio of risk$-$free asset and the market portfolio. The expected return on the market portfolio is $13\mathrm{.}5$ percent and the risk$-$free rate is $4\mathrm{.}25$ percent. How much of the funds should be in the risk$-$free asset if the portfolio has an expected return of $10$ percent?