Consider two investors: Michael is an $80-$year$-$old retiree with a risk aversion parameter of $5.$ Miranda is a $30-$year$-$old engineer with a risk aversion parameter of $3.$ There is a risk$-$free investment with a rate of return of $1$ percent and a risky investment with an expected return of $8$ percent and a standard deviation of $20$ percent.

$($a$)$ Suppose investors can pick one of the two investments but cannot combine them. Which investment will Michael prefer$$risk$-$free or risky? What about Miranda?

$($b$)$ Suppose investors can combine the risk$-$free and risky investments. What will be the optimal combination for Michael? What about Miranda?

$($c$)$ Discuss and explain the results obtained above in everyday language.