You have recently inherited $$150,000$ and have decided that you should invest the money.

You have identified three funds which seem like a good fit for your investment goals: a risk

free short$-$term bond fund $($f $),$ a long$-$term bond fund $($B$),$ and a stock market index fund

$($S$).$ Your research revealed the following information about the $3$ funds:

$1$

Expected Return: E$($r$)$ Volatility: $\backslash$sigma $($r$)$

Risk$-$Free Fund $($f $)$

$0\mathrm{.}035$

$0\mathrm{.}000$

Long$-$Term Bond Fund $($B$)$

$0\mathrm{.}060$

$0\mathrm{.}075$

Stock Market Index Fund $($S$)$

$0\mathrm{.}110$

$0\mathrm{.}18$

Correlation between B and S: $\backslash$rho BS $=0\mathrm{.}75$

$($a$)$ First you consider investing $1/3$ of your inheritance in each of the $3$ funds. What is

the expected return and volatility of this portfolio?

$($b$)$ Having taken the first two weeks of Fin$-532,$ you know that you can construct a more

efficient portfolio than simply putting an equal weight of your inheritance in each fund.

You start by constructing the Mean$-$Variance Efficient $($MVE$)$ portfolio.

$($i$)$ What are the portfolio weights in the MVE portfolio?

$($ii$)$ What is the expected return and volatility of the MVE portfolio?

$($c$)$ After considering your investment goals and risk tolerance, you$$ve decided to make sure

your portfolio volatility is no greater than $12\%.$ Given this restriction, how should you

allocate the inheritance between $($f$,$ B$,$ and S$)$ to maximize your expected return