Problem $2$ Suppose it is the start of year $2024.$ You have $$100,000$ to invest and decide

to invest in a combination of a riskless asset $(f),$ a stock index fund $(S)$ and a long$-$term

bond fund $(B),$ with the following properties:

The correlation of the returns on the long$-$term bond fund and the stock index fund is equal

to $-0\mathrm{.}3.$

$(1)$ If you only consider the risky assets, what is the global minimum variance portfolio?

What is the global minimum variance portfolio if you also consider the risk$-$free

asset?

$(2)$ What is the tangency portfolio $($i$.$e$.,$ mix$)$ of the risky assets $S$ and $B?$ Hint: Use

Solver in Excel $-$ update the Excel spreadsheet from class with the information in

the table, define the slope of the CAL, and maximize it$.$ You need to submit your

Excel with comments.

$(3)$ What dollar amounts should be invested in each of the three asset classes $f,S,$

and $B$ if you would like to achieve $($for the year $2024)$ an expected return of $10\%$

with the lowest possible standard deviation? What is the standard deviation of the

$($percentage$)$ return on this portfolio?

To answer this question, you will need to derive the standard deviation for the

portfolio that contains three assets, one of which is risk$-$free. This is how you

should proceed:

$($a$)$ Use the formula for the variance of the portfolio return in case of $N$ assets $($in our

case, three$)$ from lecture $4.$ Hint: How many quadratic terms and correlation

terms will you have? Are there any properties of the risk$-$free asset that can

help you simplify the formula?

$($b$)$ Use the property of the risk$-$free asset to simplify the formula, as we did in

lecture $5$ after the risk$-$free asset was introduced. Which terms disappear?

$($c$)$ Write down the formula that you obtain after the simplification. Which formula

from the lectures does it resemble?

When solving further, keep in mind that $w_1+w_2+w_3=1.$

$(4)$ For the year $2024,$ the realized return on $f$ was $0\mathrm{.}05(5\%),$ the realized return on $B$

was $0\mathrm{.}12(12\%)$ and the realized return on $S$ was $0(0\%).$ At the end of $2024,$ what

are your dollar holdings of $f,$ of $B$ and of $S,$ and how much money do you have in

total? What is your realized $($percentage$)$ portfolio return for year $2024?$