Let $x$ be the number represented by the last $2$ digits of your student ID number. If your

student ID number ends with $00,$ use $x=50.$

Assume that your utility is $U(R)=x*E(R)-5x*var(R),$ where $R$ denotes the annual

return on your portfolio. You identified $3$ assets you want to invest in and estimated their

annual expected returns and standard deviations to be as follows:

Therefore, if your student ID number ends with $25,$ then your utility function is $U(R)=$

$25*E(R)-125*var(R),$ and the three assets you want to invest in have the following expected

values and standard deviations:

The correlation between the two risky assets is assumed to be $0\mathrm{.}25($irrespective of what your $x$

is$).$

$(10$p$)$ Compute your optimal portfolio of the three assets, assuming that short selling is

allowed.

You want to invest $$1,000$ in the optimal portfolio you found in part $(1)$ and you want

to hold that portfolio for $10$ years, reinvesting your yearly interest in the same portfolio

every year. You estimate that the risk$-$free return will remain constant over the next $10$

years and that the annual returns on the two risky assets are normally distributed with

the means and standard deviations listed in the table above $($corresponding to your $x).$

$($a$)(10$p$)$ Run a Monte Carlo simulation to generate a set of possible returns on your

optimal portfolio for the next $10$ years. Compute how much money you will have at

the end of the $10$ years. Keep in mind that, since you are reinvesting every year in

the same portfolio, your returns will compound every year.