Jackson is a fresh actuarial graduate who just started his rst full time job in a reputable insurance company in Sydney on $1$ January $2024.$ Upon the start of his job, Jackson also decided to make an investment for a deposit to purchase his rst home in $10$ years time. Jackson has an awareness of investments of dierent types, including stocks, bonds, and some other risky asset Y $.$ He has asked you to analyze his investment strategies.

To help you with your analyses, Jackson has provided you with all necessary information. In particular:

a$.$ Salary: Jackson's salary can be decomposed into three components $($all after income tax$)$:

An annual base salary of $$60,000$ and it will increase by $6\%$ every year at the beginning of each year. He receives his salary monthly on the last day of each month $($i$.$e$.,$ the rst payment will be on $31$ January $2024).$

A sign$-$on bonus of $$20,000,$ paid with his rst salary.

Performance bonus paid at the end of each year. His bonus in year t$,$ Bt is calculated using the

following formula

Bt $=0\mathrm{.}15$At $$ It $$ exp$(1/$t$),$

where At is his base salary for year t$,$ and It is a Bernoulli random variable with success

probability p $=0\mathrm{.}8.$

$1.$ Calculate exactly $($i$.$e$.,$ without using simulation$)$

the probability that Jackson's total performance bonus $($nominal amount$)$ will be no less than $$90,000.$

the probability it is no more than $$10,000.$