$1)$ It has become more and more common for highly paid Corporate executives and even sports celebrities to defer some of their salaries to later periods. The reasons people may choose to do so varies, ranging from tax strategies to future income assurance, just to name a couple.

In December of $2023,$ Shohei Ohtani, a major league baseball superstar, signed a $10-$yr contract with the Los Angeles Dodgers $(2024-2033).$ As a surprise to many, he agreed to play for only $$2$ million a year for those $10$ years$--$well below what a player of his caliber is worth. However, the Dodgers will pay him $$68$ million per year for the years $2034$ through $2043($after his $10-$yr commitment to the Dodgers expires$).$ The cash Ohtani will realize over the terms of this contract total $$700$ million, the largest sum ever pledged to a baseball player. However, there is a time$-$value of money issue at play in this type of deferred contract$--$so the value of the contract is NOT $$700$ million.

a$)$ Calculate the value of this contract at December $31,2023$ assuming a market rate of interest of $9\%($the average annual S&P $500$ return over the last $20$years$).$ For simplicity, assume the payments are made to him at the end of each year. Ignore any potential tax impact$/$effect$.\{$Round answer to the nearest dollar$\}$

b$)$ Calculate the value of the contract at December $31,2023$ if he had instead chosen to receive the $$70$ million a year for the $10$ years that he will be playing. Again, assume payment at the end of each year. $\{$Round answer to the nearest dollar$\}$

$2)$ Assume you are graduating at the end of this year and you will begin earning a salary which finally gives you the opportunity to start saving to buy a house. You hope to be able to purchase the home after working $10$ years and your estimates are that the home you want will probably cost about $$500,000$ at that time. In order to avoid paying mortgage insurance, you will have to come up with $20\%$ in cash as a down payment. You assume interest rates will be $5\%$ over the next $10$ years.

Calculate how much will you have to save each semi$-$annual period to have the required down payment in time for the purchase. $\{$Round answer to the nearest dollar$\}[$Assume you will deposit your savings at the end of semi$-$annual period$]$