S&P $500$ Index futures: Beginning of term $$4,465(09/12/2023),$ End of term $$4,518(11/14/2023).$ a$.$ Assume that you purchased an S&P $500$ futures contract at the beginning of the school term, with the first settlement date being beyond the end of the school term. Also assume that you sold an S&P $500$ futures contract with this same settlement date at the end of the school term. Given that this contract has a value of the futures price times $$250,$ determine the difference between the dollar value of the contract you sold and the dollar amount of the contract you purchased.

b$.$ Assume that you invested an initial margin of $20$ percent of the amount that you would owe to purchase the S&P $500$ index at the settlement date. Measure your return from taking a position in the S&P $500$ index futures as follows. Take the difference determined in part $($a$)($which represents the dollar amount of the gain on the futures position$)$ and divide it by the amount you originally invested $($the amount you originally invested is $20$ percent of the dollar value of the futures contract that you purchased$).$

c$.$ The return that you just derived in part $($b$)$ is not annualized. To annualize your return, multiply it by $12/$m$,$ where m is the number of months in your school term.

d$.$ Apply the concepts discussed in Chapter $13$ to explain why your return on your S&P $500$ index futures position was low or high over the school term.