The table below shows the daily closing values of the S&P $500$ index, along with the daily prices for the S&P $500$ futures contract with maturity date September $2$nd of the same year. The futures contract size is $$250$ times the S&P $500$ index value.

The initial margin required for the futures contract is $$12,250$ per contract. The maintenance margin is $$10,000$ per contract. The risk free rate is $2\%$ per year $($continuously compounded$).$ Assume this is the rate you earn on your margin account. Note that the maintenance margin determines whether you receive a margin call. If you receive a margin call, you must increase your account balance upto the initial margin.

$($a$)$ Suppose you take a short position in $8$ futures contracts on Aug $27.$ What is your

profit$/$loss on Aug $28$ and Aug $29?$ What is the balance in your margin account at the

end of each of these days? Do you face any margin calls?

$($b$)$ Repeat part $($i$)$ for a long position of $3$ futures contracts.

$($c$)$ Using the Aug $29$ index value and futures price, what is the implied dividend yield on

the S&P $500?($Assume that the futures price is the same as the equivalent forward

price.$)$

$($d$)$ For the three days shown, the index value is larger than the futures price. Why?