Question $4$: S&P $500$ Futures Contracts $(3/10)$ Suppose the S&P $500$ index is currently $800$ on Jan $1$st$,$ the initial margin is $20\%$ and the maintenance margin is $50\%$ of the initial margin. You wish to enter into a long position for $10$ S&P $500$ futures contracts.

$(1)$ What is the contract size for S&P $500$ Futures? What is the notional value of your position?

$(2)$ What is the initial margin in dollars?

$(3)$ Suppose you earn an annual continuously$-$compounded interest rate of $5\%$ on your margin balance, what is your margin balance after $1$ month $($on Feb $1$st$)$ if the future price does not change?

$(4)$ Now suppose you do not earn any interest on your margin balance, what is your margin balance after $1$ month $($on Feb $1$st$)$ if the future price increases by $$50?$

$(5)$ Following $(4),$ you decide to close out your position on Feb $1$st$.$ What would you do and what is your return on investments $($i$.$e$.$ profits over initial margin$)?$

$(6)$ Keep assuming no interest and assume that today's $($Jan $1$st$)$ futures price is the same as the spot price, $800.$ What is the greatest S&P $500$ index futures price $1$ month from today at which will you receive a margin call?