The multiplier for the S&P$500$ index futures is $$250$ per index point. The maturity of the contract is one year. Today, you decided to buy one S&P$500$ index futures contract at the price of $5226\mathrm{.}17,$ and made an initial margin deposit of $$100,000.$

a$.$ Over the next five days, the futures contract price evolves as follows: $5014\mathrm{.}50,5033\mathrm{.}75,5023\mathrm{.}75,5039\mathrm{.}50,5042\mathrm{.}75.$ What is the gain deposited to $($or loss subtracted from$)$ the account on day $5?$

b$.$ What is the cumulative gain or loss after day $5?$

c$.$ What is the cumulative rate of return on this futures position after day $5?$