You want to set up a synthetic index fund for one year with $$1,000,000$ by buying index futures and treasury bills. The index currently stands at $$1000$ and there are $100$ units in each index futures contract. The futures price is $$1050,$ the index dividend yield is $4\%$ and the risk$-$free rate is $6\%.$

a$)$ Describe the steps that are required to match the performance of the index with the synthetic fund.

b$)$ Assuming at the end of the year the value of the index is $$1150$ compare the return of the synthetic index fund to that of the index itself. Do they match? If not why?