Here is information for the mini$-$S&P $500$ futures contract:

A portfolio manager wants to use this futures contract to hedge a $$200$ million diversified stock portfolio that mirrors the S&P $500$ to hedge against a decline in the value of the S&P $500.$ You have the following information:

Contract multiplier: $50$

Price of the futures contract at the time the hedge is placed: $4,200$

a$.$ What is the optimal number of futures contracts that the portfolio manager should take? Be sure to indicate if it is to buy or sell the futures contract.

b$.$ Suppose at the delivery date, the value of the S&P $500$ is $3,800.$ Show what the outcome for this hedge will be$.$