$2.$ In early October an insurance company portfolio manager has a stock portfolio of $$800,000$ with a portfolio beta of $1\mathrm{.}25.$ The portfolio manager plans on selling the stocks in the portfolio in December to be able to pay out funds to policyholders for annuities, and is worried about a fall in the stock market. In April, the CME Group S&P $500$ mini$=$ Futures contract index is $4136$ for a December futures contract $($$$50$ multiplier for this contract$).$

a$.$ What type of futures position should be taken to hedge against the stock market going down and how many future contracts are needed for this hedge?

Type of Position $\_\_\_\_\_\_($short or long$)$

Explain why $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

How many contracts should you get?

$[$Hint: # contracts $=[$Portfolio x Beta$]/[$ Futures Index Price x Multiplier$]$

Number of Contracts$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

b$.$ Suppose in December the stock market $($S&P$500$ index$)$ falls by $10\%$ and the S&P$500$ futures index falls by $10\%$ as well, what is the portfolio manager$$s opportunity loss $($gain$)$ on his portfolio, and his futures gain $($loss$)$ and the net hedging result?

Spot Gain or Loss $\_\_\_\_\_\_\_\_$ Futures Gain or Loss $\_\_\_\_\_\_\_$

Net Hedging Result $\_\_\_\_\_\_\_\_\_\_\_$