On May $1$ you are thinking of adding the common stock of Black & Decker to one of your bank's trust accounts. Black and Decker stock is currently trading at a price of $$137\mathrm{.}41$ per share and has a beta of $1\mathrm{.}20.$ From now until September $1$ you anticipate the stock market as a whole will go up about $3\%.$ Your bank plans to purchase $50,000$ shares on September $1$ and you are worried that general market increases will make the planned purchase prohibitively expensive. September S&P $500$ futures are currently priced at $2374.$

a$)$ What type of futures hedge should you use to reduce the risk of your position, and how many contracts should you use to fully hedge?

b$)$ How many futures contracts should you use if you did not wish to hedge fully, but instead you wished to reduce the beta of your implied position to $1\mathrm{.}0?$

c$)$ If you hedge fully, is the hedged position riskless? Why or why not? THINK!

d$)$ Suppose that on May $1,$ you put on a full hedge. On September $1,$ Black & Decker stock is priced at $$152\mathrm{.}81$ and the September S&P $500$ futures is at $2419.$ The hedge is lifted and the stock is purchased. On December $31,$ Black & Decker is priced at $$159\mathrm{.}21$and the stock is sold. What is the effective $($annualized with compounding$)$ capital gain $\%($EAR$)$ earned on the stock over the $4$ month investment period?