a$)(6$ pts$)$ The $3-$month SOFR zero rate is $2\mathrm{.}5\%$ per annum, the $6-$month SOFR zero rate is $3\%$ per annum, and the $9-$month SOFR zero rate is $3\mathrm{.}5\%$ per annum $($all continuously compounded$).$ Calculate the $3-$month forward SOFR rates for $36$ months and $69$ months, and interpret the yield curve. b$)(6$ pts$)$ A company borrows $$10$ million today for $6$ months at the $6-$month compounded SOFR plus a $1\%$ spread. The current $6-$month SOFR is $3\%,$ and the $6-$month SOFR futures price is $96\mathrm{.}80($contract size: $$1$ million$).$ Calculate the implied forward SOFR, and determine the number of contracts to hedge if SOFR is expected to rise to $4\%.$ c$)(8$ pts$)$ At $6$ months, the $6-$month compounded SOFR is $4\%,$ and the futures price is $96\mathrm{.}00.$ Compute the futures gain$/$loss$,$ the effective borrowing rate, and explain how SOFR futures mitigate risk. d$)(10$ pts$)$ The company expects to receive $$10$ million in $6$ months from a project tied to the S&P $500($current spot $=3,950,$ futures $=4,000,$ multiplier $=$ $$250).$ Design a strategy to hedge both the borrowing cost and equity exposure, aiming for a beta of $1\mathrm{.}0$ for the future cash flow. Note that the $$10$ million is a fixed receivable not yet invested, so consider its current market sensitivity relative to the S&P $500$ when determining the futures position. Calculate the combined financial impact if the S&P $500$ rises to $4,100$ and SOFR is $4\%.$