Question $3-$ Interest rate derivatives and CDS

a$)$ Derive the formula that converts an interest rate with continuous compounding into a rate with discrete compounding.

Type answer here

b$)$ Write a MATLAB function that performs the conversion in a$).$ Name this function cont$2$dis. Submit this function. Make sure to adopt the same notation as used in this unit.

c$)$ A financial institution has entered into a swap where it agreed to receive quarterly payments at a rate of $4\mathrm{.}5\%$ per annum and pay the SOFR three$-$month reference rate on a notional principal of $$100$ million. The swap now has a remaining life of $10$ months. Assume the risk$-$free rates with continuous compounding for $1$ month, $4$ months, $7$ months and $10$ months are $3\mathrm{.}8\%,4\%,4\mathrm{.}1\%$and $4\mathrm{.}4\%$ respectively. Assume also that the continuously compounded risk$-$free rate observed for the last $2$ months is $3\mathrm{.}6\%.$ Estimate the value of the swap. State whether the financial institution would receive or pay if they are to liquidate the position today. In your code, make use of the function you wrote in b$).\%$ Type your code in this box. Clearly comment your working.