You will be taking out a $3-$month loan of $200,000$ in $5$ months' time. You wish to hedge against changes in the interest rate by entering an FRA. The $5-$month libor rate is $4\%$ and the $8-$month libor rate is $5\%.$ There are $151$ days in the first $5-$month period, and $92$ days in the subsequent $3-$month period.

$($a$)$ What is the price of the FRA? $(3$ marks$)$

$($b$)$ What is the present value of the FRA at inception? $(1$ mark$)$

$($c$)$ Assume that $2$ months $(59$ days$)$ have elpased and the $3-$month and $6-$month Libor rates are, respectivley, $3\%$ and $4\%.$ What is the current vlaue of the FRA? $(3$ marks$)$

$($d$)$ The bank with which you have set up the FRA allows you to exit your position by entering into an opposite FRA. Based on your answer to part $($c$),$ discuss wether or not you would want to exit the FRA. $(1$ mark$)$