The treasurer of a small bank has borrowed funds for $1$ month at an interest rate of $4\%$ and has lent funds for $5$ months at $6\%.$ The treasurer borrowed and lent $1$ million dollars in t$0.$

a$.$ Represent the exposure on cash flow diagrams. Be specific on the size of the cash flows in each period.

b$.$ The treasurer decides to wait $1$ month and take a spot loan then. To cover his exposure, the treasurer signs a $1$x$5$ FRA.

$($i$)$ Find the break$-$even rate

$($ii$)$ What is his outflow in the FRA at the break$-$even forward rate?

$($iii$)$ At a borrowing cost at t$1=6\mathrm{.}8\%,$ What is the settlement amount in the FRA?

$($iv$)$ If the forward rate at t$0$ was $6\mathrm{.}1\%,$ what are the overall profits of this bank?