Suppose that as a loan officer for a bank you just completed a $2$ million loan to a small business. This is a bullet loan, senior unsecured, with one$-$year maturity and carries a $5\mathrm{.}5\%$ interest rate. You rate the borrower internally as B$,$ which translates into a one$-$year probability of default $($PD$)$ estimate of $3\%.$ Loss given default $($LGD$)$ for the loan is estimated as $45\%.$

a$)$ Determine the expected cash flow from the loan at maturity.

$(10$ marks$)$

b$)$ If the estimated all$-$inclusive cost of funding for the loan is $2\mathrm{.}5\%$ per year, determine the bank$$s expected profit $($in $)$ on the loan.

$(10$ marks$)$

c$)$ Now suppose that the bank funds this loan $($and possibly many others$)$ with deposits that on average have one$-$month duration. The current interest rate that the bank pays on these deposits is $1\%$ per year, but recent political developments lead the bank management to believe that interest rates on these deposits may have to be increased in a few months$$ time.

i$)$ If the bank decides to hedge its interest rate exposure, briefly state how it can utilise a $6$x$12$ forward rate agreement $($FRA$)$ for this purpose. No calculations are necessary, state whether the bank should buy the FRA or sell it $($if you cannot recall what buying and selling means in the FRA market, answer in terms of paying or receiving LIBOR$).$ Explain your reasoning clearly.

$(10$ marks$)$

ii$)$ Now suppose that the bank enters into an FRA in the direction $($buy or sell$)$ you identified in part $($i$),$ with a notional amount of $2$ million. The rate on the contract is $1\mathrm{.}5\%$ per year. At the end of the six months, when settlement day comes, $6-$month LIBOR turns out to be $2\mathrm{.}4\%$ per year. Using a day count of $180$ days for the six$-$month period, determine the payoff to the bank from the FRA on settlement day.