Question $5$: Interest Rate Swaps $(2/10)$ Consider a $1$ year fixed for floating interest

rate swap with a notional value of $$1.$ The floating leg is tied to three$-$month LIBOR with

payments every quarter and payments in the fixed leg occur semiannually. Hence, the first

floating rate payment is in $3$ months and the first fixed rate payment is in $6$ months.

Table $1$ gives you the spot LIBOR $($annual$,$ continuously compounding$)$ at time $0.$

Table $1$: Spot LIBOR

Notation Maturity In Years LIBOR

r$0,0\mathrm{.}250\mathrm{.}250\mathrm{.}100\%$

r$0,0\mathrm{.}500\mathrm{.}50\mathrm{.}350\%$

r$0,0\mathrm{.}750\mathrm{.}750\mathrm{.}600\%$

r$0,1\mathrm{.}0010\mathrm{.}850\%$

Consider LIBOR as proxy for the risk$-$free interest rate.

$($i$)$ Suppose the fixed swap rate $($annual$,$ continuously compounding$)$ is R and the realized

LIBOR interest rate $($annual$,$ continuously compounding$)$ from t$1$ to t$2$ is rt$1,$t$2($e$.$g$.$

r$0\mathrm{.}75,1$ is the interest rate from $0\mathrm{.}75$ years to $1$ year$),$ what is the payoff stream of short

position $($receiver of fixed interest$)$ of the swap?

$($ii$)$ To replicate the payoffs of $($i$),$ we can use FRAs. What are the fair forward interest rates

$($annual$,$ simple$)$ f$0,0,0\mathrm{.}25,$ f$0,0\mathrm{.}25,0\mathrm{.}50,$ f$0,0\mathrm{.}50,0\mathrm{.}75,$ and f$0,0\mathrm{.}75,1\mathrm{.}00?$ Hint: for an interest

rate, its simple term and continuous compounding term always satisfy $1+$ rsimple $\backslash$times T $=$

exp$($rcc $\backslash$times T $).$