Historical simple forward rates R$($Ti$,$ Tj$)($in fact GBP LIBOR rates recorded

during a six$-$month period in $2009)$ are given in the following table $($for

example, R$($T$1,$ T$6)=1\mathrm{.}33000\%)$:

T$0(1$ Jun$)$ T$1(1$ Jul$)$ T$2(1$ Aug$)$ T$3(1$ Sep$)$ T$4(1$ Oct$)$ T$5(1$ Nov$)$

T$1(1$ Jul$)0\mathrm{.}67063\%$

T$2(1$ Aug$)1\mathrm{.}06563\%0\mathrm{.}64938\%$

T$3(1$ Sep$)1\mathrm{.}27250\%0\mathrm{.}97500\%0\mathrm{.}58250\%$

T$4(1$ Oct$)1\mathrm{.}35125\%1\mathrm{.}17750\%0\mathrm{.}69000\%0\mathrm{.}53000\%$

T$5(1$ Nov$)1\mathrm{.}42250\%1\mathrm{.}25625\%0\mathrm{.}88250\%0\mathrm{.}56125\%0\mathrm{.}50125\%$

T$6(1$ Dec$)1\mathrm{.}48438\%1\mathrm{.}33000\%0\mathrm{.}96375\%0\mathrm{.}68000\%0\mathrm{.}50750\%0\mathrm{.}51313\%$

Use these rates to compute:

$($a$)$ The swap rate quoted on $1$ June for an interest rate swap settled

in arrears starting on $1$ June, with $6$ monthly coupon settlement

dates falling on the $1$st day of each month from July till December

inclusive, the floating leg of the swap contract paying interest at the

appropriate simple forward rate. $(20$ marks$)$

$($b$)$ The cash flow for the fixed leg and floating leg of this interest rate

swap on a $100$ principal. $(10$ marks$)$

$($c$)$ The balancing payment received or made when closing a payer position

in the above swap on a $100$ principal on $1$ September.