Suppose that $12-$month, $24-$month, $36-$month, and $48-$month and $60-$month zero rates are $4\mathrm{.}4\%,4\mathrm{.}6\%,4\mathrm{.}9\%,5\%$ and $5\mathrm{.}3\%$ per annum with continuous compounding respectively.

$($a$)$ Calculate the value of the portfolio $($you may wish to write some python or excel code to accomplish this$).$

$($b$)$ Calculate the yield to maturity of the bond if it had been purchased for $183$ just after the second coupon payment.

$($c$)$ If the interest rates shift $($upwards and downwards$)$ by $0\mathrm{.}5\%,$ calculate the new value of the portfolio in each case $($This is much easier if you wrote some codeto calculate the value of the portfolio in the previous part$).$

$($d$)$ Calculate the $\backslash$rho for this portfolio; and the $\backslash$Gamma $,$ for the option.

$($e$)$ Explain briefly $($in no more than a few sentences$)$ how you would hedge thisportfolio against making a large loss.