$3$ bonds with a face value of $200,$ paying an annual coupon of $5$ semiannually that is due to mature in $40$ months time. Assume the original term of the bond was $4$ years.A portfolio consists of the following instruments:

$3$ bonds with a face value of $200,$ paying an annual coupon of $5$ semiannually

that is due to mature in $40$ months time. Assume the original term of the bond

was $4$ years.

$1$ forward contract to buy an investment vehicle, currently worth $550,2$ years

from now that provides an income equivalent to a continuous dividend yield at

the nominal rate of $2\mathrm{.}6\%$ p$.$a$.$

$6$ European Call Option on some stock with the strike price is $420$ and an

expiry of $36$ months. The spot price of the underlying asset is $400$ and the

volatility in the market is $25\%.$

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 $($This is much easier if you wrote some code to calculate

the value of the portfolio in the previous part$).$

$($d$)$ Calculate the $$ for this portfolio; and the $,$ for the option.

$($e$)$ Explain briefly $($in no more than a few sentences$)$ how you would hedge this

portfolio against making a large loss.