You manage a bond portfolio comprising of two types ot bonds. Your portfolio has zero coupon bonds maturing in four years from today; and also $10\%$ perpetual bonds. Assume the par value of each is $$1000$ and that you hold equal numbers of each bond. Assume the yield curve is flat at $10$ percent.$($a$)$ Compute the duration of your portfolio.$($b$)$ How would the value of your portfolio change approximately if the yield were to unexpectedly increase by $9$ basis points $(0\mathrm{.}09\%)$ later today $("$approximately$"->$ don't answer this question by computing the value of the portfolio before and after the yield change!$)$Assume you invested into this portfolio only because you agreed to buy your neighbours' house for $$500,000$ in S years' time. This brings up the question whether you current portfolio weights are optimal.$($c$)$ Compute the optimal portfolio weights given this objective.$($d$)$ Compute: are the portfolio weights you computed in $($c$)$ still optimal next year?$($e$)$ Compute how many of each bond you need to hold $($round to $1$ decimal$)$