Today is $1$ July $2021.$ Joan has a portfolio which consists of two different$$types of financial instruments $($henceforth referred to as instrument A and instrument$$B$).$ Joan purchased all instruments on $1$ July $2015$ to$$create this portfolio and this portfolio is composed of $305$ units of instrument$$A and $358$ units of instrument B$.$

Instrument A is a zero$-$coupon bond with a face value of $100.$ This bond matures at par. The maturity date is $1$ January $2030.$

Instrument B is a Treasury bond with a coupon rate of j$2=3\mathrm{.}1\%$ p$.$a$.$ and face value of $100.$ This bond matures at par. The maturity date is $1$ January $2024.$

$($a$)$Calculate the current price of instrument A per $$100$ face value $($today$\text{'}$s value$).$Round your answer to four decimal places. Assume the yield rate is$$j$2=3\mathrm{.}86\%$ p$.$a$.$

$($b$)$Calculate the current price of instrument B per $$100$ face value $($today$\text{'}$s value$).$Round your answer to four decimal places. Assume the yield rate is$$j$2=3\mathrm{.}86\%$ p$.$a$.$ and Joan has just received the coupon payment.

$($c$)$What is the current duration of instrument B$?$ Express your answer in$$terms of years and round your answer to three decimal places.$$Assume the yield rate is$$j$2=3\mathrm{.}86\%$p$.$a$.$

$($d$)$Based on the price in part a and part b$,$ and the duration value in part c$,$calculate the current duration of Joan$$s portfolio. Express your answer$$in terms of years and round your answer to two decimal places.