A $10-$year U$.$S$.$ Treasury bond with a face value of $$1,000$ o$$ers a coupon of $5\mathrm{.}5\%$ per annum $($pa$),$ to be paid semi$-$annually. The reported yield to maturity is $5\mathrm{.}2\%$ pa$,$ compounded semi$-$annually.

$($a$)$ What is the PV of this bond?

$($b$)$ What is the PV if the yield to maturity increases to $5\mathrm{.}8\%$ pa$?$

$($c$)$ Explain your observations.