$(20$ points$)$ On August $19,2019,$ both two$-$year interest rate and three$-$year interest rate are around $1\mathrm{.}5\%.(1)$ Calculate the bonds price for a two$-$year coupon bond with a face value $$500$ dollars and coupons $$50$ dollars. $(2)$ Calculate the bonds price for a threeyear coupon bond with a face value $$500$ dollars and coupons $$50$ dollars. $(3)$ Suppose both interest rates increases from $1\mathrm{.}5\%$ to $4\%,$ what are the new bonds prices of two$-$year and three$-$year coupon bonds? According to the calculation results, which bond price is more sensitive to interest rate changes. $($Hints: the coupons for coupon bonds are paid annually.$)$