Answer the questions for the bond below, which pays interests semi$-$annually. The modified duration is $3\mathrm{.}9944$ years and convexity measure is $19\mathrm{.}7636$ years. $($Assume par value is $$1000).$ Coupon rate: $9\%,$ current yield to maturity: $8\%,$ maturity: $5$ years.

$(1)$ Calculate the actual price of the bond for a $50-$basis$-$point increase in interest rates $($yield changes from $8\%$ to $8\mathrm{.}5\%)$

$(2)$ Using duration, estimate the approximate price of the bond for a $50-$basis point increase in interest rates $($yield changes from $8\%$ to $8\mathrm{.}5\%)$

$(3)$ Using both duration and convexity measure, estimate the approximate price of the bond for a $50-$basis$-$point increase in interest rates $($yield changes from $8\%$ to $8\mathrm{.}5\%)$

$(4)$ Compare your results in $(2)$ and $(3)$ and explain which is closer to the actual price in $(1)$