A $4$ year Treasury Bond with a face value of $$1,000$ and an annual coupon rate of $5\mathrm{.}80\%$ has a yield to maturity of $4\mathrm{.}29\%.$ This bond makes $2($semiannual$)$ coupon payments per year and thus has $8$ periods until maturity.

$(1)$ What is the price sensitivity of a bond to changes in yield and how does that compare to the duration approximation, and compare to the duration plus convexity approximation?