$(9)$ The yield of a two year quarterly coupon bond with coupon rate $8\%$ is $9\%.$

$($i$)$ What are the price $B,$ duration $D,$ and convexity $C$ of the bond?

$($ii$)$ Use both the formula

$\frac{B_{\text{m}\text{e}\text{w},D}-B}{B}=\frac{B}{B}$~~$-Dy.$

which does not include any convexity adjustment, and the formula

$\frac{B_{\text{n}\text{e}\text{w},D,C}-B}{B}=\frac{B}{B}$~~$-Dy+\frac{1}{2}C(y{)}^2$

to find an approximate price of the bond if the yield increases by ten basis points $($i$.$e$.,$

$0\mathrm{.}001),$ fifty basis points, one percent, two percent, and four percent respectively. Also, for

each different value of the yield change, compute the actual price $B(y+y)$ of the bond.

Report the results in the following table:

$($iii$)$ For each different value of the yield change considered above, compute the relative

approximation errors, i$.$e$.,$

$\frac{|B_{\text{n}\text{e}\text{w},D}-B(y+y)|}{B(y+y)}$ and $\frac{|B_{\text{n}\text{e}\text{w},D,C}-B(y+y)|}{B(y+y)}$

of the approximations above.