The treasury has just issued a $5-$year zero coupon bond $($Price of $$100)$ with a yield to maturity of $5\%.$

a$)$Draw a graph of the price $($y$-$axis$)$ vs yield to maturity $($x$-$axis$)$ to illustrate how changes in yield may impact its price.$$

b$)$Compute Modified Duration of the bond using both a numerical approximation$-$ DV$01/$P and by using the actual formula for $($dP$/$dy$)/$P$.$

c$)$Use both the duration measures to predict the change in price if the yield to maturity increases by $100$ basis points.$$

d$)$Compute the convexity of the bond using the numerical approximation.$$

e$)$Now use both duration and convexity again to compute the change in price when rates increase by $100$ basis points.$$ Is the prediction better than what you computed in Part $($b$)?$ Why?