Based on the duration and convexity formulas you found in Problem $1,$ derive the change in bond prices $($Price$)$ for perpetual, zero$-$coupon, and coupon paying bonds as a linear approximation of thea$)$ Modified Durationb$)$ Durationc$)$ Modified Duration and Convexityd$)$ Duration and Convexity

Problem $1$

Let P$($YTM$)$ denote the bond pricing equation for perpetual, zero$-$coupon, and coupon paying bonds as a function of the yield$-$to$-$maturity $($YTM$).$

Perpetual bonds: P$($YTM$\_$p$)=$ C$/$YTM$\_$p

Zero$-$coupon bonds: P$($YTM$\_$a$)=$ F$/(1+$YTM$\_$a$){?}^m$

Coupon$-$paying bonds: $P(YT{M}_p)=\frac{C}{(1+YT{M}_P{)}^1}+\frac{C}{(1+YT{M}_p{)}^2}+$dots$+\frac{C}{(1+YT{M}_p{)}^{2m}}+$ F$/(1+$YTM$\_$p