Problem $1$Let P$($YTM$)$ denote the bond pricing equation for perpetual, zero$-$coupon, and coupon paying bonds as afunction of the yield$-$to$-$maturity $($YTM$).$Perpetual bonds: P$($YTM$\_$p$)=$ C$/$YTM$\_$pZero$-$coupon bonds: P$($YTM$\_$a$)=$ F$/(1+$YTM$\_$a$)$mCoupon$-$paying bonds: P$($YTM$\_$p$)=$ C$/(1+$YTM$\_$p$)1+$ C$/(1+$YTM$\_$p$)2+...+$ C$/(1+$YTM$\_$p$)2$m $+$ F$/(1+$YTM$\_$p$)2$mWhere, C is the coupon, m is maturity in years, YTM$\_$p is the periodic $($semi$-$annual$)$ YTM$,$ YTM$\_$a is the annual YTM$.$ For all types of bonds:a$)$ Calculate the first derivative of the pricing function, dP$($YTM$)/$dYTMb$)$ Calculate the second derivative of the pricing function, d$2$P$($YTM$)/$dYTM$2$c$)$ Calculate the Modified Duration, MD $=-(1/$P$)$ x dP$($YTM$)/$dYTMd$)$ Calculate the Duration, D $=$ MD x $(1+$YTM$)$e$)$ Calculate the Convexity, Conv $=(1/$P$)$ x d$2$P$($YTM$)/$dYTM$2$Problem $2$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