PLease help with simple explain or calculation, just want to learn sth no need to rigourous, thank youBonus Problem $1($Optional$,10$ marks$)$

In this problem, you are going to derive general formula for computing the Macaulay

duration and modified convexity of the bond,

We consider a $nT-$year coupon bond. The face value of the bond is $F$ and it pays coupon of

amount $rF$ at times $T,2T,$dots,$nT$ respectively. We let $i$ be the annual effective yield rate of

the bond.

$($a$)$ Derive a formula for the Macaulay duration of the bond. Express your answer in

terms of $r,F,n$ and $T.$ In addition, your answer must be free from summation.

$($b$)$ Derive a formula for the Modified convexity of the bond. Express your answer in

terms of $r,F,n$ and $T.$ In addition, your answer must be free from summation.

$()$Hint:

The following summation formula will be useful:

${\displaystyle \underset{k=1}{\overset{n}{}}}\frac{k}{(1+i{)}^k}=\frac{1+i-(1+i{)}^{-(n-1)}}{i^2}-\frac{n}{i(1+i{)}^n}$

${\displaystyle \underset{k=1}{\overset{n}{}}}\frac{k^2}{(1+i{)}^k}=\frac{1}{i}(1-\frac{n^2+2n+1}{(1+i{)}^n}+2{\displaystyle \underset{k=1}{\overset{n}{}}}\frac{k}{(1+i{)}^k}+{\displaystyle \underset{k=1}{\overset{n}{}}}\frac{1}{(1+i{)}^k})$

The derivation of these formula can be found from the Appendix of this assignment$)$

In this problem, you are going to derive general formula for computing the Macaulay

duration and modified convexity of the bond,

We consider a nT$-$year coupon bond. The face value of the bond is F and it pays coupon of

amount rF at times T$,2$T$,$dots,nT respectively. We let i be the annual effective yield rate of

the bond.

$($a$)$ Derive a formula for the Macaulay duration of the bond. Express your answer in

terms of r$,$F$,$n and T$.$ In addition, your answer must be free from summation.

$($b$)$ Derive a formula for the Modified convexity of the bond. Express your answer in

terms of r$,$F$,$n and T$.$ In addition, your answer must be free from summation.

$()$Hint:

The following summation formula will be useful:

$\backslash$sum$\_($k$=1)^$n $($k$)/((1+$i$)^($k$))=(1+$i$-(1+$i$)^(-($n$-1)))/($i$^(2))-($n$)/($i$(1+$i$)^($n$).)$

$\backslash$sum$\_($k$=1)^$n $($k$^(2))/((1+$i$)^($k$))=(1)/($i$)(1-($n$^(2)+2$n$+1)/((1+$i$)^($n$))+2\backslash$sum$\_($k$=1)^$n $($k$)/((1+$i$)^($k$))+\backslash$sum$\_($k$=1)^$n $(1)/((1+$i$)^($k$)))$

The derivation of these formula can be found from the Appendix of this assignment$)$