$($Duration and Convexity for General Cashflow Streams$)$ Equation $(4)$ and $(11)$

in the Lecture Notes $2$ give the duration and convexity for the coupon bonds.

This question illustrates duration and convexity for general cash flow streams.

Consider the $n-$period cash flow stream $(n1)$ depicted on Page $24$ of Lecture

Notes $2,$ with $x_0=0,x_{k}0$ for $k=1,2,$dots,$n-1,$ and $x_n>0.$ Assume that

all periods have equal length of $1$ year $($therefore$,R=r).$ Denote the present

value of this cash flow stream as

$P=P(r)=\frac{x_1}{1+r}+\frac{x_2}{(1+r{)}^2}+$cdots$+\frac{x_n}{(1+r{)}^n}.$

Required precision: $4$ digits after decimal point.

$($a$)$ The duration is defined as $D=-\frac{1+r}{P}\frac{dP}{dr}.$ Find out the expression for $w_k,$

$k=1,$dots,$n$ which satisfy the following conditions: $w_1+w_2+$cdots$+w_n=1,$

$w_{k}0,k=1,$dots,$n,$ and

$D=w_1*1+w_2*2+$cdots$+w_n*n.$

$($b$)$ The convexity is defined as $Cx=\frac{1}{P}\frac{d^{2}P}{d{r}^2}.$ Find out the expression for $u_k,$

$k=1,$dots,$n$ which satisfy the following conditions: $u_1+u_2+$cdots$+u_n=1,$

$u_{k}0,k=1,$dots,$n,$ and

$Cx=\frac{1}{(1+r{)}^2}*[u_1*1*2+u_2*2*3+$cdots$+u_n*n*(n+1)].$

$($c$)$ Now consider a $6-$period cash flow stream $x_1=0,x_2=5,x_3=0,x_4=$

$4,x_5=0,x_6=1,$ with $r=4\%.$ Calculate its present value, duration and

convexity.

$($d$)$ If the interest rate changes from $r$ to $r+r,$ the present value of the cash

flow stream changes from $P(r)$ to $P(r+r).$ The duration model predicts

$P(r+r)=P(r)-\frac{D*P}{1+r}r$

and the convexity model predicts that

$P(r+r)=P(r)-\frac{D*P}{1+r}r+\frac{1}{2}Cx*P*(r{)}^2.$

Consider the cash flow stream and $r$ in $($c$).$ If $r$ increases from $4\%$ to $5\%,$

what is the new present values predicted by the duration model and the

convexity model, respectively? Which one of them predicts more accu$-$

rately?