$($a$)$ Given the following information about the default$-$free, coupon paying yield curve,

with face value of $1000,$ derive the zero$-$coupon yield curve for years $1$ through $4$

using discrete compounding. Coupons are paid annually at the end of the year.

Solution The candidate should apply the bootstrap methodology:

$\frac{F(1+c_1)}{1+y_1}=P{V}_1=\frac{F(1+c_1)}{1+r_1}=>r_1=y_1=0\mathrm{.}03$

$\frac{F*C_2}{1+y_2}+\frac{F(1+C_2)}{(1+y_2{)}^2}=P{V}_2=\frac{F{C}_2}{1+r_1}+\frac{F(1+C_2)}{(1+r_2{)}^2}=>r_2=0\mathrm{.}0402$

and similarly $r_3=0\mathrm{.}0506$ and $r_4=0\mathrm{.}0614.$

$($b$)$ You have $1,000,000$ at the beginning of year $2.$ Using the information from the

table above, and assume that all the bonds are selling at par $($ignore the column

of YTM for this part$),$ what transaction should you undertake today to lock in the

forward rate from year $2$ to year $3,$ using the coupon paying bonds provided above?

What is the forward rate?

Solution Solving the following system of equations:

$1000x+1000y+1000z=0$

$1020x+30y+40z=0$

$1030y+40z=-1,000,000$

yields $x=-10\mathrm{.}3,y=-1010\mathrm{.}5,z=1020\mathrm{.}8.$ The payoff in year $3$ is given by

$1040z=1,061,632$ implying a forward rate from period $2$ to $3$ of $6\mathrm{.}16\%.$ I do not understand how to slove question b in the solution$$ Could youpl ease explain to me in detailed how does it works$$ Thanks