Question

You are interested in valuing a 2-year semi-annual corporate coupon bond using spot rates but there are no liquid strips available. However, you do find the following 4 comparable semi-annual bonds (below) maturing over the next 2 years. Using the bootstrap approach, calculate the 12-month spot rate.

Time remaining to maturity Coupon Bond price
6 months 0.000% 99.000
1 year 1.250% 98.000
18 months 1.500% 97.000
2 years 1.250% 96.000
a. 1.668%
b. 3.335%
c. 4.167%
d. 4.189%
e. 4.204%

Answer 1

Following are the solution to this question:

Step-by-step explanation:

Assume that
`r_1` will be a 12-month for the spot rate:

`\to 1.25 \% * (100)/(2) * 0.99 + ((1.25\% * (100)/(2)+100))/((1+(r_1)/(2))^2)=98\\\\\to (1.25)/(100) * (100)/(2) * 0.99 + (((1.25)/(100) * (100)/(2)+100))/((1+(r_1)/(2))^2)=98\\\\\to (1.25)/(2) * 0.99 + (((1.25)/(2) +100))/((1+(r_1)/(2))^2)=98\\\\\to 0.61875 + (( 0.625 +100))/(((2+r_1)/(2))^2)=98\\\\\to 0.61875 + (( 100.625))/(((2+r_1)/(2))^2)=98\\\\\to 0.61875 + (402.5)/((2+r_1)^2)=98\\\\`

`\to 0.61875 + (402.5)/((2+r_1)^2)=98\\\\\to 0.61875 -98 = (402.5)/((2+r_1)^2)\\\\\to -97.38125= (402.5)/((2+r_1)^2)\\\\\to (2+r_1)^2= (402.5)/( -97.38125)\\\\\to (2+r_1)^2= -4.13\\\\ \to r_1=3.304\%`

Assume that
`r_2` will be a 18-month for the spot rate:

`\to 1.5\% * (100)/(2) * 0.99+1.5\% * (100)/(2) * (1)/((1+ (3.300\%)/(2))^2)+((1.5\% * (100)/(2)+100))/((1+(r_2)/(2))^3)=97\\\\\to (1.5)/(100) * (100)/(2) * 0.99+(1.5)/(100) * (100)/(2) * (1)/((1+ ((3.300)/(100))/(2))^2)+(((1.5)/(100) * (100)/(2)+100))/((1+(r_2)/(2))^3)=97\\\\`

`\to (1.5)/(2) * 0.99+(1.5)/(2)* (1)/((1+ ((3.300)/(100))/(2))^2)+(((1.5)/(2) +100))/((1+(r_2)/(2))^3)=97\\\\\to 0.7425+0.75 * (1)/((1+ ((3.300)/(100))/(2))^2)+((0.75 +100))/((1+(r_2)/(2))^3)=97\\\\\to 1.4925 * (1)/((1+0.0165)^2)+((100.75 ))/((1+(r_2)/(2))^3)=97\\\\\to 1.4925 * (1)/((1.033))+((100.75 ))/((1+(r_2)/(2))^3)=97\\\\`

`\to 1.4925 * 0.96+((100.75 ))/((1+(r_2)/(2))^3)=97\\\\\to 1.4328+((100.75 ))/((1+(r_2)/(2))^3)=97\\\\\to 1.4328-97= ((100.75 ))/((1+(r_2)/(2))^3)\\\\\to -95.5672= ((100.75 ))/((1+(r_2)/(2))^3)\\\\\to (1+(r_2)/(2))^3= -1.054\\\\\to r_2=3.577\%`

Assume that
`r_3` will be a 18-month for the spot rate:

`\to 1.25\% * (100)/(2) * 0.99+1.25\% * (100)/(2) * (1)/((1+(3.300\%)/(2))^2)+1.25\%*(100)/(2) * (1)/((1+(3.577\%)/(2))^3)+(1.25\% * ((100)/(2)+100)/((1+(r_3)/(2))^4))=96\\\\`

to solve this we get
`r_3=3.335\%`