Question

Four bounds with 6, 12, 18, and 24 months of maturity and $10,000 face value are selling for $9,400, $8,900, $9,484, and $9,625 respectively. The first two are discount bonds, the third one pays 8% and the last one pays 9% /year coupon semi-annually. Using the bootstrapping method, calculate the 6, 12, 18, and 24 month zero rates.

Answer 1

6-month discount bond market value = $ 9400 and Par Value = $ 10000

Let the 6-month spot rate be = 2r1 %

Therefore, 9400 / (1+r1) = 10000, r1 = [10000/9400] -1 = 0.06383 or 6,383% per half-year

6-month spot rate = 6.383 x 2 = 12.766 % per annum

1 year discount bond = $ 8900 and Par Value = $ 10000

Let the 1 year spot rate be 2r2

Therefore, 8900 = 10000 / (1+r2)^(2)

(1+r2) = [10000/8900]^(1/2)

r2 = 0.05999 or 5.999 %

1 - year spot rate =2r2 = 2 x 5.999 = 11.998 % per annum

The 1.5 year coupon bond has an annual coupon rate of 8%, payable semi-annually. Par Value = $ 10000 and Market Value = $ 9484

Semi-Annual Coupon = 0.08 x 0.5 x 10000 = $ 400

Let the 1.5 year spot rate be = 2r3 % per annum

Therefore, 9484 = 400 / 1.06383 + 400 / (1.05999)^(2) + 10400 / (1+r3)^(3)

9484 - 375.999 - 356.0053 = 10400 / (1+r3)^(3)

(1+r3)^(3) = 10400 / 8751.9957

r3 = [(10400/8751.9957)^(1/3)] - 1 = 0.059194 or 5.919%

Therefore, 1.5 year spot rate = 2 x 5.919 = 11.838 %

2 year coupon bond has a par value of $10000, coupon payments payable semi-annually at a rate of 9% per annum, market value = $ 9625

Sem-Annul Coupon = 0.09 x 0.5 x 10000 = $ 450

Let the 2 year spot rate be 2r4

Therefore, 9625 = 450 / 1.06383 + 450 / (1.05999)^(2) + 450 / (1.0.5919)^(3) + 10450 / (1+r4)^(4)

9625 - 422.999 - 400.506 - 378.696 = 10450 / (1+r4)^(4)

8422.799 = 10450 / (1+r4)^(4)

r4 = [(10450/8422.799)^(1/4)] - 1 = 0.05539 or 5.539 % per half year

Therefore, 2 year spot rate = 2 x r4 = 11.078 %

Therefore,

6-month zero rate = 12.776 % per annum

1 year zero rate = 11.998 % per annum

1.5 year zero rate = 11.838 % per annum

2 year zero rate = 11.078% per annum

NOTE: Zero rates or discount rates for zero coupon bond of corresponding maturity is the same as spot rates for corresponding maturity.