Question

If a half-year C1 percent coupon bond (paying twice per year) is trading at C2 and a one-year C3 percent coupon bond (paying twice per year) is trading at C4, find half-year and one-year discount factors. The face value of either bond is $100. Assume semi-annual compounding. Write your answers for the following:

18. Half-year discount factor.

19. One-year discount factor.

20. Half-year spot interest rate.

21. One-year spot interest rate.

22. Forward rate for 6 to 12 months.

C1=2.25

C2=100.35

C3=8.25

C4=106.2

CAN YOU PLEASE SHOW WORK

Answer 1

Let's assume K_1/2 and K₁ as half year and full year discount factors.

If a half-year C1 percent coupon bond (paying twice per year) is trading at C2 and a one-year C3 percent coupon bond (paying twice per year) is trading at C4, find half-year and one-year discount factors. The face value of either bond is $100. Assume semi-annual compounding.

Part (18)

C₂ = PV of all the future coupon and face value of a half-year C1 percent coupon bond (paying twice per year)

Pending coupon amount = C₁ / 2 = 2.25% / 2 x Face value = 1.125% x 100 = 1.125

Face value, FV = $ 100

Both the payments are due in half year's time

Hence, C₂ = 100.35 = (C₁/2 + FV) x K_1/2 = (1.125 + 100) x K_1/2

Hence, K_1/2 = half year discount factor = 100.35 / 101.125 =  0.992336

(Please do rounding off as per your requirement)

Part (19)

C₄ = PV of two semi annual coupons and face value

Semi annual coupons = C₃ / 2 = 8.25% / 2 x FV = 4.125% x 100 = 4.125

FV = $ 100

Hence, C₄ = 106.2 = C₃ / 2 x K_1/2 + (C₃ / 2 + FV) x K₁ = 4.125 x 0.992336 + 104.125 x K₁

Hence, K₁ =  0.9806157

(Please do rounding off as per your requirement)

Part (20)

Let's assume S_0,0.5 as half year spot rate then, and assuming semi annual compounding

(1 + S_0,0.5 / 2) = 1 / K_1/2 =  1.0077230

Hence, S_0,0.5 = Half year spot rate = 0.0154459 = 1.5446%

Part (21)

Let's assume S_0,1 as one year spot rate then, and assuming semi annual compounding

(1 + S_0,1 / 2)² = 1 / K₁ = 1.019767

Hence, S_0,1 = One year spot rate =(1.019767^1/2 - 1) x 2 =  0.0196707 = 1.967071%

Part (22)

Let's assume Forward rate for 6 to 12 months as F_0.5,1 then under semi annual compounding

(1 + S_0,0.5 / 2) x (1 + F_0.5,1 / 2) = (1 + S_0,1 / 2)²

Hence, (1 + F_0.5,1 / 2) = [(1 + S_0,1 / 2)²] / (1 + S_0,0.5 / 2) = K_1/2 / K₁ = 1.01195217

Hence, F_0.5,1 = 0.023904339 = 2.3904%