Question

Current (annualised) US Treasury spot rates are as follows:

6 months	1 year	18 months	2 years
0.4%	0.5%	0.6%	0.7%

Derive six-monthly forward rates, including six- months forward rate 6 month from now - _0.5f_0.5, six-month forward rate 12 months from now - ₁f_0.5, and six-months forward rate 18 months from now - _1.5f_0.5 for the bond. Show calculations. (4.5 marks)

Answer 1

(1 + 1 year spot rate / 2)² = (1 + 6 month spot rate / 2) * (1 +  6 months forward rate 6 months from now / 2)

(1 + (0.5% / 2))² = (1 + (0.4% / 2)) * (1 +  6 months forward rate 6 months from now / 2)

(1 +  6 months forward rate 6 months from now / 2) = (1 + (0.5% / 2))² / (1 + (0.4% / 2))

(1 +  6 months forward rate 6 months from now / 2) = 1.003000

6 months forward rate 6 months from now / 2 = 0.003

6 months forward rate 6 months from now = 0.006 or 0.6%

(1 + 1.5 year spot rate / 2)³ = (1 + 1 year spot rate / 2)² * (1 +  6 months forward rate 12 months from now / 2)

(1 + (0.6% / 2))³ = (1 + (0.5% / 2))² * (1 +  6 months forward rate 12 months from now / 2)

(1 +  6 months forward rate 12 months from now / 2) = (1 + (0.6% / 2))³ / (1 + (0.5% / 2))²

(1 +  6 months forward rate 12 months from now / 2) = 1.004000

6 months forward rate 12 months from now / 2 = 0.004

6 months forward rate 12 months from now = 0.008 or 0.8%

(1 + 2 year spot rate / 2)⁴ = (1 + 1.5 year spot rate / 2)³ * (1 +  6 months forward rate 18 months from now / 2)

(1 + (0.7% / 2))⁴ = (1 + (0.6% / 2))³ * (1 +  6 months forward rate 18 months from now / 2)

(1 +  6 months forward rate 18 months from now / 2) = (1 + (0.7% / 2))⁴ / (1 + (0.6% / 2))³

(1 +  6 months forward rate 18 months from now / 2) = 1.005000

6 months forward rate 18 months from now / 2 = 0.005

6 months forward rate 18 months from now = 0.01 or 1%