Question

On March 11, 20XX, the existing or current (spot) one-year, two-year, three-year, and four-year zero-coupon Treasury security rates were as follows:

₁R₁ = 1.35%, ₁R₂ = 1.87%, ₁R₃ = 2.11%, ₁R₄ = 2.22%

Using the unbiased expectations theory, calculate the one-year forward rates on zero-coupon Treasury bonds for years two, three, and four as of March 11, 20XX.

Year 2%

Year 3%

Year 4%

Answer 1

1 year Fwd rate for Year 2 = [ (1 + 1R2 )^2 / ( 1 + 1R1) ] - 1

= [ ( 1 + 0.0187)^2 / ( 1 + 0.0135) ] - 1

= [ (1.0187^2 ) / 1.0135 ] - 1

= [ 1.03775 / 1.0135 ] - 1

= 1.0239 - 1

= 0.0239 i.e 2.39%

1 year Fwd rate for Year 3 = [ (1 + 1R3 )^3 / ( 1 + 1R2)^2 ] - 1

= [ ( 1 + 0.0211)^3 / ( 1 + 0.0187)^2 ] - 1

= [ (1.0211^3 ) / (1.0187^2) ] - 1

= [ 1.0646/ 1.03775] - 1

= 1.0259 - 1

= 0.0259 i.e 2.59%

1 year Fwd rate for Year 4 = [ (1 + 1R4 )^4 / ( 1 + 1R3)^3 ] - 1

= [ ( 1 + 0.0222)^4 / ( 1 + 0.0211)^3 ] - 1

= [ (1.0222^4 ) / (1.0211^3) ] - 1

= [ 1.0918/ 1.0646] - 1

= 1.0255 - 1

= 0.0255 i.e 2.55%