Question

Two-year Treasury bills are currently yielding 1.60%, and the rate on five-year Treasury notes is 1.63%. According to the unbiased expectation theory, what is expected three-year rate starting in year 3?

Answer 1

Expected three year rate (Forward rate for three years commencing third year) is calculated using the following formula:

                                                       (1+s₅)⁵

F₂,₅= Cube root (ie. 5-2) of    ------------- - 1

                                                      (1+s₂)²

Where F_2,5 = Forward rate for the period 3 to 5 years, s₂ = Spot rate for 2 years and s₅ = Spot rate for 5 years.

Given, Spot rate for two years=1.60% and Spot rate for 5 years= 1.63%

Substituting these values in the formula,

Forward Rate for the period 3 to 5 years= [Cube root of (1+1.63%)^5/(1+1.60%)^2]-1

= [Cube root of ((1.0163^5)/(1.016)^2)] – 1   = [Cube root of (1.0842006/1.032256)]-1

=31.050321 - 1 = 1.0164999-1 = 0.0164999

= 3V1.050321

Or, Expected three year rate starting in year 3 = 1.64999% Rounded to 1.65%