Question

Suppose that the current 1-year rate (1-year spot rate) and expected 1-year T-bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows: 1R1 = 2.54%, E(2r1) = 3.80%, E(3r1) = 4.30%, E(4r1) = 5.80% Using the unbiased expectations theory, calculate the current (longterm) rates for 1-, 2-, 3-, and 4-year-maturity Treasury securities. Plot the resulting yield curve.

Answer 1

According to the unbiased expectations theory, Interest earned by investing in a long term bond should be equal to interest earned by investing in successive short term bonds for same period.

Therefore,  (1+rate of interest on two-year bond)² = (1+ current one year spot rate) * (1+ expected one year spot rate over year two).

Given,

Current 1-year spot rate	2.54%
Expected 1-year T-bill rates over year 2	3.80%
Expected 1-year T-bill rates over year 3	4.30%
Expected 1-year T-bill rates over year 4	5.80%

So,

(1+ current longterm rate for 1-year security)1 = (1 + current 1-year spot rate)1
	Current longterm rate for 1-year treasury security = (1 + 0.0254) - 1 = 0.0254 = 2.54%

(1+ current longterm rate for 2-year security)2 = (1 + current 1-year spot rate) * (1 + expected 1-year T-bill rates over year 2)
	Current longterm rate for 2-year treasury security = (((1 + 0.0254) * (1+ 0.0380))(1/2)) - 1 = 3.17%

(1+ current longterm rate for 3-year security)3 = (1 + current 1-year spot rate) * (1 + expected 1-year T-bill rates over year 2) * (1 + expected 1-year T-bill rates over year 3)
	Current longterm rate for 2-year treasury security = (((1 + 0.0254) * (1+ 0.0380) * (1+ 0.0430))(1/3)) - 1 = 3.54%

(1+ current longterm rate for 4-year security)4 = (1 + current 1-year spot rate) * (1 + expected 1-year T-bill rates over year 2) * (1 + expected 1-year T-bill rates over year 3) * (1 + expected 1-year T-bill rates over year 4)
	Current longterm rate for 2-year treasury security = (((1 + 0.0254) * (1+ 0.0380) * (1+ 0.0430) * (1+ 0.0580))(1/4)) - 1 = 4.1%

Result:

Current longterm rate for 1-year security	2.54%
Current longterm rate for 2-year security	3.17%
Current longterm rate for 3-year security	3.54%
Current longterm rate for 4-year security	4.10%

Yield Curve: