Question

Suppose 1-year Treasury bonds yield 4.00% while 2-year T-bonds yield 4.10%. Assuming the pure expectations theory is correct, and thus the maturity risk premium for T-bonds is zero, what is the yield on a 1-year T-bond expected to be one year from now? Round the intermediate calculations to 4 decimal places and final answer to 2 decimal places.

a. 4.41

b. 3.82

c. 3.57

d. 4.20

e. 4.49

Answer 1

Under the pure expectations theory, the forward rate is representative of expected future spot rates

That is, investing now at the 1-year spot rate, and reinvesting the proceeds after 1 year at the expected 1-year spot rate 1 year from now, should yield the same return as investing now for 2 years at the 2-year spot rate

Let us say that the expected 1-year spot rate 1 year from now is R. Then :

• future value of investment if we invest now at the 1-year spot rate, and reinvest the proceeds after 1 year at the expected 1-year spot rate 1 year from now is  (1 + 4.00%) * (1 + R)
• future value of investment if we invest now for 2 years at the 2-year spot rate = (1 + 4.10%)²

(1 + 4.00%) * (1 + R) = (1 + 4.10%)²

We need to solve for R

(1 + R) =    (1 + 4.10%)² / (1 + 4.00%)

R = ((1 + 4.10%)² / (1 + 4.00%)) - 1

R = 0.0420, or 4.20%