Question

The 3-year spot interest rate is 4.15%, the 3-year forward rate expected 3 years from now has been estimated to be 4.15%. What is the other spot rate you need to know to find the forward rate given above using the pure expectations theory? Round to the nearest 0.01%. E.g., if your answer is 5.783%, record it as 5.78.

The 8-year spot interest rate is 5.44%, the 2-year spot rate is 3.96%. What is the forward rate you can find using the pure expectations theory? Round to the nearest 0.01%. E.g., if your answer is 5.78%, enter it as 5.78.
   

Answer 1

Pure Expectation Theory asserts that forward rates exclusively represent the expected future rates.

Now, for first question,

3-year spot rate is given and 3-year forward rate expected 3 year from now is given. So the other spot rate that we should know is basically 6 year spot rate.

(1 + 6yr spot rate)⁶ = (1 + 3yr spot rate)³ * (1 + 3 yr fwd rate 3 yr from now)³

First, let us understand this equation. Based on pure expectations theory, this equation means, whether you invest in a 6 year bond (investment duration = 6) or whether you invest in a 3 yr bond, and after maturity, invest the proceeds in another 3 year bond (investment duration = 3 + 3 = 6), you will end up with the same amount.

(1 + 6yr spot rate)⁶ = (1 + 4.15%)³ * (1 + 4.15)³

(1 + 6yr spot rate)⁶ = 1.2763

6 yr spot rate = 4.15% (Flat Yield Curve!!)

Using the similar approach for second case,

you are giiven 8 yr spot rate and 2 yr spot rate. the other rate you can find is 6 yr forward rate, 2 years from now.

(1 + 8yr spot rate)⁸ = (1 + 2yr spot rate)² * (1 + 6 yr fwd rate 2 yr from now)⁶

(1 + 5.44%)⁸ = (1 + 3.96%)² * (1 + 6 yr fwd rate 2 yr from now)⁶

6 yr fwd rate 2 yr from now = 5.94%