The 3-year spot interest rate is 4.15%, the 3-year forward rate expected 3 years

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.
   

Explanation / Answer

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%

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