Based on the expectation theory, the interest rate for (N-M) years but starting from year M+1 should satisfy the following equation:
(1+R)^N = (1+r)^m *(1+x)^(N-M)
Based on the simple version of the expection theory (arithematic version), the interest rate for (N-M) years staring from year M+1 should satisfy the following equation:
N*R = M*r +(N-M)*(x)
What is the Expectations Theory
The expectations theory attempts to predict what short-term interest rates will be in the future based on current long-term interest rates. The theory suggests that an investor earns the same amount of interest by investing in two consecutive one-year bond investments versus investing in one two-year bond today. The expectations theory is also known as the Unbiased Expectations Theory.
Example of Calculating Expectations Theory
Let's say that the present bond market provides investors with a three-year bond that pays an interest rate of 20 percent while a one-year bond pays an interest rate of 18 percent. The expectations theory can be used to forecast the interest rate of a future two-year bond.
(1+20%)*(1+20%)*(1+20%) = (1+18%)*(1+x)*(1+x), so X = 21.01%
Or 20%+20%+20% = 18% + x+x, so x=21% (the arithematic version)
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