SOLUTION: three-year treasury securities currently yield 6%, while 4-year treasury securities currently yield 6.5%. assume that the expectations theory holds. what does the market believe th

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Question 1177706: three-year treasury securities currently yield 6%, while 4-year treasury securities currently yield 6.5%. assume that the expectations theory holds. what does the market believe the rate will be on 1-year treasury securities three years from now?
Answer by CPhill(1959) (Show Source):
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Let's solve this problem using the expectations theory.
**Understanding the Expectations Theory**
The expectations theory states that long-term interest rates are determined by the market's expectations of future short-term interest rates. In other words, the yield on a long-term bond is the average of the expected yields on short-term bonds over the same period.
**Given Information**
* 3-year Treasury yield: 6%
* 4-year Treasury yield: 6.5%
**Applying the Expectations Theory**
Let's denote:
* r_3 = the yield on the 3-year Treasury security (6% or 0.06)
* r_4 = the yield on the 4-year Treasury security (6.5% or 0.065)
* f_4 = the expected 1-year Treasury yield three years from now (what we want to find)
According to the expectations theory:
* (1 + r_4)^4 = (1 + r_3)^3 * (1 + f_4)
**Solving for f_4**
1. Plug in the given values:
* (1 + 0.065)^4 = (1 + 0.06)^3 * (1 + f_4)
2. Calculate the terms:
* (1.065)^4 = (1.06)^3 * (1 + f_4)
* 1.2864388126 = 1.191016 * (1 + f_4)
3. Isolate (1 + f_4):
* 1 + f_4 = 1.2864388126 / 1.191016
* 1 + f_4 ≈ 1.07928
4. Solve for f_4:
* f_4 ≈ 1.07928 - 1
* f_4 ≈ 0.07928
5. Convert to percentage:
* f_4 ≈ 7.928%
**Therefore, the market believes the rate will be approximately 7.93% on 1-year Treasury securities three years from now.**