The YTM for one year bond is 5%, 2-year bonds is 4.5% and 3-year bonds is 3.5% Expectation theory is believed to be true. What are the expected one-year rates over the next three years?
The YTM for one year bond is 5%, 2-year bonds is 4.5% and 3-year bonds is 3.5% Expectation theory is believed to be true. What are the expected one-year rates over the next three years?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![### Yield to Maturity (YTM) and Expectation Theory: An Analysis
**Problem Statement:**
The YTM for one-year bond is 5%, for two-year bonds is 4.5%, and for three-year bonds is 3.5%. Expectation theory is assumed to be true. What are the expected one-year rates over the next three years?
**Formulas and Calculations:**
1. **Definition and Formula to Find Expected One-Year Rate:**
\[
i_{2t} = \frac{i_{1t} + i_{1, t+1}}{2}
\]
2. **Solving for \(i_{1, t+1}\):**
\[
i_{1, t+1} = 2i_{2t} - i_{1t} = 4\%
\]
3. **Substituting Known Values:**
\[
i_{2t} = \frac{5\% + 4\%}{2} = 4.5\%
\]
4. **General Formula for \(i_{3t}\):**
\[
i_{3t} = \frac{i_{1t} + i_{1, t+1}^e + i_{1, t+2}^e}{3}
\]
5. **Solving for \(i_{1, t+2}^e\):**
\[
i_{1, t+2}^e = 3i_{3t} - i_{1t} - i_{1, t+1}
\]
6. **Substituting Known Values:**
\[
i_{1, t+2}^e = 3(3.5\%) - 5\% - 4\% = 1.5\%
\]
7. **Calculating \(i_{3t}\) with Expectation Theory:**
\[
i_{3t} = \frac{5\% + 4\% + 1.5\%}{3} = 3.5\%
\]
**Conclusion:**
With the Expectation Theory, the expected one-year rates are calculated clearly. The final expected one-year rate over the next three years is 3.5%.
**Additional Note:**
A question](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00828718-2df8-4d22-87de-a2e53e10f924%2F0015f74d-1ca8-47a5-b8e5-5074a7eea0f2%2Fzqbbl9a_processed.png&w=3840&q=75)
Transcribed Image Text:### Yield to Maturity (YTM) and Expectation Theory: An Analysis
**Problem Statement:**
The YTM for one-year bond is 5%, for two-year bonds is 4.5%, and for three-year bonds is 3.5%. Expectation theory is assumed to be true. What are the expected one-year rates over the next three years?
**Formulas and Calculations:**
1. **Definition and Formula to Find Expected One-Year Rate:**
\[
i_{2t} = \frac{i_{1t} + i_{1, t+1}}{2}
\]
2. **Solving for \(i_{1, t+1}\):**
\[
i_{1, t+1} = 2i_{2t} - i_{1t} = 4\%
\]
3. **Substituting Known Values:**
\[
i_{2t} = \frac{5\% + 4\%}{2} = 4.5\%
\]
4. **General Formula for \(i_{3t}\):**
\[
i_{3t} = \frac{i_{1t} + i_{1, t+1}^e + i_{1, t+2}^e}{3}
\]
5. **Solving for \(i_{1, t+2}^e\):**
\[
i_{1, t+2}^e = 3i_{3t} - i_{1t} - i_{1, t+1}
\]
6. **Substituting Known Values:**
\[
i_{1, t+2}^e = 3(3.5\%) - 5\% - 4\% = 1.5\%
\]
7. **Calculating \(i_{3t}\) with Expectation Theory:**
\[
i_{3t} = \frac{5\% + 4\% + 1.5\%}{3} = 3.5\%
\]
**Conclusion:**
With the Expectation Theory, the expected one-year rates are calculated clearly. The final expected one-year rate over the next three years is 3.5%.
**Additional Note:**
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