The Wall Street Journal reports that the rate on four-year Treasury securities is 5.60 percent and the rate on five-year Treasury securities is 6.15 percent.

According to the unbiased expectations theory, what does the market expect the one-year Treasury rate to be four years from today, E(5r1)?

The unbiased expectations theory suggests that the long-term interest rates are an average of current and future expected short-term interest rates. According to this theory, the interest rate on a five-year Treasury security should be equal to the average of the current one-year rate and the expected future one-year rates over the next four years.

Given:
- The rate on four-year Treasury securities (\(r_4\)) is 5.60%.
- The rate on five-year Treasury securities (\(r_5\)) is 6.15%.

We want to find the expected one-year Treasury rate four years from today, denoted as \(E(5r1)\).

According to the unbiased expectations theory, the relationship can be expressed as:
\[
(1 + r_5)^5 = (1 + r_4)^4 \times (1 + E(5r1))
\]

Where:
- \(r_4\) is the rate on the four-year security.
- \(r_5\) is the rate on the five-year security.
- \(E(5r1)\) is the expected one-year rate four years from today.

Substituting the given rates into the equation:

\[
(1 + 0.0615)^5 = (1 + 0.0560)^4 \times (1 + E(5r1))
\]

Let's solve this equation to find \(E(5r1)\).The market expects the one-year Treasury rate four years from today (\(E(5r1)\)) to be approximately **8.38%** according to the unbiased expectations theory.

Would you like to explore any details further or have any questions? Here are some related questions that could expand on this topic:

1. How does the unbiased expectations theory differ from the liquidity premium theory?
2. What are the implications of an increasing term structure of interest rates for future economic conditions?
3. Can the unbiased expectations theory be applied to other financial instruments besides Treasury securities?
4. What are the risks associated with using the unbiased expectations theory for investment decisions?
5. How would a change in inflation expectations affect the expected future interest rates?

**Tip:** The unbiased expectations theory assumes that there are no arbitrage opportunities and that the market is efficient. It's important to consider other factors, like liquidity premiums, when analyzing interest rates.

The Wall Street Journal reports that the rate on four-year Treasury securities is 5.60 percent and the rate on five-year Treasury securities is 6.15 percent. According to the unbiased expectations theory, what does the market expect the one-year Treasury rate to be four years from today, E(5r1)?

Understand how to calculate the expected one-year Treasury rate four years from today using the unbiased expectations theory. This example uses current interest rates on four-year and five-year Treasury securities, and applies mathematical concepts like exponential equations and the unbiased expectations theory to derive the solution.

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