The Wall Street Journal reports that the rate on four-year Treasury securities is 2.1 percent and the rate on five-year Treasury securities is 2.8 percent. According to the unbiased expectations theory, what does the market expect the one-year Treasury rate to be four years from today,  E(r15)
  ?

To find the expected one-year Treasury rate four years from today, denoted as \( E(r_{1,5}) \), we can use the **unbiased expectations theory**. According to this theory, long-term interest rates are determined by market expectations of future short-term interest rates. In this case, the five-year rate is a function of the current four-year rate and the expected one-year rate in year 5.

We can express this relationship using the following formula:

\[
(1 + r_{5})^5 = (1 + r_{4})^4 \times (1 + E(r_{1,5}))
\]

Where:
- \( r_4 = 2.1\% \) is the rate on four-year Treasury securities,
- \( r_5 = 2.8\% \) is the rate on five-year Treasury securities,
- \( E(r_{1,5}) \) is the expected one-year Treasury rate four years from now.

### Step-by-step solution:

1. **Convert the rates to decimal form**:  
   \( r_4 = 0.021 \), \( r_5 = 0.028 \)

2. **Plug in the values into the equation**:

\[
(1 + 0.028)^5 = (1 + 0.021)^4 \times (1 + E(r_{1,5}))
\]

3. **Calculate the left-hand side and right-hand side**:

   Left-hand side:  
   \[
   (1 + 0.028)^5 = (1.028)^5
   \]

   Right-hand side:  
   \[
   (1 + 0.021)^4 = (1.021)^4
   \]

Now, let's calculate those values and solve for \( E(r_{1,5}) \).The calculations are as follows:

- The left-hand side of the equation \( (1 + 0.028)^5 \) equals approximately 1.1481.
- The right-hand side of the equation \( (1 + 0.021)^4 \) equals approximately 1.0867.

Now, solving for the expected one-year Treasury rate four years from today:

\[
E(r_{1,5}) = \frac{1.1481}{1.0867} - 1 \approx 0.0565 \text{ or } 5.65\%
\]

Thus, the market expects the one-year Treasury rate four years from today to be approximately **5.65%**.

Would you like further details on this calculation, or do you have any questions?

Here are 5 related questions:
1. How does the unbiased expectations theory apply to bond pricing?
2. What factors could influence changes in short-term Treasury rates?
3. How does the term premium differ from expectations in interest rate theory?
4. Can you calculate the expected two-year rate starting in three years?
5. How might an unexpected economic event affect future Treasury rates?

**Tip**: The unbiased expectations theory assumes no risk premiums, but in reality, investors often demand a premium for longer-term securities.

The Wall Street Journal reports that the rate on four-year Treasury securities is 2.1 percent and the rate on five-year Treasury securities is 2.8 percent. According to the unbiased expectations theory, what does the market expect the one-year Treasury rate to be four years from today, E(r15)?

Learn how to calculate the expected one-year Treasury rate four years from today using the unbiased expectations theory. Given the current rates on four-year (2.1%) and five-year (2.8%) Treasury securities, this problem demonstrates how to use the relationship between short-term and long-term rates. Suitable for undergraduate finance or economics students.

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