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.