Question 1177706
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.**