8. Suppose that the current one-year rate (one-year spot rate) and expected one-year T-bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows:
1R1 = 5 percent, E(2r1) = 6 percent, E(3r1) = 7.5 percent E(4r1) = 6.85 percent
To calculate the expected one-year spot rates for years 2, 3, and 4, we need to apply an equation known as the Expectations Theory. According to the Expectations Theory, the expected one-year spot rate for year n (denoted as E(nr1)) is equal to the current one-year rate plus the average expected change in one-year spot rates over the next n-1 years.
Let's calculate the expected one-year spot rates for years 2, 3, and 4 using the given information:
Year 2:
The expected one-year spot rate for year 2 (E(2r1)) is given as 6 percent.
We know that the current one-year rate (1R1) is 5 percent.
To calculate the average expected change in one-year spot rates, we subtract the current rate from the expected one-year spot rate for year 2:
Average expected change = E(2r1) - 1R1 = 6% - 5% = 1%
Therefore, the expected one-year spot rate for year 2 would be:
E(2r1) = 1R1 + Average expected change = 5% + 1% = 6%
Year 3:
The expected one-year spot rate for year 3 (E(3r1)) is given as 7.5 percent.
To calculate the average expected change, we subtract the current rate from the expected one-year spot rate for year 3:
Average expected change = E(3r1) - 1R1 = 7.5% - 5% = 2.5%
Therefore, the expected one-year spot rate for year 3 would be:
E(3r1) = 1R1 + Average expected change = 5% + 2.5% = 7.5%
Year 4:
The expected one-year spot rate for year 4 (E(4r1)) is given as 6.85 percent.
To calculate the average expected change, we subtract the current rate from the expected one-year spot rate for year 4:
Average expected change = E(4r1) - 1R1 = 6.85% - 5% = 1.85%
Therefore, the expected one-year spot rate for year 4 would be:
E(4r1) = 1R1 + Average expected change = 5% + 1.85% = 6.85%
In summary, the expected one-year spot rates for years 2, 3, and 4 are as follows:
Year 2: E(2r1) = 6%
Year 3: E(3r1) = 7.5%
Year 4: E(4r1) = 6.85%