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
Using the unbiased expectations theory, calculate the current (long-term) rates for one-, two-, three-, and four-year-maturity Treasury securiti
es.
To calculate the current (long-term) rates for one-, two-, three-, and four-year-maturity Treasury securities using the unbiased expectations theory, we need to use the expected one-year T-bill rates as our best estimate for future rates.
Given the following expected one-year T-bill rates:
E(2r1) = 6 percent
E(3r1) = 7.5 percent
E(4r1) = 6.85 percent
Using the unbiased expectations theory, the current long-term rates can be calculated as follows:
For the one-year maturity Treasury security, we can simply use the current one-year rate:
Long-term rate for one-year-maturity Treasury security = 1R1 = 5 percent
For the two-year maturity Treasury security, we can use the expected one-year T-bill rate for year 2:
Long-term rate for two-year-maturity Treasury security = E(2r1) = 6 percent
For the three-year maturity Treasury security, we can use the average of the expected one-year T-bill rates for years 2 and 3:
Long-term rate for three-year-maturity Treasury security = (E(2r1) + E(3r1)) / 2 = (6 percent + 7.5 percent) / 2 = 6.75 percent
For the four-year maturity Treasury security, we can use the average of the expected one-year T-bill rates for years 3 and 4:
Long-term rate for four-year-maturity Treasury security = (E(3r1) + E(4r1)) / 2 = (7.5 percent + 6.85 percent) / 2 = 7.175 percent
Therefore, the current long-term rates for one-, two-, three-, and four-year-maturity Treasury securities are:
- One-year-maturity: 5 percent
- Two-year-maturity: 6 percent
- Three-year-maturity: 6.75 percent
- Four-year-maturity: 7.175 percent
To calculate the current (long-term) rates for different maturity Treasury securities using the unbiased expectations theory, we need to apply the formula:
Current rate = Expected future rate / (1 + Current spot rate)
Let's calculate the current rates for one-, two-, three-, and four-year-maturity Treasury securities one by one:
1. Current one-year rate:
Given: 1R1 = 5 percent
The current rate for a one-year-maturity Treasury security is simply the given one-year spot rate.
So, the current one-year rate is 5 percent.
2. Current two-year rate:
Given: E(2r1) = 6 percent
To calculate the current two-year rate, we need the current one-year rate. Using the formula mentioned earlier:
Current two-year rate = E(2r1) / (1 + 1R1)
Substituting the given values:
Current two-year rate = 6 percent / (1 + 5 percent)
Current two-year rate = 6 percent / 1.05
Current two-year rate = 5.7143 percent
Therefore, the current two-year rate is approximately 5.7143 percent.
3. Current three-year rate:
Given: E(3r1) = 7.5 percent
To calculate the current three-year rate, we need the current one-year rate. Using the formula:
Current three-year rate = E(3r1) / (1 + 1R1)
Substituting the given values:
Current three-year rate = 7.5 percent / (1 + 5 percent)
Current three-year rate = 7.5 percent / 1.05
Current three-year rate = 7.1429 percent
Therefore, the current three-year rate is approximately 7.1429 percent.
4. Current four-year rate:
Given: E(4r1) = 6.85 percent
To calculate the current four-year rate, we need the current one-year rate. Using the formula:
Current four-year rate = E(4r1) / (1 + 1R1)
Substituting the given values:
Current four-year rate = 6.85 percent / (1 + 5 percent)
Current four-year rate = 6.85 percent / 1.05
Current four-year rate = 6.5238 percent
Therefore, the current four-year rate is approximately 6.5238 percent.
Therefore, using the unbiased expectations theory, the current rates for one-, two-, three-, and four-year-maturity Treasury securities are as follows:
- One-year rate: 5 percent
- Two-year rate: 5.7143 percent
- Three-year rate: 7.1429 percent
- Four-year rate: 6.5238 percent
1R1 = 6%
1R2 = [(1 + .06)*(1 + .07)] 1/2- 1 = 6.5%
1R3 = [(1 + .06)*(1 + .07)*(1 + .075)] 1/3- 1 = 6.8%
1R4 = [(1 + .06)*(1 + .07)*(1 + .075)*(1 + .0785)] 1/4- 1 = 7.1%