3. Unbiased Expectations Theory One-year Treasury bills currently earn 5.50 percent. You expect that one year from now, one-year Treasury bill rates will increase to 5.75 percent. If the unbiased expectations theory is correct, what should the current rate be on two-year Treasury securities?
5.62
To determine the current rate on two-year Treasury securities using the unbiased expectations theory, we need to consider the expectations of future rates for one-year Treasury bills.
According to the unbiased expectations theory, the current rate on a two-year Treasury security can be derived from the expectation of future one-year Treasury bill rates. The theory assumes that investors are indifferent between holding one-year Treasury bills or rolling them over for two years.
In this case, we expect the one-year Treasury bill rate to increase from 5.50 percent to 5.75 percent one year from now. To find the current rate on two-year Treasury securities, we need to calculate the average rate that investors would earn over the two-year period.
To calculate the average rate, we can take the geometric mean of the expected future rates. The formula for the geometric mean is as follows:
Geometric Mean = (1 + r1) * (1 + r2)^(1/n) - 1
Where:
- r1 is the expected rate for the first period (5.50%)
- r2 is the expected rate for the second period (5.75%)
- n is the number of periods (2 years)
Using the formula, we can calculate the geometric mean:
Geometric Mean = (1 + 0.055) * (1 + 0.0575)^(1/2) - 1
Calculating this out, we get:
Geometric Mean = 1.0283 - 1
Geometric Mean = 0.0283
Therefore, the current rate on two-year Treasury securities, according to the unbiased expectations theory, should be approximately 2.83 percent.
The unbiased expectations theory states that the yield on a longer-term bond should be equal to the average of the yields on shorter-term bonds expected to be issued during the holding period.
In this case, the current rate on one-year Treasury bills is 5.50%, and you expect the rate to increase to 5.75% one year from now.
To calculate the current rate on a two-year Treasury security, we need to calculate the average of the expected rates for the two years.
The expected rate for the first year is 5.50%, and the expected rate for the second year is 5.75%.
The average between these two rates is (5.50% + 5.75%) / 2 = 5.625%.
Therefore, according to the unbiased expectations theory, the current rate on a two-year Treasury security should be 5.625%.