the existing or current (spot) 1-, 2-, 3-, and 4-year zero coupon Treasury security rates were as follows:
1R1 = 0.30%, 1R2 = 0.90%, 1R3 = 1.30%, 1R4 = 1.45%
Using the unbiased expectations theory, calculate the 1-year forward rates on zero coupon Treasury bonds for years 2, 3, and 4
1R1 = 0.65%, 1R2 = 1.30%, 1R3 = 1.70%, 1R4 = 1.85%.
Using the unbiased expectations theory, calculate the 1-year forward rates on zero coupon Treasury bonds for years 2, 3, and 4
To calculate the forward rates on zero coupon Treasury bonds using the unbiased expectations theory, we need to use the given spot rates. The unbiased expectations theory states that the forward rate is equal to the expected future spot rate.
1. First, let's calculate the 1-year forward rate for year 2 using the spot rates. The formula for calculating the forward rate is:
Forward Rate = (1 + Spot Rate for Year 2)^2 / (1 + Spot Rate for Year 1) - 1
So, applying the formula, we have:
Forward Rate for Year 2 = (1 + 0.90%)^2 / (1 + 0.30%) - 1
= (1 + 0.009)^2 / (1 + 0.003) - 1
= 1.0181 / 1.003 - 1
= 1.014055 - 1
= 0.014055 or 1.4055%
Therefore, the 1-year forward rate on zero coupon Treasury bonds for year 2 is 1.4055%.
2. Similarly, let's calculate the 1-year forward rate for year 3 using the spot rates:
Forward Rate for Year 3 = (1 + Spot Rate for Year 3)^3 / (1 + Spot Rate for Year 2)^2 - 1
Substituting the spot rates, we get:
Forward Rate for Year 3 = (1 + 1.30%)^3 / (1 + 0.90%)^2 - 1
= (1 + 0.013)^3 / (1 + 0.009)^2 - 1
= 1.039217 - 1
= 0.039217 or 3.9217%
Therefore, the 1-year forward rate on zero coupon Treasury bonds for year 3 is 3.9217%.
3. Lastly, let's calculate the 1-year forward rate for year 4 using the spot rates:
Forward Rate for Year 4 = (1 + Spot Rate for Year 4)^4 / (1 + Spot Rate for Year 3)^3 - 1
Substituting the spot rates, we get:
Forward Rate for Year 4 = (1 + 1.45%)^4 / (1 + 1.30%)^3 - 1
= (1 + 0.0145)^4 / (1 + 0.013)^3 - 1
= 1.060904 - 1
= 0.060904 or 6.0904%
Therefore, the 1-year forward rate on zero coupon Treasury bonds for year 4 is 6.0904%.