On May 23, 20XX, the existing or current (spot) one-year, two-year, three-year, and four-year zero-coupon Treasury security rates were as follows:
₁R₁ = 5.25%,
₁R₂ = 5.75%,
₁R₃ = 6.25%,
₁R₄ = 6.45%
Using the unbiased expectations theory, what is the one-year forward rate on zero-coupon Treasury bonds for year 4 as of May 23, 20XX?

Question

Quizplus · Accepted Answer

The unbiased expectations theory suggests that the forward rates can be derived from the spot rates in such a way that they are an unbiased predictor of future short-term interest rates. To find the one-year forward rate for year 4, we can use the formula that equates the return from investing in a four-year bond to the return from investing in a three-year bond and then reinvesting the proceeds for one year at the forward rate we are trying to find. Mathematically, this can be represented as: $(1 + s_4)^4 = (1 + s_3)^3 \times (1 + f_{3,4})$, where $s_4$ is the spot rate for the four-year bond, $s_3$ is the spot rate for the three-year bond, and $f_{3,4}$ is the one-year forward rate from year 3 to year 4. Plugging in the given spot rates: $(1 + 0.0645)^4 = (1 + 0.0625)^3 \times (1 + f_{3,4})$. Solving for $f_{3,4}$ gives us approximately 7.05%, indicating that the expected one-year interest rate for year 4 is 7.05%.

On May 23, 20XX, the Existing or Current (Spot) One-Year