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: ${ } _ { 1 } R _ { 1 } = 5.25 \% , \quad { } _ { 1 } R _ { 2 } = 5.75 \% , \quad { } _ { 1 } R _ { 3 } = 6.25 \% , \quad { } _ { 1 } R _ { 4 } = 6.45 \%$ 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

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:

{ } _ { 1 } R _ { 1 } = 5.25 \% , \quad { } _ { 1 } R _ { 2 } = 5.75 \% , \quad { } _ { 1 } R _ { 3 } = 6.25 \% , \quad { } _ { 1 } R _ { 4 } = 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?

Quizplus · Accepted Answer

According to the unbiased expectations theory, the current long-term interest rate is equal to the geometric average of expected future short-term interest rates.

Therefore, the expected one-year spot rate in year four is:

[(1+R1)(1+R2)(1+R3)(1+F1,4)]^(1/4) = (1+R4)

Solving for the one-year forward rate,F1,4:

F1,4 = [(1+R4)^(4) / (1+R1)(1+R2)(1+R3)] - 1

Substituting the given values:

F1,4 = [(1+0.0645)^(4) / (1+0.0525)(1+0.0575)(1+0.0625)] - 1

F1,4 = 0.0705 or 7.05%

Therefore, the answer is choice C.

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