Suppose 1-year Treasury bonds yield 4.00% while 2-year T-bonds yield 5.10%. Assuming the pure expectations theory is correct, and thus the maturity risk premium for T-bonds is zero, what is the yield on a 1-year T-bond expected to be one year from now? Round the intermediate calculations to 4 decimal places and final answer to 2 decimal places.

Question

Suppose 1-year Treasury bonds yield 4.00% while 2-year T-bonds yield 5.10%. Assuming the pure expectations theory is correct, and thus the maturity risk premium for T-bonds is zero, what is the yield on a 1-year T-bond expected to be one year from now? Round the intermediate calculations to 4 decimal places and final answer to 2 decimal places.

BlackTom AI · Accepted Answer

To assess the expected 1-year yield one year from now under the pure expectations theory (and with zero maturity risk premium), we use the relationship between the 2-year yield, the current 1-year yield, and the expected 1-year forward rate one year from now.

First, convert the given annual yields to growth factors:
- Current 1-year yield y1 = 4.00% → (1 + y1) = 1.0400
- Current 2-year yield y2 = 5.10% → (1 + y2) = 1.0510

Under the expectations theory (no MRP), the 2-year compounded return should equal the product of the current 1-year return and the expected 1-year return starting next year:
(1 + y2)^2 = (1 + y1) × (1 + f1,1), where f1,1 is the forward rate for the period from year 1 to year 2.

Compute the left-hand side:
(1 + y2)^2 = (1.0510)^2 = 1.104601.

Compute the right-hand side piece with the known (1 + y1):
(1 + y1) × (1 + f1,1) = 1.0400 × (1 + f1,1).

Set equal and solve for (1 + f1,1):
1.0400 × (1 + f1,1) = 1.104601 → (1 + f1,1) = 1.104601 / 1.0400.

Perform the division: 1.104601 / 1.0400 ≈ 1.062115 → f1,1 ≈ 0.062115, i.e., 6.2115%.

Round intermediate calculations to four decimals as instructed: forward rate f1,1 ≈ 6.2115%.

Now, the yield on a 1-year T-bond one year from now, under the pure expectations theory with zero MRP, is this forward rate, approximately 6.2115%.

Evaluate the answer choices:
- 6.21%: This matches the calculated forward rate when rounded to two decimals.
- 7.39%: This would require a much higher forward rate than implied by the current 1-year and 2-year yields, so it overstates the expectation.
- 7.27%: Similar issue as 7.39%; it’s not supported by the arithmetic from the given yields.
- 5.47%: This is lower than the current 1-year yield and far from the forward rate implied by a higher 2-year yield, so it’s inconsistent with the no-MRP assumption.
- 6.09%: While closer to the forward rate, it underestimates the precise calculation of 6.2115% and would require rounding or rounding error in a way not consistent with the four-decimal intermediate calculation.

Putting it together, the correct logic yields a forward 1-year rate of about 6.2115%, which supports selecting 6.21% when rounded to two decimals.

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