One, two, and three year maturity zero-coupon bonds have yield to maturity equal to 2.23%, 3.12%, and 3.95%, respectively. According to the expectations theory, the implied two-year forward rate, one year from today is

Question

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We’re given the yields on zero-coupon bonds: 1-year y1 = 2.23%, 2-year y2 = 3.12%, and 3-year y3 = 3.95%. We need the implied 1-year forward rate that starts one year from today, denoted f1,2, under the expectations theory.

Step 1: Convert yields to gross terms.
- 1-year gross factor: 1 + y1 = 1.0223
- 2-year gross factor: 1 + y2 = 1.0312
- 3-year gross factor: 1 + y3 = 1.0395

Step 2: Use the forward-rate relationship for the 1-year forward starting at t = 1.
Under the expectations theory, the 2-year total return equals the 1-year return followed by the 1-year forward return, so:
(1 + y2)^2 = (1 + y1) * (1 + f1,2)
Thus, 1 + f1,2 = (1 + y2)^2 / (1 + y1).

Step 3: Compute (1 + y2)^2.
(1.0312)^2 = 1 + 2*(0.0312) + (0.0312)^2 = 1 + 0.0624 + 0.00097344 = 1.06337344.

Step 4: Divide by (1 + y1).
(1.06337344) / (1.0223) ≈ 1.0402 (approximately).

Step 5: Subtract 1 to get f1,2.
f1,2 ≈ 1.0402 − 1 = 0.0402, or about 4.02% (approximately).

Now, evaluate each answer option against this result:
- Option A: 4.68% — This is higher than the computed forward rate. If the forward rate were 4.68%, then (1 + y2)^2 would have to equal (1 + y1) * (1 + 0.0468) = 1.0223 * 1.0468 ≈ 1.0717, which is inconsistent with (1.0312)^2 = 1.06337344. Therefore 4.68% is not correct.
- Option B: 5.30% — This is even larger. Using 5.30% would yield (1 + y1)*(1 + 0.0530) ≈ 1.0223 * 1.0530 ≈ 1.0767, also not matching (1.0312)^2 = 1.06337344. So this is not correct.
- Option C: 4.82% — This is closer to the 4.02% calculation but still does not satisfy the exact relation: (1.0223) * (1.0482) ≈ 1.073, which is notably higher than 1.06337344. Therefore 4.82% is not consistent with the forward-rate formula given the 1-year and 2-year yields.
- Option D: 5.16% — This is even more inconsistent for the same reason; (1.0223) * (1.0516) ≈ 1.0765, again not equal to 1.06337344.

Step 6: Synthesis of the reasoning.
The core check is the forward-rate formula: forward rate f1,2 = (1 + y2)^2 / (1 + y1) − 1. Plugging in the numbers gives f1,2 ≈ 4.02%. All listed options deviate from this computed value, either by being too high (4.68%, 4.82%, 5.16%, 5.30%) or by not satisfying the fundamental relation. The mismatch indicates that none of the provided choices exactly matches the forward rate implied by the given zero-coupon yields under the pure expectations theory.

If a student must pick among the given options, they should note that the mathematically correct forward rate is approximately 4.02%, and recognize that all presented options are not exact matches to that result. The close-but-not-exact options (like 4.68% or 4.82%) illustrate how small arithmetic differences in the setup or rounding can lead to different selections, but without exact alignment to the forward-rate formula, they are not exact answers.

COMM_V 371 101-107 2025W1 Lecture 7 Practice Quiz

One, two, and three year maturity zero-coupon bonds have yield to maturity equal to 2.23%, 3.12%, and 3.95%, respectively. According to the expectations theory, the implied two-year forward rate, one year from today is

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