Below are the yield rates for different maturities (rates on the yield curve)

Maturity

1-year

2-year

3-year

Rates

6%

5%

4%

Based on the expectation hypothesis, what is the expected future interest rate between year 2 and year 3?

(please answer in %. If the answer is 8.05%, then in the box, write 8.05)

Question

Below are the yield rates for different maturities (rates on the yield curve)

Maturity

1-year

2-year

3-year

Rates

6%

5%

4%

Based on the expectation hypothesis, what is the expected future interest rate between year 2 and year 3?

(please answer in %. If the answer is 8.05%, then in the box, write 8.05)

BlackTom AI · Accepted Answer

Under the expectation hypothesis with annual compounding, multi-year spot rates equal the compounded product of the sequence of one‑year forward (expected) rates. The relationship is:
(1 + s3)^3 = (1 + s1) × (1 + f1→2) × (1 + f2→3),
and similarly (1 + s2)^2 = (1 + s1) × (1 + f1→2). We can eliminate f1→2 and solve for f2→3.

Step-by-step reasoning and arithmetic:
1) Compute the 3-year spot accumulation factor: (1 + s3)^3 = (1.04)^3.
   (1.04)^2 = 1.0816; (1.04)^3 = 1.0816 × 1.04 = 1.124864.
2) Compute the 2-year spot accumulation factor: (1 + s2)^2 = (1.05)^2 = 1.1025.
3) Divide to isolate the year‑3 one‑year forward factor:
   1 + f2→3 = (1.04)^3 / (1.05)^2 = 1.124864 / 1.1025 = 1.020284...
4) Subtract 1 to get the forward rate:
   f2→3 = 1.020284... − 1 = 0.020284... = 2.0284%.
5) Rounding to two decimal places gives 2.03%.

Why this is correct:
- The derivation follows directly from the expectation hypothesis with annual compounding; the product-of-factors form is exact for discrete annual compounding and yields the forward rate by dividing multi-year accumulation factors.
- The 2.03% value is the implied one‑year interest rate expected to prevail between year 2 and year 3 that makes the 3‑year spot rate consistent with the 1‑ and 2‑year spots.

Why other simple approaches can be misleading:
- A naive arithmetic-average approach (s3 = (s1 + f1 + f2)/3) gives f2→3 = 2.00% — close but not exact because it ignores compounding (it’s an approximation).
- Using the wrong compounding convention or miscomputing the accumulation factors would produce incorrect values (e.g., 4%, 5%, or 6% are inconsistent with the given spots).

Conclusion: The implied expected one‑year rate between year 2 and year 3 is approximately 2.03%, which matches the provided answer.

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