The term-structure of spot rates is currently as follows:

Term-structure of Spot Rates

Term (years)

Spot Rate with Annual Compounding

1

4%

2

5%

3

6%

According to the pure expectations hypothesis, what yield-to-maturity (with annual compounding) does the market expect to observe as of today on a 2-year zero issued one year from today?

Question

The term-structure of spot rates is currently as follows:

Term-structure of Spot Rates

Term (years)

Spot Rate with Annual Compounding

1

4%

2

5%

3

6%

According to the pure expectations hypothesis, what yield-to-maturity (with annual compounding) does the market expect to observe as of today on a 2-year zero issued one year from today?

BlackTom AI · Accepted Answer

Under the pure expectations hypothesis, the market’s expected yield on a 2-year zero issued one year from today equals the 2-year forward rate beginning at t = 1. This forward rate f satisfies the no-arbitrage relation that rolling over a one-year investment then investing at the forward 2-year rate for two years yields the same accumulation as investing for three years today.

Step-by-step derivation:
1. Write spot accumulations: (1 + s3)^3 is the accumulation over 3 years at the current 3-year spot s3 = 6% → (1.06)^3.
2. Rolling strategy accumulation: invest for 1 year at s1 = 4% then for 2 years at the forward rate f: (1 + s1)*(1 + f)^2 = (1.04)*(1 + f)^2.
3. Equate accumulations: (1.06)^3 = (1.04)*(1 + f)^2.
4. Solve for (1 + f)^2: (1 + f)^2 = (1.06)^3 / 1.04.

Numeric intermediate calculations:
- (1.06)^2 = 1.1236
- (1.06)^3 = 1.1236 * 1.06 = 1.191016
- Divide by 1.04: 1.191016 / 1.04 = 1.14520153846
- Take square root: 1 + f = sqrt(1.14520153846) ≈ 1.070137
- Subtract 1: f ≈ 0.070137 = 7.0137%, which rounds to 7.014%.

Why this is correct: the pure expectations hypothesis equates expected future spot rates with forward rates implied by current spot curve; the algebra above is the standard no-arbitrage link between spot and forward rates. The computed forward 2-year rate beginning in one year is therefore the expected yield-to-maturity on the 2-year zero to be issued at that time.

Why other choices are wrong (brief):
- 7.244%, 8.012%, 7.350%, 7.530% arise from incorrect algebraic manipulations (wrong exponents or dividing by the wrong spot rate) and do not satisfy (1.06)^3 = (1.04)*(1+f)^2.

Final confirmation: the expected yield-to-maturity is 7.014%, matching the provided answer.

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