Step 1: Understand the given data

We are given zero-coupon bonds (coupon = 0%) with the following prices and YTMs (which are the spot rates):

• 1-year zero: Price = 95.694 → 1-year spot rate (r₁) ≈ 4.50%
• 2-year zero: Price = 87.344 → 2-year spot rate (r₂) ≈ 7.00%
• 3-year zero: Price = 77.218 → 3-year spot rate (r₃) ≈ 9.00%

We are also given three implied forward rates:

• 1-year forward rate 1 year from today (f₁,₂) = 9.56%
• 1-year forward rate 2 years from today (f₂,₃) = 10.77%
• 2-year forward rate 1 year from today (f₁,₃) = 11.32%

The question is: Which statement is true based on consistency with the bond prices?

Step 2: Recall the no-arbitrage relationship for forward rates

Forward rates must be derived directly from the spot rates (or zero-coupon bond prices) to prevent arbitrage.

The key formulas are:

For 1-year forward rates:

• (1 + r₂)² = (1 + r₁) × (1 + f₁,₂)
• (1 + r₃)³ = (1 + r₂)² × (1 + f₂,₃)

For the 2-year forward rate starting in 1 year:

• (1 + r₃)³ = (1 + r₁) × (1 + f₁,₃)²

If a given forward rate does not satisfy the equation above, it is inconsistent with the bond prices → arbitrage opportunity exists.

Step 3: Calculate the correct (implied) forward rates from the bond prices

a) 1-year forward rate one year from today (f₁,₂)

(1 + 0.07)² = (1 + 0.045) × (1 + f₁,₂)
1.1449 = 1.045 × (1 + f₁,₂)
1 + f₁,₂ = 1.1449 / 1.045 ≈ 1.0956
f₁,₂ ≈ 9.56%

→ The given forward rate of 9.56% matches exactly. It is correct.

b) 1-year forward rate two years from today (f₂,₃)

(1 + 0.09)³ = (1 + 0.07)² × (1 + f₂,₃)
1.295029 = 1.1449 × (1 + f₂,₃)
1 + f₂,₃ = 1.295029 / 1.1449 ≈ 1.1311
f₂,₃ ≈ 13.11%

→ The given rate is only 10.77%, which is too low.

c) 2-year forward rate one year from today (f₁,₃)

(1 + 0.09)³ = (1 + 0.045) × (1 + f₁,₃)²
1.295029 = 1.045 × (1 + f₁,₃)²
(1 + f₁,₃)² = 1.295029 / 1.045 ≈ 1.2393
1 + f₁,₃ ≈ √1.2393 ≈ 1.1133
f₁,₃ ≈ 11.33%

→ The given rate of 11.32% is essentially correct (minor rounding difference).

Step 4: Evaluate the answer choices

• A: The 1-year forward rate one year from today is too low.
  → False. It matches exactly at 9.56%.

• B: The 2-year forward rate one year from today is too high.
  → False. It is correct (≈11.33%).

• C: The 1-year forward rate two years from today is too low.
  → True. The correct rate should be ~13.11%, but it is given as only 10.77%.

• D: The forward rates and bond prices provide no opportunities for arbitrage.
  → False. Because the 1y-forward-2y rate is mispriced, arbitrage exists (you can create a synthetic position using the zeros that profits from this inconsistency).

Correct Answer: C

The given 1-year forward rate two years from today (10.77%) is too low compared to what the bond prices imply (~13.11%). This inconsistency creates an arbitrage opportunity.