Answer: The 1-year forward rate two years from today is too low.

### 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.