Explanation

To calculate the six-month forward rate one year from now (i.e., the forward rate from year 1 to year 1.5), we use the relationship between spot rates and forward rates:

Step 1: Understand the timeline

• We want the forward rate from t=1 year to t=1.5 years (a 6-month period)
• This is denoted as f(1,1.5) or f(2,3) in period notation

Step 2: Use the forward rate formula The relationship between spot rates and forward rates is:

$(1 + z_3)^{3} = (1 + z_2)^{2} \times (1 + f_{2,3})$

Where:

• z₂ = spot rate for period 2 (1.0 year) = 2.10% = 0.0210
• z₃ = spot rate for period 3 (1.5 years) = 2.80% = 0.0280
• f₂,₃ = forward rate from period 2 to period 3 (what we're solving for)

Step 3: Convert to semiannual rates Since these are BEY (bond equivalent yields), they are already quoted on a semiannual bond basis, so we use them directly:

$(1 + \frac{z_3}{2})^{3} = (1 + \frac{z_2}{2})^{2} \times (1 + \frac{f_{2,3}}{2})$

Step 4: Plug in the values $(1 + \frac{0.0280}{2})^{3} = (1 + \frac{0.0210}{2})^{2} \times (1 + \frac{f_{2,3}}{2})$ $(1 + 0.0140)^{3} = (1 + 0.0105)^{2} \times (1 + \frac{f_{2,3}}{2})$ $(1.0140)^{3} = (1.0105)^{2} \times (1 + \frac{f_{2,3}}{2})$

Step 5: Calculate $1.042669 = 1.021103 \times (1 + \frac{f_{2,3}}{2})$ $1 + \frac{f_{2,3}}{2} = \frac{1.042669}{1.021103} = 1.021123$ $\frac{f_{2,3}}{2} = 0.021123$ $f_{2,3} = 0.042246 \text{ or } 4.2246\%$

Step 6: Compare to options 4.2246% is closest to 4.21% (Option C).

Alternative calculation method: $f_{2,3} = 2 \times \left[ \frac{(1 + \frac{z_3}{2})^{3}}{(1 + \frac{z_2}{2})^{2}} - 1 \right]$ $f_{2,3} = 2 \times \left[ \frac{(1.0140)^{3}}{(1.0105)^{2}} - 1 \right]$ $f_{2,3} = 2 \times [1.021123 - 1] = 2 \times 0.021123 = 0.042246$

Why not the other options:

• Option A (2.10%): This is simply the 1-year spot rate, not the forward rate
• Option B (3.64%): This might result from incorrect calculation or using annual compounding instead of semiannual

Key Concept: Forward rates represent the expected future interest rate between two future periods, implied by current spot rates. They are calculated using the no-arbitrage principle in the yield curve.