Concept Overview

This question tests your understanding of the relationship between spot rates and implied forward rates. A fundamental concept in fixed income is the "no-arbitrage" condition. This principle states that investing in a series of shorter-term bonds (using spot and forward rates) should yield the exact same return as investing in a single long-term bond, provided the time horizons are identical.

For any investment period, the return from rolling over shorter-term investments must equal the return of a long-term zero-coupon bond. In this specific problem, we are looking at a 6-year investment horizon. We can reach this 6-year total return through two different paths that must be equal:

1. Path 1: Invest for 2 years at the 2-year spot rate, then reinvest for the next 4 years at the given 4-year forward rate.
2. Path 2: Invest for 4 years at the 4-year spot rate, then reinvest for the next 2 years at the unknown 2-year forward rate.

Calculation Steps

The general formula equating the two paths for a total of $T$ years is: $(1 + S_A)^A \times (1 + IFR_{A, B-A})^{B-A} = (1 + S_B)^B$

For this question, we set up the equality where the total time is 6 years ($2+4 = 4+2$):

$(1 + S_2)^2 \times (1 + {}_2f_4)^4 = (1 + S_4)^4 \times (1 + {}_4f_2)^2$

Where:

• $S_2 = 1.0\%$ (2-year spot rate)
• ${}_2f_4 = 3.0\%$ (4-year forward rate, 2 years from now)
• $S_4 = 2.5\%$ (4-year spot rate)
• ${}_4f_2 =$ ? (The 2-year forward rate, 4 years from now—this is what we need to find)

Step 1: Calculate the left-hand side (The 2-year + 4-year path) $(1.01)^2 \times (1.03)^4$ $= 1.0201 \times 1.1255$ $\approx 1.1481$

Step 2: Set up the equation with the right-hand side $`$1.14`81 = (1.025)^4 \times (1 + {}_4f_2)^2 $$

Calculate the known part of the right-hand side: $(1.025)^4 \approx 1.1038$

Step 3: Solve for the unknown rate $`$1.14`81 = 1.1038 \times (1 + {}_4f_2)^2 $$ (1 + {}_4f_2)^2 = \frac{1.1481}{1.1038} $$ (1 + {}_4f_2)^2 \approx 1.0401 $$

To find the annual rate, take the square root (because this is a 2-year rate): $`$1` + {}_4f_2 = \sqrt{1.0401} \approx 1.0199 $$ {}_4f_2 \approx 0.0199 \text{ or } 1.99% $$

Rounding to the nearest whole number gives us 2%.

Explanation of Options

• Option A (2%): Correct. This is the correct result derived from equating the total return of the two investment paths spanning 6 years. By calculating the geometric linking of the known rates and solving for the missing forward rate, we arrive at approximately 1.99%.

• Option B (3%): Incorrect. This option is incorrect because it simply copies the value of the forward rate given in the problem prompt (the 4-year forward rate, two years from today). It fails to recognize that forward rates change over different time periods depending on the slope of the yield curve.

• Option C (4%): Incorrect. This option likely results from a common calculation error. After computing $\frac{1.1481}{1.1038} \approx 1.0401$, a student might mistakenly treat the decimal .0401 as the percentage rate (4%) and forget to take the square root to convert the 2-year total return into an annualized rate.

Reference Answer

The correct answer is A.