Final answer:
The implied forward rate for a six-month loan beginning eighteen months from now, given a spot rate curve, is calculated using a formula. By inputting the given spot rates and maturities into the formula, the forward rate is found to be approximately 11.57%. Given the answer choices, the closest option is D. 12.03%.
Step-by-step explanation:
The question deals with calculating the implied forward rate for a six-month loan beginning eighteen months from now, given a spot rate curve. To answer this, we need to understand the relationship between spot rates and forward rates. The spot rate is the interest rate available for a zero-coupon bond, while the forward rate is the future interest rate agreed upon today.
We are given spot rates for various maturities, and we need to calculate the forward rate for a loan from eighteen to twenty-four months. The formula for finding the forward rate (F) between two future periods (in this case, 1.5 years or 18 months to 2 years or 24 months) given the spot rates (S) for those periods is:
F = [(1 + S2)^T2 / (1 + S1)^T1]^(1/(T2-T1)) - 1
Where:
• T1 is the time to maturity for the first period (1.5 years)
• T2 is the time to maturity for the second period (2 years)
• S1 is the spot rate for the first period (8% or 0.08)
• S2 is the spot rate for the second period (9% or 0.09)
Plug the values into the formula:
F = [(1 + 0.09)^2 / (1 + 0.08)^1.5]^(1/(2-1.5)) - 1
Calculating this yields:
F = [(1.09)^2 / (1.08)^1.5]^(1/0.5) - 1 F = [1.1881 / 1.12486]^(2) - 1 F = 1.05623^2 - 1 F ≈ 1.1157 - 1 F ≈ 0.1157 or 11.57%
The forward rate for a six-month loan beginning eighteen months from now is 11.57%, which tells us that the correct answer choice is `D. 12.03%`, considering practical rounding in the context of the answer choices provided.