Final answer:
The rates with continuous compounding are 4.88%, 5.41%, 5.63%, and 5.8%. The forward rate for the 6-month period beginning in 18 months is 1.46%. The value of an FRA that promises to pay 6.5% on a principal of $1 million for the 6-month period starting in 18 months is $960,300.
Step-by-step explanation:
(a) To find the rates with continuous compounding, we can use the formula: r_continuous = ln(1 + r_semiannual)
Using this formula:
r_continuous (6-month) = ln(1 + 0.05) = 4.88%
r_continuous (12-month) = ln(1 + 0.055) = 5.41%
r_continuous (18-month) = ln(1 + 0.0575) = 5.63%
r_continuous (24-month) = ln(1 + 0.06) = 5.8%
(b) The forward rate for the 6-month period beginning in 18 months can be calculated using the formula: forward rate = [(1 + r2_continuous)^n2 / (1 + r1_continuous)^n1] - 1
Using this formula:
forward rate = [(1 + 0.058)^2 / (1 + 0.0575)^1] - 1 = 1.46%
(c) The value of an FRA that promises to pay 6.5% (compounded semiannually) on a principal of $1 million for the 6-month period starting in 18 months can be calculated using the formula: FRA value = [1 + (forward rate - FRA rate) * T] * principal
Using this formula:
FRA value = [1 + (0.0146 - 0.065) * (6/12)] * $1,000,000 = $960,300.