Find the periodic payment that will amount to the given sum under the given conditions: S=$150,000; interest is 6% compounded semiannually; payments are made at the end of each semiannual period for 11 years.
To find the periodic payment that will amount to the given sum under the given conditions, we can use the formula for the future value of an ordinary annuity.
The formula for the future value (FV) of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value (the desired sum, S)
P = Periodic Payment (unknown)
r = Interest rate per period (compounded semiannually, so divide the annual interest rate by 2)
n = Number of periods (11 years, so multiply by 2 to account for semiannual periods)
Let's plug in the values into the formula:
S = $150,000
r = 6% / 2 = 0.06 / 2 = 0.03 (interest rate per semiannual period)
n = 11 * 2 = 22 (number of semiannual periods)
150,000 = P * [(1 + 0.03)^22 - 1] / 0.03
To isolate P, let's multiply both sides of the equation by 0.03:
150,000 * 0.03 = P * [(1 + 0.03)^22 - 1]
4,500 = P * [(1 + 0.03)^22 - 1]
Now, let's calculate [(1 + 0.03)^22 - 1]:
(1 + 0.03)^22 - 1 = (1.03)^22 - 1 ≈ 1.813615 - 1 ≈ 0.813615
Let's substitute this value back into the equation:
4,500 = P * 0.813615
To solve for P, divide both sides of the equation by 0.813615:
P = 4,500 / 0.813615 ≈ $5,523.65
So, the periodic payment that will amount to $150,000 under the given conditions is approximately $5,523.65.