What is the size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 6.6%, compounded semiannually, so that the account will have a future value of $140,000 at the end of 17 years?
To find the size of the payments that must be deposited at the beginning of each 6-month period, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value
P = Payment
r = Interest rate per period
n = Number of periods
In this case, since the payments are made semiannually, we'll use the semiannual interest rate and the number of semiannual periods.
Given:
FV = $140,000
r = 6.6% (or 0.066 as a decimal)
n = 17 years * 2 (semiannual periods per year) = 34 periods
Substituting the values into the formula:
$140,000 = P * [(1 + 0.066)^34 - 1] / 0.066
Now, we need to solve for P. Rearranging the equation, we have:
P = $140,000 * 0.066 / [(1 + 0.066)^34 - 1]
Using a calculator or spreadsheet software, evaluate the right-hand side of the equation to find the value of P.
P ≈ $1,962.64
Therefore, the size of the payments that must be deposited at the beginning of each 6-month period is approximately $1,962.64.