Supposed that your paychecks are issued on the 1st and the 15th of each month, and that you deposit a portion of each paycheck into an account that earns 8% interest. How much would you need to deposit from each paycheck in order to have $50,000 in the account in 6 years?
Question
Supposed that your paychecks are issued on the 1st and the 15th of each month, and that you deposit a portion of each paycheck into an account that earns 8% interest. How much would you need to deposit from each paycheck in order to have $50,000 in the account in 6 years?
Solution
To solve this problem, we need to use the formula for the future value of an annuity due, which is a series of equal payments made at the beginning of each period. The formula is:
FV = P * [(1 + r/n)^(nt) - 1] * (1 + r/n) / (r/n)
where: FV = future value of the annuity P = amount deposited each period r = annual interest rate n = number of compounding periods per year t = number of years
We know that FV = $50,000, r = 8% or 0.08, n = 24 (since there are 24 pay periods in a year), and t = 6. We want to solve for P.
First, let's simplify the right side of the equation:
(1 + r/n)^(nt) = (1 + 0.08/24)^(24*6) = 1.718186
Subtract 1 and multiply by (1 + r/n):
[(1 + 0.08/24)^(24*6) - 1] * (1 + 0.08/24) = 0.718186 * 1.003333 = 0.720519
Divide by (r/n):
0.720519 / (0.08/24) = 216.1557
So, the equation becomes:
$50,000 = P * 216.1557
Solving for P, we get:
P = 231.38
So, you would need to deposit approximately 50,000 in the account in 6 years.
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