How much money should be deposited today in an account that earns 7% compounded semiannually so that it will accumulate to $15,000 in three years?
We can use the formula for compound interest to solve this problem:
A = P(1 + r/n)^(nt)
Where:
A = final amount ($15,000)
P = principal amount (initial deposit)
r = annual interest rate (7% = 0.07)
n = number of times interest is compounded per year (2 for semiannually)
t = number of years (3)
Substituting the given values into the formula, we have:
$15,000 = P(1 + 0.07/2)^(2 * 3)
$15,000 = P(1 + 0.035)^6
$15,000 = P(1.035)^6
$15,000 = P(1.222419)
Now, we can isolate P by dividing both sides of the equation by (1.222419):
P = $15,000 / (1.222419)
P ≈ $12,276.68
Therefore, approximately $12,276.68 should be deposited today in order to accumulate to $15,000 in three years.