How much money should be deposited today in an account that earns 4% compounded semiannually so that it will accumulate to $15,000 in three years?
To calculate the amount of money that should be deposited today, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
- A is the future value of the account ($15,000)
- P is the amount to be deposited today
- r is the annual interest rate (4% or 0.04)
- n is the number of times the interest is compounded per year (2, since it is compounded semiannually)
- t is the number of years (3)
Plugging in the given values, we have:
$15,000 = P(1 + 0.04/2)^(2*3)
Simplifying the equation:
$15,000 = P(1 + 0.02)^6
Taking the sixth root on both sides:
(1.02)^6 = P
1.126416 = P
Therefore, approximately $11,264.16 should be deposited today in order to accumulate to $15,000 in three years at a 4% compounded semiannually.