To find out how much money should be invested today, we need to calculate the present value (PV) of a future amount of $15,000 that will be available in 6 years, given an annual interest rate of 3% compounded monthly.
We can use the compound interest formula for this purpose:
[tex]\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \][/tex]
Where:
- \( PV \) is the present value (the amount we want to find).
- \( FV \) is the future value, which is $15,000.
- \( r \) is the annual interest rate, which is 3% or 0.03.
- \( n \) is the number of compounding periods per year, which is 12 (for monthly compounding).
- \( t \) is the number of years the money is invested, which is 6.
Let's plug the given values into the formula step-by-step:
1. Identify the annual interest rate \( r = 0.03 \).
2. Identify the number of compounding periods per year \( n = 12 \).
3. Identify the number of years \( t = 6 \).
4. Identify the future value \( FV = 15000 \).
First, calculate the monthly interest rate:
[tex]\[ \frac{r}{n} = \frac{0.03}{12} = 0.0025 \][/tex]
Next, calculate the total number of compounding periods:
[tex]\[ nt = 12 \times 6 = 72 \][/tex]
Now substitute these values into the compound interest formula:
[tex]\[ PV = \frac{15000}{(1 + 0.0025)^{72}} \][/tex]
Calculating the compound factor:
[tex]\[ (1 + 0.0025)^{72} \approx 1.1970 \][/tex]
Finally, divide the future value by the compound factor:
[tex]\[ PV = \frac{15000}{1.1970} \approx 12531.87 \][/tex]
Therefore, you should invest approximately [tex]$12,531.87 today in an account that earns 3% interest compounded monthly in order to have $[/tex]15,000 in 6 years.
