To determine the future value of regular investments made at the end of each quarter over a 12-year period with an annual interest rate of 6%, we need to use the Future Value of an Annuity formula. Here’s how you can calculate it step by step:
1. Identify the parameters provided:
- Investment per quarter (P): [tex]$380
- Number of years (years): 12
- Quarters per year: 4
- Annual interest rate (annual_interest_rate): 6% or 0.06
2. Calculate the total number of investment periods (n):
Since there are 4 quarters in a year and the investment is made for 12 years:
\[
n = \text{years} \times \text{quarters per year} = 12 \times 4 = 48
\]
3. Determine the quarterly interest rate (r):
The annual interest rate is 6%. To find the quarterly interest rate:
\[
r = \frac{\text{annual interest rate}}{\text{quarters per year}} = \frac{0.06}{4} = 0.015
\]
4. Apply the Future Value of an Annuity formula:
The formula for the future value of an annuity is given by:
\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Substituting the known values:
\[
FV = 380 \times \left( \frac{(1 + 0.015)^{48} - 1}{0.015} \right)
\]
5. Calculate the inside of the parentheses first (1 + r)^n:
\[
(1 + 0.015)^{48} \approx 2.0667
\]
6. Subtract 1 from the result:
\[
(1 + 0.015)^{48} - 1 \approx 2.0667 - 1 = 1.0667
\]
7. Divide by the quarterly interest rate (r):
\[
\frac{1.0667}{0.015} \approx 71.1133
\]
8. Multiply by the investment per quarter (P):
\[
FV = 380 \times 71.1133 \approx 27023.054
\]
9. Round to the nearest cent:
\[
FV \approx 27023.05
\]
10. Final amount:
The lump sum after 12 years, based on this investment strategy, will be approximately $[/tex]26,434.78.
