Final answer:
Using the formula for the future value of an annuity, after 5 years Joe will have $2,123.64 in his savings account by depositing $400 annually at a 3% interest rate. This calculation illustrates the power of compound interest, showing how regular savings grow over time thanks to interest being earned on both the initial deposits and the accumulated interest.
Step-by-step explanation:
When it comes to understanding how savings and interest work together, the concept of compound interest is fundamental. Joe is depositing $400 annually into a savings account with a 3% interest rate. To calculate the total amount in the account after 5 years, we need to apply the formula for the future value of an annuity:
FV = P \times \left(\frac{\left(1 + r\right)^n - 1}{r}\right)
Where:
- FV is the future value of the annuity
- P is the annual payment ($400)
- r is the annual interest rate (0.03)
- n is the number of years (5)
Substituting the values we get:
FV = 400 \times \left(\frac{\left(1 + 0.03\right)^5 - 1}{0.03}\right)
FV = 400 \times \left(\frac{\left(1.03\right)^5 - 1}{0.03}\right)
FV = 400 \times \left(\frac{1.159274 - 1}{0.03}\right)
FV = 400 \times 5.3091
FV = $2,123.64
The power of compounding is evident when you consider that the interest is being earned not only on the initial deposits but also on the accumulated interest from previous periods. This is why starting early and regularly saving can have such a significant impact on the growth of savings over time.