Final answer:
To determine the amount accumulated after making monthly deposits of $400 for 30 years at a 5% APR, the future value of an annuity formula can be used. This exercise highlights the benefits of regular contributions and the power of compound interest over time.
Step-by-step explanation:
If you deposit $400 at the end of each month into a retirement account that pays 5% APR, you're essentially making a series of regular contributions to an account that grows with compound interest. To calculate the future value of these contributions, we can use the future value of an annuity formula, which for monthly contributions can be written as:
FV = P × {[(1 + r)^n - 1] / r} × (1 + r)
Where:
FV is the future value of the annuity.
P is the periodic payment amount.
r is the monthly interest rate (annual rate divided by 12).
n is the total number of payments (months).
Given the 5% annual interest rate, the monthly interest rate is 0.05/12. Since you will be depositing monthly for 30 years, the total number of deposits would be 30 × 12.
After applying these values to the formula, you'll find the total amount accumulated in the retirement account after 30 years. This exercise is an excellent illustration of the power of compound interest and the benefits of making regular contributions over a long period.
To visualize the impact of compound interest more clearly, let's compare the total amount accumulated with regular monthly contributions to a one-time lump-sum investment early in life. For instance, a $3,000 deposit at a 7% real annual rate of return would grow significantly over 40 years, showing that the sooner you start saving and investing, the more you can benefit from compound interest.