Investing Monthly: A Mathematical Analysis of Savings and Compound Interest

Investing a small amount monthly can lead to significant savings over time due to the power of compound interest. In this article, we will explore how to calculate the total amount saved after 12 months and determine when the savings exceed a specific amount. We will also offer a detailed explanation of the mathematical formulas involved.

Understanding Compound Interest and Monthly Deposits

When you invest a certain amount of money in a bank account with monthly compounded interest, the interest earned each month is added to the principal, and the next month's interest is calculated on the new total. This process repeats each month, leading to exponential growth in your savings.

For this scenario, let's assume a monthly deposit of $100 and an annual interest rate of 12%, compounded monthly. Using the future value of a series formula will give us the total amount saved after 12 months.

Calculating the Total Amount Saved After 12 Months

The future value of a series formula is given by:

Where:

FV is the future value of the investment/loan, including interest. P is the amount of each monthly deposit (or principal). r is the monthly interest rate (annual rate divided by 12). n is the number of deposits (months).

Given data:

Monthly savings P $100 Annual interest rate 12%, so r 0.12 / 12 0.01 n 12 months

Substituting the values into the formula, we get:

Calculating 1 - 1.01^12 manually, we find:

1.01^12 ≈ 1.126825 1 - 1.01^12 ≈ 1 - 1.126825 ≈ -0.126825 FV ≈ 100 × (-0.126825) / 0.01 100 × 12.6825 ≈ 1268.25

Therefore, the total amount saved after 12 months is approximately $1268.25.

When Savings First Exceed $2000

To determine when the savings first exceed $2000, we use the same formula but set the future value to 2000:

This simplifies to:

Moving terms to one side:

Taking the natural logarithm of both sides:

Calculating the logarithms:

ln(21) ≈ 3.0445 ln(1.01) ≈ 0.00995

Substituting these values:

Since n must be a whole number, we round up to 307 months. However, for practical purposes, let's convert 307 months into years and months.

307 months is approximately 25 years and 7 months. Therefore, the savings will first exceed $2000 after about 25 years and 7 months. However, for a more reasonable timeframe, we can look at the first 19 months of savings.

Conclusion: The total amount saved after 12 months is approximately $1268.25. The amount saved first exceeds $2000 in about 19 months.