Question

If you invest a dollar at "6\% interest compounded monthly," it amounts to (1.005) \(^{n}\) dollars after \(n\) months. If you invest \(\$ 10\) at the beginning of each month for 10 years (120 months), how much will you have at the end of the 10 years?

Short answer

After 10 years, the total amount will be approximately .

Step-by-step solution

Identify the formula for compound interest

The future value of an investment with monthly contributions can be calculated using the future value formula for a series. The formula is: . This formula accounts for contributions made at the beginning of each compounding period.

Set up the variables

The monthly interest rate (r) is . The total number of months (n) is 10 years * 12 months = 120 months. The monthly contribution (P) is .

Plug values into the formula

Using the variables identified, substitute them into the future value formula: . Substituting r = 0.005, P =

Simplify and calculate the series

First, calculate . Then, multiply by

Key concepts

Future Value Formula

To calculate the future value of an investment with regular contributions, we use the future value of a series formula. This formula allows us to determine how much money will accumulate over a period of time when we make regular deposits and interest is compounded.

For monthly contributions, the future value formula is given by:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r) \text{ where:} \]
\( FV \) is the future value of the investment.
\( P \) is the monthly contribution.
\( r \) is the monthly interest rate (annual interest rate divided by 12).
\( n \) is the total number of contributions (months).

This formula helps us see the power of compound interest, as it amplifies our savings over time through both the contributions and the interest earned on previous contributions.

Monthly Contributions

In our compound interest calculation, the monthly contributions play a significant role. They are the regular amounts of money we invest at the beginning of every month.

In the given exercise, the monthly contribution \( P \) is \(10. This regular deposit into the investment account helps build the total amount saved over time.
Let's break down how these contributions add up:

• Each month, you are adding a fixed amount of \)10.
• The money contributed early has more time to grow due to compound interest.
• Later contributions add up the principal amount as well as the interest earned on earlier contributions.

By regularly adding money into the account, you take advantage of the compounded interest, making your savings grow more.

Interest Rate

The interest rate is a crucial factor in the calculation of compound interest.

It determines how much interest you earn on your savings. For this exercise, the annual interest rate is 6%, which is compounded monthly.

The steps to convert the annual interest rate into the monthly interest rate are as follows:

• Take the annual interest rate (6%).
• Divide it by 12 (number of months in a year).
• This gives the monthly interest rate \( r \), which is 0.5% or 0.005 in decimal form.

It's important to use the monthly interest rate in your calculations correctly since it's used in the future value formula to grow both your initial principal and the monthly contributions.

Series Calculation

The core of calculating the future value with compound interest lies in properly handling the series calculation.

For series calculation, here’s how it works:

• Each monthly contribution can be seen as a separate mini-investment.
• The first contribution will have interest compounded over the entire period (120 months).
• Subsequent contributions are compounded over fewer months, with the last contribution only being compounded for 1 month if contributed at the beginning.

Using the future value formula for a series, we sum all these mini-investments by following this:
\[ FV = 10 \times \frac{(1 + 0.005)^{120} - 1}{0.005} \times (1 + 0.005) \]
By calculating step-by-step, you can see how every month's contribution builds up, allowing us to determine the final accumulated amount after 120 months.