Problem 1 - Applied Calculus

Question

A yearly deposit of \(\$ 1000\) is made into a bank account that pays \(8.5 \%\) interest per year, compounded annually. What is the balance in the account right after the \(20^{\text {th }}\) deposit? How much of the balance comes from the annual deposits and how much comes from interest?

Step-by-Step Solution

Answer

The balance is $49,936.27; $20,000 from deposits and $29,936.27 from interest.

Step 1 - Understanding Compound Interest Formula

The formula to calculate the future value of a series of cash flows in compound interest is \[ F = P \left(\frac{(1 + r)^n - 1}{r}\right) \]where \( F \) is the future value, \( P \) is the annual deposit, \( r \) is the annual interest rate, and \( n \) is the number of years.

Step 2 - Calculating Total Future Balance

Using the formula: \( P = \\(1000 \), \( r = 0.085 \), and \( n = 20 \), we calculate the total future balance:\[ F = 1000 \left(\frac{(1 + 0.085)^{20} - 1}{0.085}\right) \]Calculating further:\[ F = 1000 \left(\frac{4.99362741 - 1}{0.085}\right) \approx \\) 49,936.27 \]

Step 3 - Determine Amount from Deposits

Since each year \\(1000 is deposited and this happens over 20 years, the total contribution from the deposits is:\[ \text{Total Deposits} = 1000 \times 20 = \\) 20,000 \]

Step 4 - Calculating Amount from Interest

To find the amount that comes from interest, we subtract the total deposits from the future balance:\[ \text{Interest Earned} = F - \text{Total Deposits} = 49,936.27 - 20,000 = \$ 29,936.27 \]

Key Concepts

Future Value Calculation

The future value (FV) calculation is an essential part of understanding compounded growth over time. It helps us determine how much a series of periodic cash flows will amount to after a certain period, considering a specific interest rate. Essentially, the future value tells us what our investments are worth in the future.
To compute the future value of annual deposits made into an account with compound interest, we use the following formula:

• \( F = P \left( \frac{(1 + r)^n - 1}{r} \right) \)

Here:

• \( F \) is the future value of the deposits after \( n \) years.
• \( P \) represents the annual deposit made each year.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( n \) is the number of years the deposits are made.

By substituting the given values into the formula, we find that depositing \( \\(1000 \) annually at an interest rate of \( 8.5\% \) for \( 20 \) years will amount to approximately \( \\)49,936.27 \). This calculation forecasts the balance right after the 20th deposit.

Annual Deposits

Annual deposits are regular contributions made into a savings or investment account. These consistent deposits play a significant role in the growing future balance, especially when compounded annually.
In the exercise, a fixed amount of \( \\(1000 \) is deposited each year for 20 years. This regularity simplifies the process since the total contribution from these deposits can be easily calculated by multiplying the annual deposit by the number of years:

• \( \text{Total Deposits} = 1000 \times 20 = \\)20,000 \)

Understanding the importance of such incremental investments is crucial because it highlights how consistent contributions can significantly boost the future value over time, even without considering the added advantage of compound interest. Consistent annual deposits form the backbone of any long-term growth strategy.

Interest Earned Calculation

When it comes to growth in investments or savings, the magic of compound interest comes into play. The interest earned is essentially the extra amount gained due to interest being applied on not just the initial deposits but also on the accumulated interest over time.
To find out how much of the final balance results from interest earnings, subtract the initial total deposits from the future value:

• \( \text{Interest Earned} = F - \text{Total Deposits} \)
• \( \text{Interest Earned} = 49,936.27 - 20,000 = \\(29,936.27 \)

In this case, \( \\)29,936.27 \) of the final account balance comes solely from the interest accrued over 20 years. This impressive figure highlights the power of compounded growth, showing the potential significant returns when compounding works on your side. Interest earned can greatly surpass the simple addition of deposits, emphasizing the importance of investing over long periods. This concept is central to building wealth through savings and investment.