Problem 63 - Precalculus

Question

How much money should you deposit into a money market account that pays \(5 \%\) a year compounded continuously to have \(\$ 38,000\) in the account in 20 years?

Step-by-Step Solution

Answer

Deposit approximately \( \$13,981.93 \).

Step 1: Understand Continuous Compounding Formula

The formula for continuous compounding is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after \( n \) years, including interest. \( P \) is the principal amount (the initial amount of money), \( r \) is the annual interest rate (decimal), \( t \) is the time in years, and \( e \) is the base of the natural logarithms, approximately equal to 2.71828.

Step 2: Identify Given Values

From the problem, we have \( A = 38000 \), the future value of the account; \( r = 5\% = 0.05 \), the annual interest rate; \( t = 20 \) years, the time period.

Step 3: Rearrange the Formula to Solve for P

To find the principal \( P \), rearrange the continuous compounding formula to \( P = \frac{A}{e^{rt}} \). This allows us to solve for the initial deposit required to reach \( A \) after \( t \) years at the given rate \( r \).

Step 4: Calculate \( e^{rt} \)

First, calculate the exponent: \( rt = 0.05 \times 20 = 1.0 \). Then calculate \( e^{1.0} \). Using the approximate value of \( e = 2.71828 \), we get \( e^1 = 2.71828 \).

Step 5: Compute Principal \( P \)

Substitute \( A = 38000 \) and \( e^{1.0} = 2.71828 \) into the formula for \( P \): \( P = \frac{38000}{2.71828} \). Calculate this to find \( P \approx 13981.93 \).

Step 6: Solution Conclusion

The principal amount you should deposit in the account to have \( \\(38,000 \) in 20 years, with 5% annual interest compounded continuously, is approximately \( \\)13,981.93 \).

Key Concepts

Compound Interest

Compound interest refers to the concept where the interest earned on an initial principal amount is reinvested to earn more interest. As time passes, this process continues, leading to an increase in the overall amount in the account. It is a beneficial financial strategy because:

• The interest is calculated not only on the initial amount but also on the accumulated interest from previous periods.
• This results in interest leading to more interest, creating a compounding effect.
• Over time, the initial investment grows exponentially rather than linearly.

In the context of continuous compounding, which is slightly different from the more common periodic compounding, interest adds instantaneously, allowing the amount to grow at a faster rate.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a curve on a graph. In many real-world scenarios, including finance, exponential growth can be observed when quantities grow by a consistent percentage of the whole.

• An intuitive example of this is money in a bank account using compound interest.
• For instance, if you start with a principal amount and it grows due to compound interest, each year you'll see not only the original principal but also the interest it has earned accumulating further interest.
• Exponential growth is characterized by not having a fixed increment but rather an increasing rate as the base amount grows.

When compounding continuously, like in our exercise, the growth is even smoother, as the formula uses the natural exponential function \(e\), facilitating this rapid increase.

Financial Mathematics

Financial mathematics integrates mathematical concepts into finance to solve various problems and optimize financial decisions. Using continuous compounding in our exercise is a part of financial mathematics. This involves using the formula \( A = Pe^{rt} \) to determine how investments grow:

• It's crucial for tasks such as calculating the value of investments, planning savings, and understanding loans.
• In our exercise, it helps determine how much to deposit today to achieve a future financial goal given a set interest rate and time period.
• It also assists in comparing different financial products based on how they calculate interest.

Understanding these concepts allows investors, financial analysts, and regular individuals to make informed decisions about their savings and investments, ensuring they maximize returns over time.