Problem 68 - College Algebra with Corequisite Support

Question

An investment account with an annual interest rate of \(7 \%\) was opened with an initial deposit of $$\$ 4,000$$ Compare the values of the account after 9 years when the interest is compounded annually, quarterly, monthly, and continuously.

Step-by-Step Solution

Answer

Continuous compounding yields the highest account value after 9 years.

Step 1: Understanding Compound Interest Formula

The formula for compound interest is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \(A\) is the amount of money accumulated after \(n\) years, including interest. \(P\) is the principal amount (initial deposit), \(r\) is the annual interest rate (decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the number of years the money is invested. The continuous compound interest is given by \[ A = Pe^{rt} \].

Step 2: Calculate Annual Compounding

For annual compounding, use \(n = 1\). Substitute \(P = 4000\), \(r = 0.07\), \(n = 1\), and \(t = 9\) into the compound interest formula: \[ A = 4000 \left(1 + \frac{0.07}{1}\right)^{1 \times 9} = 4000 \times (1.07)^9 \].Calculate \( (1.07)^9 \) and then multiply by \(4000\).

Step 3: Calculate Quarterly Compounding

For quarterly compounding, use \(n = 4\). Substitute \(P = 4000\), \(r = 0.07\), \(n = 4\), and \(t = 9\) into the formula: \[ A = 4000 \left(1 + \frac{0.07}{4}\right)^{4 \times 9} = 4000 \times (1.0175)^{36} \].Calculate \( (1.0175)^{36} \) and multiply by \(4000\).

Step 4: Calculate Monthly Compounding

For monthly compounding, use \(n = 12\). Substitute \(P = 4000\), \(r = 0.07\), \(n = 12\), and \(t = 9\) into the formula: \[ A = 4000 \left(1 + \frac{0.07}{12}\right)^{12 \times 9} = 4000 \times (1.0058333)^{108} \].Calculate \( (1.0058333)^{108} \) and multiply by \(4000\).

Step 5: Calculate Continuous Compounding

For continuous compounding, use the formula \( A = Pe^{rt} \). Substitute \(P = 4000\), \(r = 0.07\), and \(t = 9\) into the formula: \[ A = 4000 \times e^{0.07 \times 9} = 4000 \times e^{0.63} \].Calculate \( e^{0.63} \) and multiply by \(4000\).

Step 6: Compare the Results

Calculate the accumulated values for each compounding method:- Annual: \(A \approx 7396.72\)- Quarterly: \(A \approx 7453.48\)- Monthly: \(A \approx 7484.56\)- Continuously: \(A \approx 7521.09\)Compare and observe that the account balance is highest with continuous compounding, followed by monthly, quarterly, and finally annual compounding.

Key Concepts

Annual Compounding

Annual compounding is one of the simplest forms of compound interest. With this method, interest is calculated once per year. When using annual compounding, you apply the interest rate to the initial principal only once every 12 months.

This can be represented by the formula:

• \( A = P \left(1 + r\right)^t \)

In this formula:

• \( P \) is the initial deposit (e.g., \(4,000)
• \( r \) is the annual interest rate (expressed as a decimal)
• \( t \) is the number of years the money is invested

For example, if you invest \)4,000 at an annual interest rate of 7% for 9 years, your calculation becomes:

• \( A = 4000 \times (1 + 0.07)^{9} \)

Calculating \( (1.07)^9 \) and multiplying by $4,000 will give you the future value of your investment under annual compounding.

Quarterly Compounding

With quarterly compounding, the interest is calculated four times a year. This means the frequency of compounding is higher than with annual compounding, leading to more interest accrual.

The formula for quarterly compounding is:

• \( A = P \left(1 + \frac{r}{4}\right)^{4t} \)

In this scenario:

• \( r/4\) is the interest rate per quarter.
• \( 4t \) indicates the total number of compounding periods over the years.

For a principal of \(4,000 at 7% annually, compounded quarterly for 9 years:

• \( A = 4000 \left(1 + \frac{0.07}{4}\right)^{36} \)

This means you'll compute \( (1.0175)^{36} \) and multiply that result by \)4,000 to see how much your money grows with quarterly compounding. These more frequent calculations lead to a slightly higher result than annual compounding.

Monthly Compounding

Monthly compounding increases the compounding frequency even more, occurring twelve times each year. This allows interest to be calculated on the accumulated principal at the end of every month.

The formula used is:

• \( A = P \left(1 + \frac{r}{12}\right)^{12t} \)

In this approach:

• \( r/12 \) represents the monthly interest rate.
• \( 12t \) is the total number of months over the investment period.

For the same \(4,000 investment, the calculation is:

• \( A = 4000 \left(1 + \frac{0.07}{12}\right)^{108} \)

Here, you'll calculate \( (1.0058333)^{108} \) and then multiply by \)4,000. The result gives an idea of how the accumulated value changes when compounded monthly. It's notably larger than with quarterly or annual compounding.

Continuous Compounding

Continuous compounding differs from the other methods because the interest is compounded an infinite number of times per year. This means the compounding happens continuously, offering the quickest way for interest to accumulate for an investment.

The continuous compounding formula is:

• \( A = Pe^{rt} \)

In this context:

• \( e \) is the mathematical constant, approximately 2.71828.
• \( rt \) is the product of the annual interest rate and time.

For a \(4,000 initial investment at 7% for 9 years, you'd calculate:

• \( A = 4000 \times e^{0.63} \)

After determining \( e^{0.63} \) and multiplying the result by \)4,000, you'll discover that continuous compounding produces the maximum amount of interest. This method maximizes growth and minimizes time lost without compounding, making it appealing for growing investments rapidly.