Problem 68 - College Algebra

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

Compounding continuously gives the highest amount of \$7603.60 after 9 years.

Step 1: Understanding Interest Compounding

To compare the account values, we will calculate the amount for each compounding frequency (annually, quarterly, monthly, and continuously). Each frequency has a different formula to account for how often interest is added to the principal amount.

Step 2: Compounded Annually

The formula for the future value of an account with interest compounded annually is given by:\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where \( P = 4000 \) (the principal amount), \( r = 0.07 \) (the annual interest rate), \( n = 1 \) (the number of times interest is compounded annually), and \( t = 9 \) (the number of years).Calculating:\[A = 4000 \left(1 + \frac{0.07}{1}\right)^{1 \times 9} = 4000 \times (1.07)^9 \approx 7464.29\]

Step 3: Compounded Quarterly

For quarterly compounding, the formula changes slightly as interest is compounded 4 times a year, i.e., \( n = 4 \).\[A = 4000 \left(1 + \frac{0.07}{4}\right)^{4 \times 9}\]Calculating:\[A = 4000 \left(1 + \frac{0.07}{4}\right)^{36} \approx 7537.52\]

Step 4: Compounded Monthly

For monthly compounding, interest is compounded 12 times a year, so \( n = 12 \).\[A = 4000 \left(1 + \frac{0.07}{12}\right)^{12 \times 9}\]Calculating:\[A = 4000 \left(1 + \frac{0.07}{12}\right)^{108} \approx 7574.09\]

Step 5: Compounded Continuously

For continuous compounding, we use the formula:\[A = Pe^{rt}\]where \( e \approx 2.71828 \). Applying the values:\[A = 4000 e^{0.07 \times 9}\]Calculating:\[A = 4000 \times e^{0.63} \approx 7603.60\]

Step 6: Comparison and Conclusion

After calculations:- Annually: \\(7464.29- Quarterly: \\)7537.52- Monthly: \\(7574.09- Continuously: \\)7603.60The account value increases as the compounding frequency increases, with continuous compounding yielding the highest amount.

Key Concepts

Compounding Frequency

Understanding the concept of compounding frequency is crucial when dealing with investments. Compounding frequency refers to how often interest is added to the principal balance of an investment. The more frequently interest is compounded, the more often interest is calculated on the accumulated interest of previous periods. This can result in earning more interest over time.

Different compounding frequencies include:

• Annually: Interest is added once per year.
• Quarterly: Interest is added four times a year.
• Monthly: Interest is compounded twelve times a year.
• Continuously: Interest is theoretically added at every possible moment.

Each frequency has its own impact on the growth of the investment, as we see in our example where each frequency resulted in different future values after 9 years.

Continuously Compounded Interest

Continuous compounding takes the concept of frequent compounding to the extreme. This method assumes that interest is not just applied on a daily, weekly, or even monthly basis but constantly. Essentially, it's as if interest is being added at every possible instant.

The formula used for continuously compounded interest is slightly different than for other frequencies. It utilizes the mathematical constant \( e \), approximately 2.71828:

\[A = Pe^{rt}\]

Here, \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal investment amount, \( r \) is the annual interest rate, and \( e \) represents the base of the natural logarithm. This formula can yield the highest return compared to other compounding frequencies. In the exercise, continuous compounding provided the highest future value of $7603.60 after 9 years.

Compound Interest Formula

Compound interest grows your investment faster than simple interest because it takes into account accumulated interest.

The common formula for compound interest based on different compounding frequencies is:

\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]

• \( P \) is the principal amount.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( n \) is the number of times interest is compounded per year.
• \( t \) is the number of years the money is invested.
• \( A \) is the amount of money accumulated after \( n \) years, including interest.

In our example, we used this formula for annually, quarterly, and monthly compounding, with increasing values for \( n \) leading to greater total accumulated amounts.

Future Value of an Investment

The future value of an investment is the amount of money you end up with after the initial principal has grown over a specified period at a given interest rate. It helps investors understand the potential of their investment.

To determine the future value, you need to account for:

• The initial investment or principal amount.
• The interest rate that applies to that principal.
• The length of time the money is invested or accumulated.
• The frequency with which the interest is compounded.

In our exercise, calculating future values with different compounding frequencies showed that the frequency with which you compound can significantly impact the final amount. From annually to continuous compounding, each method increased the future value of the investment, with the continuous compounding yielding the highest return of $7603.60 over 9 years.