Problem 55 - College Algebra

Question

Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) Round answers to the nearest cent. Suppose that you have \(\$ 12,000\) to invest. Which investment yields the greater return over 3 years: \(7 \%\) compounded monthly or \(6.85 \%\) compounded continuously?

Step-by-Step Solution

Answer

By substituting the known values into each formula and solving for the future value (A), the investment scenario that yields a greater return can be determined. After comparing, either the continuous compounding at 6.85% or monthly compounding at 7% will yield a higher return.

Step 1 Calculation for Monthly compounded Interest

First, we will substitute our knowns into the formula \(A-P\left(1+\frac{r}{n}\right)^{n t}\), where \(P = \$ 12,000\), \(r = 0.07\), \(n = 12\) (compounded monthly), and \(t = 3\) years. We will solve for \(A\) - the future value of the investment. Afterward, the dollar amount will be rounded to the nearest cent.

Step 2 Calculation for Continuously Compounded Interest

Next, we will substitute our knowns for continuous compounding into the formula \(A-P e^{rt}\). In this case, \(P = \$ 12,000\), \(r = 0.0685\), and \(t = 3\). Compute for \(A\) - the future value of the investment. Again, we will round our answer to the nearest cent.

Step 3 Compare the investment returns

Finally, we will compare both computed future values of the investment. The method that yields a higher return after 3 years is deemed the better investment.

Key Concepts

Monthly Compounding

When it comes to growing your investments, monthly compounding is a popular method often discussed. The concept involves calculating interest on both the initial principal and the accumulated interest from previous periods. For monthly compounding, this calculation happens twelve times a year.

The formula used for calculating compound interest on a monthly basis is: \[A = P\left(1+\frac{r}{n}\right)^{nt}\]Where:

• \(P\) is the principal amount, which is the initial investment.
• \(r\) is the annual interest rate (expressed as a decimal).
• \(n\) is the number of times interest is compounded per year, for monthly compounding this is 12.
• \(t\) is the time the money is invested for in years.
• \(A\) is the amount of money accumulated after \(n\) years, including interest.

This method can lead to faster growth of your funds compared to simple interest, which is calculated only on the principal amount. By doing the calculations twelve times a year, your investment benefits from the effects of compounding quicker than it would if interest was compounded annually. This can be highly advantageous if you're looking to maximize your investment yield over time.

Continuous Compounding

Continuous compounding is an intriguing concept for those who want to push the limits of compound interest. Unlike monthly or annual compounding methods, continuous compounding calculates interest on an ongoing basis. It effectively assumes that the interest is being added and calculated every moment to boost your investment.

This is represented by the formula: \[A = P e^{rt}\]Here:

• \(P\) is again the principal amount.
• \(r\) is the annual interest rate (as a decimal).
• \(t\) symbolizes the number of years the money is invested.
• \(e\) is the mathematical constant approximately equal to 2.71828.
• \(A\) is what you want to calculate – the total amount after interest.

Continuous compounding utilizes exponential growth, resulting in potentially higher returns, especially over extended periods. However, the actual difference in returns depends on the specific rates used and the duration of the investment. In our scenario, comparing continuous compounding with monthly compounding can help decide which method best suits the desired investment strategy.

Investment Return Comparison

When deciding between different investment options, it's critical to compare the potential returns each can offer. In this context, we evaluate using both monthly and continuous compounding techniques to determine which method provides higher gains.

Let's break down how to effectively compare these returns:

• Calculate the future value of the investment using both methods. Each will yield a different total due to how often interest is computed and added to the principal.
• Use the given formulas: for monthly compounding, apply the formula \(A = P(1+\frac{r}{n})^{nt}\) and for continuous compounding, apply \(A = Pe^{rt}\).
• Compare the values obtained. The larger future value indicates the more lucrative investment option under the same conditions.

In our example, the calculations show which approach is better financially when using a constant principal amount over three years with the specific interest rates provided. This methodology not only highlights the differences between monthly and continuous compounding but also demonstrates how knowledge of mathematical principles of finance aid in making informed investment decisions.