Problem 44 - College Algebra

Question

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{r t}\) to solve. Round answers to the nearest cent. Suppose that you have \(\$ 6000\) to invest. Which investment yields the greatest return over 4 years: \(8.25 \%\) compounded quarterly or \(8.3 \%\) compounded semiannually?

Step-by-Step Solution

Answer

The investment offering 8.25% compounded quarterly yields a slightly greater return over the 4 year period than the investment offering 8.3% compounded semiannually; hence it is the preferred option in this case. After 4 years, the first investment will amount to \$9185.69, while the second will amount to \$9183.34.

Step 1: Calculate the Return for the First Investment

The first investment will compound quarterly, or four times a year (n = 4), with an interest rate of 8.25% (r = 0.0825), and over a course of four years (t = 4). Using the formula for compound interest, \(A=P\left(1+\frac{r}{n}\right)^{n t}\), and substituting in these values, the total amount returned from the first investment option can be calculated as follows: \(A_{1} = \$6000 \left(1+ \frac{0.0825}{4}\right)^{4 * 4} = \$9185.69 \).

Step 2: Calculate the Return for the Second Investment

The second investment compounds semiannually, or twice a year (n = 2), with an interest rate of 8.3% (r = 0.083), and over four years (t = 4). Using the same compound interest formula as before, and substituting these values this time, the total amount returned from the second investment option can be calculated as: \(A_{2} = \$6000 \left(1+ \frac{0.083}{2}\right)^{2 * 4} = \$9183.34\).

Step 3: Compare the Two Investments

The goal is now to determine which investment option provides the greater return. By simply comparing \$9185.69 (from the first investment) and \$9183.34 (from the second investment), it can be observed that the first investment option yields a slightly larger return.

Key Concepts

Compound Interest Formula

Understanding the compound interest formula is crucial for anyone looking to invest money and see it grow over time. Simply put, compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.

The standard formula to calculate compound interest is:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\].
In this formula,

• \(A\) represents the future value of the investment/loan, including interest,
• \(P\) is the principal amount (the initial amount of money),
• \(r\) stands for the annual interest rate (in decimal),
• \(n\) is the number of times that interest is compounded per year, and
• \(t\) denotes the time the money is invested or borrowed for, in years.

When the interest is compounded, it means that the interest earned each period is added to the principal, so that the balance doesn't merely grow linearly, but exponentially. This can lead to a significant impact on the growth of the investment over time, especially for long-term investments.

The exercise provided illustrates a practical application of the formula, showing how different compounding frequencies can affect the total return on an investment.

Investment Return Calculation

Calculating the return on an investment involves determining how much money you will have at the end of the investment period, given the principal amount, interest rate, and compounding frequency. The essential goal is to understand how your money grows over time.

In the sample exercise, we're comparing two investment scenarios with different compounding intervals. The process involves:

• Identifying the variables: principal, interest rate, compounding frequency, and time.
• Inserting those variables into the compound interest formula.
• Calculating the future value for each scenario.
• Assessing which investment gives a higher return by comparing the future values.

This calculation demonstrates a key principle in finance: compounding frequency can make a difference even when the rates are very close, like in the scenario of 8.25% compounded quarterly versus 8.3% compounded semiannually. By performing these steps, we see that the investment with a higher compounding frequency yields a slightly greater return, a conclusion that can guide decisions for investors seeking the highest growth for their savings.

Exponential Functions

Exponential functions play a fundamental role in modeling situations where growth or decay accelerates over time, like in the case of calculating compound interest. To understand exponential functions, consider any function that can be represented in the form:
\[f(x) = ab^{x}\].
In this equation,

• \(a\) is a constant that signifies the starting value,
• \(b\) is the base of the exponential function representing the growth (if greater than 1) or decay (if between 0 and 1) factor,
• \(x\) corresponds to the exponent, which typically stands for time.

When the base, \(b\), is greater than 1, the function models growth; the larger the value of \(b\), the steeper the growth. With compound interest, each period of growth is a factor of the previous one, making it an exponential function. This exponential nature means that small changes in the rate or the frequency of compounding can significantly affect the outcome over time, as evidenced by the textbook exercise.