Problem 56 - 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 \(\$ 6000\) to invest. Which investment yields the greater return over 4 years: \(8.25 \%\) compounded quarterly or \(8.3 \%\) compounded semiannually?

Step-by-Step Solution

Answer

After calculating the future values and comparing them, the investment that yields the larger future value will provide the greater return over 4 years.

Step 1: Convert Percentage Rates to Decimal Form

Firstly, the percentages need to be converted into decimal form. For \(8.25\%\), the decimal form is \(0.0825\). For \(8.3\%\), the decimal form is \(0.083\).

Step 2: Calculate Future Value for the Investment Compounded Quarterly

For the \(8.25\%\) compounded quarterly, apply the formula \(A = P \left(1+ \frac{r}{n}\right)^{n t}\). Here, - \(P = \$6000\) (principal amount)- \(r = 0.0825\) (annual interest rate in decimal form)- \(n = 4\) (number of times interest applied per time period - quarterly implies 4 times a year)- \(t = 4\) (the time the money is invested for in years)Then,\[A = 6000 \left(1+\frac{0.0825}{4}\right)^{4 * 4}\]

Step 3: Calculate Future Value for the Investment Compounded Semiannually

For the \(8.3\%\) compounded semiannually, the formula \(A = P \left(1+ \frac{r}{n}\right)^{n t}\) is applied again, but with different parameters. Here, - \(P = \$6000\) - \(r = 0.083\)- \(n = 2\) (compounded semiannually implies 2 times a year)- \(t = 4\) So,\[A = 6000 \left(1+\frac{0.083}{2}\right)^{2 * 4}\]

Step 4: Compare the Results

Finally, once both future values are calculated, they are then compared. The investment with the larger future value yields the greater return over the 4 year period.

Key Concepts

Future Value

Future value is the amount of money an investment will grow to after applying interest over a specific period. When dealing with compound interest, future value calculations help quantify how much an investment will grow over time. In our exercise, we're using two different interest rates and compounding frequencies, and our goal is to determine the future value for each scenario.
The formula to calculate future value with compound interest is \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where:

• \( A \) is the future value.
• \( P \) stands for the principal amount (initial investment).
• \( r \) represents the annual interest rate in decimal form.
• \( n \) is the number of times interest is compounded per year.
• \( t \) denotes the time in years.

It's essential to convert interest rates from percentage to decimal form by dividing by 100. In our given example, this involves applying different rates and frequencies to determine which plan offers the higher future value over 4 years.

Interest Rate

The interest rate is a critical factor in the calculation of future value. It indicates the percentage of the principal amount that is paid as interest to the investor. The formula for calculating future value is highly sensitive to the rate chosen, as even a small difference in the rate can lead to different investment outcomes.
In our exercise, we have two interest rates:

• 8.25% compounded quarterly
• 8.3% compounded semiannually

Before using these rates in calculations, they must be converted into decimal form. This is done by dividing by 100:

• 8.25% becomes 0.0825
• 8.3% becomes 0.083

These decimal rates are then used in the compound interest formula, helping us find out how the initial investment grows with these rates over a period.

Compound Frequency

Compound frequency refers to how often interest is applied to an investment balance within a year. The frequency of compounding can significantly affect the growth of an investment. The more frequently interest is compounded, the more interest will be added on, and thus, the more growth is seen in the investment over a fixed period.
In our scenario, we have two different compounding frequencies:

• Quarterly compounding means that interest is calculated and added four times a year.
• Semiannual compounding means it is calculated and added twice a year.

The formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] uses \( n \) to represent the frequency of compounding. This affects the exponent \( nt \), demonstrating the power of compound frequency in enhancing the investment value over time.

Investment Comparison

The ultimate purpose of analyzing future value, interest rate, and compound frequency is to compare different investment options. By calculating the future values for different rates and frequencies, we can determine which investment is more beneficial.
In this example, using the formula for compound interest, we compute future values for both investments:

• At 8.25% compounded quarterly
• At 8.3% compounded semiannually

After calculating, compare these future values to determine the better investment. The one with the higher future value represents a better return on investment over the specified period, letting you choose the optimal option based on growth potential.