Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the compound interest formulas, \(A=P\left(1+\frac{r}{n}\right)^{n-1}\) and \(A=P e^{n t},\) to solve Exercises \(39-42 .\) 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

The option which yielded the greater return after all calculations will be the final short answer, this will depend on the results of the calculations done in steps 1 and 2.

Step 1: Calculating final amount for 8.25% compounded quarterly

Insert $6000 for P, 0.0825 for r, 4 for n (compounded quarterly implies 4 periods per year), and 4 for t into the compound interest formula. It will look like this: \(A=6000(1+0.0825/4)^{4*4}\)Next, simplify and calculate this expression to find the final amount after four years with the first investment option.

Step 2: Calculating final amount for 8.3% compounded semiannually

Now do the same for the second investment option. Insert $6000 for P, 0.083 for r, 2 for n (compounded semiannually implies 2 periods per year), and 4 for t into the compound interest formula. It will look like this: \(A=6000(1+0.083/2)^{2*4}\)Again, simplify and calculate this expression to find the final amount after four years with the second investment option.

Step 3: Comparing the two investment options

Once the calculations in Step 1 and Step 2 are done, compare the final amounts for each option. The investment option resulting in a higher final amount is the one that yields a greater return over four years.

Key Concepts

Investment Options

When it comes to growing your savings, understanding various investment options is crucial. The exercise you're dealing with involves choosing between two specific scenarios: investing money at a quarterly compound rate versus a semiannual compound rate.

Investment options vary widely and can include stocks, bonds, mutual funds, or even different savings accounts. The goal is to maximize the return on your investment, which is where interest rates and compounding frequency come into play.

• Higher interest rates usually equate to higher returns, but they often come with increased risk.
• Compounding frequency refers to how often the interest is calculated and added to your investment. More frequent compounding typically leads to more significant growth.

Understanding these dynamics will help you make informed decisions about where to put your money for optimal growth.

Quarterly Compounding

Quarterly compounding means that the interest on your investment is calculated and added to your principal every three months. This frequent addition of interest allows your investment to grow at a faster pace than if it were compounded less frequently.

For example, in the provided exercise, we have an annual interest rate of 8.25% that is compounded quarterly. To apply the quarterly compounding formula, you need to:

• Divide the annual interest rate by four (since there are four quarters in a year).
• Calculate the total number of times the interest is compounded over the investment period (four times per year over four years would be sixteen times).
• Plug these numbers into the compound interest formula to find the final amount.

The benefit of a quarterly compounding plan is in its potential to increase your returns more compared to less frequent compounding schedules.

Semiannual Compounding

Semiannual compounding, in contrast, involves calculating and adding the interest to your investment every six months. This means interest is compounded twice a year, which is less frequent than quarterly compounding.

In this scenario with an 8.3% annual rate, the process involves:

• Dividing the annual interest rate by two (as there are two compounding periods each year).
• Determining the total number of compounding instances which, for four years, is eight times.
• Utilizing these values in the compound interest formula to compute the final investment value.

While semiannual compounding might not yield as high of a return as its quarterly counterpart within the same interest rate, it's crucial to compare both the rate and frequency to make an informed financial decision.

Interest Formulas

Interest formulas are mathematical expressions used to calculate the growth of an investment over time. In the exercise, two main formulas are used. Both cater to discrete compounding rather than continuous.

• The standard compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Here, \(P\) is the principal, \(r\) is the annual interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the time in years.
• This formula considers both the rate and frequency of compounding, allowing you to predict the amount of interest earned over a given period.

Understanding these formulas is essential as they help in predicting potential gains from different investment choices, enabling wise financial decisions.