Problem 43 - 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 \(\$ 12,000\) to invest. Which investment yields the greatest return over 3 years: \(7 \%\) compounded monthly or \(6.85 \%\) compounded continuously?

Step-by-Step Solution

Answer

Compute step 2 and step 3, then compare the results. The investment with the higher final amount is the one that yields the greatest return over 3 years.

Step 1: Identify the parameters for each investment

For the investment compounded monthly, the following parameters are given: \n - Principal P = $12,000 \n - Annual interest rate r = 7% = 0.07 \n - Number of times the interest is compounded per year n = 12 (monthly) \n - Number of years t = 3. \nFor the investment compounded continuously, the parameters are: \n - Principal P = $12,000 \n - Annual interest rate r = 6.85% = 0.0685 \n - Number of years t = 3.

Step 2: Calculate the final amount for the first investment

Substitute P = $12,000, r = 0.07, n = 12 and t = 3 into the compound interest formula. \n A = $12,000 * (1 + 0.07 / 12)^(12 * 3) \n Calculate the above expression to find A.

Step 3: Calculate the final amount for the second investment

Substitute P = $12,000, r = 0.0685 and t = 3 into the continuous compounding formula. \n A = $12,000 * e^(0.0685 * 3) \n Calculate the above expression to find A.

Step 4: Compare the results

Compare the amounts calculated in steps 2 and 3. The investment with the higher final amount yields the greatest return.

Key Concepts

Continuous Compounding

In the world of finance, investors are constantly looking for ways to grow their funds. That's where continuous compounding comes into play. This concept refers to the mathematical limit that compound interest can reach if it's calculated and added to the principal balance an infinite number of times, theoretically every moment. The formula that represents this process is often quoted as:

\[ A = Pe^{rt} \] where:

• \(A\) is the future value of the investment/loan, including interest,
• \(P\) is the principal investment amount (the initial deposit or loan amount),
• \(e\) is the base of the natural logarithm (approximately equal to 2.71828), sometimes referred to as Euler's number,
• \(r\) is the annual interest rate (decimal),
• \(t\) is the time the money is invested or borrowed for, in years.

Continuous compounding is the extreme case of compounding frequency. Unlike simple or standard compound interest, where interest is calculated periodically, continuous compounding calculates interest in an ongoing way. This formula is particularly important when dealing with investments or loans that specify a continuously compounded rate, such as the example of the 6.85% continuously compounded investment from the exercise.

Compound Interest Formula

When it comes to growing savings, compound interest is an essential concept for students and investors alike to understand. The standard compound interest formula, which differs from continuous compounding, is expressed as:

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

Breaking Down the Formula

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

With compound interest, the interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. For the exercise in question, understanding this formula is crucial to calculate the return of a 7% interest rate compounded monthly over 3 years. One must plug the respective values into the formula to find the future value of the investment.

Annual Interest Rate

An annual interest rate is the percentage increase in money that can be earned or paid on an investment or loan over a year. This rate is critical in both of our given formulas as it determines how much interest will be applied to the principal sum. It is commonly presented in a percentage form, but when used in formulas for calculations, it is converted to a decimal by dividing the percentage by 100.

For example, a 7% annual interest rate is expressed as 0.07 in our formulas. Remember to always convert the percentage into a decimal when substituting it into the compound interest formulas.

Dealing with different compounding frequencies alters how the annual interest rate is applied, and this is where \(n\) in the standard compound interest formula plays a vital role; it adjusts the rate to reflect the number of times compounding occurs per year. For continuous compounding, however, the rate is used as-is in the exponential function. Therefore, understanding the role of the annual interest rate and its application in compounding scenarios is a fundamental concept for any student tackling financial mathematics.