Problem 47 - College Algebra and Calculus: An Applied Approach

Question

MAKE A DECISION: CASH SETTLEMENT You invest a cash settlement of \(\$ 10,000\) for 5 years. You have a choice between an account that pays \(6.25 \%\) interest compounded monthly with a monthly online access fee of \(\$ 5\) and an account that pays \(5.25 \%\) interest compounded continuously with free online access. Which account should you choose? Explain your reasoning.

Step-by-Step Solution

Answer

By following the step-by-step solution, the future value of both accounts can be calculated. The account to choose would be the one that provides the highest future value after 5 years.

Step 1: Understanding the Compound Interest Setup

Firstly distinguish between the two offers: a monthly compounded interest account with a 6.25% annual rate and a $5 monthly fee, and an account with continuously compounded 5.25% interest with no fee. The formula for compound interest is \(A = P(1 + r/n)^(nt)\) where: \(A\) is the future value of the investment/loan including interest, \(P\) is the principal investment amount, \(r\) is annual interest rate (in decimal), \(t\) is time the money is invested for in years, and \(n\) is number of times that interest is compounded per year.

Step 2: Apply the Compound Interest Formula

Next, apply the formula for Account 1. The given values are: \(P = $10,000\), \(r = 0.0625\) (6.25% in decimal form), \(t = 5\) years, \(n = 12\) times/year. However, remember to subtract the monthly fee times the total number of months from the result, which equals the sum $5 * 12 * 5.

Step 3: Understand the Setup for Continuously Compounded Interest

For the second account, since interest is compounded continuously, a different interest formula is going to be used, which is \(A = Pe^(rt)\) where: \(e\) is Euler's number and is approximately equal to 2.71828.

Step 4: Apply the Continuously Compounded Interest Formula

Next, apply this formula for account 2. The given values are \(P = $10,000\), \(r = 0.0525\) (5.25% in decimal), \(t = 5\).

Step 5: Compare the Future Values

Finally, compare the future values of both accounts. Choose the account that gives the highest future value after 5 years. This is the account that should be chosen.

Key Concepts

Monthly Compounded Interest

When interest is compounded monthly, the investment grows by adding the interest after every month. This means, every month, a little bit of interest is added to your principal amount, which then further earns interest in the following months. The formula to calculate the future value of an investment with monthly compounded interest is given by:\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]Where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (initial investment).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times the compound interest is applied per year (12 for monthly).
• \(t\) is the number of years the money is invested or borrowed.

Remember, if there are any additional fees, like a monthly online access fee of $5, you need to subtract this from the total accumulated amount at the end.

Continuously Compounded Interest

Continuously compounded interest is a fascinating concept where the interest is calculated at an infinite number of times per year. Essentially, the interest is added for each and every moment, leading to slightly more interest accumulated compared to regular compounding periods. The formula utilized for continuously compounded interest is:\[A = Pe^{rt}\]Where:

• \(A\) stands for the future value of the investment/loan, including interest.
• \(P\) is the principal investment amount.
• \(r\) is the annual interest rate (decimal form).
• \(t\) represents the time in years.

The beauty of continuous compounding is that it leverages Euler's Number \(e\), approximately 2.71828, which allows for a more constant growth trajectory. One of the key attractions of continuous compounding is often its simplicity, as there are no periodic compounding adjustments involved, making it an attractive option for many investors.

Future Value Formula

The future value formula is crucial in deciding between investment options. It determines how much a present amount of money is going to be worth in the future, which is essential for planning and making informed financial decisions. Let's examine both formulas here:- For regular compounding (like monthly): \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]- For continuous compounding: \[ A = Pe^{rt} \]Both formulas aim to answer the same question: What will my investment be worth "t" years from now? Whether you're using regular or continuous compounding, understanding these formulas allows you to predict and compare the growth of your investments, taking different compounding frequencies into account.

Investment Comparison

In deciding between different investment options, comparing the future values using related formulas is key. In our example, two different accounts have distinct characteristics:

• Account 1 offers a higher interest rate at 6.25% compounded monthly but has a monthly fee.
• Account 2 offers a lower rate at 5.25% but is compounded continuously with no fees.

For a fair comparison: 1. Calculate the future value for both accounts. 2. Account for extra fees or benefits, such as the absence of a fee or convenience features.
In essence, even if the monthly compounded interest's rate seems higher initially, additional charges, like the $5 monthly fee, can lessen its attractiveness. On the other hand, continuously compounded interest may accumulate less at first glance but can be more beneficial in the absence of added costs. With these calculations and facts in mind, a thorough analysis shows that after considering all factors, the account with the highest future value after all deductions might be the best choice.