Question

MAKE A DECISION: SALES COMMISSION You invest a sales commission of \(\$ 12,000\) for 6 years. You have a choice between an account that pays \(4.85 \%\) interest compounded monthly with a monthly online access fee of \(\$ 3\) and an account that pays \(4.25 \%\) interest compounded continuously with free online access. Which account should you choose? Explain your reasoning.

Short answer

After calculating the future value of both accounts, the account with the higher future value should be chosen. The calculations and comparisons will provide the correct decision.

Step-by-step solution

Understand the Scenarios

There are two different investment choices. First choice offers an annual interest rate of 4.85% compounded monthly with a $3 monthly online access fee. Second choice offers a continuously compounded annual interest rate of 4.25% with no access fee.

Calculate Future Value for First Account

For the first choice, using the formula for the future value (FV) of a compound interest account, \(FV = P(1+ r/n)^{nt}\), where P represents the principal amount (\$12,000), r is the annual interest rate (4.85% or 0.0485), n is the number of times that interest is compounded per year (12), and t is the money deposited for number of years (6). But remember, we should also subtract the online access fee (\$3/month * 12 months * 6 years) from the result.

Calculate Future Value for Second Account

For the second choice, using the formula for the future value (FV) in a continuously compounded interest account, \(FV = Pe^{rt}\), where P is the principal amount, r is the annual interest rate (4.25% or 0.0425), e is the base of the natural logarithm (approx. 2.71828), and t is the time the money is invested for.

Compare the Results

Once the future values for both accounts have been calculated, they should be compared. The account with the higher future value should be chosen for investment due to its higher return

Making Decision

Make the decision based on the comparison of the future values of the two accounts. The account with the higher resulting value will be affordable and beneficial.

Key concepts

Future Value Calculation

Future value calculation is a key concept in finance focusing on understanding how much an investment will grow over a given period of time when interest is applied. It is essential for individuals saving for goals like retirement or education funding, as well as for businesses looking at the future value of their investments.

Consider a sales commission of \$12,000 invested for 6 years, the future value (FV) with monthly compounding interest can be determined using the formula:
\[ FV = P(1 + \frac{r}{n})^{nt} \]
where P represents the initial principal balance, r is the annual interest rate in decimal form, n is the number of times interest is applied per year, and t is the time in years. In this scenario, the account has an annual interest rate of 4.85%, compounded monthly, which requires subtracting the cost of the monthly access fee to determine the actual earnings.

It's important to consider that fees can significantly impact the effective return on investment, which the above calculation reflects by subtracting access fees over the investment period. This example underscores the importance of scrutinizing all potential costs associated with an investment account.

Continuously Compounded Interest

Continuously compounded interest represents the mathematical limit of an investment's growth as it earns interest upon interest at an infinitely small time scale. It provides the greatest possible amount an investment might reach, assuming constant reinvestment of interest.

The formula for calculating the future value of investment with continuous compounding is
\[ FV = Pe^{rt} \]
where P is the principal amount, r is the annual interest rate in decimal form, e is Euler's number (approximately 2.71828), and t is the time in years. Using this formula, an investment with a principal of \$12,000 at an annual interest rate of 4.25% over 6 years would grow based on the essence of continuous growth.

When comparing different investment options, remember that the absence of fees can be favorable, as evident in the scenario where the second account offers continuously compounded interest with no additional charges for access, emphasizing a key advantage over its counterpart that compounds interest monthly with a fee.

Comparing Investment Accounts

When comparing investment accounts, one must consider several factors including interest rates, compounding frequency, and any associated fees. Different accounts offer varying terms, and it's crucial to calculate the overall future value to determine which is more beneficial.

For instance, when deciding between two accounts - one with a higher interest rate and monthly fees and another with a slightly lower interest rate but no fees and continuous compounding - you must do the math to see which results in the highest end value. The previous calculations highlight that while the nominal interest rate is important, the effective rate after considering compounding frequency and fees ultimately determines the best choice. This process often requires careful analysis and comparison, as demonstrated in the presented exercise where the monthly compounding account is offset by a fee, possibly making the continuous compounding account more attractive despite its lower nominal rate.

Determining which investment option is better necessitates a full understanding of how each account compounds interest and the long-term impact of any fees. By conducting a thoughtful analysis using the future value calculations, investors can make informed decisions that align with their financial goals and maximize their returns.