Question

Many retirement funds charge an administrative fee each year equal to \(0.25 \%\) on managed assets. Suppose that Alexx and Spenser each invest \(\$ 5,000\) in the same stock this year. Alexx invests directly and earns \(5 \%\) a year. Spenser uses a retirement fund and earns 4.75\%. After 30 years, how much more will Alexx have than Spenser?

Short answer

After 30 years, Alexx will have approximately $2407.63 more than Spenser.

Step-by-step solution

Calculate the future value of Alexx's investment

Using the compound interest formula, we can calculate the future value of Alexx's investment after 30 years. In this case, PV = $5,000, r = 0.05, n = 1, and t = 30. \[FV_A = 5000(1 + 0.05/1)^{1\times30}\] Calculate the future value: \[FV_A = 5000(1.05)^{30}\] \[FV_A \approx 21634.60\]

Calculate the future value of Spenser's investment

Similarly, we can calculate the future value of Spenser's investment after 30 years. In this case, PV = $5,000, r = 0.0475, n = 1, and t = 30. \[FV_S = 5000(1 + 0.0475/1)^{1\times30}\] Calculate the future value: \[FV_S = 5000(1.0475)^{30}\] \[FV_S \approx 19226.97\]

Find the difference between Alexx's and Spenser's investments

Now, we can find the difference between the future values of Alexx's and Spenser's investments: \[Difference = FV_A - FV_S\] Substitute the values calculated in steps 1 and 2: \[Difference = 21634.60 - 19226.97\] \[Difference \approx 2407.63\] After 30 years, Alexx will have approximately $2407.63 more than Spenser.

Key concepts

Future Value Calculation

Understanding the future value calculation is crucial when planning your financial goals. It tells you what an investment made today will be worth at a certain date in the future, considering a specific interest rate. The formula to calculate future value using compound interest is:
def[^1]:[^1]: $$ FV = PV \times (1 + \frac{r}{n})^{n \times t} $$ Where:

• \textbf{FV} is the future value of the investment
• \textbf{PV} is the present value or the initial amount invested
• \textbf{r} is the annual interest rate (in decimal form)
• \textbf{n} is the number of times that interest is compounded per year
• \textbf{t} is the number of years the money is invested for

To make it student-friendly, consider an example: If you invest $1,000 at an interest rate of 5% compounded annually, over 10 years, your future value will be calculated as follows: def[^2]:[^2]: $$ FV = 1000 \times (1 + 0.05)^{10} $$ This formula is a cornerstone in understanding how investments grow over time, and it's the same formula used in the textbook exercise with Alexx and Spenser's investments.

Retirement Fund Investment

When it comes to saving for retirement, a retirement fund investment can be a wise choice. These funds are designed to grow over a long period, taking advantage of compound interest. As with any investment, they may have fees associated with them, like the administrative fee in our example, which can affect the overall return. It’s important to understand the impact of these fees.
For instance, an administrative fee can seem small as a percentage, but over time it can significantly reduce the amount of money you have at retirement. In our example, Spenser is investing through a retirement fund with a 0.25% fee, which reduces the effective annual return to 4.75%. Over 30 years, this seemingly minor fee has a noticeable impact on the final amount Spenser will receive. Here is a quick glance at the difference a fee can make when considering retirement fund investments:

• Without fee: 5% annual interest
• With 0.25% fee: 4.75% annual interest

It's a stark reminder to factor in all costs when planning your retirement strategy.

Investment Returns Over Time

The power of compound interest shines when evaluating investment returns over time. The longer the duration, the greater the effect compound interest has on the investment's growth, often referred to as the 'miracle of compounding.' In our exercise, with each passing year, the earned interest is added to the principal, and from that point on, interest is earned on the new sum. It’s a cycle that leads to exponential growth of your investments.

Considering our example, after 30 years, Alexx earns more due to a higher interest rate, compounded annually, without the deduction of fees. This demonstrates how a slightly higher interest rate, when compounded over many years, can lead to a significantly larger sum.
For students, it's vital to apply this concept to long-term savings goals, such as retirement. Start investing early, remain aware of fees and rates, and use the compound interest formula to project future savings. For instance, adjusting your investment strategy in your 20s to aim for a higher return rate, even by half a percentage point, could result in a substantially larger retirement fund by the time you decide to retire.