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 $866.07 more than Spenser, considering their respective annual growth rates and the retirement fund's annual administrative fee.

Step-by-step solution

Annual Percentage Growth Calculation for Alexx

Alexx invests directly and earns an annual percentage growth of 5%. Let the initial principal amount be denoted by `P` and the number of years `n=30`. So, the growth formula for Alexx's investment can be described as: A = P * (1 + r)^n where A is the amount after 30 years, r is the annual interest rate = (5/100) = 0.05.

Annual Percentage Growth Calculation for Spenser

Spenser uses a retirement fund and earns an annual growth percentage of 4.75\%. However, there's an administration fee of 0.25\% each year. So, the effective rate Spenser earns is 4.75-0.25 = 4.5\%. Let the same formula be used for Spenser's investment, i.e., A = P * (1 + r)^n In this case, r = (4.5/100) = 0.045.

Calculate Alexx's and Spenser's Final Amounts

For Alexx: P = $5000 r = 0.05 n = 30 A_Alexx = P * (1 + r)^n => A_Alexx = 5000 * (1 + 0.05)^30 For Spenser: P = $5000 r = 0.045 n = 30 A_Spenser = P * (1 + r)^n => A_Spenser = 5000 * (1 + 0.045)^30

Calculate the Difference between Alexx's and Spenser's investments

To find the difference, subtract A_Spenser from A_Alexx: Difference = A_Alexx - A_Spenser => Difference = (5000 * (1 + 0.05)^30) - (5000 * (1 + 0.045)^30) Now, compute the values and find the difference: Difference = (5000 * (1.05)^30) - (5000 * (1.045)^30) Difference ≈ $866.07 So, after 30 years, Alexx will have approximately $866.07 more than Spenser.

Key concepts

Compound Interest

Understanding compound interest is essential for anyone looking to invest in a retirement fund. It refers to the process where the interest earned on an investment is reinvested, allowing subsequent interest to be earned on the initial principal and the accrued interest. This can significantly boost the growth of your investment over time.

Take the case of Alexx and Spenser. Alexx's investment grows at the rate of 5% per year. If we use the formula for compound interest, which is \(A = P \times (1 + r)^n\), where \(P\) is the principal amount (\$5000), \(r\) is the annual interest rate (expressed as a decimal), and \(n\) is the number of years; Alexx's investment growth can be calculated. This formula reflects the exponential growth of the investment due to compound interest, where not just the original principal earns interest, but the accumulated interest does as well.

Investment Growth Calculation

To understand how much an investment will grow, one can use the compound interest formula explained above. This calculation is crucial for planning retirement savings because it helps investors to estimate the future value of their investments.

The growth of Alexx’s and Spenser's investments is impacted by the interest rates and the administrative fee that Spenser pays. Alexx's investment grows at a rate of 5%, while Spenser’s grows at 4.75% but effectively at 4.5% after the fee. By applying their respective rates to the compound interest formula over 30 years, one can determine the future value of their investments. Alexx’s investment would grow as follows: \(A_{Alexx} = 5000 \times (1 + 0.05)^{30}\), while Spenser’s investment calculation would slightly differ: \(A_{Spenser} = 5000 \times (1 + 0.045)^{30}\). This step-by-step calculation provides a clear picture of how small differences in interest rates and fees can have a significant impact over time.

Retirement Savings

Planning for retirement savings involves not only putting money aside but also making informed investment decisions that account for factors such as compound interest, administration fees, and the time value of money. As shown in the example of Alexx vs. Spenser, even a small administrative fee can affect the total savings over a long period. Saving early and understanding these factors can make a substantial difference in the amount accumulated at retirement.

The exercise demonstrates that, because of the difference in net growth rates, after 30 years, Alexx will end up with approximately an extra $866.07 compared to Spenser. This highlights the power of compound interest and the importance of considering fees when choosing a retirement fund. The earlier and more carefully one begins to save and invest for retirement, taking advantage of the benefits of compound interest, the more financially secure they will be in their retirement years.