Problem 67 - College Algebra with Corequisite Support

Question

Alyssa opened a retirement account with \(7.25 \%\) APR in the year 2000. Her initial deposit was \(\$ 13,500\). How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?

Step-by-Step Solution

Answer

With monthly compounding, \(\$75,535.61\); continuous gives \(\$75,812.67\); \(\$277.06\) more with continuous.

Step 1: Understanding the Problem

We need to calculate the future value of Alyssa's retirement account with two different compounding methods over 25 years (from 2000 to 2025). The initial deposit is \(\$13,500\), and the annual percentage rate (APR) is \(7.25\%\). First, we will calculate the amount with monthly compounding, and then with continuous compounding.

Step 2: Calculating Future Value with Monthly Compounding

The formula for future value with monthly compounding is given by \[A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(P = 13,500\), \(r = 0.0725\), \(n = 12\), and \(t = 25\) years. We solve:\[A = 13,500 \left(1 + \frac{0.0725}{12}\right)^{12 \times 25} = 13,500 \times (1.00604)^{300} \]Calculating further gives us the future value, \(A \approx \$75,535.61\).

Step 3: Calculating Future Value with Continuous Compounding

The formula for future value with continuous compounding is \[A = Pe^{rt} \]where \(P = 13,500\), \(r = 0.0725\), and \(t = 25\). Substituting in the values, we have:\[A = 13,500 \times e^{0.0725 \times 25} \]Calculating this, we find \(A \approx \$75,812.67\).

Step 4: Determining the Difference in the Two Methods

The difference in the final amounts from the two compounding methods is:\[\\(75,812.67 - \\)75,535.61 = \\(277.06\]Alyssa would make an additional \(\\)277.06\) with continuous compounding compared to monthly compounding.

Key Concepts

Future Value

The future value of an investment is the amount the investment is worth after a certain period. This calculation helps in understanding the growth potential of the invested money over time. In Alyssa's case, the future value is determined by the amount in her retirement account after 25 years. To find the future value, we must consider the principal amount, the interest rate, and the compounding frequency. These factors together let us predict how much money Alyssa will have at her desired time, considering the interest rate of 7.25%. This rate indicates how fast her money grows annually.
Understanding future value is crucial because it informs financial decisions related to savings and investments. If Alyssa knows how much she will have in 2025, she can plan her retirement more effectively.

• It helps in long-term financial planning.
• Adjusts for inflation and potential risks.
• Provides insight into yield from different compounding methods.

Continuous Compounding

Continuous compounding is a situation where the compounding of interest happens endlessly within a given period. Unlike other compounding methods that might be yearly or monthly, continuous compounding means interest is calculated and added back to the principal continuously. The future value with continuous compounding can be very slightly higher than those with monthly compounding due to the perpetual application of interest.
The formula used for continuous compounding is given by \(A = Pe^{rt}\), where \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is time in years. Here, \(e\) is the mathematical constant approximately equal to 2.71828. It's exciting to see how small differences accumulate into a noticeable sum.

• Presents the maximum return possible from an investment.
• Ideal for theoretical and precise calculations.
• Often applied in natural growth processes and financial models.

Monthly Compounding

When using monthly compounding, interest is calculated each month and added to the total investment. This means that 12 times a year, the interest rate is divided by 12 and applied to the total amount to compute the new balance. The future value with monthly compounding is determined by the formula \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where \(P\) is the initial investment, \(r\) is the annual interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the total number of years.
Monthly compounding allows more frequent accumulation, which can lead to more noticeable growth compared to annual compounding. However, it slightly trails behind continuous compounding in returns.

• More common than continuous compounding in real-world scenarios.
• Reflects the typical banking practices for savings accounts.
• Balances accuracy with practicality in financial calculations.