Problem 108 - Essential Calculus

Question

Consider making monthly deposits of \(P\) dollars in a savings account at an annual interest rate \(r .\) Use the results of Exercise 106 to find the balance \(A\) after \(t\) years if the interest is compounded (a) monthly and (b) continuously. $$ P=\$ 75, \quad r=5 \%, \quad t=25 \text { years } $$

Step-by-Step Solution

Answer

The balance after 25 years with a monthly deposit of $75 and an annual interest rate of 5% when interest is compounded monthly will be \$78896.67 while continuously compounded interest will result in a balance of \$79001.45.

Step 1: Calculate amount with compounded monthly interest

First let's calculate the future value for the case when the interest is compounded monthly. Here, \( n = 12 \) because there are 12 months in a year, \( P = \$75 \), \( r = 0.05 \) because 5% has to be expressed as a decimal, and \( t = 25 \). Substituting the values into the formula, we get: \( A = \$75 \times \left(1+\frac{0.05}{12}\right)^{(12\times 25)} \) = \$78896.67.

Step 2: Calculate amount with continuously compounded interest

Now let's calculate the future value when the interest is compounded continuously. The future value formula for the case is \( A = Pe^{rt} \). Substituting our values into the formula: \( A = \$75 \times e^{(0.05 \times 25)} \) = \$79001.45.

Step 3: Compare the results

In this exercise, although the difference is not stark, compound interest continuously gives slightly higher returns compounding monthly. So, if given a choice, opt for continuous compounding for higher returns.

Key Concepts

Continuous Compounding

Continuous compounding involves calculating and adding interest to an account's balance continuously, rather than at regular intervals like daily, monthly, or yearly. This concept uses the mathematical constant 'e', which is approximately 2.718. The formula for calculating continuous compounding is:\[ A = P e^{rt} \]Here:

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

When you compound interest continuously, you assume that you are adding interest not just at the end of each time period, but at an infinite number of moments within that period. This results in slightly higher returns compared to other compounding methods.
In the given problem, where \( P = \\(75 \), \( r = 0.05 \), and \( t = 25 \), continuous compounding yields a future value of approximately \( \\)79001.45 \). This demonstrates how continuous compounding subtly enhances growth over a long-term investment.

Monthly Compounding

Monthly compounding is a more common form of compounding interest where interest is calculated and added to the account balance at the end of each month. Each time the interest is added, you begin earning interest on that previously accumulated interest. The formula for monthly compounding is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount.
• \( r \) is the annual interest rate (decimal).
• \( n \) represents the number of compounding periods per year (12 for monthly compounding).
• \( t \) is the time the money is invested for in years.

The method effectively grows your money by adding interest at regular, frequent intervals. In the original problem, using the formula for \( P = \\(75 \), \( r = 0.05 \), \( n = 12 \), and \( t = 25 \), the future value is calculated to be \( \\)78896.67 \).
This approach is popular in savings accounts and loans, primarily due to its frequent interest additions, which can compound quite quickly.

Future Value Calculation

The future value calculation is used to determine how much an investment made today will grow over a specified time period with a specified rate of return. It helps in understanding the growth potential of your investments.
The future value depends heavily on the method of compounding chosen. As seen in the original exercise, compounding method choices greatly influence the final amount. With monthly compounding, the future value of deposits was \( \\(78896.67 \) and with continuous compounding, it was \( \\)79001.45 \).
This calculation is particularly useful in financial planning for goals like retirement, education savings, or significant purchases. It lets you gauge how much you should invest now to achieve a desired financial target in the future. By understanding future value and compounding, you can make more informed decisions and comparisons between different investment options.