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. $$ P=\$ 75, \quad r=5 \%, \quad t=25 \text { years } $$

Short 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-by-step solution

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.

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.

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.