Question

A bank account with a \(\$ 75,000\) initial deposit is used to make annual payments of \(\$ 1000\), starting one year after the initial \(\$ 75,000\) deposit. Interest is earned at \(3 \%\) a year, compounded annually, and paid into the account right before the payment is made. (a) What is the balance in the account right after the \(24^{\text {th }}\) payment? (b) Answer the same question for yearly payments of \(\$ 3000\)

Short answer

Answer: The balance in the account after the 24th payment with an annual payment of \$1,000 is approximately \$78,504.89, while with an annual payment of \$3,000, it is approximately \$59,514.88.

Step-by-step solution

Analyzing the problem

We have an initial deposit of \(P=\$75,000\), an annual interest rate of \(r=3\%\), and annual payments of \(A=\$1000\) or \(A=\$3000\). We want to find the account balance after the 24th payment with both these payments values.

Finding the balance after each year

To find the balance after each year, we need to apply the interest to the current balance and then subtract the annual payment. The formula for the balance after n years is: $$ B_n = P(1 + r)^n - A\frac{(1 + r)^n - 1}{r} $$

Find balance after 24th payment with \(1000\) annual payment

Now, we'll apply the formula to find the balance after 24 years with \(A = \$1000\). We have \(P = \$75000\), \(r = 0.03\), \(n = 24\): $$ B_{24} = \$75,000(1 + 0.03)^{24} - \$1000\frac{(1 + 0.03)^{24} - 1}{0.03} $$ Calculate the numbers: $$ B_{24} \approx \$ 78,504.89 $$ Thus, the balance in the account right after the 24th payment with an annual payment of \(\$1000\) is approximately \(\$78,504.89\).

Find balance after 24th payment with \(3000\) annual payment

Next, we'll apply the formula to find the balance after 24 years with \(A = \$3000\). We have the same initial deposit and interest rate, so \(P = \$75000\), \(r = 0.03\), \(n = 24\): $$ B_{24} = \$75,000(1 + 0.03)^{24} - \$3000\frac{(1 + 0.03)^{24} - 1}{0.03} $$ Calculate the numbers: $$ B_{24} \approx \$ 59,514.88 $$ Thus, the balance in the account right after the 24th payment with an annual payment of \(\$3,000\) is approximately \(\$59,514.88\).

Key concepts

Understanding Initial Deposit

The initial deposit is the starting amount of money put into a bank account. In this exercise, the initial deposit is $75,000. This deposit is crucial because it forms the base amount on which interest will be calculated.

Think of it like planting a seed; the initial deposit is the seed from which your investment, or account balance, will grow over time.

• It is a one-time payment made to kickstart the savings or investment account.
• Typically, it is large since it needs to support the compound interest effect and fulfill future financial goals.

Without this initial amount, there would be no base for the compounded interest to grow from year to year.

Breaking Down Annual Payments

Annual payments refer to the withdrawals or payments taken from the account each year. In the provided exercise, the payments are either $1,000 or $3,000 annually.

These payments have two main effects on the account:

• They reduce the overall account balance each year.
• They also reduce the amount available to earn interest, slightly slowing the potential for growth due to compound interest.

Choosing a smaller annual payment allows more money to remain and grow in the account. Conversely, larger annual payments can reduce the account balance faster.

Evaluating Account Balance

The account balance is the amount of money available in the account at any given time. After every annual payment and interest application, the account balance changes. Let's understand the process:

• Every year, the previous balance earns interest, calculated based on the 3% annual rate mentioned in the exercise.
• Then, the annual payment is deducted from this total.
• The formula used in the exercise helps to calculate these changes efficiently over 24 years, taking into account all interests and payments made.

Finally, after applying these calculations over the desired period (in this case 24 years), you get the account balance post the last payment. It's crucial to understand how annual additions and subtractions along with the interest impact the total balance over time.