Problem 5 - Precalculus Mathematics for Calculus

Question

Find the amount of an annuity that consists of 20 annual payments of \(\$ 5000\) each into an account that pays interest of \(12 \%\) per year.

Step-by-Step Solution

Answer

The amount of the annuity is approximately \( \$360,262.50 \).

Step 1: Identify the annuity formula

For this annuity problem, we will use the future value of an ordinary annuity formula, which is given by: \[FV = P \times \frac{(1 + r)^n - 1}{r}\]where \( FV \) is the future value of the annuity, \( P \) is the annual payment (\(5000\) in this case), \( r \) is the annual interest rate expressed as a decimal (\(0.12\)), and \( n \) is the number of payments (\(20\)).

Step 2: Convert the interest rate

The interest rate for the account is given as 12\% per year. To use it in the annuity formula, we must convert this percentage into a decimal. This is done by dividing by 100: \[0.12 = \frac{12}{100}\]

Step 3: Calculate the growth factor

Next, we calculate the growth factor by raising \(1 + r\) to the power of \(n\):\[(1 + r)^n = (1 + 0.12)^{20}\]

Step 4: Compute the growth factor

Using a calculator, evaluate \((1.12)^{20}\):\[(1.12)^{20} \approx 9.6463\]

Step 5: Compute the future value of the annuity

Now substitute the values back into the annuity formula to find \(FV\):\[FV = 5000 \times \frac{9.6463 - 1}{0.12}\]Calculate \[\frac{9.6463 - 1}{0.12} = \frac{8.6463}{0.12} \approx 72.0525\].Then,\[FV = 5000 \times 72.0525 \approx 360,262.50\]

Step 6: Final Result

After computing all parts of the formula, the future value of the annuity, or the amount accumulated after 20 annual payments each of \( \\(5000 \) at \( 12\% \) interest per year, is approximately \( \\) 360,262.50 \).

Key Concepts

ordinary annuity

An ordinary annuity is a series of equal payments made at the end of consecutive periods over a fixed length of time. In the context of personal finance and investments, this might involve regular deposits into a retirement fund or similar financial product. Ordinary annuities are common financial vehicles that people use to plan for future expenses. Usually, the payments are made on a regular schedule, such as monthly, quarterly, or annually. In this exercise, the annuity consists of 20 annual payments, each of $5,000. Because these payments are made annually and consistently, it qualifies as an ordinary annuity. A key characteristic of ordinary annuities is that interest is not compounded until the end of the payment period. This means if you make a payment today, you start earning interest from this point forward. In the example provided, each of the $5,000 payments will start accruing interest at the end of the payment period, aligning with the nature of an ordinary annuity. This distinction affects how the future value of the annuity is calculated.

annual interest rate

The annual interest rate is a critical component in the calculation of future values in annuities. It represents the percentage charged on the principal by a lender for the use of its money, or the percentage earned on the annuity payments by the investor. In this problem, the annual interest rate is given as 12%. To use this rate effectively in formulas, it must be converted from a percentage to a decimal. Therefore, you divide by 100, changing 12% into 0.12. This decimal form is essential when performing calculations involving growth factors or compound interest. It’s important to ensure accuracy in this conversion because a mistake in the rate can lead to a significantly incorrect future value calculation. The annual interest rate determines how much interest accumulates over each period before payments are made. In this situation, the rate of 12% per year enables the annuity to grow substantially over the 20-year period.

future value calculation

Calculating the future value of an annuity allows us to determine how much money will accumulate from constant deposits made over time, considering the effects of compounded interest. To find the future value of an ordinary annuity, the main formula used is:\[FV = P \times \frac{(1 + r)^n - 1}{r}\]Where:- \(FV\) is the future value of the annuity,- \(P\) is the amount of each payment,- \(r\) is the annual interest rate (as a decimal),- and \(n\) is the total number of payments.In our scenario, substitutes are made as follows: each payment \(P\) is \(5,000, the rate \(r\) is 0.12, and the total number of payments \(n\) is 20. Using these figures, we first calculate the growth factor by evaluating \((1 + 0.12)^{20}\). Once this is done, we use the future value formula to determine exactly how much the payments will grow over the payment period.Thus, the future value, calculated as \)360,262.50, indicates the amount gathered after making these regular deposits under 12% interest, emphasizing the power of compound interest in multiplying savings over time.