Problem 82 - College Algebra

Question

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. To offer scholarship funds to children of employees, a company invests \(\$ 15,000\) at the end of every three months in an annuity that pays \(9 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.

Step-by-Step Solution

Answer

The company will have an approximate value in scholarship funds at the end of ten years. The total interest earned for this period is the difference between the future value and the total investments made which will be computed in step 4.

Step 1: Identifying Variables

Identify and assign variables based on the question. Let P equals $15000, r equals 0.09, n equals 4 (as the interest is compounded every three months in a year) and t equals 10 years.

Step 2: Compute the future value of annuity

Insert the identified values into the future value of an annuity formula to calculate the total funds at the end of ten years. The formula becomes \(FV = 15000 \times [ (1 + 0.09/4)^{4*10} - 1 ] / (0.09/4)\). Solve this to get the future value of the annuity.

Step 3: Calculate the Total Investment

Calculate the total amount invested by the company over the 10 years by multiplying the quarterly investment by the total number of quarters in 10 years. This is done as Total Investment = $15000 * 4 * 10.

Step 4: Determine the Interest

The interest earned can be found by subtracting the total investment from the future value of the annuity. This is done as Interest = Future Value - Total Investment.

Key Concepts

Future Value

The future value of an annuity is the amount of money you can expect to have in the future, after regularly investing over a period of time. In this case, the company wants to know how much the scholarship fund will grow after investing \(15,000 at the end of every three months for 10 years.

The concept of future value is central to understanding how investments grow over time. By using the future value formula for annuities \[ FV = P imes \frac{(1 + r/n)^{nt} - 1}{r/n} \] where:

• \( P \) is the amount of each annuity payment (e.g., \)15,000),
• \( r \) is the annual interest rate (e.g., 9%),
• \( n \) is the number of times the interest is compounded per year (e.g., quarterly means 4), and
• \( t \) is the total number of years (e.g., 10 years),

we calculate the total scholarship fund the company will have at the end of 10 years.

Compounded Interest

Compounded interest refers to the process where the investment earns not only on the original principal but also on the accumulated interest from previous periods. This effect significantly increases the future value of an investment over time.

In our problem, the interest on the investment is compounded quarterly, meaning it is calculated and added to the investment four times a year. Each time the interest is added, it becomes part of the new principal, which earns interest in the next period.

The formula for compounded interest used in this context is: \[ (1 + r/n)^{nt} \] This helps us understand how much the investment grows due to the power of compound interest. In simple terms, it is "interest on interest," making this a powerful tool for long-term wealth accumulation. We can see that with compounding, even small differences in the interest rate or period can lead to significant differences in the final amount.

Interest Rate

An interest rate represents the percentage increase on the invested amount over a specific period. In our scenario, the interest rate is 9% annually.

It's important to note how the interest rate is applied within different compounding periods. To understand its effect, this annual rate is often converted into a rate that matches the compounding periods (quarterly in this case). This is done by dividing the annual rate by the number of compounding periods per year.

• The quarterly interest rate is thus \( \frac{0.09}{4} \).

Understanding the rate's periodic conversion aids in accurate calculations of the final investment value. The interest rate, particularly when compounded, has a huge impact on the growth of investments. Higher rates can significantly enhance accumulated returns, highlighting the importance of hybrid strategies focusing not only on high rates but also on the compounding frequency.

Quarterly Investments

Quarterly investments refer to regular payments made every three months into an investment account, as seen in the company's strategy. By investing this way, investors can take advantage of the benefits of regular contributions and compound interest.

In this problem, the company invests $15,000 at the end of each quarter. This means there are four investments per year, totaling 40 quarterly investments over 10 years.

Regular investments help in building a habit of saving and investing. Additionally, they provide a buffer against market volatility, as money invested at different times may yield varying interest, leading to dollar-cost averaging. This strategy allows the company to amass a scholarship fund of significant size through consistent, periodic investments while leveraging compounding to enhance returns.