Problem 82 - Algebra and Trigonometry

Question

Use the formula for the value of an annuity to solve Exercises 77–84. 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

a. The company will have approximately \$931,506 in scholarship funds at the end of ten years. b. The interest earned is approximately \$331,506.

Step 1: Understand and identify given values

Here, the initial principal or investment amount for every quarter is \$15,000. The annual interest rate is 9%, and this is compounded quarterly. Further, the investment duration is 10 years.

Step 2: Calculate total number of periods

Since the investment is compounded quarterly (four times a year) and the total duration is 10 years, the total number of periods would be \(N = 4 \times 10 = 40\).

Step 3: Convert annual interest rate to quarterly interest rate

Since the interest is compounded quarterly, we need to divide the annual interest rate by 4 to find the quarterly interest rate. So, it will be \(r = 9 \% / 4 = 2.25 \% = 0.0225\) in decimal.

Step 4: Calculate the future value of the annuity

The formula to calculate the future value of an ordinary annuity is given by \(FV = P \times \frac{((1 + r)^N - 1)}{r}\). Substituting the values we get, \(FV = 15000 \times\frac{(1 + 0.0225)^{40} - 1}{0.0225}\). Calculating it further, we get \(FV \approx \$931,506\).

Step 5: Calculate the interest

The interest earned is the final value minus the total invested. The total invested is the regular investment amount times the number of periods which is \$15000 \times 40 = \$600000. Thus, the interest earned is \(\$931,506 - \$600,000 = \$331,506\).

Key Concepts

Future Value of Annuity

The future value of an annuity is the total value of a series of regular payments at the end of a specified period, taking into account compound interest. Imagine an annuity as a piggy bank you keep adding to every period (monthly, quarterly, etc.), and each contribution grows due to interest.

For example, if a company is investing $15,000 every quarter for ten years, with a 9% annual interest rate compounded quarterly, future value helps us determine how much the company will have in the end. To calculate this, we apply the future value formula for an annuity:
\[ FV = P \times \frac{{(1 + r)^N - 1}}{r} \]
where \( P \) is the payment amount per period, \( r \) is the interest rate per period, and \( N \) is the total number of periods. Through the step by step solution, we've learned that this formula helps us peek into the future and estimate the benefits of regular investments over time.

Compound Interest

Compound interest is what makes the money tree grow faster by earning interest on interest—like a snowball rolling downhill and getting bigger over time. It's different from simple interest, which only grows based on the initial principal. The magic of compound interest becomes noticeable over long periods, as the accumulated interest also begins to earn interest.

In our annuity example, a 9% annual interest compounded quarterly means that the interest isn’t just calculated and added at year’s end; instead, it’s applied four times a year. Every quarter, the investment gains some interest, and that extra amount is then factored into the next quarter's interest calculation. It's interest on top of interest, snowballing each period, which is an essential concept for growing investments.

Quarterly Compounding

Quarterly compounding occurs when the interest is calculated and added to the principal four times a year. In financial terms, a quarter refers to a three-month period. With this method, an annual interest rate is divided into four to get the quarterly rate.

For instance, a 9% annual rate becomes 2.25% each quarter. More frequent compounding means the investment grows faster than it would with annual compounding because each quarter's interest becomes part of the principal for the next period’s interest calculation, increasing the effect of compound interest.

To truly appreciate the power of quarterly compounding, let's visualize each quarter's added interest like a brick added to a growing structure—the more often we add bricks (compounding periods), the quicker the structure (investment) reaches the skies (financial goals).