Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the amount (future value) of each ordinary annuity. $$\$ 1000$$ /year for 10 yr at \(10 \% /\) year compounded annually

Step-by-Step Solution

Answer

The future value of this ordinary annuity, after 10 years at a 10% interest rate compounded annually, is approximately \$15,937.

Step 1: Identify the given values

We are given the following values: - Annual payment (P): $1000 - Interest rate (r): 10% per year = 0.1 - Number of years (n): 10 years

Step 2: Use the future value of annuity formula

Now we will use the future value of annuity formula to find the total value of the annuity after 10 years: \[FV = P \times \frac{(1 + r)^n - 1}{r}\] Plugging in the given values, we get: \[FV = 1000 \times \frac{(1 + 0.1)^{10} - 1}{0.1}\]

Step 3: Calculate the future value

Let's now perform the calculations inside the brackets first: (1 + 0.1) = 1.1 \ (1.1)^10 ≈ 2.5937 Next, subtract 1 from 2.5937: 2.5937 - 1 = 1.5937 Now, divide 1.5937 by 0.1: 1.5937 ÷ 0.1 = 15.937 Finally, multiply the result by the annual payment ($1000): 15.937 x 1000 = $15937

Step 4: Express the final answer

The future value of this ordinary annuity, after 10 years at a 10% interest rate compounded annually, is approximately \$15,937.

Key Concepts

Ordinary Annuity

An ordinary annuity refers to a series of equal payments made at regular intervals over a period of time. In our exercise, we're dealing with an ordinary annuity because payments are made annually. To understand this concept better, let's break it down:

• An annuity features a fixed sum of money (here it's $1000) paid each year.
• The payments occur at the end of each period (year), which defines it as "ordinary."
• This contrasts with an annuity due, where payments are made at the beginning of each period.

Ordinary annuities are very common in financial products like retirement savings plans and certain insurance products. They help individuals to determine how much they can accumulate over time and plan their finances accordingly.

Compounding Interest

Compounding interest is a critical concept when understanding how money grows over time, especially in investments like annuities. It refers to the process of earning interest on both the initial principal and the accumulated interest from previous periods. This is how it applies:

• In our case, the interest is compounded annually, meaning the interest calculation occurs once each year.
• Each year, the earned interest is added to the principal, which increases the principal for the next year's interest calculation.
• The formula for calculating the future value of an annuity already includes this compounding aspect.

Compounding interest can significantly increase the amount of money in an annuity over time, as seen in the final future value of $15,937 from the original exercise.

Mathematics Education

Understanding annuities and compounding interest is a vital part of mathematics education, particularly in financial mathematics. These concepts provide students with practical skills for managing personal finances and making informed economic decisions.

• Concepts like ordinary annuities and compound interest are crucial for understanding loans, mortgages, and savings as they involve similar financial calculations.
• Students learn to apply mathematical formulas in real-life situations, reinforcing their analytical and problem-solving skills.
• Having a grasp of these ideas prepares students for more advanced financial studies or careers in fields like accounting and finance.

Overall, teaching about annuities and compounding interest not only helps develop mathematical proficiency but also fosters educated and financially savvy individuals.