Problem 2 - Algebra and Trigonometry

Question

Annuity Find the amount of an annuity that consists of 24 monthly payments of \(\$ 500\) each into an account that pays 8\(\%\) interest per year, compounded monthly.

Step-by-Step Solution

Answer

The annuity amounts to approximately $13,088.78 after 24 months.

Step 1: Understanding the Problem

We have 24 monthly payments of $500 each, and the interest rate is 8% per year. Since the interest is compounded monthly, we need to find out how it affects our calculations.

Step 2: Identify the Formula

To find the future value of an annuity with monthly payments, we use the formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where \( FV \) is the future value of the annuity, \( P \) is the monthly payment, \( r \) is the monthly interest rate, and \( n \) is the total number of payments.

Step 3: Calculate the Monthly Interest Rate

Since the interest rate is given as 8% per year, we divide this by 12 to get the monthly interest rate. \[ r = \frac{0.08}{12} = 0.0066667 \]

Step 4: Substitute Known Values

We substitute the given values into the annuity formula: \( P = 500 \), \( r = 0.0066667 \), and \( n = 24 \).

Step 5: Plug Values into the Formula

Now plug the values into the annuity formula: \[ FV = 500 \times \frac{(1 + 0.0066667)^{24} - 1}{0.0066667} \]

Step 6: Compute the Future Value

First calculate \( (1+0.0066667)^{24} \), then subtract 1, then divide by \( 0.0066667 \), and finally, multiply the result by 500 to find the future value of the annuity. This will give us the total amount accumulated after 24 months.

Step 7: Final Calculation

After computing, the future value \( FV \) comes out to approximately \( 13,088.78 \).

Key Concepts

Compound Interest

Compound interest is the process where the interest earned on an amount of money grows based on the principal amount and any accumulated interest over time. Think of it as the snowball effect in finance.

• Uncomplicated principle: Interest is earned on the interest already generated, as well as on the initial principal.
• Frequent compounding: In our problem, the interest is compounded monthly. This means we're calculating interest 12 times a year, each time on a slightly higher amount.
• Impact on savings: Over time, compound interest can significantly boost your savings, given enough time and a stable interest rate. It's important to understand its benefits, especially when planning long-term investments or savings such as annuities.

Understanding how compound interest works is essential for correctly calculating the total accumulation in financial products like annuities.

Future Value

The future value (FV) is a term used to describe the total amount of money that will be accumulated after a series of cash flows (such as regular deposits) are invested over time at a certain interest rate.

• This concept is essential when calculating the worth of an annuity, as it determines what your total balance will be at a future date.
• In the given exercise, the annuity's FV is determined using a specific formula: \[FV = P \times \frac{(1 + r)^n - 1}{r}\] where each symbol holds significant importance. Here, the variable \( P \) represents the monthly payment amount.
• The future value is pivotal in deciding how much money you can expect to have, based on your consistent deposits and the effects of compound interest.

This measure helps you assess the ultimate impact of your investment or savings strategy, accounting for steady growth over time.

Monthly Payments

Monthly payments refer to fixed amounts regularly deposited into an account or invested. These payments are critical in financing plans, such as annuities, and ensure dependable growth over time through consistent cash flows.

• Stability in contribution: With monthly payments, you're developing a disciplined saving regime where you add a specific amount at regular intervals.
• Each payment contributes to increased compound growth. The sooner and more frequently you invest, the more the compound interest can act on your investments.
• In this exercise, the monthly payment was determined to be \( \$500 \). Over 24 months, this creates a foundational principal, which grows through compound interest to achieve the future value of the annuity.

By understanding how these payments interact with other parts of the annuity calculation, you gain insight into how regular contributions help maximize financial growth.

Interest Rate

Understanding interest rates is crucial when calculating investments like annuities. The interest rate determines how quickly your money will grow over time.

• When the exercise talks about an 8% annual interest rate, it signifies the rate at which your account will grow in one year.
• This annual interest rate needs adjustment when compounded more frequently. Compounding monthly means the rate is divided by 12, giving us a monthly rate in this scenario.
• The formula used adjusts the overall growth factor: \[r = \frac{0.08}{12} = 0.0066667\]
• Smarter financial decisions are driven by understanding interest rates, helping you to compare potential investments or savings vehicles effectively.

By knowing how much you're relatively gaining or being charged for borrowing, you can strategize better and plan effectively for the future.