Problem 8 - Precalculus Mathematics for Calculus

Question

Find the amount of an annuity that consists of 40 annual payments of \(\$ 2000\) each into an account that pays interest of \(5 \%\) per year.

Step-by-Step Solution

Answer

The amount of the annuity is approximately $241,604.

Step 1: Understand the Annuity Formula

The amount of an annuity can be found using the formula for the future value of an annuity: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]where \(FV\) is the future value, \(P\) is the periodic payment, \(r\) is the interest rate per period, and \(n\) is the number of periods. In this problem, \(P = 2000\), \(r = 0.05\), and \(n = 40\).

Step 2: Substitute Values into the Formula

Using the values from the problem statement, substitute into the annuity formula:\[ FV = 2000 \times \frac{(1 + 0.05)^{40} - 1}{0.05} \]

Step 3: Calculate the Expression for \\(1 + r\\) Raised to \\(n\\)

First, calculate \( (1 + 0.05)^{40} \):\[ (1 + 0.05)^{40} = 1.05^{40} \approx 7.0401 \]

Step 4: Evaluate the Complete Formula for Future Value

Substitute \(7.0401\) back into the formula:\[ FV = 2000 \times \frac{7.0401 - 1}{0.05} \]Calculate the subsequent operations:\[ FV = 2000 \times \frac{6.0401}{0.05} \]\[ FV = 2000 \times 120.802 \approx 241,604 \]

Step 5: Interpret the Result

The future value of the annuity after making 40 annual payments of \(2000\) dollars at a \(5\%\) interest rate is approximately \(241,604\) dollars.

Key Concepts

Future Value

Future value is a financial concept that refers to the amount of money an investment or a series of cash flows will be worth at a particular time in the future. It helps you understand how much your current investments will grow over time. The future value considers factors like periodic payments, interest rates, and the length of the investment period. In the context of an annuity, the future value means how much all your yearly contributions will be worth in the future.

To reiterate, the future value of an annuity can be calculated using the formula:

• \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]

Where:

• \(FV\) is the future value of the annuity.
• \(P\) is the payment per period.
• \(r\) represents the interest rate per period.
• \(n\) is the number of periods.

In our example, with regular payments into an account over 40 years, we use future value to understand the eventual worth of these payments.

Interest Rate

The interest rate is essentially the cost of borrowing money or the return on investment for savings. It's usually expressed as a percentage and can significantly affect the total accumulation or cost over time. In our scenario, it's the return each year on the money that is deposited into the account.

An interest rate can be simple or compound:

• Simple Interest: Earned only on the principal amount.
• Compound Interest: Earned on both the principal and interest that accumulates from prior periods, leading to exponential growth.

For the annuity problem, a compound interest rate of 5% is applied annually. This means every year, the interest is calculated based on the current balance, including interest accrued in the previous years.

Annuities in Mathematics

Annuities involve making a series of regular payments over a set period of time. This financial concept is central in planning for retirement, insurance products, and investments. Mathematically, an annuity is evaluated by looking at its future or present value, depending on the situation.

There are several key types of annuities:

• Ordinary Annuity: Payments are made at the end of each period, as shown in our problem.
• Annuity Due: Payments are made at the beginning of each period.

Our exercise example looks at an ordinary annuity, as the $2000 payments occur annually at the end of each year. By using the future value formula for annuities, one can compute the total value after all payments have been made.

Compound Interest

Compound interest is a powerful concept in finance that occurs when interest is added to the principal sum, so the added interest also earns interest from then on. This leads to growth at an accelerating rate, making compound interest extremely beneficial for savings and investments over time.

Here's how it works:

• Interest is calculated on the initial principal.
• Each time interest is added to the principal, the overall value increases.
• Future interest calculations are based on this increasing value, resulting in exponential growth.

In the annuity problem, a 5% compound interest rate is applied yearly to the total accumulated amount. This compounding effect significantly ramps up the total future value, as seen by the impressive amount accumulated after 40 years of consistent payments.