Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 150 ;\) total deposits: 24 ; interest rate: \(3 \%\), compounded monthly

Step-by-Step Solution

Answer

The annuity value is approximately $3707.85.

Step 1: Understand the Annuity Formula

The formula to determine the future value of an annuity compounded monthly is given by:\[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where:- \( FV \) is the future value of the annuity.- \( P \) is the monthly deposit amount.- \( r \) is the monthly interest rate.- \( n \) is the total number of deposits.

Step 2: Identify Known Values

We know:- \( P = 150 \) (monthly deposit)- The annual interest rate is \( 3\% \), so the monthly rate \( r = \frac{3}{100 \times 12} = 0.0025 \).- Total number of deposits \( n = 24 \).

Step 3: Plug Known Values Into the Formula

Using the formula:\[ FV = 150 \times \frac{(1 + 0.0025)^{24} - 1}{0.0025} \]

Step 4: Calculate Future Value

First, calculate \((1 + 0.0025)^{24} \):- Approximation: \( (1 + 0.0025)^{24} \approx 1.0618 \).Then use the values in the formula:\[ FV = 150 \times \frac{1.0618 - 1}{0.0025} \approx 150 \times 24.72 \approx 3707.85 \]

Step 5: Conclude the Solution

The future value of the annuity, after making 24 monthly deposits of \\(150 each with an interest rate of 3%, compounded monthly, is approximately \\)3707.85.

Key Concepts

Future Value of Annuity

The future value of an annuity refers to how much money you will have at the end of a savings plan when you make consistent investments over time. In this context, an annuity is a series of identical payments made at regular intervals. When you make these payments into a savings or investment account, the amount you end up with after a specific period of time is called the future value of the annuity.

To find this future value, we use a special formula:

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

Where:

• \(FV\) is the future value of the annuity.
• \(P\) is the monthly deposit amount.
• \(r\) is the monthly interest rate.
• \(n\) is the total number of payments.

By inserting all these variables into the formula, it allows us to determine the accumulated investment after the given period, considering the interest applied.

Compound Interest

Compound interest is a key element in calculating the future value of an annuity. Unlike simple interest, which only accrues on the original principal, compound interest accumulates on both the initial principal and the interest that has been added to it over previous periods.

This means your investment can grow more quickly over time. The mathematical formula for compound interest is part of the overall annuity calculation for future value:

• It involves the term \((1 + r)^n\) in the annuity formula.
• The result gives an estimation of how much the investment is compounded over time.

Each time that interest is added to the account (for example, monthly), the overall amount on which future interest will be calculated increases slightly. This compounding effect can significantly raise the value of an investment over long periods.

Monthly Deposits

Monthly deposits refer to the consistent, regular payments made into an investment account over a specified duration. In our example, these are the $150 deposits made every month.

Understanding monthly deposits is crucial because these regular contributions are what help to build the future value of your annuity. Over time, the steady influx of money allows more funds to start generating compound interest. Here are a few points to remember:

• Regular deposits increase the principal on which interest is calculated.
• Even small monthly contributions can accumulate significantly thanks to compound interest.
• Consistency in making deposits is key to maximizing the future value of the annuity.

Making deposits monthly rather than yearly gives the opportunity for more frequent compounding, potentially leading to higher returns.

Interest Rate Calculation

Calculating the correct interest rate involves breaking down the annual rate first. In many cases, as shown in this exercise, accounts compound interest monthly, so it's necessary to convert the annual interest rate into a monthly rate.

To calculate the monthly interest rate from an annual rate, you follow this process:

• First, express the annual rate as a decimal by dividing by 100.
• Then, divide this result by 12, since there are 12 months in a year.
• For example, a 3% annual rate becomes \(0.03/12\) giving us a monthly rate of \(0.0025\).

Understanding how to break down an annual interest rate into a monthly one is essential, as it ensures accurate calculations for monthly compounding scenarios. This correct monthly rate is then used in the annuity formula to determine the future value accurately.