Problem 48 - College Algebra

Question

Determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 450 ;\) total deposits: \(60 ;\) interest rate: \(4.5 \%\), compounded quarterly

Step-by-Step Solution

Answer

The annuity's future value is approximately $9720.

Step 1: Understand the Annuity Formula

The annuity formula for future value (FV) when compounded periodically is \( FV = P \frac{(1 + r/n)^{nt} - 1}{r/n} \), where \( P \) is the periodic deposit, \( r \) is the annual interest rate, \( n \) is the number of compounding periods per year, and \( t \) is the total number of years.

Step 2: Assign Given Values

We're given \( P = 450 \), the interest rate \( r = 0.045 \) (as a decimal), a total of \( 60 \) deposits, and quarterly compounding, hence \( n = 4 \). Let's find \( t \), the total number of years. Since there are 5 years (because \( 60 / 12 = 5 \)), \( t = 5 \).

Step 3: Substitute Into the Formula

Substitute these values into the annuity formula: \[ FV = 450 \frac{(1 + 0.045/4)^{4 \times 5} - 1}{0.045/4} \].

Step 4: Calculate Components

First calculate \( 1 + 0.045/4 = 1.01125 \). Then, raise it to the power of \( 4 \times 5 = 20 \): \( 1.01125^{20} \approx 1.243 \).

Step 5: Compute the Future Value

Apply the calculations: \( FV = 450 \frac{1.243 - 1}{0.01125} \). Compute \( 1.243 - 1 = 0.243 \), then \( 0.243 / 0.01125 \approx 21.6 \). Multiply \( 450 \times 21.6 \approx 9720 \).

Step 6: Verify Calculation

Verify each step to ensure no computational errors. Recalculate each component as necessary to confirm that \( FV \approx 9720 \) is correct.

Key Concepts

Compound Interest

Compound interest is a crucial concept in finance and investments. Unlike simple interest, which is calculated only on the principal amount, compound interest works by adding the interest earned back into the principal. This means that from each period onwards, the interest is calculated on the new, larger total. Essentially, you earn interest on both your initial deposit and the interest that accumulates over time.

This can be visualized as a snowball effect—this effect is what makes compound interest extremely powerful. The formula for calculating the compound interest for any given period is:

• \[ A = P(1 + \frac{r}{n})^{nt} \]

where \( A \) is the amount of money accumulated after \( n \) years, including interest.
The \( P \) represents the principal amount, \( r \) the annual interest rate, \( n \) the number of times that interest is compounded per year, and \( t \) the number of years the money is invested for.

Understanding how compound interest works is key to maximizing the returns on your investment, especially when saving for long-term goals like retirement or education.

Future Value Calculation

The future value calculation of an investment or annuity is a way of determining how much a series of regular deposits will grow over a period of time given a specific interest rate.

To compute the future value of an annuity, we use the annuity formula:

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

Here, \( FV \) denotes the future value of the annuity, \( P \) is the payment amount per period, \( r \) is the annual interest rate expressed as a decimal, \( n \) is the number of compounding periods per year, and \( t \) is the time in years.

The term \( \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}} \) is crucial as it shows the effect of interest compounding over time on multiple periodic deposits. Thus, calculating future value is pivotal in understanding how much your money will grow if invested wisely.

Compounding Periods

Compounding periods refer to how frequently the interest is applied to the principal balance. The more frequent the compounding, the more times per year interest is added, increasing the total amount being multiplied by the interest rate each period.

• Annual compounding means the interest is added once per year.
• Semi-annual compounding happens twice a year.
• Quarterly compounding occurs four times annually.
• Monthly, daily, and even continuous compounding are also possible, each adding the interest more frequently than the last.

The number of compounding periods \( n \) directly affects the outcome of any investment and how quickly it can grow. The formula for compound interest incorporates \( n \) to show how the frequency of compounding contributes to the overall future value or total accrued amount. Understanding compounding periods is essential for evaluating the true earning potential of any annuity or investment product. It emphasizes the importance of not only knowing the interest rate but also how often that interest is calculated and added to the balance.