Problem 78 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 4000\) is invested in an account for which interest is compounded quarterly, find the amount of the investment at the end of 5 years for the following interest rates. $$ \begin{array}{ll}{\text { (a) } 6 \%} & {\text { (b) } 6 \frac{1}{2} \%} \\\ {\text { (c) } 7 \%} & {\text { (d) } 8 \%}\end{array} $$

Step-by-Step Solution

Answer

(a) $5398.12, (b) $5563.37, (c) $5733.69, (d) $5989.54.

Step 1: Understand the Compound Interest Formula

The formula for compound interest is given by \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.

Step 2: Identify Values for Calculation

Given that the principal \(P\) is $4000, compounded quarterly means \(n=4\), and \(t=5\) years for each rate provided. The interest rates need to be converted into decimals by dividing the percentage by 100: (a) 0.06, (b) 0.065, (c) 0.07, (d) 0.08.

Step 3: Calculate for Each Interest Rate

Using the values from Step 2, we plug them into the compound interest formula to find \(A\) for each interest rate.**For 6% (a)**: \[ A = 4000 \left( 1 + \frac{0.06}{4} \right)^{4 \times 5} = 4000 \left( 1 + 0.015 \right)^{20} \] Use calculator to find \( A \approx 5398.12 \).**For 6.5% (b)**: \[ A = 4000 \left( 1 + \frac{0.065}{4} \right)^{4 \times 5} = 4000 \left( 1 + 0.01625 \right)^{20} \] Use calculator to find \( A \approx 5563.37 \).**For 7% (c)**: \[ A = 4000 \left( 1 + \frac{0.07}{4} \right)^{4 \times 5} = 4000 \left( 1 + 0.0175 \right)^{20} \] Use calculator to find \( A \approx 5733.69 \).**For 8% (d)**:\[ A = 4000 \left( 1 + \frac{0.08}{4} \right)^{4 \times 5} = 4000 \left( 1 + 0.02 \right)^{20} \] Use calculator to find \( A \approx 5989.54 \).

Step 4: Present the Final Amounts

The final amounts for each interest rate are approximately: (a) \(\(5398.12\), (b) \(\)5563.37\), (c) \(\(5733.69\), and (d) \(\)5989.54\).

Key Concepts

Investment Growth

Investment growth through compound interest is a powerful concept in finance. It's the increase in the value of an investment due to the addition of interest earned on the principal as well as on accumulated interest from previous periods. This process of earning interest on interest leads to exponential growth over time, which can significantly increase the amount of money saved or invested.

For example, when you invest $4000 in an account, over time, not only does the principal amount gain interest, but the interest amount also accrues additional interest. This effect is amplified as the number of compounding periods increases, resulting in potentially higher returns.

• The initial amount, also called the principal, acts as the foundation for growth.
• Interest, when compounded, accelerates this growth by generating earnings on both the principal and the interest that has already been added.
• Over longer periods, even small changes in interest rates can lead to significant differences in the final amount through the power of compounding.

Understanding investment growth through compound interest can help you strategize better for savings and retirement plans, highlighting the importance of starting investments early and choosing the right financial products.

Interest Rates

Interest rates play a crucial role in determining how much an investment will grow. They are usually expressed as an annual percentage and can significantly affect the total interest that's compounded over the life of an investment. The rates offered by banks or investment firms can vary, influencing how fast your money can grow.

In the context of our example, we have four different annual interest rates: 6%, 6.5%, 7%, and 8%. Each of these impacts the final value of the investment differently when compounded quarterly.

• At a rate of 6%, the investment grows steadily, reaching approximately $5398.12 after 5 years.
• With a slight increase to 6.5%, the amount rises to about $5563.37.
• A 7% rate results in $5733.69, showing the compounding effect's power.
• Finally, an 8% rate significantly boosts the investment to $5989.54.
  
  Higher interest rates generally lead to more substantial investment growth due to higher compounded returns over time. However, it's essential to evaluate the stability and risk associated with higher rates in different investment vehicles.

Quarterly Compounding

Quarterly compounding means that interest is calculated and added to the principal four times a year. This is opposed to simple interest, where interest is only added once per year. Compounding more frequently can lead to more total interest accrued, as interest is calculated on an ever-increasing principal amount.

In our example, the interest is compounded quarterly, which means every three months. This leads to more opportunities for the interest to be added to the principal, enhancing the investment's growth.

• Each quarter, the interest applied to the investment increases the principal amount.
• This new principal then earns additional interest in the following quarters.
• More compounding periods (like quarterly) traditionally result in more growth than fewer periods, such as annually.

The formula used to calculate compound interest, \(A = P \left( 1 + \frac{r}{n} \right)^{nt}\), reflects this by utilizing \(n=4\) to represent the quarterly compounding frequency. This illustrates how the frequency of compounding can substantially affect the end value of an investment, particularly over extended periods.