Problem 54 - Algebra and Trigonometry

Question

Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 5000\) for 10 years at an interest rate of \(6.5 \%\) if the money is a. compounded semiannually, b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Step-by-Step Solution

Answer

The accumulated values of the investment are: a. \$8952.14 when compounded semiannually, b. \$8999.61 when compounded quarterly, c. \$9031.21 when compounded monthly, and d. \$9038.27 when compounded continuously.

Step 1: Compute When Compounded Semiannually

Using the compound interest formula with \(n = 2\), we get: \(A = 5000 (1 + \frac{0.065}{2})^{2*10} = 8952.14\). So when compounded semiannually, the accumulated value is \$8952.14.

Step 2: Compute When Compounded Quarterly

Using the compound interest formula with \(n = 4\), we get: \(A = 5000 (1 + \frac{0.065}{4})^{4*10} = 8999.61\). So when compounded quarterly, the accumulated value is \$8999.61.

Step 3: Compute When Compounded Monthly

Using the compound interest formula with \(n = 12\), we get: \(A = 5000 (1 + \frac{0.065}{12})^{12*10} = 9031.21\). So when compounded monthly, the accumulated value is \$9031.21.

Step 4: Compute When Compounded Continuously

Using the continuous compound interest formula, we get: \(A = 5000 e^{0.065*10} = 9038.27\). So when compounded continuously, the accumulated value is \$9038.27.

Key Concepts

Accumulated Value

When you invest money, the amount it grows to over time is referred to as the "accumulated value." This value accounts for not just the initial amount you put in but also includes all the interest it has earned.
In simple terms, if you start with an investment of $5,000, the accumulated value is what your investment has become after interest has been added for a specified period. The longer your money stays invested, the more the interest can grow.

• It includes the principal (original amount).
• Adds all the interest earned over time.

The key to increasing your accumulated value is understanding how your money can be compounded over time. Different compounding frequencies can affect how fast your investment grows.

Interest Rate

The interest rate is a significant factor in determining how your investment grows. It is expressed as a percentage and indicates how much interest your money earns over a year.

• For instance, a 6.5% interest rate means each year your investment earns 6.5% of its total value.
• If your rate is higher, your money grows more quickly.

Interest rates can be fixed or variable. A fixed rate stays the same for the duration of the investment, providing stability and predictability. On the other hand, a variable rate can change, offering potential for higher returns when rates increase.

Investment

Investment refers to the initial sum of money you decide to put away in hopes that it will grow over time. In this scenario, you start with $5,000 and plan to let it accumulate over 10 years.
Choosing the right investment option is critical. Consider factors like:

• The interest rate offered.
• The compounding frequency, which affects how often interest is calculated and added to your account.
• The risk factor, depending on whether your investment is in a secure setting or more volatile environment.

The goal is to choose an investment strategy that aligns with your financial objectives, risk tolerance, and time horizon.

Compounding Frequency

Compounding frequency dictates how often interest is added to the principal of your investment. It plays a crucial role in determining the growth of your accumulated value.
Here are the four types of common compounding frequencies:

• Semiannually: Interest is compounded twice a year. It leads to a larger accumulated value than annual compounding.
• Quarterly: Interest is compounded every three months. This generally results in a greater accumulated value than semiannual compounding.
• Monthly: Interest is compounded each month, making it grow even faster.
• Continuously: Interest is compounded constantly. This leads to the highest growth possible, exploiting the full power of compounding.

Remember, the greater the frequency of compounding, the more often interest is applied, leading to a higher accumulated value.