Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finance A deposit of \(\$ 7500\) in a mutual fund reaches a balance of \(\$ 11,752.45\) after 10 years. What annual interest rate on a certificate of deposit compounded monthly would yield an equivalent return?

Step-by-Step Solution

Answer

The annual interest rate compounded monthly would yield an equivalent return of approximately 4.5%

Step 1: Understand the Compound Interest Formula

The formula for compound interest is as follows: \( A = P(1 + r/n)^{nt} \) where: A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year and t is the time in years.

Step 2: Insert the Given Values

Start by inserting the given values into the formula. It's known that: \( P = \$7500 \), \( A = \$11752.45 \), \( n = 12 \) (for monthly compounding), and \( t = 10 \) years. The equation now is: \( \$11752.45 = \$7500 (1 + r/12)^{12*10} \)

Step 3: Isolate the Unknown

Isolate r by dividing both sides by \$7500, then take 12th root, and finally subtract 1. These steps give the formula for r as: \( r = 12 * ((\$11752.45/\$7500)^(1/120) - 1) \)

Step 4: Calculate r

After plugging in the numbers into the formula, r is calculated as: \( r = 12 * ((\$11752.45/\$7500)^(1/120) - 1) \approx 0.045 \) .

Key Concepts

interest rate calculation

Interest rate calculation is a fundamental concept in financial mathematics. It involves determining the rate at which interest is earned on an investment or paid on a loan over a specific period. In the context of compound interest, the interest rate is usually represented annually, even if the compounding occurs more frequently.

To find the interest rate, you need to use the compound interest formula:

• \( A = P(1 + r/n)^{nt} \)

Here:

• \( A \) is the total amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (the initial amount of money).
• \( r \) is the annual interest rate (in decimal form).
• \( n \) is the number of times interest is compounded per year.
• \( t \) is the time in years.

To isolate the interest rate \( r \), you usually rearrange the formula to solve for \( r \) after substituting known values for \( A \), \( P \), \( n \), and \( t \). Understanding how to manipulate and solve this equation is key to mastering interest rate calculations.

monthly compounding

Monthly compounding plays a crucial role in financial growth, affecting how interest accumulates over time. When interest is compounded monthly, it means that the interest earned each month is added to the principal, and then, the next month's interest is calculated on this new sum.

This frequency of compounding accelerates the growth of investments:

• More frequent compounding periods lead to the principal amount growing more quickly.
• The compound interest formula adapts to monthly compounding by setting \( n = 12 \), indicating that interest is compounded 12 times per year.

Monthly compounding can significantly impact the future value of an investment, thus making it an essential aspect of financial calculations. Understanding how monthly compounding differs from annual or quarterly compounding helps in making informed decisions about investment choices.

financial mathematics

Financial mathematics is a specialized area that applies mathematical methods and techniques to solve financial problems. One of its primary applications is calculating future values of investments through concepts such as compound interest.

Core techniques in financial mathematics include:

• Developing formulas that relate to time value of money, such as the compound interest formula.
• Calculating present and future values of cash flows to help inform financial decisions.
• Evaluating different investment options through various interest compounding frequencies, like monthly or annually.

Financial mathematics provides the foundation for many financial analyses, allowing individuals and companies to forecast outcomes and make rational financial choices. It's essential for understanding how money grows over time, determining the attractiveness of different financial products, and managing risks associated with investments.