Question

In the following exercises, substitute the given quantitics into the indicated formula from technology and finance. Use the formula for compound interest, \(y=a(1+n)^{t},\) to find, to the nearest dollar, the amount \(y\) to which a principal \(a\) of \(\$ 9570\) will accumulate in \(t=5\) years at a compound interest rate \(n\) of \(6.75 \%\).

Short answer

After substituting the values into the compound interest formula and calculating, the amount to which the principal accumulates is approximately \(y \$13,439\).

Step-by-step solution

Understand the compound interest formula

The formula for compound interest is given by \(y = a(1+n)^t\), where \(y\) is the future value of the investment, \(a\) is the principal amount, \(n\) is the annual interest rate (expressed as a decimal), and \(t\) is the number of years the money is invested or borrowed.

Convert the interest rate to a decimal

Convert the annual interest rate from a percentage to a decimal by dividing by 100. For an interest rate of 6.75%, this will be \(n = \frac{6.75}{100} = 0.0675\).

Substitute the given values into the formula

Substitute \(a = 9570\), \(n = 0.0675\), and \(t = 5\) years into the compound interest formula to calculate the future value:\[y = 9570(1 + 0.0675)^5\].

Calculate the compound interest

Calculate the value inside the parentheses first, then raise it to the power of 5, and finally multiply by the principal:\[y = 9570 \times (1 + 0.0675)^5 \approx 9570 \times (1.0675)^5\].

Round to the nearest dollar

Use a calculator to find the approximate value to the nearest dollar. This gives you the amount \(y\) to which the principal will accumulate.

Key concepts

Financial Mathematics

At the heart of enabling transactions and growth in the economic world is the discipline of financial mathematics. It uses advanced mathematical techniques to analyze and solve problems related to finance, such as identifying fair prices for financial products, managing risk, and maximizing profits.

One fundamental element of financial mathematics is the application of the compound interest formula, which allows us to calculate the growth of an investment over time. It's integral for anyone involved in finance to understand how to use this formula, as it helps in making informed decisions about investments, loans, and savings. In our given exercise, for instance, we determine the accumulated amount of an initial deposit, or principal, after a specified time at a given interest rate—a classic example of compounding.

When it comes to solving problems like these, we follow several steps. Initially, you would clarify the terms of the problem and convert percentages to decimals, as we did in converting the interest rate of 6.75% to 0.0675. Subsequent steps involve mathematics: substituting our known values into the compound interest formula to calculate the future value of the investment.

To ensure students can follow along, it's vital to emphasize clarity in each step, highlight the transformation of the formula with substituted values, and encourage the use of calculators for accurate findings.

Exponential Growth

Exponential growth is a process that increases quantity over time at a rate proportional to its current value. In finance, this concept explains how investments grow faster as they earn interest on both the initial principal and the accumulated interest from prior periods—this is what we call compound interest.

In our textbook exercise, the formula for compound interest, \( y = a(1+n)^t \), showcases exponential growth. The term \( (1+n)^t \) includes an exponent, indicating that the original amount is not just multiplied by a fixed rate, but is exponentially increased by the rate over each time period (years, in this case).

Understanding exponential growth is essential because it can significantly impact financial planning and investment decisions. With knowledge of how compound interest works, students can understand and predict the future value of their investments and make more strategic choices about where to allocate their funds. For example, they can see how a small increase in the interest rate can vastly increase the future value, due to the exponential nature of the growth.

Time Value of Money

The time value of money is a concept in finance that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This principle underpins the core reason why people invest—it's better to receive money now rather than later.

Our calculation in the exercise demonstrates the time value of money through compound interest. It tells us that \( \$9570 \) invested today at a rate of 6.75% will be worth more in 5 years. This can be attributed to the opportunity of the principal amount to earn interest over the given period.

Recognizing the time value of money influences how we view investments, savings, and loans. For example, it impacts retirement planning (how much to save now for a future goal) and loan repayments (understanding amortization schedules). It's crucial for students to comprehend that every financial decision made today has implications for future value, which can be quantified using the compound interest formula.