Compound Interest Formula

Definition

The compound interest formula is a mathematical expression that calculates the future value of an investment or loan when interest is compounded over time. It is a fundamental concept in finance and economics, with applications in areas such as personal finance, business investments, and economic modeling.

5 Must Know Facts For Your Next Test

1. The compound interest formula is $A = P(1 + r/n)^{nt}$, where $A$ is the future value, $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.
2. Compound interest leads to exponential growth, where the investment or loan grows at a faster rate over time compared to simple interest.
3. The more frequently the interest is compounded (i.e., the higher the value of $n$), the greater the future value of the investment or loan.
4. Compound interest is a key concept in the study of exponential functions, as the formula can be expressed in the form of an exponential equation.
5. The compound interest formula is also used in the analysis of infinite geometric series, which is a topic covered in the study of series and their notations.

Review Questions

• Explain how the compound interest formula is related to the study of exponential functions.
  • The compound interest formula, $A = P(1 + r/n)^{nt}$, can be expressed in the form of an exponential equation, where the future value $A$ is the dependent variable, and the time $t$ is the independent variable. The base of the exponent, $(1 + r/n)$, represents the growth factor, which determines the rate of exponential growth. This connection between the compound interest formula and the characteristics of exponential functions is a key concept in the study of 4.2 Graphs of Exponential Functions.
• Describe how the compound interest formula is used in the analysis of infinite geometric series, as covered in 11.4 Series and Their Notations.
  • The compound interest formula can be used to model the behavior of infinite geometric series, which are sequences where each term is a constant multiple of the previous term. Specifically, the formula $A = P(1 + r/n)^{nt}$ can be rewritten in the form of a geometric series, $A = P \sum_{i=0}^{n-1} (1 + r/n)^i$, where the sum represents the infinite series. This connection allows for the analysis of the convergence and sum of these series, which is a key topic in 11.4 Series and Their Notations.
• Evaluate the impact of the compounding frequency (the value of $n$) on the future value of an investment using the compound interest formula.
  • The compounding frequency, represented by the value of $n$ in the compound interest formula, has a significant impact on the future value of an investment. As the compounding frequency increases (i.e., $n$ becomes larger), the future value of the investment grows at a faster rate. This is because the interest earned in each compounding period is reinvested and earns additional interest in the subsequent periods, leading to exponential growth. Understanding the relationship between compounding frequency and future value is crucial in making informed financial decisions, as it allows for the comparison of different investment options and the optimization of compound interest earnings.

Related terms

Simple Interest: Simple interest is the interest earned on the principal amount of an investment or loan, without any compounding of the interest over time.

Exponential Growth: Exponential growth refers to the rapid increase of a quantity over time, where the growth rate is proportional to the current value of the quantity.

Time Value of Money: The time value of money is the concept that money available at the present time is worth more than the same amount of money available in the future, due to its potential earning capacity.