Compound Interest

Compound interest is interest on an investment or loan that is added to the principal after accruing, thus allowing additional interest to accrue on the new amount. For investors, compound interest is a valuable tool used to create wealth, in that it enables investors to transform a small initial investment into a much larger sum over time. This is especially important in regard to retirement savings, which may remain in a compound interest–generating account for many decades. However, compound interest is also used by some lenders for their own financial benefit but to the detriment of individuals who have taken out loans. Over time, the interest owed on a loan is compounded and is thus added to the loan principal. Because of this, the holder of the loan may ultimately owe significantly more than the initial amount borrowed. Credit card debt and some student loan debt are among the various types of debt that can increase dramatically due to compound interest.

Background

Since the invention of money, individuals with funds to spare have loaned portions of their funds to individuals without. At times, lenders and borrowers carried out this process without introducing any additional fees: one individual would borrow a certain sum from another, and he or she would later pay back that same sum. In many cases, however, lenders charged interest, or a fee based on a percentage of the money borrowed. Interest has long been a controversial subject in much of the world, with some ancient societies banning the charging of interest. The practice of charging interest became known as usury, although in later centuries, that word came to mean the practice of charging unreasonable interest rates.

Compound interest, also known as compounding or accumulating interest, likewise developed in ancient times and proved far more controversial than its more straightforward counterpart, which is commonly known as simple interest. The Roman Empire banned the charging of compound interest, and that prohibition later found its way into numerous European and Middle Eastern societies. Over time, however, many lenders and governments recognized the benefits of compound interest, and societies began to loosen the longstanding restrictions on it. In the Islamic world, the charging of interest remained largely forbidden; in much of Europe, however, the charging of simple or compound interest became a standard procedure. Tough restrictions on the use of interest remained in place in the United States through the nineteenth century, but by the early twentieth century, they had loosened significantly. In the early twenty-first century, lenders throughout the country charge simple interest or compound interest, and maximum permissible interest rates vary from state to state.

Compound interest has long been a topic of interest among mathematicians, who for centuries have studied its effects in a theoretical context. The thirteenth-century mathematician Leonardo of Pisa (ca. 1170–ca. 1240), better known as Fibonacci, notably discussed simple and compound interest calculations in his Liber abaci (ca. 1202), a foundational text of modern mathematics. More than five centuries later, the British mathematician and scientist Isaac Newton (1643–1727) published compound interest formulas in the Arithmetica Universalis (1707). Because of the laws concerning interest, many mathematicians focused primarily on the ways in which compound interest affected the value of savings and investments. Their work in that area would prove crucial to the later understanding of compound interest as a wealth-building tool rather than a mere aspect of moneylending.

Overview

To understand compound interest, one must first understand simple interest, its more common and, as the name suggests, simpler counterpart. Simple interest is a fee paid on a loan or deposit, typically calculated as a percentage of the total principal. If a loan has an initial principal of $10,000 and an interest rate of 2 percent per year, and the borrower does not make any payments during the year, by the end of the year he or she will owe $10,200, $200 more than the amount originally borrowed. Similarly, if an individual deposits $10,000 into a savings account with the same interest rate, he or she will end the year having gained $200. Simple interest accrues based solely on the principal and subsequent deposits and would thus not be affected by the $200 increase over the course of the year.

Compound interest, on the other hand, is added to the principal in accordance with the loan or account’s compounding schedule. In some instances, interest compounds every day, while in others, it is a weekly, monthly, or yearly occurrence. After the interest compounds, thus becoming part of the principal, further interest will accrue based on that new amount. For example, consider what would happen if one of the accounts in the previous examples were subject to compound interest rather than simple interest. Assuming that the interest compounds only once per year, the end-of-year result for the first year would remain $10,200. The following year, however, the 2 percent interest would accrue based not on the initial principal of $10,000 but on the new principal of $10,200.

To calculate compound interest, an individual may use one of several approaches. Compound interest may be calculated by hand, as it was by mathematicians from Fibonacci to Newton and beyond. One formula widely used in such calculations is compound interest = [P (1 + r)ⁿ] − P. In that equation, P represents the principal, or starting amount; r represents the interest rate divided by the number of compounding periods each year; and n represents the number of compounding periods each year multiplied by the number of years.

For example, if an individual deposits $15,000 into a high-yield savings account that offers a yearly interest rate of 1.05 percent, compounded monthly, one would calculate the compound interest accrued over one year by solving the equation (15,000 × 1.000875¹²) − 15,000. Assuming the individual has made no additional deposits or withdrawals over the course of the year and has not been charged any bank fees, he or she will have earned $158.26 in compound interest over the course of the year, resulting in an end-of-year total balance of $15,158.26.

Compound interest may also be calculated through the use of compound interest tables or online calculators. The latter is perhaps the easiest method, as it allows anyone, regardless of mathematical ability, to enter the necessary data into a simple form and receive an accurate result. Although many compound interest calculators are available on the Internet, one can best ensure the accuracy of one’s calculations by selecting a calculator offered by a reputable and authoritative web site. The web site Investor.gov, for instance, is managed by the US Securities and Exchange Commission and features an easy-to-use interest calculator as well as other useful tools and information.

For many investors, compound interest is a valuable tool for the generation of wealth. By depositing funds in an account that accrues compound interest, investors can ensure that their wealth continues to grow even without the addition of further funds. Depositing additional funds into the account on a regular basis allows the principal to grow at an even faster rate. Compound interest is particularly important in regard to retirement savings, as such long-term savings may remain in an account for decades. Over time, the money in the account builds upon itself, resulting in significantly greater returns.

The following scenario provides an example of the benefits of compound interest to long-term savings: an individual opens a retirement savings account that offers an interest rate of 5 percent per year, compounded monthly. He or she begins with an initial deposit of $100 and deposits an additional $100 per month for forty years. Using an interest calculator, one can see that by the end of the forty-year term, the account in question (assuming no additional deposits, withdrawals, or bank fees) will contain a total of about $153,000—about $105,000 more than the $48,000 the individual deposited over the forty-year term.

Although compound interest is beneficial to many Americans from a savings standpoint, it can prove detrimental when put to use by lenders. Many loans accrue only simple interest; however, certain types of loans accrue compound interest, thus increasing the amount of money owed over time. In such cases, compound interest is sometimes referred to as capitalized interest. Among the lenders who frequently use compound interest are credit card companies and some private student loan providers.

The following example shows the effect of compound interest on a loan: while preparing to attend college, an individual takes out an unsubsidized $10,000 loan from a private student loan provider. The loan has an interest rate of 2.25 percent, compounded daily. The student plans to graduate in four years and is not required to begin repaying the loan until then; however, the loan will accrue interest while he or she is in school. Using an interest calculator, one can see that by the end of four years, the total amount due will be $10,941.71, nearly $1,000 more than the amount originally borrowed.

Bibliography

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Fibonacci. Fibonacci’s Liber Abaci. Trans. Laurence Sigler. New York: Springer, 2003. Print.

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