### What is Compound Interest?

Simple interest refers to interest earned only on the principal, usually denoted as a fixed percentage of the principal. Determining a single interest payment is as simple as multiplying the interest rate with the principal. Simple interest is seldom ever used in real world applications of interest.

On the other hand, compound interest is interest earned on both the principal and on the accumulated interest. Because interest is also earned on interest, earnings compound over time like an exponentially-growing, avalanching snowball. Compound interest is widely used for interest calculations on many things including mortgages, auto loans, banking, and much more.

In order to determine whether interest is compounded or not in the U.S., the Truth in Lending Act (TILA) requires that lenders disclose all pertinent loan information to borrowers, including whether interest accrues simply or in compounded fashion. Another way to determine whether interest is simple or compounded is to look at the repayment schedule for the loan. In the case of simple interest, each year's interest payment and the total amount owed will be the same. If the interest is compounded, each year's interest payment will be different.

### Practical Ways to Use Compound Interest

To start with, any form of savings that doesn't earn interest, such as cash or many checking accounts, will not benefit from compound interest. Common funds that benefit from compound interest include savings accounts, stocks (with reinvested dividends), and some of the more common retirement plans such as 401(k)s and IRAs.

Compound interest can be highly financially rewarding. The longer that interest is allowed to compound for any investment, the greater the growth. While this is true for all investments, retirement investments are the main financial instruments that people use to take full advantage of compound interest. As a simple example, a person at age 19 decides to invest $2,000 every year for eight years at an 8% interest rate. Suddenly, they decide to halt annual payments, but allow the funds to grow uninterrupted until they reach the age of 65. With an initial investment of only $16,000 over eight years, their funds will have grown to almost $430,000 for use in retirement! And all this without paying a single cent for 39 years. This is due in large part to the nature of compound interest.

While compound interest is very effective at growing wealth, it can also work against you if you have any debt that is subject to compound interest. This is why compound interest can be described by some as a double-edged sword. Putting off or prolonging outstanding debt will increase the total interest owed. As such, it is as important to ensure that debts are paid off quickly as it is to put money into a retirement account early to allow it the maximum amount of time to grow.

### Factors that Work Against Compound Interest

**Tax**—If any taxation is to be applied, the rate and timing of taxation will affect the magnitude of compounding interest. The less that taxation is involved, the greater the magnitude of compounding because of fewer reductions in the balance of the investment.

**Fees**—In the case of long-term investments such as a retirement account, even a fee as low as 1% will have a significant impact on the end result. 1% vs 0.5% may not feel like much over the course of 1 or 2 years, but when saving for retirement, it can mean the difference between retiring at different ages.

### Different Compounding Frequencies

Interest can be compounded on any given frequency schedule, and the calculator allows the conversion between compounding frequencies of daily, bi-weekly, semi-monthly, monthly, quarterly, semi-annually, annually, and continuously (infinitely many number of periods). The interest rates of savings accounts and Certificate of Deposits (CD) tend to be compounded annually. Home mortgage loans, home equity loans, and credit card accounts tend to be compounded monthly.

### History of Compound Interest

There is evidence from ancient texts that compound interest was first used 4400 years ago by the Babylonians and Sumerians, two of the earliest civilizations in human history. However, their application of compound interest was quite different from what is widely used today. In their application, 20% of the principal amount was accumulated until the interest was equal to the principal, which was then added to the principal. Historically, simple interest was mostly considered legal. However, certain societies didn't grant the same legality to compound interest, labeling it as usury. For example, it was severely condemned by Roman law, and both Christian and Islamic texts have described it as a sin. Nevertheless, compound interest has been in use ever since.

The equation for continuously compounding interest, which is the mathematical limit that compound interest can reach, utilizes something called Euler's Constant, also known as e. Although e is widely used today in many areas, it was discovered when Jacob Bernoulli was studying compound interest in 1683. The mathematician understood that, within a specified finite time period, the more compounding periods involved, the faster the compounding principal was able to grow. It didn't matter whether it was in intervals of years, months, days, hours, minutes, seconds, or nanoseconds, each additional period generated higher returns (for the lender). Bernoulli discerned that this sequence eventually approached a limit he defined as e, which describes the relationship between the plateau and the interest rate when compounding.

### Compound Interest Formulas

###### Basic Compound Interest

Basic formula for compound interest:

A |

A_{0} : principal amount, or initial investment

A_{t} : amount after time t

r : interest rate

n : number of compounding periods, usually expressed in years

The following is an example of $1,000 in a savings account for two years advertised at 6% APY compounded once a year. Use the equation above to find the total due at maturity:

A_{t} = $1,000 × (1 + 6%)^{2} = $1,123.60 |

For other compounding frequencies (such as monthly, weekly, or daily), the situation calls for the formula below.

A_{t} = A_{0} × (1 + r/n)^{nt} |

A_{0} : principal amount, or initial investment

A_{t} : amount after time t

n : number of compounding periods in a year

r : interest rate

t : number of years

Assume that the $1,000 in the savings account in the previous example comes with a 6% rate with interest accumulated daily. The interest earned every day is:

6% ÷ 365 = 0.0164384% |

Using the formula above, it is possible to find the value at the end.

A |

$1,127.49 will be the end value of a 2-year savings account containing $1,000 that has a 6% interest rate compounded daily.

**Rule of 72**

The Rule of 72 is a shortcut to determine how long it'll take for a specific amount of money to double, given a fixed return rate that is compounded annually. It can be used for any investment, as long as there is a fixed rate that involves compound interest. Simply divide the number 72 by the annual rate of return and the result of this is how many years it'll take. As an example, $100 with a fixed rate of return of 8% will take around 9 (72 divided by 8) years to become $200. Note that "8" is used to denote 8%, not "0.08". Keep in mind that the Rule of 72 disregards any investment fees, management fees, and trading commissions, and doesn't account for losses incurred from taxes paid on investment gains. It is best used as a rough guideline.

**Continuous Compound Interest**

Continuously compounding interest represents the mathematical limit that compound interest can reach within a specified time period. The continuous compound equation is as follows:

A |

A_{0} : principal amount, or initial investment

A_{t} : amount after time t

r : interest rate

t : number of years

e : mathematical constant e, ~2.718

Say for instance, we wanted to find the maximum interest that could possibly be earned on the $1,000 savings account in two years.

Using the equation above:

A |

From the 3 examples provided it can be seen that the shorter the compounding frequency, ceteris paribus, the higher the interest earned. It can be seen however, that above a certain compounding frequency, the interest gained is marginal, particularly on smaller principals.