# What Is Compound Interest?

## Compound Interest – What Is It?

It’s the interest on a loan or deposit that’s calculated using both the initial principal and the accumulated interest from previous periods. Compound interest (or compounding interest) Compound interest, which is sometimes referred to as “interest on interest,” is thought to have originated in Italy in the 17th century and will cause a sum to grow faster than simple interest, which is only calculated on the principal amount.

If interest is compounded more frequently, the interest accrues at a faster rate, so the more periods of compounding there are, the more interest accrues. Because of this, over the same period, the compound interest earned on \$100 compounded at 10% annual rate will be less than the interest earned on \$100 compounded at 5% semi-annually. Compounding is sometimes referred to as the “miracle of compound interest” because of the increasing positive returns that can be generated by the interest-on-interest effect.

## The Most Important Things to Remember

• When a deposit or loan is repaid, the original principal and all accrued interest are recalculated to arrive at the new interest rate.
• It is calculated by multiplying a sum of money by one plus the annual interest rate multiplied by the number of compound periods minus one, and then multiplying that sum by one.
• There are a variety of ways in which interest can be compounded, from continuous to daily to annual.
If you are calculating compound interest, the number of compounding periods is critical.

## Compound Interest: A Basic Concept

The annual interest rate is raised to the number of compound periods minus one and the initial principal amount is multiplied by this number. Once the result has been computed, we deduct the loan’s initial principal.

Compound interest is calculated using the following formula: Compound interest is the difference between the present value of the principal and the future value of the principal and interest (or present value)

To put it another way,

• Compound interest = total amount of principal and interest in future (or future value) less principal amount at present (or present value)

= [P (1 + i)n] – P

= P [(1 + i)– 1]

Where:

P stands for “principal.”

I = nominal annual interest rate expressed as a percentage

Number of compounding periods is denoted by the notation n.

Consider a 5%-per-year, three-year loan for \$10,000. The interest is compounded annually. For how much money would I have to pay in interest? In this instance, the answer is:

a total of \$1,576.25 is equal to \$10,000 [(1 + 0.05)3 – 1].

As a result of compound interest,

This is due to the fact that interest accrued in previous periods is included in the current period’s compound interest. This loan has a three-year interest payment of \$1,576.25, but the interest rate is not constant over that time period, as it would be with simple interest. In the table below, you can see how much interest you’ll owe at the end of each year.

Long-term returns can be significantly enhanced by compound interest. Over the course of ten years, a deposit of \$100,000 earning 5% simple annual interest would yield \$50,000 in total interest, while a deposit of \$10,000 earning 5% annual compound interest would yield \$62,889.46 in total interest over the same time frame. The total interest would rise to \$64,700.95 if the compounding period were paid monthly over the same 10-year period at 5% compound interest.

## Schedules for Compound Interest

From daily to annually, there is no limit to how often interest can be compounded. Compounding frequency schedules for financial instruments are standard.

Savings accounts at banks typically use a daily compounding schedule. Compounding occurs on a daily, monthly, or semiannual basis for CDs, and on a daily basis for money market accounts. The most common compounding schedule for home mortgages, home equity loans, personal business loans, and credit card accounts is monthly.

The period of time it takes for interest to be added to an account is also subject to change. Compounding interest daily is possible, but crediting interest only happens once a month. In order for the interest to begin earning interest on the account, it must be credited or added to the current balance first.

Continuous compounding interest is also offered by some banks, which means that the interest on a principal can be increased at any time. Unless you want to deposit money and withdraw it the same day, it doesn’t accrue any more interest than daily compounding.

The investor or creditor will benefit from more frequent interest compounding. Borrowers, on the other hand, have the opposite problem.

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## Periods of Compounding

If you are calculating compound interest, the number of compounding periods is critical. Compound interest grows in proportion to the number of periods during which it is compounded.

The following table illustrates the impact of compounding periods on a 10-year amortization of a \$10,000 loan at a 10% annual interest rate.

## Particulars to Keep in Mind

The time value of money and the Rule of 72, both critical concepts in investing, are intimately linked to compound interest.

Taking into account the monetary value of time

For investors who want to maximize their income and wealth allocation, understanding the time value of money and the exponential growth created by compounding is essential.

Here are the formulas for calculating future and present values:

In other words, FV is equal to PV (1 +i)n and PV is equal to FV/(1 +i)n

For example, a \$10,000 investment that grows at a rate of 5% per year for three years:

One thousand and fivety-five dollars

There are 3 x \$10,000 = \$11,576.25

Five percent discount rate: \$11,576.25 in present value, discounted for three years:

equals \$11,576.25 multiplied by 1 + 0.05 = \$11,576.25

When multiplied by 3, the result is \$11,576.25.

In this case, the discount factor is equal to the reciprocal of 1.157625, or 0.8638376.

## Consideration of the 72-Hour Rule

At a given interest rate or rate of return I the so-called Rule of 72 calculates the approximate time over which an investment will double. It can only be used to compound interest once a year.

An investment with a 6% annual rate of return will double in 12 years, for example. In nine years, an investment with an annual return of 8% will have doubled in value.

## As a percentage of GDP (CAGR)

Most financial applications use the compound annual growth rate (CAGR) to calculate a single growth rate over a period of time.

What is the compound annual growth rate (CAGR) if your investment portfolio grew from \$10,000 to \$16,000 over the course of five years? Assuming that PV = -\$10,000, FV = \$16,000, and t = 5, then the variable I has to be calculated. It can be determined using a financial calculator or Excel that I = 9.86%.

Your \$10,000 initial investment (PV) is shown with a negative sign because it represents a cash outflow. This is in accordance with the cash flow convention. In order to find I in the equation above, the signs of PV and FV must be in opposition.

## Real-World Applications of CAGR

The CAGR is widely used to calculate returns on stock, mutual funds, and investment portfolios over time. To determine whether a mutual fund manager or portfolio manager has outperformed the market’s rate of return over a period of time, the CAGR is also utilized. Over the course of a five-year period, for example, a market index returned a total of 10%, but the manager of a mutual fund only generated annual returns of 9%.

Over long periods of time, long-term investment portfolios can benefit from calculating their expected growth rate using the CAGR. Take a look at these examples:

Risk-averse investors are content with a 3% annual return on their investments. The \$100,000 she has now will grow to \$180,611 in 20 years if she continues to invest. In contrast, a risk-averse investor who expects a 6% annual return on her portfolio would see \$100,000 grow to \$320,714 in 20 years if she invested \$100,000.

Example 2: The CAGR can be used to estimate how much money is needed to save for a specific goal. Assuming a 4% annual return on their savings, a married couple who are saving \$50,000 for a down payment on a condo in 10 years would need to save \$4,165 per year. In order to save \$3,975 per year, they would have to take a little risk and expect a CAGR of 5%.

CAGR can also be used to demonstrate the advantages of investing early in life rather than later. A 25-year-old who wants to save \$1 million by the time he or she reaches 65 would need to save \$6,462 per year, assuming a CAGR of 6%. However, if you were to save \$18,227 for a 40-year-old to achieve the same goal, you would need to save nearly three times as much.

In economic data, CAGRs are also frequently seen. Here’s a case in point: In 1980, China’s per capita GDP was \$193, and in 2012, it was \$6,091. Over the course of 32 years, how much has per capita GDP grown? In this case, the growth rate I is an impressive 11.4%.
Compounding: The Good and the Bad

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In spite of Albert Einstein’s apocryphal claim that compounding is the eighth wonder of the world or man’s most important invention, compounding can work against consumers who have high-interest debt, such as credit card debt, which compounding can compound. Over the course of a year, a \$20,000 credit card balance would accrue \$4,388 in compound interest, or about \$365 per month, if the interest rate on the card was 20%.

When it comes to your investments, compounding can work in your favor and can be a powerful tool for wealth creation. In addition to mitigating wealth-eroding factors, such as rising costs of living, inflation, and decreased purchasing power, compounding interest can help.

Compound interest can be reaped by investors through the use of mutual funds, which is one of the simplest methods available. By choosing to reinvest your mutual fund dividends, you’re essentially buying more stock in the company. The value of the investment in the fund will continue to grow as more and more shares are purchased over time.

Consider a mutual fund investment with a \$5,000 initial investment and a \$2,400 annual contribution. Future value of the fund is \$798,500 if annual returns of 12% are maintained for 30 years. It is the difference between the amount of money you put into an investment and the amount of money you will get out of it in the future. Over the course of 30 years, if you contribute \$77,000, or \$200 per month, you will end up with \$721,500 in the bank.

Even if the money is in a tax-advantaged account, the interest income is usually taxed at a standard rate based on the taxpayer’s tax bracket.

## Investments that earn interest on a regular basis

When an investor uses a brokerage account’s reinvestment plan, they are essentially putting their money to work through the magic of compound interest.

With a zero-coupon bond, investors can also reap the benefits of compounding interest. It’s not uncommon for traditional bond issues to pay out interest payments to investors on a regular basis in the form of a check, which prevents interest from compounding. Instead of paying interest to investors, zero-coupon bonds are purchased at a discount and grow in value over time. Compounding is used by the issuers of zero-coupon bonds to raise the bond’s value until it reaches its full maturity price.

When it comes time to pay back a loan, compounding can help you save money. For example, if you pay half your mortgage twice a month instead of once a month, your amortization period will be reduced and you will save a significant amount of interest.

## Compound Interest: A Step-by-Step Guide

Fear not, math aficionados: Compounding can be calculated using simple tools. Exponent functions are available on a wide variety of calculators, both on the go and in the office.

## Excel’s Compound Interest Calculation.

In Microsoft Excel, there are three ways to perform more complex compounding tasks.

Compound interest is first calculated by multiplying the new balance for each year by the interest rate. To see how much money you’ll have at the end of five years after putting \$1,000 in an interest-bearing savings account earning 5% annually, use the formula below.

“Year” is entered in cell A1 and “Balance” is entered in cell B1. Cells A2 through A7 should be filled in with the dates from 0 to 5. Cell B2 needs to contain the number “1000” because the account balance at the start of the year is \$1,000. Place a = in cell B3, followed by a * in cell B2. In cell B4, type in “=B3*1.05” and keep going until you get to cell B7. In cell B7, the expression “=B6*1.05” is entered. Finally, the sum of \$1,276.28 in cell B7 represents your five-year savings account balance. Subtracting \$1,000 from \$1,276.28 yields \$276.28 as the compound interest value.

To calculate compound interest using a fixed formula is the second method. P is the initial amount, I is the yearly interest rate, and n is the number of periods over which the interest accrues. The compound interest formula is ((P*(1+i)n)-P). Then, in cells A1 and B1, enter “Principal value” and “1000,” respectively. Cell A2 should be populated with “Interest rate” and “.05” in cell B2. “Compound periods” and “5” should be entered into cells A3 and B3 respectively. The compound interest in cell B4 can be calculated by entering “=(B1*(1+B2)B3)-B1”, resulting in a value of \$276.28.

A macro function can also be used to calculate compound interest. Start the Visual Basic Editor, which can be found in the developer tab, first. Select Module from the Insert menu. Then, in the first line, type “Function Compound Interest (P as Double, I as Double, N as Double) as Double”. Hit the tab key and type “Compound Interest = (P*(1+i)n) – P” on the second line. Enter “End Function” on the module’s third line. In order to calculate the compound interest rate, you’ve created a macro function. Enter “Compound interest” into cell A6 and then type “=Compound Interest(B1, B2, B3)” into cell A7. That works out to a total of \$276.28, which is in line with the previous two values.
Various Other Online Calculators for Compound Interest

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Several online compound interest calculators are available, as are numerous handheld calculators.

You can use Financial-free Calculators.com’s compound interest calculator to calculate interest on your savings from daily to yearly. There’s a choice of continuous compounding and actual calendar start and end dates can also be entered. Results show interest earned, future value, annual percentage yield (APY) (a measure that includes compounding), and daily interest after the necessary calculation data is entered.
A free online compound interest calculator is available at Investor.gov, a website operated by the US Securities and Exchange Commission (SEC). For calculating earnings where additional monthly savings are being deposited, this calculator is fairly simple, but it does allow for monthly additional deposits to the principal.
In addition to the basic interest rate calculator, TheCalculatorSite.com offers a few advanced options. An inflation-adjusted increase to the monthly deposits or withdrawals can also be included in the calculations.

## Is There a Way to Tell if My Interests are Combined?

Under the Truth in Lending Act (TILA), prospective borrowers must be made aware of their loan’s terms and how interest is accrued, as well as whether interest is simply compounded.

Additionally, lenders must disclose their annual percentage rates (APRs) as part of the Truth-in-Lending Act (TILA). The APR is a simple interest rate derived from all of your loan’s finance charges, including interest and fees. If the interest rate and APR have a significant difference, one of two things is likely to happen: Compound interest may be part of your loan, or you may be required to pay substantial loan fees on top of the interest. When it comes to the same type of loan, the APR range varies widely between lenders due to the financial institution’s fees and other costs. ‘

Interest rates are determined by a variety of factors, such as your credit score. Interest rates on loans for people with good credit are significantly less than those for people with poor credit.

## Defintion of Compound Interest: What Is It?

Over time, interest on a bank account, loan, or investment will grow exponentially rather than linearly. This is known as compound interest. The word “compound” is the key to understanding the concept.

If you invest \$100 in a company that pays you a 10% dividend every year, you’ll get back \$100. Reinvesting dividends into new shares is an option, but you can also choose to keep your dividends as cash. Your initial \$100 investment will begin to grow in value over time as the dividends are reinvested and compounded.

## Compound interest is a boon to whom?

There are many different types of “investors,” and the term “investor” has a wide range of meaning. In the case of banks, for example, they reap the benefits of compound interest when they lend money and then reinvest that money into making more loans. Compound interest is a benefit to depositors as well when they receive interest on their bank accounts, bonds, or other investments

The term “compound interest” includes the word “interest,” but the concept applies to more than just bank accounts and loans, such as credit cards and mortgages.

## How Much Money Can You Make With Compound Interest?

Yes. Compound interest is, in fact, one of the most potent wealth-creating mechanisms ever devised. Compound interest has been used by merchants, lenders, and other businesspeople for literally thousands of years. For example, over 4,000 years ago, students in Babylon were taught the mathematics of compound interest using clay tablets.

One of the wealthiest people in the world today is Warren Buffett, who achieved this status by meticulously compounding the returns on his investments over a long period of time. For the foreseeable future, it is likely that people will be using compound interest to build wealth in some form or another.