# Compound Interest: Definition, Formulas, and Solved Example

Banking

Interest is one of the best things about saving money. Even if you’re only earning 0.5% on your money, that 0.5% can add up. Every dime of that interest also earns interest.

That process is known as compound interest. Your interest earns interest as you go. As interest rates increase, you’re better able to earn money on the money you’ve set aside. But compound interest works in the other direction as well. If you’ve borrowed money, your borrowed money will earn interest that compounds.

Here are a few things to know about compound interest.

## What is compound interest?

Compound interest refers to the way interest builds on your savings or debt. As you earn interest, it’s added to the original amount, known as the initial principal, and future interest is applied to the principal plus all interest earned.

The opposite of compound interest is simple interest. With simple interest, the interest is calculated only against the initial principal amount, no matter how much interest you accrue over the course of the account.

## How Compound Interest Works

You may have heard it referred to as “the magic of compound interest.” Why is it magic? If you’re earning a small amount of interest, over time, compound interest can boost your balance significantly. This happens even if you don’t add money to the account.

But how that interest is calculated can vary from one account to another. There are four major ways interest is compounded:

• Annually: With this method, interest is compounded annually, which means your interest is applied once a year.
• Quarterly: When compounded quarterly, interest is compounded four times during the year. That means you’ll earn interest in the first quarter, the interest in the second quarter will be based on that slightly elevated amount, and so on.
• Monthly: As with quarterly compounding, when interest is compounded monthly, your interest is calculated every month, with the following month’s interest based on that higher amount.
• Daily: Some accounts are compounded daily, which gives you the advantage of boosting your principal sum every day for a slight increase in interest.

The difference between an annual rate and a daily rate can be subtle, depending on the interest rate on your account. So, it’s important to use a calculator to compare if you’re trying to decide between different account options.

## Formula for Periodic Compounding Rate

You’ll find plenty of compound interest calculators out there, but it can help to understand the formula for compounded interest. When compounding interest, lenders multiply the current value to the interest rate, then divide it by the number of compounding periods. So, the formula would be:

Current value x interest rate รท number of compounding periods = future value.

This is where the different compounding periods come into play. If your interest is compounded daily, that third figure will be 365, while interest that’s compounded quarterly will be 4.

## How does compound interest grow over time?

The best thing about compound interest is that you can enjoy significant returns over a longer period of time. You don’t even have to add to the initial amount to see gains. That makes it a great low-risk addition to an investor’s portfolio.

Say you put \$1,000 into a high-yield savings account at the age of 20. You enjoy a steady 3% APY for the life of the account, but that rate is compounded annually. In 30 years, that beginning balance will have risen to \$2,427.26. That’s without putting another dime into the account.

What happens if your interest follows a daily compounding schedule? There’s not much of a difference. After 30 years, your initial investment will have grown to \$2,459.51, which is only \$32.25 more than if your interest were compounded annually.

## How to Calculate Compound Interest

You can use the above formula to determine exactly how much interest you’ll earn with compounding. But there’s a much easier way.

A compound interest calculator can help you get a feel for just how much you can earn over time. Here are three to consider:

With a compound interest calculator, you can toy with various annual interest rates and toggle between a quarterly or annual compounding period to determine exactly the terms you need. You can also note just how much small variances in interest rate can change your overall interest accrued.

## Who benefits from compound interest?

Just as a compound interest formula can work for you when you’re saving or investing, it can also work against you when you take a loan. Credit card companies and lenders let compound interest work to earn them more interest on the money you borrow.

This is another area where a compound interest calculator can help. You can crunch the numbers and compare what one lender or credit card provider is offering against competitors. If you’ll be borrowing thousands or tens of thousands of dollars, the small difference in interest calculated daily vs. annually can save you money.

## Applying the Formula for Compound Interest

Once you’ve used the formula to calculate compound interest, it’s time to figure out how to make that interest work for you. You can throw some money in a savings account and wait for the accumulated interest to build, but lower rates that come with these account types will ensure interest grows slower than if you’d put the money elsewhere.

Here are a few vehicles that will help you start earning money using compound interest:

• High-yield savings accounts: You can often find a better annual interest rate on this type of savings account, but be aware that it will vary based on the market.
• Money market account: If you want your money to be accessible, a money market account can be a way to earn a higher interest amount without locking your funds down for a fixed time period.
• Savings bonds: With a savings bond, you lend money to a company or government entity in exchange for a fixed annual interest rate.
• Certificates of deposits: Although CDs do require you to agree to lock your funds away for six months to five years, they generally offer a higher annual rate than savings accounts and money markets.
• Investments: Investment returns can be higher, but they also come with a greater risk. With the right investment advice, though, you can not only enjoy the benefits of compounding interest, but you can reinvest any earnings to boost your dividends.

## Compound Interest Investments

The stock market can be a great way to take a principal balance and watch it grow. Look for low-risk dividend stocks that will grow slowly, over time. While these types aren’t known for earning money at the same rates as more volatile stocks, you won’t lose sleep worrying about what the stock market is doing.

Another low-risk, high-yield investment option is a REIT. Short for real estate investment trust, this savings vehicle pays dividends that you then reinvest, and you earn dividends on the reinvested amount. This offers compounding benefits similar to a savings option.

## What’s the difference between simple and compound interest?

Compounding does not apply to all interest earned. It varies from lender to lender. Some car loans and short-term personal loans use something called simple interest. You’ll also find this type of interest on your student and mortgage loans.

With simple interest, the interest is based solely on the principal amount. If you borrow \$1,000, you’ll only ever pay interest on that \$1,000. Some savings accounts use simple interest rather than a compound interest formula. This means any interest you earn from your savings will be eliminated.

## How to Determine Interest Amount

The total interest you accumulate will consist of the interest gained today, in addition to the interest accrued from prior periods. As long as you know the principal sum, you can determine exactly how much you’ve earned between the time you moved the money into the account and today.

The formula for determining the lump sum you’ve earned on your investment or savings is as follows:

Current value – initial amount = total interest earned.

If you’ve continued to put money into the account since your initial investment, you’ll need to factor that in, as well. Calculating compound interest on your investment requires first deducting any sporadic or monthly contribution in addition to your initial deposit.

Current value – initial amount – additional money contributed = total interest earned.

## What is continuous compounding?

Most interest is calculated using what’s known as continuous compounding periods. Those will be daily, monthly, quarterly, or yearly. The interest rate is applied to the previously earned interest only as it hits that milestone.

However, there is a concept known as continuous compounding. This type of compounding uses a different type of compounding frequency. In theory, this compounding takes place constantly. Continuous compounding is used when you’re trying to calculate how much you’ll earn on your money over an indefinite period of time.

## Formula for Continuous Compounding

The easiest way to calculate continuous compound interest is to use an online calculator. There are several available, including one from Omni Calculator and one from Math Warehouse.

But it can help to know how the formula is calculated. Continuous compound interest follows this formula:

Present Value x e (i x t) = Future Value

In this formula, e is the mathematical constant, and it is given the value of 2.7183. A mathematical constant is a number that has been assigned to a formula. Interest rate is represented by “i” in the formula, and you multiply it by the length of time, or “t,” that the interest will be earned.

## What is half-yearly compounding?

Although it’s not quite as common as daily, monthly, quarterly, or annual interest, half-yearly compounding is an option. With this type of compound interest, every six months, the interest is calculated and compounded. So, the last half of the year, you’re earning interest at a higher rate than in the first half. This can help your balance grow more quickly than with an annual rate.

You may see this type of compounding more often when you’re paying interest than when you’re earning it. Lenders might call it semi-yearly instead of half-yearly. Either way, the interest added to your balance will grow quicker than if you went with a loan that calculates annually.

## Formula for Half-Yearly Compounding

As with other types of compounding, half-yearly compound interest follows a formula. In this case, the numbers are added together, then divided by the number of compounding periods, which is two.

The easiest way to calculate half-yearly compounding is to use a calculator. You can find a half-yearly compound interest calculator at The Calculator Site or Calculator Soup. Simply choose semi-annually from the options.

If you need the formula for half-yearly compounding, it is as follows:

Present Value x {1+(i/2)/100)^t = Future Value

In this case, “i” represents interest, while “t” equals the number of semi-annual periods the loan or savings will be compounding.

## Continuing Contributions and Compounding

As powerful as compound interest can be, nothing will boost the future value of your savings and investments like adding funds to the account. As you continue to deposit funds over the weeks, months, and years, the interest you’re earning grows because the percentage is applied to a higher amount.

So if you initially deposited \$1,000, without adding another dime, you can increase to \$2,427.26 over 30 years at 3% interest, compounded annually. But if you add just \$10 a month over that 30-year period, your earnings will more than triple, with an ending balance of \$8,136.31.

How does it grow so fast? That’s the magic of compound interest. Initially, you’re earning interest on \$1,000, but the next month, you’re earning interest on \$1,010. If interest is compounded daily, each day the interest you earned the previous day is added to that total. For annually compounded interest, at the end of the first year, you’re earning interest on \$1,120, plus the interest you’ve earned on that \$1,120.

## How can I tell if interest Is compounded?

When you’re shopping for a new account, there’s a quick way to determine whether interest is compounded or not. You’ll see three letters after the rate, and those three letters will signal one of two types:

• APR: Short for annual percentage rate, this is a simple interest rate that only offers interest on the principal balance.
• APY: Annual percentage yield indicates that interest is compounded at least annually, if not daily, weekly, monthly, or quarterly.

## When to Avoid Compound Interest

Compound interest isn’t always a benefit to you, the consumer. That’s why you’ll see “APY” listed after interest rates offered on investments and savings accounts. Financial institutions will boldly list “APR” as the rate you’re getting when you borrow money.

If you end up with a loan where interest compounds, you’ll pay more over the course of the loan. With each month that borrowed amount accrues interest, you’ll owe total compound interest on both the principal and the total accrued interest so far. This additional interest charged can add up over time.

## Compound Interest and Retirement Planning

When it comes to setting money aside for retirement, annual or monthly compounding can be your friend. It allows you to set your money aside without worrying about losing it with stock market fluctuations. If you start early, you can sometimes earn more money over multiple decades, thanks to compound interest, than with other investments that carry more risk.

Some experts use compounding as a wealth creation tool. In that case, compounding happens when an investor puts all earned dividends back into investments to keep those dollars working. Over time, an investor can substantially increase an original principal using the funds earned on it.

## Compound Interest and Taxability

As with simple interest, your compound interest counts as taxable income. At tax time, you’ll receive a form from each of your financial institutions detailing the interest you’ve earned during the tax year. You’re expected to claim those funds on your taxes.

One exception is if you’ve parked your money in a tax-deferred account like a Traditional IRA. The funds will grow, tax free, throughout your lifetime, with the interest compounding as you go. But when you take the funds out later, you’ll owe taxes on all that interest. During retirement planning, a compound interest calculator can help you come up with the total earnings so you can begin to estimate how much you’ll owe.

## Interest Rates and Compounding Interest

As important as features like compounding frequency can be, it’s also important to look at the interest rate you’re earning. Even one or two percentage points can make a big difference in the interest you’d get on the same amount, especially if you’re earning that interest over a 20- or 30-year period.

The same goes when you’re paying interest. That one or two percentage points can save you money when interest is compound rather than simple. For a large monthly payment like a car loan or mortgage, this is especially important.

## Interest Rate Fluctuations and Compound Interest

Unless you lock in an interest rate when you take a loan or deposit funds into a new account, your rate will fluctuate over the years. In fact, rates have been on the rise in 2023, which is good news if you’re investing and saving, but bad news if you’re looking for a mortgage or personal loan.

If you’re earning compound interest on your money, though, interest rate hikes can be good news. The new, higher rate will be applied to your principal plus all previously earned interest. If it drops, you won’t lose the interest you earned when the rate was higher, which means you’ll continue to benefit from that extra amount for the duration of the account.

## When to Avoid Compound Interest

When it comes to a compound interest rate, you can only stand to benefit from it if you’re saving or investing. That money you put in will grow exponentially faster than if you had an account that earned simple interest. It becomes a matter of choosing an interest rate with the compound periods that will benefit you most.

However, if you get investment advice, you might find you’re steered toward other savings vehicles. Compounding works best when you have plenty of time for your funds to grow. If, for instance, you’re trying to save for retirement using compound interest, starting at the age of 20 will be better than starting at 50. For that reason, experts might steer you toward government bonds or mutual funds.

But compounding is worth avoiding if you’re the one paying the interest. When you’re looking for a new credit card or personal loan, pay attention to how interest is applied to your balance and look for one that will keep your interest as low as possible. You might, in this instance, find that a loan that uses simple interest or one that only compounds annually works best.

## Making the Most of Compounding

What’s the secret to making money on compounding? Time. If you can put your money into an account with compound interest when you’re in high school or young adulthood, then leave it there until retirement, you’ll get the maximum benefit. But it’s never too late to make money on compound interest.

The key to getting the most out of any savings or investment vehicle is to look for the best terms to start. Try to find the best compound interest rate available that also meets your needs, but also consider other features. If you have to give a little on interest rate on an account that’s compounded monthly instead of annually, you might be able to recover the small hit you’re taking on interest.

Whether you’re investing money or setting funds aside in a savings account, compound interest will get you farther, faster than simple interest. The key is to choose a compound interest rate that will maximize your earnings and regularly contribute to that account to give growth a boost. If you’re investing, reinvesting dividends can help your earnings grow in the same way compound interest benefits your savings vehicles.

Meet the author

Stephanie Faris is a professional finance writer with more than a decade of experience. Her work has been featured on a variety of top finance sites, including Money Under 30, GoBankingRates, Retirable, Sapling, and Benzinga.