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Discrete Compounding

Continuous compounding, special considerations, the bottom line.

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Discrete Compounding vs. Continuous Compounding: What's the Difference?

Discrete Compounding vs. Continuous Compounding: An Overview

People invest with the expectation of receiving more than what they invested. That added amount is commonly referred to as interest. Depending on the investment, interest can compound differently. The most common ways interest accrues is through discrete compounding and continuous compounding.

Discrete compounding and continuous compounding are closely related terms. Discretely compounded interest is calculated and added to the principal at specific intervals (e.g., annually, monthly, or weekly). Continuous compounding uses a natural log-based formula to calculate and add back accrued interest at the smallest possible intervals.

Interest can be compounded discretely at many different time intervals. Discrete compounding explicitly defines the number of and the distance between compounding periods. For example, an interest that compounds on the first day of every month is discrete.

There is only one way to perform continuous compounding—continuously. The distance between compounding periods is so small (smaller than even nanoseconds) that it is mathematically equal to zero.

Key Takeaways

• Compounding occurs when interest is paid not only on account balances but on previously-paid sums of interest.
• This "interest on interest" can lead to increasingly large returns over time, and has been heralded as the "miracle" or "magic" of compound interest.
• How often interest is paid on interest matters, as the more often it is paid, the more it will generate over time.
• Discrete compounding refers to payments made on balances at regular intervals such as weekly, monthly, or yearly.
• Continuous compounding yields the largest net return and computes (using calculus) interest paid hypothetically at every moment in time.

If the interest rate is simple (no compounding takes place), then the future value of any investment can be written as:

 F V = P ( 1 + r m ) m t where: F V = Future value P = Principal ( r / m ) = Interest rate m t = Time period \begin{aligned} &FV = P (1+ \frac{r}{m})^{mt}\\ &\textbf{where:}\\ &FV = \text{Future value}\\ &P = \text{Principal}\\ &(r/m) = \text{Interest rate}\\ &mt = \text{Time period}\\ \end{aligned} ​ F V = P ( 1 + m r ​ ) m t where: F V = Future value P = Principal ( r / m ) = Interest rate m t = Time period ​ 

Compounding interest calculates interest on the principal and accrued interest. When interest is compounded discretely, its formula is:

 FV = P ( 1 + r m ) m t where: t = The term of the contract (in years) m = The number of compounding periods per year \begin{aligned} &\text{FV} = \text{P} (1+ \frac{r}{m})^{mt}\\ &\textbf{where:}\\ &t = \text{The term of the contract (in years)}\\ &m = \text{The number of compounding periods per year}\\ \end{aligned} ​ FV = P ( 1 + m r ​ ) m t where: t = The term of the contract (in years) m = The number of compounding periods per year ​ 

Continuous compounding introduces the concept of the natural logarithm. This is the constant rate of growth for all naturally growing processes. It's a figure that developed out of physics.

The natural log is typically represented by the letter e. To calculate continuous compounding for an interest-generating contract, the formula needs to be written as:

F V = P × e r t FV=P\times e^{rt} F V = P × e r t

Many credit cards compound daily, resulting in extremely high credit card balances that are difficult to pay off. Make sure you're aware of how your credit card calculates interest and aim to pay off your balance every month to avoid increasing levels of debt.

Interest rates impact people in different ways. For example, an investor wants to earn the most interest possible, as it brings more of a return to their initial investment amount. A borrower wants the least amount of interest possible, because that makes the cost of borrowing lower. More interest on borrowed money ends up making it more expensive.

As an individual, you want to ensure that you are finding the best interest profile for yourself. In the case of an investor, they would benefit from compounding rather than simple interest , because simple interest calculates interest only on the principal amount. While this is good for borrowers, it is bad for investors.

As an investor, compounding is always the best scenario; however, if you can receive continuous compounding over discrete compounding, that is even better.

What Is an Example of Compounding Interest?

Compounding interest is interest earned on interest. For example, say you invest $5,000 that earns 5% every year. After the first year, you would have earned $250. This would be added to your initial investment of $5,000, for a new balance of $5,250. In the second year, you would earn 5% not on the $5,000, but on the $5,250. So in year two, you would've earned $262.50 in interest. Your new balance amount would be $5,512.50. In year three, interest would be calculated on the new balance of $5,512.50. You can see how with compounding interest you earn more interest over time.

What Are the Different Amounts of Time Interest Can Be Compounded?

Interest can be compounded at any time. It is dependent on who is determining the compounding intervals. Interest can be compounded daily, weekly, monthly, or annually. The more often it is compounded, the more interest is earned, and the faster your money grows.

Is Simple Interest Good?

Simple interest is good for a borrower because it calculates interest on just the principal amount, not on the interest amount that has accrued, making the cost of borrowing lower for a borrower. Simple interest is not good for an investor, as interest is only earned on the principal amount, but not on the accumulated interest, which would earn more money faster.

Compounding refers to how interest is calculated on interest on an investment. The two most common methods, discrete compounding and continuous compounding, will have different outcomes on the return of an investment.

Continuous compounding adds more interest, so it is better for investors, whereas discrete compounding adds less. However, all forms of compounding are better for investors than simple interest, which only calculates interest on the principal amount.

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Continuously Compounded Interest

Formula for Continuously Compounded Interest

To calculate continuously compounded interest use the formula below. In the formula, A represents the final amount in the account that starts with an initial ( principal ) P using interest rate r for t years . This formula makes use of the mathemetical constant e .

Continuously Compounded Interest is a great thing when you are earning it! Continuously compounded interest means that your principal is constantly earning interest and the interest keeps earning on the interest earned!

Practice Problems

If you invest $1,000 at an annual interest rate of 5% compounded continuously, calculate the final amount you will have in the account after five years.

If you invest $500 at an annual interest rate of 10% compounded continuously, calculate the final amount you will have in the account after five years.

If you invest $2,000 at an annual interest rate of 13% compounded continuously, calculate the final amount you will have in the account after 20 years.

If you invest $20,000 at an annual interest rate of 1% compounded continuously, calculate the final amount you will have in the account after 20 years.

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• A Visual Guide to Simple, Compound and Continuous Interest Rates

Interest rates are confusing, despite their ubiquity. This post takes an in-depth look at why interest rates behave as they do.

Understanding these concepts will help understand finance (mortgages & savings rates), along with the omnipresent e and natural logarithm . Here’s our cheatsheet:

• P = principal, your initial investment (i.e., \$1,000)
• r = interest rate (i.e., 5% per year)
• n = number of time periods (i.e., 3 years)

And a quick calculator to convert APR to APY:

Why the fuss?

Interest rates are complex. Like Roman numerals and hieroglyphics, our first system “worked” but wasn’t quite ideal.

In the beginning, you might have had 100 gold coins and were paid 12% per year (percent = per cent = per hundred — those Roman numerals still show up!). It’s simple enough: we get 12 coins a year. But is it really 12?

If we break it down, it seems we earn 1 gold a month: 6 for January-June, and 6 for July-December. But wait a minute — after our June payout we’d have 106 gold in July, and yet earn only 6 during the rest of the year? Are you saying 100 and 106 earn the same amount in 6 months? By that logic, do 100 and 200 earn the same amount, too? Uh oh.

This issue didn’t seem to bother the ancient Egyptians, but did raise questions in the 1600s and led to Bernoulli’s discovery of e (sorry math fans, e wasn’t discovered via some hunch that a strange limit would have useful properties). There’s much to say about this riddle — just keep this in mind as we dissect interest rates:

• Interest rates and terminology were invented before the idea of compounding. Heck, loans were around in 1500 BC, before exponents, 0, or even the decimal point! So it’s no wonder our discussions can get confusing.
• Nature doesn’t wait for a human year before changing . Interest earnings are a type of “growth”, but natural phenomena like temperature and radioactive decay change constantly, every second and faster. This is one reason why physics equations model change with “e” and not “$(1+r)^n$”: Nature rudely ignores our calendar when making adjustments.

Learn the Lingo

As a result of these complications, we need a few terms to discuss interest rates:

• APR (annual percentage rate): The rate someone tells you (“12% per year!”). You’ll see this as “r” in the formula.
• APY (annual percentage yield): The rate you actually get after a year, after all compounding is taken into account. You can consider this “total return” in the formula. The APY is greater than or equal to the APR.

APR is what the bank tells you, the APY is what you pay (the price after taxes, shipping and handling, if you get my drift). And of course, banks advertise the rate that looks better.

Getting a credit card or car loan? They’ll show the “low APR” you’re paying, to hide the higher APY. But opening a savings account? Well, of course they’d tout the “high APY” they’re paying to look generous.

The APY (actual yield) is what you care about, and the way to compare competing offers .

Simple Interest

Let’s start on the ground floor: Simple interest pays a fixed amount over time . A few examples:

• Aesop’s fable of the golden goose: every day it laid a single golden egg. It couldn’t lay faster, and the eggs didn’t grow into golden geese of their own.
• Corporate bonds: A bond with a face value of \$1000 and 5% interest rate (coupon) pays you \$50 per year until it expires. You can’t increase the face value, so \$50/year is what you will get from the bond. (In reality, the bond would pay \$25 every 6 months).

Simple interest is the most basic type of return . Depositing \$100 into an account with 50% simple (annual) interest looks like this:

You start with a principal (aka investment) of \$100 and earn \$50 each year. I imagine the blue principal “shoveling” green money upwards every year.

However, this new, green money is stagnant — it can’t grow! With simple interest, the \$50 just sits there. Only the original \$100 can do “work” to generate money.

Simple interest has a simple formula: Every period you earn P * r (principal * interest rate). After n periods you have:

This formula works as long as “r” and “n” refer to the same time period. It could be years, months, or days — though in most cases, we’re considering annual interest. There’s no trickery because there’s no compounding — interest can’t grow.

Simple interest is useful when:

• Your interest earnings create something that cannot grow more . It’s like the golden goose creating eggs, or a corporate bond paying money that cannot be reinvested.
• You want simple, predictable, non-exponential results . Suppose you’re encouraging your kids to save. You could explain that you’ll put aside \$1/month in “fun money” for every \$20 in their piggybank. Most kids would be thrilled and buy comic books each month. If your last name is Greenspan, your kid might ask to reinvest the dividend.

In practice, simple interest is fairly rare because most types of earnings can be reinvested. There really isn’t an APR vs APY distinction, since your earnings can’t change: you always earn the same amount per year.

Really Understanding Growth

Most interest explanations stop there: here’s the formula, now get on your merry way. Not here: let’s see what’s really happening.

First, what does an interest rate mean? I think of it as a type of “speed” :

• 50 mph means you’ll travel 50 miles in the course of an hour
• r = 50% per year means you’ll earn 50% of your principal in the course of a year. If P = \$100, you’ll earn \$50/year (your “speed of money growth”).

But both types of speed have a subtlety: we don’t have to wait the full time period!

Does driving 50 mph mean you must go a full hour? No way! You can drive “only” 30 minutes and go 25 miles (50 mph * .5 hours). You could drive 15 minutes and go 12.5 miles (50 mph * .25 hours). You get the idea.

Interest rates are similar. An interest rate gives you a “trajectory” or “pace” to follow. If you have \$100 at a 50% simple interest rate, your pace is \$50/year. But you don’t need to follow that pace for a full year! If you grew for 6 months, you should be entitled to \$25. Take a look at this:

We start with \$100, in blue. Each year that blue contributes \$50 (in green) to our total amount. Of course, with simple interest our earnings are based on our original amount, not the “new total”. Connecting the dots gives us a trendline: we’re following a path of \$50/year. Our payouts look like a staircase because we’re only paid at the end of the year, but the trajectory still works.

Simple interest keeps the same trajectory: we earn “P*r” each year, no matter what (\$50/year in this case). That straight line perfectly predicts where we’ll end up.

The idea of “following a trajectory” may seem strange, but stick with it — it will really help when understanding the nature of e.

One point: the trajectory is “how fast” a bank account is growing at a certain moment. With simple interest, we’re stuck in a car going the same speed: \$50/year, or 50 mph. In other cases, our rate may change, like a skydiver: they start off slow, but each second fall faster and faster. But at any instant , there’s a single speed, a single trajectory.

(The math gurus will call this trajectory a “derivative” or “gradient”. No need to hit a mosquito with the calculus sledgehammer just yet.)

Basic Compound Interest

Simple interest should make you squirm. Why can’t our interest earn money? We should use the bond payouts (\$50/year) to buy more bonds. Heck, we should use the golden eggs to fund research into cloning golden geese!

Compound growth means your interest earns interest . Einstein called it “one of the most powerful forces in nature”, and it’s true. When you have a growing thing, which creates more growing things, which creates more growing things… your return adds up fast.

The most basic type is period-over-period return, which usually means “year over year”. Reinvesting our interest annually looks like this:

We earn \$50 from year 0 – 1, just like with simple interest. But in year 1-2, now that our total is \$150, we can earn \$75 this year (50% * 150) giving us \$225. In year 2-3 we have \$225, so we earn 50% of that, or \$112.50.

In general, we have (1 + r) times more “stuff” each year. After n years, this becomes:

Exponential growth outpaces simple, linear interest, which only had \$250 in year 3 (100 + 3*50). Compound growth is useful when:

• Interest can be reinvested , which is the case for most savings accounts.
• You want to predict a future value based on a growth trend . Most trends, like inflation, GDP growth, etc. are assumed to be “compoundable”. Yearly GDP growth of 3% over 10 years is really $(1.03)^10 = 1.344$, or a 34.4% increase over that decade.

Interest as a Factory

The typical interpretation sees money as a “blob” that grows over time. This view works, but sometimes I like to see interest earnings as a “factory” that generates more money:

Here’s what’s happening:

• Year 0: We start with \$100.
• Year 1: Our \$100 creates a \$50 “bond”.
• Year 2: The \$100 generates another \$50 bond. The \$50 generates a \$25 bond. The total is 50 + 25 = 75, which matches up.
• Year 3: Things get a bit crazy. The \$100 creates a third \$50 bond. The two existing \$50 bonds make \$25 each. And the \$25 makes a 12.50.
• Years 4 to infinity: Left as an exercise for the reader. (Don’t you love that textbook cop out?)

This is an interesting viewpoint. The \$100 just mindlessly cranks out \$50 “factories”, which start earning money independently (notice the 3 blue arrows from the blue principal to the green \$50s). These \$50 factories create \$25 factories, and so on.

The pattern seems complex, but it’s simpler in a way as well. The \$100 has no idea what those zany \$50s are up to: as far as the \$100 knows, we’re only making \$50/year.

So why’s this viewpoint useful?

• You can separate the impact of the parent (\$100) from the children. For example, at Year 3 we have \$337.50 total. The parent has earned \$150 (“3 * 50% * \$100 = \$150”, using the simple interest formula!). This means the various “children” have contributed \$337.50 – \$150 – \$100 = \$87.50, or about 1/3 the total value.
• Breaking earnings into components helps understand e. Knowing more about e is a good thing because it shows up everywhere.

And besides, seeing old ideas in a new light is always fun. For one of us, at least.

Understanding the Trajectory

Oh, we’re not done yet. One more insight — take a look at our trajectory:

With simple interest, we kept the same pace forever (\$50/year — pretty boring). With annually compounded interest, we get a new trajectory each year .

We deposit our money, go to sleep, and wake up at the end of the year:

• Year 1: “Hey, waittaminute. I’ve got \$150 bucks! I should be making \$75/year, not \$50!”. You yell at your banker, crank up the dial to \$75/year, and go to sleep again.
• Year 2: “Hey! I’ve got \$225, and should be making \$112.50 per year!”. You scream at your bank and get the rate adjusted.

This process repeats forever — we seem to never learn.

Compound Interest Revisited

Why are we waiting so long? Sure, waiting a year at a time is better than waiting “forever” (like simple interest), but I think we can do better. Let’s zoom in on a year:

Look at what’s happening. The green line represents our starting pace (\$50/year), and the solid area shows the cash in our account. After 6 months, we’ve earned \$25 but don’t see a dime! More importantly, after 6 months we have the same trajectory as when we started. The interest gap shows where we’ve earned interest, but stay on our original trajectory (based on the original principal). We’re losing out on what we should be making.

Imagine I took your money and returned it after 6 months. “Well, ya see, I didn’t use it for a full year, so I don’t really owe you any interest. After all, interest is measured per year. Per yeeeeeaaaaar. Not per 6 months.” You’d smile and send Bubba to break my legs.

Annual payouts are man-made artifacts, used to keep things simple. But in reality, money should be earned all the time. We can pay interest after 6 months to reduce the gap:

Here’s what happened:

• We start with \$100 and a trajectory of \$50/year, like normal
• After 6 months we get \$25, giving us \$125
• We head out using the new trajectory: 50% * \$125 = \$62.5/year
• After 6 months we collect 62.5/year times .5 year = 31.25. We have 125 + 31.25 = 156.25.

The key point is that our trajectory improved halfway through, and we earned 156.25, instead of the “expected” 150. Also, early payout gave us a smaller gap area (in white), since our \$25 of interest was doing work for the second half (it contributed the extra 6.25, or \$25 * 50% * .5 years).

For 1 year, the impact of rate r compounded n times is:

In our case, we had $(1 + 50\%/2)^2$. Repeating this for t years (multiplying t times) gives:

Compound interest reduces the “dead space” where our interest isn’t earning interest . The more frequently we compound, the smaller the gap between earning interest and updating the trajectory.

Continuous Growth

Clearly we want money to “come online” as fast as possible. Continuous growth is compound interest on steroids: you shrink the gap into oblivion, by dividing the year into more and more time periods:

The net effect is to make use of interest as soon as it’s created. We wait a millisecond, find our new sum, and go off in the new trajectory. Except it’s not every millisecond: it’s every nanosecond, picosecond, femtosecond, and intervals I don’t know the name for. Continuous growth keeps the trajectory perfectly in sync with your current amount.

Read the article on e for more details (e is a special number, like pi, and is roughly 2.718). If we have rate r and time t (in years), the result is:

If you have a 50% APR, it would be an APY of $e^(.50)$ = 64.9% if compounded continuously. That’s a pretty big difference! Notice that e takes care of the icky parts, like dividing by an infinite number of periods.

Why’s this useful?

• Most natural phenomena grow continuously . As mentioned earlier, physical phenomena grows on its own schedule: radioactive material doesn’t wait for the Earth to go around the Sun before deciding to decay. Any physical equation that models change is going to use $e^rt$.
• $e^rt$ is the adjustable, one-size-fits-all exponential . It sounds strange, but e can even model the jumpy, staircase-like growth we’ve seen with compound interest. We’ll get into this in a later article.

Most interest discussions leave e out, as continuous interest is not often used in financial calculations. (Daily compounding, $(1 + r/365)^365$, is generous enough for your bank account, thank you very much. But seriously, daily compounding is a pretty good approximation of continuous growth.)

The exponential e is the bridge from our jumpy “delayed” growth to the smooth changes of the natural world.

A Few Examples

Let’s try a few examples to make sure it’s sunk in. Remember: the APR is the rate they give you, the APY is what you actually earn (your true return).

• Is a 4.5 APY better than a 4.4 APR, compounded quarterly? You need to compare APY to APY. 4.4% compounded quarterly is $(1 + 4.4\%/4)^4 = 4.47% $, so the 4.5% APY is still better.
• Should I pay my mortgage at the end of the month, or the beginning? The beginning, for sure. This way you knock out a chunk of debt early, preventing that “debt factory” from earning interest for 30 days. Suppose your loan APY is 6% and your monthly payment is \$2000. By paying at the start of the month, you’d save \$2000 * 6% = \$120/year, or \$3600 throughout a 30-year mortgage. And a few grand is nothing to sneeze at.
• Should I use several small payments, or one large payment? . You want to pay debt off as early as possible. \$500/week for 4 weeks is better than \$2000 at the end of the month. Each payment stops a few weeks’ worth of interest. The math is a bit tricker, but think of it as 4 \$500 investments, each getting different return. In a month, the first payment saves 3 week’s worth of interest: $500 * (1 + daily rate)^21$. The next saves 2 weeks: $500 * (1 + daily rate)^14$. The third saves a week $500 * (1 + daily rate)^7$ and the last payment doesn’t save any interest. Regardless of the details, prepayment will save you money.

The general principle: When investing, get interest paid early, so it can compound. When borrowing, pay debt early to prevent that interest from compounding.

Onward and Upward

This is a lot for one sitting, but I hope you’ve seen the big picture:

• The interest rate (APR) is the “speed” at which money grows .
• Compounding lets you adjust your “speed” as you earn more interest . The APR is the initial speed; the APY is the actual change during the year.
• Man-made growth uses $(1+r)^n$ , or some variant. We like our loans to line up with years.
• Nature uses $e^{rt}$ . The universe doesn’t particularly care for our solar calendar.
• Interest rates are tricky. When in doubt, ask for the APY and pay debt early.

Treating interest in this funky way (trajectories and factories) will help us understand some of e’s cooler properties, which come in handy for calculus. Also, try the Rule of 72 for a quick way to compute the effect of interest rates mentally (that investment with 6% APY will double in 12 years). Happy math.

Other Posts In This Series

• An Intuitive Guide To Exponential Functions & e
• Demystifying the Natural Logarithm (ln)
• Common Definitions of e (Colorized)
• Understanding Exponents (Why does 0^0 = 1?)
• Using Logarithms in the Real World
• How To Think With Exponents And Logarithms
• Understanding Discrete vs. Continuous Growth
• What does an exponent really mean?
• Q: Why is e special? (2.718..., not 2, 3.7 or another number?)

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Compound Interest Calculator

How to use the compound interest calculator, interest rate definition, what is the compound interest definition, simple vs. compound interest, compounding frequency, compound interest formula, compound interest examples, example 1 – basic calculation of the value of an investment, example 2 – complex calculation of the value of an investment, example 3 – calculating the interest rate of an investment using the compound interest formula, example 4 – calculating the doubling time of an investment using the compound interest formula, compound interest table, additional information, behind the scenes of compound interest calculator.

This compound interest calculator is a tool to help you estimate how much money you will earn on your deposit . In order to make smart financial decisions, you need to be able to foresee the final result. That's why it's worth knowing how to calculate compound interest. The most common real-life application of the compound interest formula is a regular savings calculation.

Read on to find answers to the following questions:

• What is the interest rate definition?
• What is the compound interest definition, and what is the compound interest formula?
• What is the difference between simple and compound interest rates?
• How to calculate compound interest?
• What are the most common compounding frequencies?

Our compound interest calculator is a versatile tool that helps you forecast the growth of your investments over time. To effectively use it, follow these instructions:

Enter initial balance : Start by inputting the amount you have initially invested or saved.

Input interest rate : Type in the annual interest rate your investment will earn.

Set the term : Determine the number of years and months over which you want the investment to grow.

Select Compounding Frequency : Choose how often the interest will be compounded. Options range from annually to daily.

Additional deposits : Decide if you'll make additional deposits. If so, specify the amount, how often, whether these are at the beginning or end of the compounding period, and their annual growth rate.

Review results : The calculator will display the final balance, total compound interest, and the breakdown of interest earned on the initial balance and additional deposits. It will also show the total principal amount and the sum of additional deposits made over the term.

Visualization : You can choose to represent your balance growth visually by selecting a bar graph, pie chart, table, or a combined chart and table view.

For example, with an initial balance of $1,000 and an 8% interest rate compounded monthly over 20 years without additional deposits, the calculator shows a final balance of $4,926.80. The total compound interest earned is $3,926.80.

Whether for personal savings, retirement planning, or educational investments, this calculator offers the foresight needed to make informed financial decisions.

Read on to learn more about the magic of compound interest and how it's calculated.

In finance, the interest rate is defined as the amount charged by a lender to a borrower for the use of an asset . So, for the borrower, the interest rate is the cost of the debt, while for the lender, it is the rate of return.

Note that in the case where you make a deposit into a bank (e.g., put money in your savings account), you have, from a financial perspective, lent money to the bank. In such a case, the interest rate reflects your profit.

The interest rate is commonly expressed as a percentage of the principal amount (outstanding loan or value of deposit). Usually, it is presented on an annual basis, which is known as the annual percentage yield (APY) or effective annual rate (EAR).

Generally, compound interest is defined as interest that is earned not solely on the initial amount invested but also on any further interest . In other words, compound interest is the interest on both the initial principal and the interest that has been accumulated on this principal so far. Therefore, the fundamental characteristic of compound interest is that interest itself earns interest . This concept of adding a carrying charge makes a deposit or loan grow at a faster rate.

You can use the compound interest equation to find the value of an investment after a specified period or estimate the rate you have earned when buying and selling some investments. It also allows you to answer some other questions, such as how long it will take to double your investment.

We will answer these questions in the examples below.

You should know that simple interest is something different than the compound interest . It is calculated only on the initial sum of money. On the other hand, compound interest is the interest on the initial principal plus the interest which has been accumulated.

Most financial advisors will tell you that compound frequency is the number of compounding periods in a year. But if you are not sure what compounding is, this definition will be meaningless to you… To understand this term, you should know that compounding frequency is an answer to the question How often is the interest added to the principal each year? In other words, compounding frequency is the time period after which the interest will be calculated on top of the initial amount .

For example:

• Annual (1/Yr) compounding has a compounding frequency of one ,
• Quarterly (4/Yr) compounding has a compounding frequency of four ,
• Monthly (12/Yr) compounding has a compounding frequency of twelve .

Note that the greater the compounding frequency is, the greater the final balance. However, even when the frequency is unusually high, the final value can't rise above a particular limit.

As the main focus of the calculator is the compounding mechanism, we designed a chart where you can follow the progress of the annual interest balances visually. If you choose a higher than yearly compounding frequency, the diagram will display the resulting extra or additional part of interest gained over yearly compounding by the higher frequency . Thus, in this way, you can easily observe the real power of compounding.

The compound interest formula is an equation that lets you estimate how much you will earn with your savings account. It's quite complex because it takes into consideration not only the annual interest rate and the number of years but also the number of times the interest is compounded per year.

The formula for annual compound interest is as follows:

• F V \mathrm{FV} FV – Future value of the investment, in our calculator it is the final balance
• P P P – Initial balance (the value of the investment);
• r r r – Annual interest rate (in decimal);
• m m m – Number of times the interest is compounded per year ( compounding frequency ); and
• t t t – Numbers of years the money is invested for.

It is worth knowing that when the compounding period is one ( m = 1 m = 1 m = 1 ), then the interest rate ( r r r ) is called the CAGR (compound annual growth rate): you can learn about this quantity at our CAGR calculator .

• Do you want to understand the compound interest equation?
• Are you curious about the fine details of how to calculate the compound interest rate?
• Are you wondering how our calculator works?
• Do you need to know how to interpret the results of compound interest calculation?
• Are you interested in all possible uses of the compound interest formula?

The following examples are there to try and help you answer these questions. We believe that after studying them, you won't have any trouble with understanding and practical implementation of compound interest.

The first example is the simplest, in which we calculate the future value of an initial investment.

You invest $10,000 for 10 years at the annual interest rate of 5%. The interest rate is compounded yearly. What will be the value of your investment after 10 years?

Firstly let’s determine what values are given and what we need to find. We know that you are going to invest $ 10000 \$10000 $10000 – this is your initial balance P P P , and the number of years you are going to invest money is 10 10 10 . Moreover, the interest rate r r r is equal to 5 % 5\% 5% , and the interest is compounded on a yearly basis, so the m m m in the compound interest formula is equal to 1 1 1 .

We want to calculate the amount of money you will receive from this investment. That is, we want to find the future value F V \mathrm{FV} FV of your investment.

To count it, we need to plug in the appropriate numbers into the compound interest formula:

The value of your investment after 10 years will be $16,288.95.

Your profit will be F V − P \mathrm{FV} - P FV − P . It is $ 16288.95 − $ 10000.00 = $ 6288.95 \$16288.95 - \$10000.00 = \$6288.95 $16288.95 − $10000.00 = $6288.95 .

Note that when doing calculations, you must be very careful with your rounding. You shouldn't do too much until the very end. Otherwise, your answer may be incorrect. The accuracy is dependent on the values you are computing. For standard calculations, six digits after the decimal point should be enough.

In the second example, we calculate the future value of an initial investment in which interest is compounded monthly.

You invest $10,000 at the annual interest rate of 5%. The interest rate is compounded monthly. What will be the value of your investment after 10 years?

Like in the first example, we should determine the values first. The initial balance P P P is $ 10000 \$10000 $10000 , the number of years you are going to invest money is 10 10 10 , the interest rate r r r is equal to 5 % 5\% 5% , and the compounding frequency m m m is 12 12 12 . We need to obtain the future value F V \mathrm{FV} FV of the investment.

Let's plug in the appropriate numbers in the compound interest formula:

The value of your investment after 10 years will be $ 16470.09 \$16470.09 $16470.09 .

Your profit will be F V − P \mathrm{FV} - P FV − P . It is $ 16470.09 − $ 10000.00 = $ 6470.09 \$16470.09 - \$10000.00 = \$6470.09 $16470.09 − $10000.00 = $6470.09 .

Did you notice that this example is quite similar to the first one? Actually, the only difference is the compounding frequency. Note that only thanks to more frequent compounding this time you will earn $ 181.14 \$181.14 $181.14 more during the same period: $ 6470.09 − $ 6288.95 = $ 181.14 \$6470.09 - \$6288.95 = \$181.14 $6470.09 − $6288.95 = $181.14 .

Now, let's try a different type of question that can be answered using the compound interest formula. This time, some basic algebra transformations will be required. In this example, we will consider a situation in which we know the initial balance, final balance, number of years, and compounding frequency, but we are asked to calculate the interest rate. This type of calculation may be applied in a situation where you want to determine the rate earned when buying and selling an asset (e.g., property) that you are using as an investment.

Data and question You bought an original painting for $2,000. Six years later, you sold this painting for $3,000. Assuming that the painting is viewed as an investment, what annual rate did you earn?

Solution Firstly, let's determine the given values. The initial balance P P P is $ 2000 \$2000 $2000 and final balance F V \mathrm{FV} FV is $ 3000 \$3000 $3000 . The time horizon of the investment is 6 6 6 years, and the frequency of the computing is 1 1 1 . This time, we need to compute the interest rate r r r .

Let's try to plug these numbers into the basic compound interest formula:

We can solve this equation using the following steps: Divide both sides by 2000 2000 2000 :

Raise both sides to the 1/6 th power:

Subtract 1 1 1 from both sides:

Finally solve for r r r :

In this example you earned $1,000 out of the initial investment of $2,000 within the six years, meaning that your annual rate was equal to 6.9913%.

As you can see this time, the formula is not very simple and requires a lot of calculations. That's why it's worth testing our compound interest calculator, which solves the same equations in an instant, saving you time and effort.

Have you ever wondered how many years it will take for your investment to double its value? Besides its other capabilities, our calculator can help you to answer this question. To understand how it does it, let's take a look at the following example.

Data and question

You put $1,000 in your savings account. Assuming that the interest rate is equal to 4% and it is compounded yearly, find the number of years after which the initial balance will double.

The given values are as follows: the initial balance P P P is $ 1000 \$1000 $1000 and final balance F V \mathrm{FV} FV is 2 ⋅ $ 1000 = $ 2000 2 \cdot \$1000 = \$2000 2 ⋅ $1000 = $2000 , and the interest rate r r r is 4 % 4\% 4% . The frequency of the computing is 1 1 1 . The time horizon of the investment t t t is unknown.

Let's start with the basic compound interest equation:

Knowing that m = 1 m = 1 m = 1 , r = 4 % r = 4\% r = 4% , and F V = 2 ⋅ P \mathrm{FV} = 2 \cdot P FV = 2 ⋅ P we can write:

Which could be written as:

Divide both sides by P P P ( P P P mustn't be 0 0 0 !):

To solve for t t t , you need take the natural log ( ln ⁡ \ln ln ), of both sides:

In our example, it takes 18 years (18 is the nearest integer that is higher than 17.67) to double the initial investment.

Have you noticed that in the above solution, we didn't even need to know the initial and final balances of the investment? It is thanks to the simplification we made in the third step ( Divide both sides by P P P ). However, when using our compound interest rate calculator, you will need to provide this information in the appropriate fields. Don't worry if you just want to find the time in which the given interest rate would double your investment; just type in any numbers (for example, 1 1 1 and 2 2 2 ).

It is also worth knowing that exactly the same calculations may be used to compute when the investment would triple (or multiply by any number, in fact). All you need to do is just use a different multiple of P in the second step of the above example. You can also do it with our calculator.

Compound interest tables were used every day before the era of calculators, personal computers, spreadsheets, and unbelievable solutions provided by Omni Calculator 😂. The tables were designed to make the financial calculations simpler and faster (yes, really…). They are included in many older financial textbooks as an appendix.

Below, you can see what a compound interest table looks like.

Using the data provided in the compound interest table, you can calculate the final balance of your investment. All you need to know is that the column compound amount factor shows the value of the factor ( 1 + r ) t (1 + r)^t ( 1 + r ) t for the respective interest rate (first row) and t (first column). So to calculate the final balance of the investment, you need to multiply the initial balance by the appropriate value from the table.

Note that the values from the column Present worth factor are used to compute the present value of the investment when you know its future value.

Obviously, this is only a basic example of a compound interest table. In fact, they are usually much, much larger, as they contain more periods t t t various interest rates r r r and different compounding frequencies m m m ... You had to flip through dozens of pages to find the appropriate value of the compound amount factor or present worth factor.

With your new knowledge of how the world of financial calculations looked before Omni Calculator, do you enjoy our tool? Why not share it with your friends? Let them know about Omni! If you want to be financially smart, you can also try our other finance calculators.

Now that you know how to calculate compound interest, it's high time you found other applications to help you make the greatest profit from your investments:

To compare bank offers that have different compounding periods, we need to calculate the Annual Percentage Yield, also called Effective Annual Rate (EAR). This value tells us how much profit we will earn within a year. The most comfortable way to figure it out is using the APY calculator , which estimates the EAR from the interest rate and compounding frequency.

If you want to find out how long it would take for something to increase by n%, you can use our rule of 72 calculator . This tool enables you to check how much time you need to double your investment even quicker than the compound interest rate calculator.

You may also be interested in the credit card payoff calculator , which allows you to estimate how long it will take until you are completely debt-free.

The depreciation calculator enables you to use three different methods to estimate how fast the value of your asset decreases over time.

Tibor Pál, a PhD in Statistical Methods in Economics with a proven track record in financial analysis, has applied his extensive knowledge to develop the compound interest calculator.

Inspired by his own need to calculate long-term investment returns and simplify the process for others, Tibor created this tool. It's designed to help users plan their financial future, whether for retirement, saving for a home, or understanding the potential growth of their investments.

Tibor has extensively used this calculator in various projects, allowing him to project financial outcomes accurately and advise on investment strategies. It's become an essential tool for anyone needing to calculate the future value of their investments, considering different compounding frequencies and additional contributions.

Trust in the compound interest calculator is grounded in our rigorous standards of accuracy and reliability. Financial experts have thoroughly vetted it to ensure it meets the practical needs of both individual investors and financial professionals.

What's compound interest?

Compound interest is a type of interest that's calculated from both the initial balance and the interest accumulated from prior periods. Essentially you can see it as earning interest from interest .

What's the difference between simple and compound interest?

While simple interest only earns interest on the initial balance , compound interest earns interest on both the initial balance and the interest accumulated from previous periods .

How do I calculate compound interest?

To calculate compound interest is necessary to use the compound interest formula , which will show the FV future value of investment (or future balance):

FV = P × (1 + (r / m)) (m × t)

This formula takes into consideration the initial balance P , the annual interest rate r , the compounding frequency m , and the number of years t .

How long does it take for $1,000 to double?

With a compounding interest rate, it takes 17 years and 8 months to double (considering an annual compounding frequency and a 4% interest rate). To calculate this:

Use the compound interest formula:

Substitute the values . The future value FV is twice the initial balance P , the interest rate r = 4% , and the frequency m = 1 :

2P = P × (1 + (0.04 / 1)) (1 × t) 2 = (1.04) t

Solve for time t :

t = ln(2) / ln(1.04) t = 17.67 yrs = 17 years and 8 months

Compound growth

Millionaire, phillips curve.

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Effective Annual Interest Rate

Interest rate that accounts for compounding effects and reflects the total annualized return on an investment.

Josh has extensive experience private equity, business development, and investment banking. Josh started his career working as an investment banking  analyst for Barclays  before transitioning to a private equity role Neuberger Berman. Currently, Josh is an Associate in the Strategic Finance Group of Accordion Partners, a  management consulting  firm which advises on, executes, and implements value creation initiatives and 100 day plans for Private Equity-backed companies and their financial sponsors.

Josh graduated Magna Cum Laude from the University of Maryland, College Park with a Bachelor of Science in Finance and is currently an  MBA  candidate at Duke University Fuqua School of Business with a concentration in Corporate Strategy.

Prior to becoming a Founder for Curiocity, Hassan worked for Houlihan Lokey as an Investment Banking Analyst focusing on sellside and buyside  M&A , restructurings, financings and strategic advisory engagements across industry groups.

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What is the Effective Annual Interest Rate?

Understanding effective annual interest rate.

• What influences EAR? 

Importance of Effective Annual Rate

Ear example.

• Steps For Calculating Effective Annual Interest Rate (EAR)
• Steps For Calculating EAR By Texas BA II Plus Financial Calculator
• What Interest Rate Is Charged By Banks?

Effective Annual Interest Rate (EAR) FAQ

An effective annual interest rate is the actual return on an investment or savings account when the rate is adjusted for compounding over a given period. Simply put, it is the actual percentage rate investor earns or pays in a year, considering  compounding.

EAR is an effective tool for evaluating interest payable or earnings for a loan/debt or investment. It incorporates the effect of compounding interest over a given time.

The formula contains two major components: the annual interest rate, also called Annual Percentage Return (APR) or Nominal Interest Rate, and the number of compounding periods.

The formula is as follows:

EAR = ((1+i/n)^n)-1

• i = Stated Annual Rate, Nominal Interest Rate, Annual Percentage Return
• n = Number of compounding periods

Effective Annual Rate Formula = (1 + r/ n) n - 1

The EAR quotes the real interest rate associated with an investment or loan. Again, it has two major components:

• The nominal interest rate
• The number of compounding periods

The most astonishing feature is that it considers that as the number of compounding periods increases, the effective interest rate gets higher.

Key Takeaways

The effective annual interest rate (EAR) is the actual return an investment considering compounding over a period.

EAR is calculated by incorporating both nominal interest rate (APR) and compounding periods, impacting real returns.

Different compounding frequencies influence EAR, with higher frequencies leading to greater annual equivalent rates (AER).

EAR aids in evaluating true returns and costs, crucial for financial planning, avoiding overestimation or underestimation, and comparing offers accurately.

EAR calculations guide investment decisions, adjusting nominal rates for compounding; it's used for loans, savings, and investment planning, ensuring accurate ROI calculations.

Assume that you have two loans, each with a 10% nominal interest rate, one compound annually and the other compound quarterly (four times a year). 

Even though both the loans have a stated annual interest rate of 10%, the effective annual interest rate of the loan that compounds four times a year will be higher. 

It is also known as the effective interest rate (EIR), annual equivalent rate (AER), or effective rate. 

The effective annual rate formula is used to differentiate the actual Internal Rate of Return for an interest rate that may or may not compound multiple times over a given period. 

As mentioned before, it can be very well used in comparing different kinds of investment opportunities or loan facilities taken under different capital structures.

But, the concept of compounding has its limits. There is a ceiling to the phenomenon. The limit of compounding is reached when it occurs an infinite number of times. The concept of such recurring compounding is called continuous compounding. 

An investment with a 10% interest rate, compounded continuously, will have an effective rate of 10.52%. The formula for continuous compounding is as follows:

  e^(i) - 1

Where e is approximately equal to 2.71828. The continuous rate is calculated by raising the e to the power of the interest rate and subtracting one. For this example, the equation will be: e^(10.52%) - 1

What influences EAR? 

Let's demonstrate the concept of compounding with a simple example. Credit card users tend to shop for groceries with a short-term loan from the bank. The bank charges a fee reflected as an interest rate (APR).

If the bill is not paid in full every month, one can end up paying interest not only on the principal amount but also on the interest that accrued in the previous month. This concept is called compounding interest. It can be fruitful if one is earning interest, not the other way round. 

Since credit card nominal rates do not reflect compounding interest, one needs to calculate the effective annual interest rate to discover the real interest rate on both the interest payment and compounding period. 

Similarly, it is true for investments and interest-bearing accounts to evaluate compounding interest earnings or gains. 

Lastly, the frequency of compounding determines EAR. The higher the frequency of compounding, the greater the annual equivalent rate (AER). Hence, an account that compounds monthly interest will have greater annual interest than an account that is compounded semi-annually. 

The effective annual interest rate is an important tool in evaluating the real return on an investment or effective interest rate for a loan. Getting a better understanding of it is crucial for such evaluations.

Say, for example, you are creating a retirement strategy. The aim is to come up with a figure that you need to save monthly to achieve your goal for what you want once you retire or leave the workforce. If the compounding effect of interest is not catered for, you may overestimate or underestimate what you need to save. Hence, calculating the EAR would give you a better estimate of what needs to be saved every month.

It is an important tool because, without it, the borrowers would underestimate the cost of debt or the cost of a loan. And investors may tend to overestimate the actual expected earnings on the investment, such as corporate bonds.

Referring to the second question, a bank may choose to advertise a loan with its nominal and effective rates. However, the nominal rate does not suggest compounding the interests that are part of the loan. The EAR is the real percentage return. This is why it is important to understand the concept of this financial tool. It will assist you in comparing multiple offers.

Consider a bank that offers you two investment opportunities of equal deposits of $10,000 at 12% and 12.2% stated interest rates.

Investment A compounds the 12% interest daily, and investment B compounds the 12.2% interest on a semiannual basis. The question is, which investment is better? 

For both investment opportunities, the bank advertised the nominal interest rate. You now have to calculate the effective annual interest rate by adjusting the nominal rate for the number of compounding periods. For both investments, the period is one year. 

The formula is 

Effective Annual Interest Rate: (1+(Nominal Rate/No. of compounding periods))^(number of compounding periods) - 1

For investment A: (1+(12%/365))^(365) - 1 = 12.74%

For investment B: (1+(12.2%/2)^(2) - 1 = 12.57%

As evident in the example, investment B has a higher stated nominal rate, but the effective annual interest rate is relatively lower than that of investment A.

Even though investment A has a lowered stated interest rate. This is because investment B has fewer compounding periods and hence a lower real rate. 

If an investor had to choose between the two investments, he/she would choose the investment with a higher effective annual interest rate.  

1. Debt Example  

Assume you have $5000 of the outstanding balance on a credit card with an APR of 20%. A common mistake would be to think that you would pay $1000 as interest over one year. But the bad news is that a credit card compounds interest daily, so you will need to account for the compounding concept. 

Replacing variables with numbers in the equation: 

(1+(0.20/365)^(365) - 1 

(1.0005)^365 - 1 

Hence, the EAR would be 22%. This means that the total interest that you will need to pay is $1100, not $1000.

2. Investment Example  

Though the concept applies in this manner, the terminologies used may vary. Assume that you now want to invest in a savings account with an annual percentage yield (APY) of 15%. Notice that we changed the terminology from a return to yield representing the interest rate effectively. 

So now you have invested in a savings account offering an interest rate of 15% compounded semiannually. 

(1+(0.20/2))^(2) - 1

(1.10)^(2) - 1

The real percentage rate on the deposit of $5000 is 21%. You invested $5000, and you would earn $1050 in gains instead of $1000. It helps in calculating the optimal ROI for an investment.

Steps for calculating Effective Annual Interest Rate (EAR) 

A detailed process for calculating the annual equivalent rate using the formula is given below. If you do not have the two components, the annual percentage rate and the number of periods, you can use the APR calculator to find the rate.

1. Determine the stated interest rate  

The stated return, also called the nominal rate, is stated with the loan or investment proposal. A common example would be: 10% interest charged quarterly. 

2. Calculate the number of the compounding period

The compounding period is defined as the number of periods the interest rate compounds over time. Normally, it is monthly, quarterly, or semi-annually. To give you a slightly better idea: 

Daily compounding = 365 compounding periods 

Weekly Compounding = 52 compounding periods 

Bi-weekly compounding = 26 compounding periods 

Monthly compounding = 12 compounding periods 

Quarterly compounding = 4 compounding periods 

Semi-annually = 2 compounding periods 

3. Apply the EAR formula  

EAR = (1 + (i/ n))^n - 1  

Example: To calculate the effective annual rate of a loan with an annual interest rate of 24%, charged quarterly: 

n = 4 

(1+(0.24/4))^4 - 1

The EAR is: 26.2%

Steps for Calculating EAR by Texas BA II Plus Financial Calculator 

This section explains using a financial calculator to calculate the effective rate. This section is of great importance for college students.  

The first step is to press the 2nd, the first button in the second row, and 2. This would show NOM, also called nominal rate. The nominal rate is the interest rate stated by the bank.

Enter the stated nominal rate. 

Press the down arrow, and you will see EFF and effective rate. It will be zero. The reason why it is zero is that we don’t know it yet. 

Press the down arrow again, and you will see C/Y, representing the number of compounding periods. For example, if the question asks you to compound quarterly, then the compounding per year would be 4. 

Enter the compounding period. 

Press the up arrow. The screen will display EFF, and press CPT. This would display the effective annual rate. This is the interest rate after accounting for the compounding effect. 

What interest rate is charged by banks? 

Banks tend to advertise nominal interest rates, which are the stated interest rate, instead of the effective annual interest rate. This tactic is applied to make consumers believe that they will have to pay lower interest. 

For example, for a loan with a stated interest rate of 25% compounded quarterly, the banks would advertise 25% instead of 27.4%. 

Though this might not be the case when banks are paying interest on the consumer’s savings account or deposit, in this case, the effective rate is advertised to attract more customers.

For example, if a deposit with the stated interest rate is 15% compounded monthly, the banks will advertise 16.1% instead of 15%.

Let us understand that can an effective annual interest rate be lowered by improving credit? 

Lenders, majorly banks, determine interest rates based on one’s creditworthiness, and the lower the credit score, the higher the real rate can be. One must review his/her credit history to work on credit. This will improve the chances of scoring a lower rate when applying for a loan.

The nominal interest rate does not take into account compounding. It is the rate stated by the financial institution. 

The compound interest rate is the real percentage rate on both the principal amount and the accumulated interest from previous periods on a deposit or loan. The frequency of compounding greatly affects the compound interest rate. 

Yes, essentially the effective interest rate (EIR) and the effective annual rate are the same. The terminologies are used interchangeably. 

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Interest calculation for 5 years

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What is compound interest?

The concept of compound interest, or 'interest on interest', is that accumulated interest is added back onto your principal sum, with future interest being calculated on both the original principal and the already-accrued interest. This compounding effect causes investments to grow faster over time, much like a snowball gaining size as it rolls downhill.

When you combine the power of interest compounding with regular, consistent investing over a sustained period of time, you end up with a highly effective growth strategy for accelerating the long-term value of your savings or investments.

To demonstrate the effect of compounding, let's take a look at an example chart of an initial $1,000 investment. We'll use a 20 year investment term at a 10% annual interest rate, to keep things simple. As you compare the compound interest line to those for standard interest and no interest at all, you can see how compounding boosts the investment value.

So, how important is compound interest? Just ask Warren Buffett, one of the world's most successful investors:

How is compound interest calculated?

Compound interest is a method that boosts your investment by adding earned interest back to the principal. This generates additional interest in the periods that follow, which accelerates your investment growth. So, when you earn interest on an investment, it isn’t just earned on the original amount; it’s calculated on the growing balance, including previously earned interest.

We'll explore an example later, to illustrate how it works. But before we do that, let's take a look at the formula.

Understanding the formula

Compound interest is calculated using the compound interest formula: A = P(1+r/n)^nt . For annual compounding, multiply the initial balance by one plus your annual interest rate raised to the power of the number of time periods (years). This gives a combined figure for principal and compound interest.

Let's break the compound interest formula down into its individual parts:

• A = the future value of the investment
• P = the principal balance
• r = the annual interest rate (decimal)
• n = number of times interest is compounded per year
• t = the time in years
• ^ = ... to the power of ...

How to calculate monthly compound interest

Here's how to calculate monthly compound interest using our compound interest formula. Monthly compound interest means that our interest is compounded 12 times per year:

• Divide your annual interest rate (decimal) by 12 and then add one to it.
• Raise the resulting figure to the power of the number of years multiplied by 12.
• Multiply your step 2 result by your principal balance (P).
• Deduct the principal balance from your step 3 result if you want just the interest.

As a formula, it looks like this:

A = P(1 + r/12)^12t

In our article about the compound interest formula , we go through the process of how to use the formula step-by-step, and give some real-world examples of how to use it.

Using our interest calculator

You can use our compound interest calculator to do all the formula work for you. It'll tell you how much you might earn on your savings, investment or 401k over a period of years and months based upon a chosen number of compounds per year.

Simply enter your initial investment (principal amount), interest rate, compound frequency and the amount of time you're aiming to save or invest for. You can include regular deposits or withdrawals within your calculation to see how they impact the future value.

See also: Simple Interest Calculator | Loan Calculator With Extra Payments

What will $10,000 be worth in 20 years?

We've discussed what compound interest is and how it is calculated. So, let's now break down interest compounding by year, using a more realistic example scenario. We'll say you have $10,000 in a savings account earning 5% interest per year, with annual compounding. We'll assume you intend to leave the investment untouched for 20 years. Your investment projection looks like this...

$10,000 invested at a fixed 5% yearly interest rate, compounded yearly, will grow to $26,532.98 after 20 years. This means total interest of $16,532.98 and a return on investment of 165%.

These example calculations assume a fixed percentage yearly interest rate. If you are investing your money, rather than saving it in fixed rate accounts, the reality is that returns on investments will vary year on year due to fluctuations caused by economic factors.

It is for this reason that the risk management strategy of diversification is widely recommended by industry experts.

Compounding with additional deposits

Combining interest compounding with regular deposits into your savings account, SIP , Roth IRA or 401(k) is a highly efficient saving strategy that can really boost the growth of your money in the longer term.

Looking back at our example from above, if we were to contribute an additional $100 per month into our investment, our balance after 20 years would hit the heights of $67,121, with interest of $33,121 on total deposits of $34,000.

As financial institutions point out, if people begin making regular investment contributions early on in their lives, they can see significant growth in their savings further down the road as their interest snowball gets larger and they gain benefit from Dollar-cost or Pound-cost averaging. 2

Note that you can include regular weekly, monthly, quarterly or yearly deposits in your calculations with our interest compounding calculator at the top of the page .

Where to invest for compound interest

The question about where to invest to earn the most compound interest has become a feature of our email inbox, with people thinking about mutual funds, ETFs, MMRs and high-yield savings accounts and wanting to know what's best.

We at The Calculator Site work to develop quality tools to assist you with your financial calculations. We can't, however, advise you about where to invest your money to achieve the best returns for you. Instead, we advise you to speak to a qualified financial advisor for advice based upon your own circumstances.

There are also some excellent articles from renowned financial websites that list ways to invest for compound interest. Here are two of the best articles, to help with your research:

• 13 Best Compound Interest Investments . WealthUp (author: Riley Adams).
• Accounts That Earn Compounding Interest . Motley Fool (author: Adam Levy).

Let's cover some frequently asked questions about our compound interest calculator.

When is my interest compounded?

With savings and investments , interest can be compounded at either the start or the end of the compounding period. If additional deposits or withdrawals are included in your calculation, our calculator gives you the option to include them at either the start or end of each period.

Can I include regular withdrawals?

You can include regular withdrawals within your compound interest calculation as either a monetary withdrawal or as a percentage of interest/earnings. This can be used in combination with regular deposits.

You may, for example, want to include regular deposits whilst also withdrawing a percentage for taxation reporting purposes. Or, you may be considering retirement and wondering how long your money might last with regular withdrawals.

What is the effective annual interest rate?

The effective annual rate (also known as the annual percentage yield ) is the rate of interest that you actually receive on your savings or investment after compounding has been factored in.

When interest compounding takes place, the effective annual rate becomes higher than the nominal annual interest rate. The more times the interest is compounded within the year, the higher the effective annual interest rate will be.

What is RoR/TWR?

Within our compound interest calculator results section, you will see either a RoR or TWR figure appear for your calculation. You may be wondering what these are, so let's take a look.

The Rate of Return (RoR) is the percentage return on your investment over the entire investment term. We calculate it by taking the Initial investment figure away from the Final value, dividing the resulting figure by the Initial investment and then multiplying it by 100. The formula looks like this: 3

If you include regular deposits or withdrawals in your calculation, we switch to provide you with a Time-Weighted Return (TWR) figure .

The TWR figure represents the cumulative growth rate of your investment. It is calculated by breaking out each period's growth individually to remove the effects of any additional deposits and withdrawals. The TWR gives you a clearer picture of how your investment might have performed if you hadn't made extra deposits or withdrawn funds, allowing you to better assess its overall performance. You can learn more about TWR in this article by The Balance .

If you want to head back up to the calculator results area, you can click the link here . If you have any feedback or questions about the RoR or TWR, please contact us .

Before you go...

Here's a final thought . If you want to roughly calculate compound interest on a savings figure, without using a calculator, you can use a formula called the rule of 72 . The rule of 72 helps you estimate the number of years it will take to double your money. The method is simple - just divide the number 72 by your annual interest rate. We use a version of the rule of 72 in our calculator.

For example, let's say you're earning 3% per annum. Divide 72 by 3, which will give you 24. So, in about 24 years, your initial investment will have doubled. If you're receiving 6% then your money will double in about 12 years. All using the power of compound interest.

I hope you found this calculator and article useful. Many of the features in my compound interest calculator have come as a result of user feedback, so if you have any comments or suggestions, I would love to hear from you .

• Using online compound interest tools to improve financial literacy , The Journal of Economic Education.
• How to take advantage of compound growth . Canada Life.
• Rate of return . CFI Education Inc.
• Warren Buffett . The Giving Pledge.

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