Albert Einstein is reputed to have said that compound interest is the eighth wonder of the world. It’s easy to see why. Continuous growth from an ever-growing base is the fundamental reason investing is so compelling a practice.
Meanwhile, many everyday financial situations involve simple interest. For example, when you’re dealing with mortgages or car loans where interest does not compound, you can just worry about simple interest. That’s because usually in these cases, interest is only added to your outstanding principal balance—generally a good thing, as you don’t want to owe more.
Compounding has the potential to grow the value of an asset more quickly than simple interest. It can rapidly increase the amount of money you owe on some loans, since your interest grows on top of both your unpaid principal as well as previous interest charges. In the world of investment however, compounding returns are a key concept to understand.
Ahead, we’ll break down some of the important basics of interest, including the main differences between simple versus compound interest, as well as what continuous compounding is.
What Is Simple Interest?
In basic terms, simple interest is the amount of money you are able to earn after you have initially invested a certain amount of money, referred to as the principal. Simple interest works by adding a percentage of the principal—the interest—to the principal, which increases the amount of your initial investment over time.
When you put money into an average savings account, chances are you are accruing a small amount of simple interest.
APY is the annual rate of return that accounts for compounding interest. APY assumes that the funds will be in the investment cycle for a year, hence the name “annual yield.”
If your interest rate is low, you might be missing out on cash that could otherwise be in your pocket. And it may be worthwhile to look into other types of accounts that could earn you more interest.
Simple Interest Formula
Calculating interest is important for figuring out how much a loan will cost. Interest determines how much you have to pay back beyond the amount of money you are lent.
The simple interest formula is I = Prt, where I = interest to be paid, r is the interest rate, and t is the time in years.
So if you’re taking out a $200 loan at a 10% rate over one year, then the interest due would be 200 x .1 x 1 = $20.
But let’s say you want to know the whole amount due, as that’s what you’re concerned about when taking out a loan. Then you would use a different version of the formula:
P + I = P(1 + rt)
Here, P + I is the principal of the loan and the interest, which is the total amount needed to pay back. So to figure that out you would calculate 200 x (1 + .1 x 1), which is 200 x (1 + .1), or 200 x 1.1, which equals $220.
Example of Simple Interest
For example, let’s say you were to put $1,000 into a savings account that earned an interest rate of 1%. At the end of a year, without adding or taking out any additional money, your savings would grow to $1,010.00.
In other words, multiplying the principal by the interest rate gives you a simple interest payment of $10. If you had a longer time frame, say five years, then you’d have $1,050.00.
Though these interest yields are nothing to scoff at, simple interest rates are often not the best way to grow wealth. Since simple interest is paid out as it is earned and isn’t integrated into your account’s interest-earning balance, it’s difficult to make headway.
So each year you will continue to be paid interest, but only on your principal—not on the new amount after interest has been added.
What Is Compound Interest?
Most real-life examples of growth over time, especially in investing and saving, are more complex. In those cases, interest may be applied to the principal multiple times in a given year, and you might have the loan or investment for a number of years.
In this case, interest compounds, meaning that the amount of interest you gain is based on the principal plus all the interest that has accrued. This makes the math more complicated, but in that case the formula would be:
A = P x (1 + r/n)^(nt)
Where A is the final amount, P is the principal or starting amount, r is the interest rate, t is the number of time periods, and n is how many times compounding occurs in that time period.
Example of Compound Interest
So let’s take our original $200 loan at 10% interest but have it compound quarterly, or four times a year.
So we have:
200 x (1 + .1 / 4)^(4×1)
200 x (1 + .025)^4
200 x (1.025)^4
200 x 1.10381289062
The final amount is $220.76, which is modestly above the $220 we got using simple interest. But surely if we compounded more frequently we would get much more, right?
More Examples of Compound Interest
Let’s look at two other examples: compounding 12 times a year and 265 times a year.
For monthly interest we would start at:
200 x (1 + .1/12)^(12×1)
200 x (1 + 0.0083)^12
200 x 1.00833^12
200 x 1.10471306744
If we were to compound monthly, or 12 times in the one year, the final amount would be $220.94, which is greater than the $220 that came from simple interest and the $220.76 that came from the compound interest every quarter. And both figures are pretty close to $221.03.
Simple interest: $220
Quarterly interest: $220.76
Monthly interest: $220.94
Continuously compounding interest: $221.03.
Notice how we get the biggest proportional jump from one of these interest compoundings to another when we go from simple interest to quarterly interest, compared to less than 20 cents when we triple the rate of interest to monthly.
But we only get 18 cents more by compounding monthly instead of quarterly, and then only 9 cents more by going from monthly to as many compoundings as theoretically possible.
What Is Continuous Compounding?
Continuous compounding calculates interest assuming compounding over an infinite number of periods—which is not possible, but the continuous compounding formula can tell you how much an amount can grow over time at a fixed rate of growth.
Continuous Compounding Formula
Here is the continuous compounding formula:
A = P x e^rt
A is the final amount of money that combines the initial amount and the interest
P = principal, or the initial amount of money
e = the mathematical constant e, equal for the purposes of the formula to 2.71828
r = the rate of interest (if it’s 10%, r = .1; if it’s 25%, r = .25, and so on)
t = the number of years the compounding happens for, so either the term or length of the loan or the amount of time money is saved, with interest.
Example of Continuous Compounding
Let’s work with $200, gaining 10% interest over one year, and figure out how much money you would have at the end of that period.
Using the continuously compounding formula we get:
A = 200 x 2.71828^(.1 x 1)
A = 200 x 2.71828^(.1)
A = 200 x 1.10517084374
A = $221.03
In this hypothetical case, the interest accrued is $21.03, which is slightly more than 10% of $200, and shows how, over relatively short periods of time, continuously compounded interest does not lead to much greater gains than frequent, or even simple, interest.
To get the real gains, investments or savings must be held for substantially longer, like years. The rate matters as well. Higher rates substantially affect the amount of interest accrued as well as how frequently it’s compounded.
While this math is useful to do a few times to understand how continuous compounding works, it’s not always necessary. There are a variety of calculators online.
The Limits of Compound Interest
The reason simply jacking up the number of periods can’t result in substantially greater gains comes from the formula itself. Let’s go back to A = P x (1 + r/n)^(nt)
The frequency of compounding shows up twice. It is both the figure that the interest rate is divided by and the figure, combined with the time, that the factor that we multiply the starting amount is raised to.
So while making the exponent of a given number larger will make the resulting figure larger, at the same time the frequency of compounding will also make the number being raised to that greater power smaller.
What the continuous compounding formula shows you is the ultimate limit of compounding at a given rate of growth or interest rate. And compounding more and more frequently gets you fewer and fewer gains above simple interest. Ultimately a variety of factors besides frequency of compounding make a big difference in how much savings can grow.
The rate of growth or interest makes a big difference. Using our original compounding example, 15% interest compounded continuously would get you to $232.37, which is 16.19% greater than $200, compared to the just over 10% greater than $200 that continuously compounding at 10% gets you. Even if you had merely simple interest, 15% growth of $200 gets you to $230 in a year.
Understanding the ways in which interest rates can work both for and against you is an important step in helping to secure your future financial stability.
If you’re interested in investing and making your money work harder for you, then identifying interest types and finding ways to earn as much interest as possible could be the difference in thousands of dollars over the course of your life.
The bottom line, though, is that the longer you invest, the more time you have to weather the ups and downs of the stock market, and the more time your earnings have to compound. Just as investing early helps you take advantage of compound interest, so does investing regularly.
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