What Is Compound Interest?

It’s believed that Albert Einstein once called compound interest the eighth wonder of the world: “He who understands it, earns it. He who doesn't, pays it.” This article will explain compound interest and teach you how to calculate it, as you look to grow your wealth.

To understand compound interest, let’s take a step back. Interest is, in the plainest terms, the cost of borrowing money. When you borrow money, you pay interest: a set percentage owed on top of the principal you borrowed. When you lend money, you receive interest: a set percentage received on top of your principal. Compound interest occurs when that initial interest compounds along with your principal.

By way of example: If your principal investment of $100 receives 10% interest per annum, you have $110 at the end of the year. That $110 then receives 10% interest per annum, leaving you with $121 at the end of the second year. That $121 then receives 10% interest per annum, so you’ll have $133.10 at the end of the third year. In other words, the amount of interest you receive is growing each year, compounding along with your principal and exponentially growing your wealth.

How does compound interest work?

Compound interest works by using the continual multiplication—or compounding—of an amount of money, over a set number of cycles, within a set period of time. The interest paid is added to your original pile of money, on which the next interest payment is calculated. You are earning interest on interest, and your initial wealth begets more and more wealth as a result.

Each time interest is paid, it’s added to your principal—the money you started with—helping that pile of money grow. Interest for the next compounding period is then calculated on this larger principal. This compounding continues for the duration of the agreement, or can continue with no end date.

Origin of compound interest

As early as the 13th century, European mathematicians created massive tables and grids of numbers designed to solve questions such as: How much should a person be charged in exchange for a contract directing payment of an annual sum for each year of the person’s life? These tables were also used to calculate annuities, loan rates, and many types of contracts.

In certain contracts, interest due on a loan may automatically be added to the loan balance. So just as compounding can work in favor of an investor who is able to save, the opposite can be true for those who borrow. For example, a credit card balance can swiftly grow almost out of control because interest charges can pile up at an accelerating rate.

How to calculate compound interest

The central mathematical concept of compounding hinges on the idea of future value. What is something going to be worth in the future, and, conversely, what should something be worth today, to pay out, or receive, a certain sum, sometime in the future?

Here’s a basic compound interest calculator that shows examples of how compound interest works.

Consider a customer who deposits $1,000 into a savings account that earns 2% per year. We’ll calculate the compound interest for a variety of time periods.
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Similarly, using the example above, consider that your $1,000 was used to purchase a stock that paid a 2% annual dividend, which is paid quarterly. If the value of the stock stayed constant at $1,000, your investment value would be the same.

Compound interest formula

Investments that generate compound interest rely on calculations involving four components:

1. The initial principal (for example, that pile of money you invested at the start).
2. The interest rate (the cost of the money or the dividend yield).
3. The number of times interest or dividends are paid during the life of the investment. If the annual interest rate is 2% per year, and payments are made quarterly (every three months), you’re already earning compound interest on your principal after the very first payment or dividend.
4. And finally, the time periods covered by the investment or agreement.
   

Here’s a breakdown of the formula where P is the principal, or present value of $1,000, earning an interest rate of 2% (r for rate) per year, with payments made every three months—or four times a year—(n is the number of times per period interest is paid), for 10 years (t for the time period).

This is expressed as the formula:

A = P(1 + r/n)nt

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Where A is the total amount of principal and accrued interest.

Calculating compound interest in Excel

Let’s use a spreadsheet to automate and simplify our calculations:
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First, in our example, there are 40 payment periods, four per year for 10 years. We put that number in cell B2. We know the annual interest rate is 2%, so we enter that in cell B3. There are no payments being made other than the initial principal invested of $1,000, so we enter 0 for payments. Now we consider that the investor puts $1,000 into the account, and therefore the present value is entered as a negative number of -1,000.

We don’t know what the future value is, so we leave that blank. Finally, we enter 4 for the number of periods payments will be made each year.

Now we can use the spreadsheet formula FV for future value. Click in cell C6 and type the formula =FV(B3/B7, B2, B4, B5) and hit return.

You will see that the FV function has done the calculation for you. You can also adjust all of the inputs to apply different time periods, principal amounts, or rates of interest. It’s a tool to compute compound interest in a variety of scenarios.

Now, let’s plug our numbers into the compound interest formula and check the accuracy of our Excel function

First, convert r, or rate, from a percent to a decimal.

r = r/100

r = 2/100

r = 0.02 rate per year

Then solve the equation for A, based on the formula cited above:

A = 1,000.00(1 + 0.02/4)(4)(10)

A = 1,000.00(1 + 0.005)(40)

A = $1,220.79

A = P + I where

P (principal) = $1,000.00

I (interest) = $220.79

A, the total amount accrued, principal plus interest, with compound interest on a principal of $1,000.00 at a rate of 2% per year, compounded four times per year over 10 years is $1,220.79.

You can see that compounding makes your money grow faster because the interest is calculated on the accumulated interest as well as your original principal. Compound interest creates a snowball effect, as your original investment, plus the interest earned from those investments, grow hand-in-hand.

Simple vs. compound interest

Now let’s compare what our investment value would be using the simple interest formula versus the compound interest formula.

Simple interest is only calculated on the principal amount—it doesn’t include interest accrued over time.

We’ll use the same example of $1,000 invested at 2% a year, with four payments per year for 10 years. Here’s the simple interest formula:

A = P (1+rt)

A = the total accrued amount (the principal + the interest)
P = the principal of $1,000
I = the interest rate of 2% which is $20, or $5 every quarter
r = rate of interest per year expressed in decimal form
t = the time period involved in months or years

A = $1,000(1 + 0.02*10)
A = $1,200 after 10 years

You would earn just $1,200 with simple interest over the same period of time, while with compound interest you accumulate 10% more, or $1,220.79.  

Compound interest is traditionally associated with money held in savings and checking accounts at banks, but because we’re in a period of extended low-interest rates, the benefits of compounding at a bank are pretty negligible. But dividend-paying stocks can provide similar advantages.

How does compound interest work with stocks and mutual funds?

Dividend-paying stocks and mutual funds can enable you to earn compound interest on your investments, as long as a dividend reinvestment plan is available and selected.

Dividend reinvestment plans work by using the dividends you receive from your investment to purchase additional shares. In other words, when your investment in Company X pays out dividends, the dividend reinvestment plan takes that capital to purchase more Company X shares. As long as Company X’s stock continues to pay dividends, you end up with more shares—and more dividends paid —each cycle.

The advantages of compounding are even greater when the stocks, or other interest-bearing assets, are held in a tax-advantaged account, such as an IRA (Individual Retirement Account).

Tax-advantaged accounts allow investors to shield the compounding value of your investments from tax liabilities, until funds are withdrawn.
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By deferring your tax liability, your principal may be able to grow larger and faster. The dividends paid on the accumulated investment in a pretax account will be greater than if they were paid on aftertax holdings. This, in turn, leads to a faster accumulation of wealth.

The bottom line

Now that you understand more about compound interest and how to calculate it, it should be clear how a consistent savings plan, leaving income in an account where it is compounded, is a fundamental tenet to growing your wealth. And the more time compounding has to work its magic, the more money you’ll have.