compound interest: the arithmetic behind the promise

In short

A friend of mine once told me he felt behind on saving, and I said I understood that feeling well. Compound interest is the plain arithmetic of earning interest on your interest, and it rewards time more than almost anything else. If you are holding a savings account or a retirement fund, the same math is working on your side right now. Starting earlier matters more than starting big. Small sums, left alone for many years, grow in ways that surprise people. Fees and long gaps between deposits work against you, so watch for both. Check the rate on your own account today, and then do the simple sums in this guide.

The whole of it

What it is

I once watched an old farmer plant a single row of corn, and he said the second year's crop came from the first year's seed. Compound interest works like that. You put in money, and it earns interest. Next period, the interest earns interest too. The base keeps growing, so each round pays a little more than the last.

Simple interest is the plainer cousin. It pays only on the money you first put in. If you deposit $1,000 at 5 percent simple interest, you earn $50 each year, every year, and no more. Compound interest pays $50 the first year, but the second year it pays 5 percent on $1,050. That is the whole difference, and it is a bigger one than it looks.

A word about the word rate. The rate is the percent the account pays for a stretch of time, usually a year. The compounding period is how often the bank adds the interest to your balance. It might be monthly, daily, or once a year. Each time the interest gets added, the next round of interest is figured on a larger sum.

How it works

You have probably seen the formula in a textbook, and it looks colder than it is. The final amount equals the starting amount times one plus the rate, raised to the number of periods. Raised to just means multiplied by itself that many times. So $1,000 at 5 percent for 3 years is $1,000 times 1.05 times 1.05 times 1.05.

Let me walk it slowly. After year one, $1,000 times 1.05 is $1,050. After year two, $1,050 times 1.05 is $1,102.50. After year three, $1,102.50 times 1.05 is $1,157.63, rounded to the cent. Simple interest would have given you $1,150. The extra $7.63 is interest earned on interest. It is small now. Give it thirty years and it is not small at all.

Now add regular deposits, which is how most of us save. Each deposit starts its own clock. The money you put in this month has the longest time to grow, and the money you put in next month has a little less. This is why a steady habit beats a lucky lump sum that arrives late.

Compounding more often helps a little. A rate of 5 percent compounded monthly means the bank adds 5 percent divided by 12 each month, which is about 0.4167 percent. Over one year, $1,000 grows to $1,051.16 that way, compared with $1,050 compounded once. The gap is tiny. The rate and the time matter far more than the schedule.

The numbers, and where to find yours

If you are wondering which numbers are yours, start with the account itself. Your bank or plan lists the rate on your statement or on its website. Federal rules require banks to disclose it. The Truth in Savings Act, enforced through Regulation DD, requires a bank to state an annual percentage yield, or APY, on deposit accounts. The APY already folds in the effect of compounding, so it is the fair figure for comparing one account with another.

If you hold a loan, look for the annual percentage rate, or APR, which the Truth in Lending Act requires lenders to disclose. For loans, compounding works against the borrower, and the same arithmetic applies in reverse. The Consumer Financial Protection Bureau explains both terms on its website, and that is a good primary source to bookmark.

For retirement accounts, the return is not promised. A savings account pays a stated rate. An investment account rises and falls. The plan statement shows what actually happened. The yearly contribution limit for a retirement account is set by law and changes, so the current figure is $7,500 for 2026 (source, checked 10/4/2026), and the IRS publishes it on irs.gov. Check that page for the year you are in.

A worked example

Let me tell you about a woman named Marisol. She is 25, earns $52,000 a year, and has a savings account paying 4 percent a year, compounded once a year. Her employer does not offer a match, so she saves on her own. She decides to put away $200 a month, which is $2,400 a year.

To keep the arithmetic honest, we will use a year by year approach with a deposit at the end of each year of $2,400. After year one, she has $2,400. After year two, $2,400 times 1.04 is $2,496, plus the new $2,400, which is $4,896. After year three, $4,896 times 1.04 is $5,091.84, plus $2,400, which is $7,491.84. You can see the habit building.

The shortcut for this kind of steady saving is a formula, and here are the inputs. Each year's deposit is $2,400, the rate is 0.04, and the years are 30. The growth factor is 1.04 raised to the 30th power, which is about 3.2434. Subtract 1 to get 2.2434. Divide by 0.04 to get 56.085. Multiply by $2,400 and you get about $134,604.

Now count what she put in herself. That is $2,400 times 30 years, which is $72,000. The rest, about $62,604, is interest. Nearly half the final balance came from the interest earning interest, not from her paycheck.

Here is the part that always makes me smile. Suppose Marisol had waited and started at 35 instead. Same $2,400 a year, same 4 percent, but only 20 years. The growth factor is 1.04 to the 20th power, about 2.1911. Subtract 1 to get 1.1911, divide by 0.04 to get 29.778, and multiply by $2,400. That is about $71,467. She would have put in $48,000 herself. Ten years of waiting cost her roughly $63,000 at the end. Same effort each year. Time did the rest.

Where it goes wrong

I have made my share of mistakes with money, so I speak as a fellow traveler. The first stumble is expecting a steady rate where none is promised. A savings account may hold its rate, but an investment account does not. A fund can lose value in a bad year, and then the math runs backward. A drop of 20 percent needs a gain of 25 percent just to get even, because the loss comes off a larger number.

The second stumble is fees. A fee of 1 percent a year sounds small. But it comes out of the balance every year, so it also removes all the future growth that money would have earned. Over thirty years, that small fee can take a big bite. Look at what your plan charges, and the plan documents list it.

The third is inflation. A balance of $134,604 sounds grand, but prices rise over thirty years, and the money buys less than it does today. The Bureau of Labor Statistics publishes the Consumer Price Index, which tracks price changes. It is worth knowing the dollars you see are not the dollars you will spend.

The fourth is taxes. Interest in a regular account is taxed as income in the year it is earned, which shrinks the amount left to compound. Some retirement accounts delay or remove that tax, and the rules differ by account type. The IRS explains each one.

The fifth is the gap. Skipped deposits and early withdrawals break the chain. Pulling money out early costs you the interest that money would have earned for every year after. It hurts twice.

Questions to answer before you leave this page

Do you know the APY on the account where your savings sit right now, and where on your statement would you find it? Have you looked at what fees your plan charges each year, and how much of your balance they take over a long stretch? If you started a small habit this month, say $50 or $100, what would the steady deposit look like on your paycheck, and could you live with it? Do you know which accounts delay or remove tax on growth, and which one fits your situation? And when you picture yourself thirty years from now, what would you like that money to do for you?