Compound Interest, With the Numbers Shown

The formula fits on one line. What it does across thirty years does not fit in your intuition, which is why the tables below are worth more than the rule of thumb.

By Hutu Laurentiu · Published 1 November 2026 · 10 min read

The formula, term by term

Compound interest is what happens when the return on an investment starts earning a return of its own. The standard expression is A equals P times (1 + r/n) raised to the power n times t. P is the starting amount, r is the annual rate as a decimal, n is how many times per year interest is credited, and t is the number of years.

Put $10,000 into that at 7 percent compounded annually for thirty years. The multiplier is 1.07 to the thirtieth power, which is 7.6123, and the ending value is $76,122.55. The investment earned $66,122.55 on a starting balance of $10,000, which is the entire point of the exercise.

The term doing most of the work is the exponent. Over thirty years, moving from 5 percent to 9 percent, an 80 percent increase in the rate, roughly triples the ending balance. Adding ten more years at a fixed 7 percent multiplies it by another 1.97. Time sits in the exponent, and that is why a decade is worth more than several points of return.

Table one: the horizon does most of the work

The table sets out a single $10,000 deposit, left alone, at three rates and three horizons. Every value is exact arithmetic on the stated inputs, with no adjustment for inflation, taxes, or fees.

Read across the bottom row and the gap between 5 percent and 9 percent is a factor of three at thirty years, and that factor applies to the whole balance rather than to the gain. Read down the 7 percent column instead: the value nearly doubles from year 10 to year 20 and nearly doubles again from year 20 to year 30, even though the same 7 percent is applied the whole way through.

Where the growth actually comes from

Break the 7 percent column into decades and the distribution is lopsided. In years one through ten the balance grows by $9,671.51. In years eleven through twenty it grows by $19,025.33. In years twenty-one through thirty it grows by $37,425.71, which is more than the first two decades combined.

At 9 percent the effect is sharper: $13,673.64, then $32,370.47, then $76,632.68. The last decade produces more gain than the first two together, and it does so on a schedule that was set before the money was deposited.

This is the mechanism behind the advice to start early, and it is worth understanding precisely. Starting earlier does not just add years of contributions. It shifts the entire balance into the part of the curve where compounding is fastest, and that shift is worth more than any plausible improvement in return.

Put another way, a 7 percent account does not grow at a steady 7 percent in dollars. It grows at 7 percent of a balance that is larger every year, so the dollar gain accelerates whether or not the rate ever changes. Anyone who has watched a retirement balance sit flat for years and then move quickly has seen this and mistaken it for the market changing.

Table two: why frequency matters less than time

The n in the formula gets a lot of attention from product marketing. On $10,000 at 7 percent for thirty years, with the nominal rate held constant and only the crediting frequency changed, here is what it is worth:

Going from annual to daily compounding adds 7.3 percent to the ending balance. Going from thirty years to forty at the same annual compounding adds $73,622.03, which is more than thirteen times as much. Frequency is a real effect and it is not nothing, but it is small next to the calendar, and it is not something you can buy by choosing a product. The account sets it. You do not.

The useful conclusion is that comparing a daily-compounded account at 4.00 percent to a monthly-compounded account at 4.10 percent should send you to the annual percentage yield rather than to the compounding language on the front of the brochure. The APY already converts frequency into a number you can compare.

Table three: contributions versus compounding

Most real saving happens through regular contributions rather than a single deposit. With $500 going in every month at 7 percent and monthly compounding, the split between what you put in and what the account produced changes shape over time.

At ten years, growth is 44 percent of contributions. At twenty years it is 117 percent. At thirty years it is 239 percent, which means the account has earned more than twice what was paid into it. The pattern matches the lump-sum case, spread across years rather than concentrated at the start.

This is also why a late start is expensive in a way that a smaller contribution is not. Doubling the monthly contribution raises the thirty-year total, but it does not move any of the early money onto the steep part of the curve. Contribution size multiplies the result. The start date decides which curve is being multiplied.

The Rule of 72 and how far off it drifts

The Rule of 72 estimates doubling time by dividing 72 by the annual rate. At 7 percent it gives 10.29 years against an exact answer of 10.24. At 9 percent it gives 8.0 years against 8.04. At 5 percent it gives 14.4 against 14.21. In the ranges people usually apply it to, the error is a few weeks.

It drifts at the extremes. At 2 percent the rule says 36 years and the exact answer is 35.0. At 20 percent the rule says 3.6 years and the exact answer is 3.80, an error of more than two months. The rule is a mental arithmetic shortcut for mid-single-digit rates, and it is accurate enough there that the drift is worth ignoring.

The more useful version is to invert it. Instead of asking how long it takes to double, ask what rate is required to double in a given number of years: divide 72 by the years. Doubling in twelve years needs roughly 6 percent. Doubling in thirty needs roughly 2.4 percent, which is about the rate a savings account has to beat just to keep pace with moderate inflation.

Two cautions apply. The rule describes a single deposit with no additions, so it will not tell you when a portfolio you keep contributing to will double. And it measures doubling rather than growth: at 4 percent the money doubles in eighteen years and quadruples in thirty-six, which is why the rule is a starting point for mental arithmetic rather than a substitute for the formula.

Real versus nominal: what the number in your account is worth

Every figure above is nominal. None of them accounts for the fact that a dollar in 2056 buys less than a dollar today. Divide the nominal result by the cumulative inflation factor to get purchasing power in today's money.

At 7 percent nominal and 3 percent inflation, $10,000 grows to $76,122.55 after thirty years, but 1.03 raised to the thirtieth power is 2.4273, so the real value is $31,361.48 in today's dollars. The real annual return is not 4 percent, the simple subtraction. It is 1.07 divided by 1.03 minus 1, which is 3.88 percent.

That gap between 4.00 and 3.88 looks pedantic, and it compounds like everything else. Over thirty years on a $10,000 real balance the two rates diverge by about $1,104, and it generalizes: the wider the rate and the longer the period, the further subtract-and-divide drifts from the correct calculation.

Taxes and fees, the part the charts omit

Compound interest illustrations almost always assume no tax and no cost. Both are large enough to change decisions. Take the $10,000 at 7 percent for thirty years: the gain is $66,122.55, and a 15 percent capital gains rate on that gain is $9,918.38. The after-tax ending value is $66,204.17 rather than $76,122.55, a reduction of about 13 percent.

Fees compound the same way returns do. A 1 percent annual expense ratio drops a 7 percent gross return to 6 percent net, and $10,000 becomes $57,434.91 instead of $76,122.55. That is $18,687.64 less, or roughly a quarter of the ending balance, for a cost that sounds like one percentage point. On $500 a month for thirty years the same 1 percent costs $107,728 in ending value.

In a taxable account the drag is annual and unavoidable, which is why tax-advantaged accounts exist. In a retirement account the drag still applies to fees, and nothing about the tax wrapper protects you from it. The fee is the only part of the equation you can negotiate before you invest.

What compounding does not promise

Compounding is not symmetric. A 30 percent loss requires a 43 percent gain to recover, because 0.70 times 1.43 is approximately 1.00, and the arithmetic does not care that the loss felt worse. A portfolio that drops 20 percent in one year and gains 20 percent the next does not break even; it ends at 0.96 of where it started.

Sequence matters too. Two portfolios with identical average returns can end in very different places depending on when the bad years land, particularly if money is being withdrawn along the way. The clean upward curve in any illustration is a statement about a constant rate, and constant rates do not exist in markets.

None of that makes the tables wrong. It makes them conditional. They describe what happens to a fixed rate applied for a fixed period, which is precisely what a savings account, a certificate of deposit, or a bond held to maturity delivers. For anything with a variable return, the tables describe a tendency rather than an outcome, and the honest projection is a range rather than a line.