Compound Interest Year by Year, the Arithmetic That Rewards the Patient Saver

Somebody, somewhere, will tell you that compound interest is the eighth wonder of the world and that a famous physicist said so. The attribution has no good evidence behind it, and the physicist is not needed. Compound interest is not a miracle. It is multiplication, and multiplication has been available to schoolchildren for centuries.

That is a better story than the miracle. A wonder you must believe in. Arithmetic you can check. This post works the arithmetic out year by year, so that you can follow every number with a pencil. Every rate below is an illustration, chosen because it is round and plausible, and not a prediction of what any investment will earn. Nothing here is advice about your own money. Returns vary, they can be negative, and no one can promise a figure.

The formula

Terms first. Principal (P) is the money you start with. The rate (r) is the annual interest rate, written as a decimal, so 7 percent is 0.07. Years (n) is the time the money is left to grow. A is the amount at the end.

If interest is added once a year and left in the account, then each year the balance is multiplied by 1 plus the rate. After n years:

A = P(1 + r)^n

That is the entire theory. The word compound means that interest is earned on the interest. Simple interest, by contrast, pays only on the original principal: $10,000 at 7 percent simple interest earns $700 every year, forever, for a total of $7,000 over ten years. Compound interest earns $700 in the first year, but in the second year it earns 7 percent on $10,700, which is $749. The extra $49 is interest on interest. It looks small. It is not small for long.

Ten years, one line at a time

Here is $10,000 left alone at a hypothetical 7 percent a year, every figure rounded to the cent.

Look at the interest column. In year 1, $700. In year 10, $1,286.92, more than 80 percent higher, though the household has not added a cent. The balance did the work. After ten years, the account holds $19,671.51, which is $9,671.51 of growth on a $10,000 start. Simple interest would have produced $17,000. The difference, $2,671.51, is the compounding.

The formula agrees. 1.07 raised to the tenth power is about 1.967151, and $10,000 times that is $19,671.51. Extend it: after 20 years, $38,696.84. After 30, $76,122.55. After 40, $149,744.58. The money doubles roughly every ten years, and the later doublings are enormous in dollars because the base is larger. Most of the growth lives at the end.

That is the lesson, and it is a lesson about time, not about cleverness. Nobody in this table made a decision after year 1.

The Rule of 72

You can estimate doubling time in your head. Divide 72 by the annual rate in percent, and the result is roughly the number of years it takes for money to double. At 7 percent, 72 divided by 7 is about 10.3 years. The exact answer, the number of years for which 1.07 raised to that power equals 2, is about 10.24. The rule is close enough for a conversation over coffee.

At 6 percent, 72 divided by 6 is exactly 12 years; the true value is about 11.9. At 3 percent it gives 24 years. The rule loses accuracy for very high or very low rates, but between about 4 and 12 percent it is good. Its virtue is that it makes the cost of a lower rate visible. One percentage point looks harmless. At 6 percent instead of 7, doubling takes about a year and a half longer, and over thirty years the difference compounds into a very large gap, as the section on fees shows.

Starting early versus starting late

Now the comparison that makes the case. Two hypothetical savers, both earning the same illustrative 7 percent a year, both saving $5,000 at the end of each year.

Saver A starts at 25, saves $5,000 a year for ten years (ages 25 through 34), then stops forever and leaves the money alone. A deposits $50,000 in total.

Saver B waits until 35, then saves $5,000 a year for thirty years (ages 35 through 64). B deposits $150,000 in total, three times as much.

Balances are shown on the birthday indicated, after the deposit for the year just ended.

At 65, Saver A, who contributed for ten years and put in $50,000, has about $525,872. Saver B, who contributed for thirty years and put in $150,000, has about $472,304. The early saver finishes ahead by roughly $53,568, having put in a third as much.

Two details in the table repay a second look. First, at age 45, Saver B has $69,082.24, which is exactly what Saver A had at 35: ten deposits of $5,000 at 7 percent. B is ten years behind, and A's money has been compounding for those ten years. Second, A's ten deposits of $5,000 growing to $69,082.24 at 35 are then multiplied by 1.07 thirty more times. That is the whole advantage: $69,082.24 times 1.07 to the thirtieth power is $525,871.63.

I want to be careful about what this does and does not show. It does not show that the early saver is better than the late one as a person, or that one should never begin late. A saver who starts at 35 and continues is far better off than one who never starts. It does not prove that every investment earns 7 percent: it does not, and in many years it does not come close. What it does show is the shape of compounding, and that the shape rewards time disproportionately. Each year earlier is worth more than each year later is, and the very first years are the most valuable of all, because they have the longest to run.

The steelman, and the answer

The best objection to all this is practical. Few people at 25 have $5,000 a year to spare. They have student loans, a first apartment, perhaps a child. Telling a twenty-five-year-old to save for retirement is easy for those who already have. Wouldn't it be wiser to pay off debts and build a cushion first, and begin to save for retirement at 35, when income is higher?

It is a good objection, and it deserves a serious answer. Part of the answer is that it is correct in its core. A household that has high-rate debt and no emergency reserve should put those first; that argument is made in Why the Emergency Fund Comes Before Every Other Money Goal You Have, and the question of which debts to attack is worked in Debt Snowball or Debt Avalanche? Two Payoff Plans Run on the Same Four Debts. Compounding does not repeal prudence.

But the objection misses two things. First, the table does not require $5,000 a year. Scale every number by a tenth and the story is the same: $500 a year from 25 to 34 grows to about $52,587 at 65, while $500 a year from 35 to 64 grows to about $47,230. Starting small is not the same as not starting. Second, an employer's matching contribution, where one is offered, is typically an immediate return on the worker's own contribution. Whether and how a given plan offers one is a question for the plan documents, and the general mechanics are in How a 401(k), a Traditional IRA and a Roth IRA Really Work, in Twelve Questions.

My verdict: start with whatever amount you can, even a small one, once the basics are covered, and raise it when you can. The habit is worth more than the amount, and the early years are worth more than the later ones.

The price of waiting one year

The comparison can be made sharper. Suppose Saver A's whole schedule is pushed back by a single year, so that every deposit and every year of growth is one year later, and the end date stays at 65. Each dollar then has one fewer year of growth, and one fewer year at 7 percent means dividing by 1.07. Saver A's $525,871.63 becomes $525,871.63 divided by 1.07, which is $491,468.81. One year of delay cost $34,402.82 at the finish line, for a saver who deposited exactly the same cash.

The cost of a year of delay is not constant, and that is the unsettling part. It is a fixed percentage of the final balance, about 6.5 percent, so the dollar cost rises as the balance rises. A single $5,000 deposit made at 35 is worth $38,061.28 at 65 under these assumptions (5,000 times 1.07 to the thirtieth power). Made at 36, it is worth $35,571.29. The one-year delay costs $2,489.99 on a single deposit. The larger the sum, the larger the penalty for waiting.

Nobody should read this as a scolding. A person who is 40 and has not begun cannot go back. The arithmetic says only that the best day to start was earlier and the second-best day is the one at hand, and that is a claim about numbers, not about character.

How often interest is added

One more detail, because people ask. In the examples above, interest is added once a year. Real accounts often add it more frequently, such as monthly or daily, and the more often it is added, the more the balance grows, though by less than most people expect. At the same stated 7 percent, $10,000 compounded monthly for ten years becomes about $20,096.61, against $19,671.51 compounded yearly. Compounded daily, it becomes about $20,136.18. The gap between yearly and daily compounding is $464.67 over ten years, which is real but modest next to the effect of the rate or the time.

The practical point is that the stated rate and the effective rate can differ. When you compare accounts or loans, compare the annual percentage yield or the annual percentage rate as the law requires them to be disclosed, and not a headline number. The same arithmetic that rewards a saver also governs a borrower: the interest on a credit card compounds against you, and the myths about carrying a balance are tested in Carrying a Balance Builds Credit and Six Other Borrowing Myths, Tested.

Fees: the quiet subtraction

The figures above assume no costs. Real investments have expenses, and an annual fee comes straight out of the return. A fee of 1 percent a year does not sound like much. Do the arithmetic.

Take $10,000 growing at an illustrative 7 percent for 30 years: $76,122.55. Now suppose a 1 percent annual fee reduces the net return to 6 percent. After 30 years, $10,000 times 1.06 to the thirtieth power is $57,434.91. The difference is $18,687.64, or about 24.5 percent of the fee-free result, taken by a charge that was 1 percent a year.

The figure is worth remembering, because it is not obvious. A 1 percent fee is one-seventh of a 7 percent return, and compounding does the rest. It is the same arithmetic that makes the early saver's money so productive, turned against the saver. Fees compound too. The lesson is not that every fee is a rip-off, since some services are worth their cost, but that you should know what you are paying and ask what you are getting for it. The place to find an investment's expenses is in its prospectus or plan disclosure.

Inflation: the number that matters is the real one

The second subtraction is not a charge at all. It is the rising cost of living. A dollar in 30 years will buy less than a dollar today, and the amount you end up with should be measured against what it will buy, not against what it says.

Suppose, hypothetically, that prices rise by 3 percent a year. After 30 years, prices are 1.03 to the thirtieth power, which is about 2.427 times their present level. So the $76,122.55 from the fee-free example is worth, in present-day purchasing power, $76,122.55 divided by 2.427, or about $31,361. The nominal growth is 661 percent. The real growth, in this illustration, is about 214 percent. Still large, but far from what the headline number implies.

The "real" rate of return, to be exact about it, is approximately the nominal rate minus the inflation rate. More precisely, it is 1.07 divided by 1.03, minus 1, which is about 3.88 percent. At that real rate, the doubling time is about 18.2 years, compared with 10.2 at the nominal one. Neither is a prediction; the inflation rate of the next thirty years is not known to anyone. But a plan that ignores inflation is a plan made for a world that does not exist.

This is also why inflation is a moral issue and not a merely technical one. When governments spend beyond their means and the currency loses value, they impose, in effect, a tax on every saver. Nobody voted for it. A family that spent forty years building a nest egg has a direct interest in a government that keeps its books. The country's own accounts are on display at the national debt clock, and they are not encouraging.

What the old moralists knew

None of this is new. Benjamin Franklin, in the maxims gathered in 1758 as The Way to Wealth, preached the virtue of small, steady habits, and the compounding table is his sermon in numbers. Benjamin Franklin's The Way to Wealth, Read Line by Line for Modern Savers takes the text apart line by line. The thrifty habits that built this country's middle class were not glamorous. They were a long series of small deposits, made year after year by people who did not expect to be thanked.

The history of how Americans came to have savings vehicles at all is told in From Savings Banks to Roth Accounts, a Timeline of How Americans Learned to Save.

The case in brief

1. Compound interest is multiplication: A = P(1 + r)^n, with interest earned on previously earned interest.
2. At an illustrative 7 percent, $10,000 becomes $19,671.51 in ten years and $76,122.55 in thirty, and the Rule of 72 estimates the doubling at about ten years.
3. Saver A, who deposits $5,000 a year from 25 to 34 and stops, has about $525,872 at 65, while Saver B, who deposits $5,000 a year from 35 to 64, has about $472,304, despite contributing three times as much.
4. A 1 percent annual fee and a 3 percent inflation rate both subtract from the result; in the examples above they cut a $76,122.55 outcome to $57,434.91 or to about $31,361 in present-day purchasing power.
5. The rates are illustrations, not predictions, and none of this is personal advice; the durable lesson is that time and steadiness do most of the work.

What you can do: look up the expense ratio or fee on one account you own, and write down what 1 percent a year would cost you over thirty years using the formula above.