Compounding, Worked Year by Year Instead of Assumed
Three people, three ages, one practice rate. The smallest deposit finishes largest. Here is the month-by-month arithmetic that makes that happen.
Compounding is one sentence: growth is worked out on the whole balance, not on the deposit. A deposit joins a balance, and the next growth is charged on that balance including everything earlier deposits already earned. That is the entire engine. What makes it surprising is not the rule but the arithmetic it produces over long runs, and the only way to see that is to run it rather than assume it. On the practice table below, the smallest deposit in the family finishes with the largest balance, and the only variable is years.
The worked example: three people, one table
Three members of an invented household, three different ages, three different monthly amounts, and every jar finishing at the same age. Simulated one month at a time, with the balance rounded to the cent every month.
| Person | Age | Monthly | Years | Paid in | Finishes at |
|---|---|---|---|---|---|
| The youngest | 11 | $5.00 | 60 | $3,600.00 | $55,938.26 |
| The parent | 34 | $25.00 | 37 | $11,100.00 | $52,718.78 |
| The grandparent | 62 | $100.00 | 9 | $10,800.00 | $15,073.31 |
Read the last two columns together, because that is where the whole lesson sits. The youngest hands over $3,600.00 across an entire life and finishes at $55,938.26, which means $52,338.26 of that balance was added by the years rather than by the person.
The parent deposits five times as much every month and puts in $7,500.00 more in total, and finishes $3,219.48 behind. Twenty-three years is what that gap is made of, and nothing else. The grandparent, at twenty times the youngest person's deposit with nine years to work in, still ends with more than went in, and starting was still worth doing.
What one dollar became
| Person | Paid in | Finished at | Multiple |
|---|---|---|---|
| The youngest | $3,600.00 | $55,938.26 | 15.54x |
| The parent | $11,100.00 | $52,718.78 | 4.75x |
| The grandparent | $10,800.00 | $15,073.31 | 1.40x |
Identical arithmetic in all three rows. Identical rate. The only variable is years, and the multiples run from 1.40x to 15.54x.
What is actually happening inside one month
- A balance exists from every month before this one.
- The deposit arrives, the same amount as last month, added on top.
- Growth is worked out on the whole balance, not on the deposit that just landed.
- It is added in, and from now on that growth is part of what growth is charged on.
- Repeat, unchanged, for as many months as anybody leaves it alone.
Step three is the entire engine and it is the one that is easy to miss. The deposit is an input. The balance is what gets paid on.
Why the early years feel like nothing
Because with a small balance there is barely anything for growth to be charged against, so nearly all the movement is money carried in by hand. That is not a broken start. It is the only start there is.
Then there is a month, different for every set of numbers, when the growth added is larger than the deposit added. On this practice run that lands around month 120, roughly year 10. After it, the balance is built more by what is already there than by what is carried in. Nothing is announced. It simply passes.
Almost everybody who abandons an arrangement like this abandons it before that month, which is worth knowing in advance rather than discovering afterwards.
The doubling clock
A rough shortcut: divide 72 by the rate and you get the number of years a balance takes to double. At the practice 7% used here that is 10.29 years.
- The youngest buys about 5.8 doublings across 60 years.
- The parent buys about 3.6 across 37 years.
- The grandparent buys about 0.9 across 9 years.
The last doubling is always the biggest one, because it doubles the largest balance the arrangement ever had. That is the whole reason years matter more than amounts over long runs, stated as a single sentence.
The same engine, running the other way
Everything above works identically on money you owe. Interest is charged on the balance, and unpaid interest joins the balance, and the next charge is worked out on the larger number. That is why a minimum payment can leave a debt running for years while a larger payment collapses it, and why the direction of the arithmetic matters more than its size.
If that side is the live question in your household, the arithmetic is worked out on real balances in two payoff orders on one practice wall, where one month of a practice card is taken apart in slow motion.
Three levers, and only one of them is large
A long-run balance is produced by three inputs: how much goes in, what rate applies, and how many years run. It is tempting to treat them as equally important. The practice table says otherwise.
- Amount. The grandparent deposits twenty times the youngest and finishes with roughly a quarter as much. Amount is the weakest lever over long runs.
- Rate. Real and powerful, and almost entirely outside anybody's control. It is also the one every advertisement is about.
- Years. The lever that produced $55,938.26 from $3,600.00, and the only one a household genuinely decides, by starting.
That ranking is uncomfortable, because years is the lever you cannot get back and the one nobody sells. It is also the reason this chapter sits in the first book of the series rather than a later one.
What the rate is not
It is not a promise, a forecast, or a description of what anything does. Every figure on this page uses a steady invented 7% because a steady rate is the only way to isolate the effect of time, and isolating it is the point of the exercise. A real arrangement does not deliver a steady rate, and any page that implies otherwise is drawing a picture rather than describing the world.
What the arithmetic can honestly show you is the mechanism, so that when you look at any real thing you know which questions to ask: how long, at what cost, on what balance. For anything about your own money, a licensed professional who can see your situation is the right person, and investor.gov publishes free educational material from a federal regulator.
The cost side of the same picture
Because growth is charged on the balance, so is anything that comes out of the balance. A small annual cost, deducted every year from a growing number, compounds against you exactly as growth compounds for you. That is not an argument about any product; it is the same arithmetic, run with a minus sign.
The published fee illustration, computed shows what 0.75% a year does across 20 years on $100,000.00, and the answer is $29,199.52. Same engine, opposite direction, and both are worth being able to see.
Where this belongs in a household
Behind a cushion, in almost every version of the arithmetic. An arrangement that gets interrupted by a car repair in year three is an arrangement that never reaches the crossover month, and the practice table above is entirely a story about not being interrupted. Sizing a starter cushion is the unglamorous step that makes the interesting one survivable.
Same monthly amount, same rate, ten years and thirty years. The gap between the two answers is the thing this page is about, and computing it yourself lands harder than reading anybody's table.
Where to start
This chapter is in the first book of the series, and that book is free. Six short lessons, one household, and the whole map of a money life in one sitting. Take it and decide after. No card, no trial, and it is the real book rather than a sample.
If it does not click, keep your money and keep the book. It is yours either way, and the single most useful thing on this page is the arithmetic, which costs nothing at all.
Questions people actually ask
What is compound interest in plain English?
Growth is worked out on the whole balance, including everything earlier deposits already earned, rather than on the deposit alone. A deposit is an input. The balance is what gets paid on, and it keeps getting bigger.
Why do small early deposits beat larger late ones?
Time. On the practice table here, $5.00 a month for 60 years finishes at $55,938.26 while $25.00 a month for 37 years finishes at $52,718.78, even though the second person put in $7,500.00 more in total. History, not a promise.
What is the rule of 72?
A shortcut for how long a balance takes to double: divide 72 by the rate. At the practice 7% used here that is 10.29 years, which buys about 5.8 doublings across sixty years and about 0.9 across nine. The last doubling is always the biggest one.
Why do the early years feel like nothing is happening?
Because with a small balance there is barely anything for growth to be charged against, so nearly all the movement is money carried in by hand. On the practice run, growth added does not exceed the deposit added until around month 120, which is roughly year 10.
Is this investment advice?
No. This is general financial education showing arithmetic. It is not financial advice, not investment advice, and not a recommendation about anything. The 7% used here is an invented practice rate held steady so the shape can be seen. Real returns move, and history is not a promise.