Most explanations of compound interest are about saving, and most explanations of debt are about borrowing, and the two are almost never put on the same page. Which is a shame, because they are the same sum.
Understand it once and three separate problems collapse into one.
The tool Compound Interest Watch savings grow, and see the year the interest starts outpacing what you put in.One sum, two directions
Compounding is a loop, and the whole of it is two steps:
- Take what the balance is now and work out the interest on it.
- Add that interest to the balance. Repeat.
That is all. The second step is the one that matters: the interest joins the balance, so next time round there is more to charge interest on. Interest earns interest, which is the entire idea and the reason the word exists.
Now notice that nothing in those two steps says who owns the balance.
If the balance is yours, that loop is the best thing in your finances. Money you did not deposit starts contributing, and eventually contributes more than you do.
If the balance is owed, the identical loop is the worst thing in your finances. Interest joins the debt, and next month you are charged interest on interest you never spent.
The savings calculator and the credit card calculator are running the same arithmetic. The only difference is which side of it you are standing on.
The one shortcut worth memorising
You cannot do compound arithmetic in your head. You can do this, and it is close enough to be useful.
Divide 72 by the rate, and that is roughly how many years the balance takes to double.
At 6%, about twelve years. At 8%, about nine. At 12%, about six. It works on the way up and on the way down: a debt at 18% that nothing is paid against doubles in about four years, and that is the same calculation.
It is an approximation, so it is worth knowing where it holds. It is at its most accurate between roughly 5% and 12% — which is where most of the rates people actually meet sit — and it drifts at the extremes, overstating the wait at very low rates and understating it at very high ones.
Useful either way. It is not meant to be exact; it is meant to let you sanity-check a number somebody has just told you, without a calculator.
Why it feels like nothing is happening, and then it does
The shape of compounding is the same in both directions: almost flat for a long time, then unmistakably steep. That shape is what makes it counter-intuitive, because the early part is the part you experience for years.
Saving, this is why the first few years feel like a waste of effort. The balance is small, so the interest on it is small, so nothing appears to be happening. Nothing is wrong: the interest is simply proportional to a balance that has not grown yet.
Borrowing, it is the same curve and the same trap. A small balance carried on a card costs a little, which is easy to live with. The balance grows slowly, then the growth itself starts growing, and the point at which it becomes alarming arrives quite suddenly after a long stretch of it seeming manageable.
In both cases the interesting part of the curve is beyond where most people stop looking. That is precisely why running the numbers out to the end is worth the two minutes — for savings and for a balance you are carrying.
The asymmetry nobody mentions
Here is the uncomfortable arithmetic. The rate you are charged on debt is, as a rule, several times the rate you are paid on savings. Both are the same loop, but one of them runs much faster.
You do not need a recommendation to see what that implies about which side of the loop deserves attention first — the two numbers say it themselves. Put the rate you are being charged next to the rate you are being paid, and the comparison is the answer.
The number that is not the rate
This one catches almost everybody, and it is worth thirty seconds.
An advertised annual rate is not what gets applied to your balance each month. A card quoted at 24% a year is charged in monthly or daily slices, and because each slice compounds on the last, twelve of them come to slightly more than 24% over the year, not exactly it.
Two practical consequences:
- The gap widens as the rate rises. At savings-account rates it is a rounding error. At credit-card rates it is real money.
- Comparing two quoted rates is not always comparing like with like, because how often interest is applied differs between products, and the headline number may not say.
This is why the totals a calculator produces can differ slightly from a statement. The formula is the same; how often the loop runs is not always what you assumed.
What the arithmetic does not know
Every one of these calculations is exact about the sum and silent about everything else. Worth stating plainly, because a confident-looking number invites more trust than it has earned.
- Inflation. A balance that grows 5% a year while prices rise 3% has grown by about 2% in what it buys. The figure on the screen is not adjusted for that.
- Tax. Interest earned is usually taxable, and the rules differ by country and by account.
- Fees. An account fee, a platform charge or an annual card fee comes off the result and is not in the formula.
- The rate staying still. Almost none of them do, over the timespans these calculations cover.
- Anything about your situation. The sum does not know your income, your obligations, or what else you might do with the money.
These pages are arithmetic, not advice — nothing here recommends a rate, a product or a course of action, and none of us is licensed to. What the arithmetic is good for is seeing the shape of a decision clearly before taking it to somebody who is.