Compounding: why repeated growth multiplies instead of adding

0:00Every percentage change is worked out on the value at the start of the period. So one question decides everything that follows. Is that value allowed to move? Freeze it, and the same amount gets added every time, and the total walks up a straight line. Let it move, and each period multiplies by the same factor, and the total curves away. Same rate. Same number of years. Two different answers.

0:30And the gap between them is not a mystery. It can be counted, coin by coin. Start with one year. Six thousand deposited at ten per cent. Ten per cent of six thousand is six hundred, so the deposit becomes six thousand six hundred. Or say it in one move. A rise of ten per cent is a multiplication by one point one. Six thousand times one point one. The same six thousand six hundred.

1:04For one year, adding the interest and multiplying by the factor agree exactly. They agree at one year for every rate there is, and that is precisely why the wrong rule survives school. Now three years, and the answer depends on which deposit was taken out. In the first, the interest is handed back to you at the end of every year, and the deposit itself comes back at the end.

1:33In the second, the interest is left in, so it joins the pile and earns alongside it. That second one is called compounding, and it is the only difference between them. Not the rate. Not the years. Not the amount put in. Take the interest out. Year one begins at six thousand and pays six hundred. Year two begins at six thousand. Year three begins at six thousand. The base never moves, so the interest is identical every single year.

2:09Six hundred three times is eighteen hundred, and with the six thousand back that is seven thousand eight hundred. A straight line, in equal steps. Now leave it in. Year one begins at six thousand, adds six hundred, ends at six thousand six hundred. Year two begins at six thousand six hundred, adds six hundred and sixty, ends at seven thousand two hundred and sixty. Year three begins there, adds seven hundred and twenty-six, and ends at seven thousand nine hundred and eighty-six.

2:47Now look hard at those three payments. Six hundred, six hundred and sixty, seven hundred and twenty-six. Every one of them is ten per cent. Of six thousand, of six thousand six hundred, of seven thousand two hundred and sixty. Nothing about the rate changed. It is ten per cent all the way down. Only the number it was taken of moved. Seven thousand eight hundred against seven thousand nine hundred and eighty-six.

3:20As a share of what went in, that is a hundred and thirty per cent against a hundred and thirty-three point one. Or as multipliers, one point three against one point three three one. And there is the whole difference, written out twice. One adds a tenth three times. The other multiplies by one point one three times. Adding the rate, or multiplying the factor. That is the entire argument.

3:51So where do the extra one hundred and eighty-six come from? Year one pays six hundred either way. Nothing in it. Year two pays sixty more, and that sixty is ten per cent of the first year's six hundred. Year three pays a hundred and twenty-six more. Sixty and a hundred and twenty-six make one hundred and eighty-six. All of it, accounted for. Multiply out one point one cubed and the same money appears in four pieces. The six thousand, then eighteen hundred, then one hundred and eighty, then six.

4:30The eighteen hundred is the plain interest. The hundred and eighty and the six are interest earned by interest. Written generally, take the interest out and the total is the principal, times one plus the rate times the time. Leave it in and the total is the principal, times one plus the rate, raised to the time. Now watch what those two do to a pair of banks. Twenty thousand at ten per cent for two years. Or twenty thousand at five per cent for four.

5:06Without compounding both come to twenty-four thousand exactly, because the rate times the time is the same in each. With compounding they part. The first gives twenty-four thousand two hundred. The second gives twenty-four thousand three hundred and ten point one two five. The smaller rate over more periods ends up further ahead. And here is the belief that needs breaking. Compounding does not automatically win. Twelve and a half thousand borrowed at twelve per cent for three years, with no compounding, costs four thousand five hundred in interest.

5:49The same sum at ten per cent compounded costs four thousand one hundred and thirty-seven and a half. The higher rate is dearer by three hundred and sixty-two and a half, and it is still dearer after four years. It takes until the fifth year for the compounded ten per cent to overtake. Over three years the higher rate simply has more to work with. Compounding is a mechanism, not an advantage. Whether it wins depends on the rate, and on how long you wait.

6:24The same machine runs downhill without changing a single part. A television bought at twenty-one thousand loses five per cent in a year. Take the loss off, or multiply by nought point nine five. Nineteen thousand nine hundred and fifty, either way. A village of one thousand two hundred and fifty loses about a tenth of its people every decade. Multiply by nought point nine three times and you get nine hundred and eleven point two five.

6:58But people cannot be halved, so round at each decade instead. One thousand one hundred and twenty-five, then one thousand and thirteen, then nine hundred and twelve. Nine hundred and twelve is not nine hundred and eleven point two five rounded off. It is the answer to a different procedure. Both are honest. One rounds once at the end, the other rounds three times on the way, and to the nearest ten they agree at nine hundred and ten.

7:33None of this is really about money. Any repeated percentage change works the same way. A colony of mice rises five per cent one month, falls two the next, and falls three the next. Five minus two minus three is nothing at all, so the colony should end exactly where it started. Multiply the factors instead. One point nought five, times nought point nine eight, times nought point nine seven.

8:04Nought point nine nine eight one three. Below one. The colony ends smaller than it began. Bus fares up three per cent one year and four the next are not up seven. They are up seven point one two. And that nought point one two is the four per cent acting on the three already added. The same interest on interest, wearing a different coat. Percentages of a group nest the same way. On an excursion, forty per cent of the students are in one year group, and sixty per cent of those are girls.

8:41That is not sixty per cent of the excursion. Sixty per cent of forty per cent is twenty-four per cent of everybody. Finally, put the two side by side and let them run. A thousand at ten per cent. Without compounding it doubles in exactly ten years. With compounding, the eighth year is the first to clear two thousand. One of those is a straight line and the other is a curve, and now you can see which is which.

9:15Straight lines add. Curves multiply. And the only thing separating them is whether the number underneath the percentage was allowed to move. That was the question at the start, and it was the whole question.