Compound interest: the ten years you cannot buy back

Two people, one illustrative rate, round numbers all the way through. The first adds a thousand a year for ten years, then stops forever. The second starts ten years later and never stops — thirty payments against ten, three times as much money in. Forty years on, the one who quit is still ahead. That is not a trick and it is not advice. It is arithmetic, and almost nobody's gut believes it. Here is the engine. Simple growth adds the same amount every year, so its picture is a straight line. Compound growth adds a percentage of a number that keeps getting bigger — growth on growth — and its picture bends. At seven percent, chosen because it is round, simple growth turns one into three point eight over forty years. Compounding turns one into almost fifteen. The first year earns seven hundred on ten thousand. The fortieth year earns nine thousand seven hundred and ninety six, at the same rate, on money nobody added to. You can do this in your head. Divide seventy two by the rate and you get the doubling time. Six percent doubles in twelve years, eight percent in nine, twelve percent in six. The exact answer is the log of two over the log of one plus the rate, and the honest numerator is sixty nine point three. But seventy two divides cleanly by two, three, four, six, eight, nine and twelve, and that is the whole reason it won. It is near exact at eight percent, wrong by under one percent from six to ten, and off by seven percent at twenty four — where it claims a doubling sooner than the arithmetic gives. Now the whole ledger. The early saver put in ten thousand dollars and finishes with a hundred and five thousand — ten dollars and fifty two cents back for every dollar in. The late saver put in thirty thousand and finishes with ninety four thousand — three dollars and fifteen cents per dollar. The missing ten years were not ten years of payments. They were the ten years with the longest left to compound. And the late saver does draw level in the end — in year sixty one. Fifty one more years of paying, to undo a ten year head start. The same engine runs against you, and that is the honest other half. A card quoted at twenty four percent a year compounds monthly. Twenty four over twelve is two percent a month, and two percent a month compounds to twenty six point eight two percent a year — nearly three points above the number printed on the page. At that rate a balance doubles in thirty five months. And a minimum payment set at two percent of the balance is exactly the interest: on five thousand dollars, a hundred dollars owed in interest, a hundred dollar payment, and the balance does not move at all. A hundred and fifty a month clears it in four point six years. Two hundred and fifty clears it in two point one. Your intuition adds. Money multiplies. The whole gap between those two is where a late start hides, and where a carried balance hides too.