Math intuitions
Compounding
Returns earned on past returns: a steady rate applied to a growing base turns small repeated gains into large ones, mostly at the end.
The arithmetic of compound interest is medieval; Luca Pacioli's Summa (1494) already carries a rule of thumb for it. Berkshire Hathaway's shareholder letters are the best-known modern demonstration.
Compounding is what happens when each period's gain is added to the base that the next period's gain is computed on. Put 1,000 at 7 percent a year and the first year earns 70. The second year earns 7 percent of 1,070, which is 74.90, and so on. After 30 years the balance is 1,000 times 1.07 to the power 30, about 7,612. If instead the 70 were paid out each year and never reinvested, 30 years would give 3,100. The difference of about 4,500 is interest on interest. The effect is small in the early years, which is why it is easy to dismiss, and most of the growth arrives late: the same account passes 2,000 after 10.2 years and 4,000 after 20.5, then adds its final 1,000 in about two years.
The mechanism is the base, and the base can be anything that feeds its own growth. Ward Cunningham's 1992 WyCash report, the origin of the term technical debt, says shipping first-time code is like going into debt: a little speeds development if it is paid back promptly with a rewrite, and every minute spent afterwards on not-quite-right code counts as interest. Skills compound because what you already know lowers the cost of the next thing. Trust compounds because each kept promise makes the next one cheaper to accept. The rate matters more than it looks. Berkshire Hathaway's 2024 letter reports a compounded annual gain in per-share market value of 19.9 percent from 1965 to 2024, against 10.4 percent for the S&P 500 with dividends. Less than double the rate, over sixty years, produced an overall gain of 5,502,284 percent against 39,054 percent: about 140 times as much.
Compounding needs the base to survive. A loss of 50 percent takes a 100 percent gain to repair, so a strategy that compounds at 20 percent and blows up once a decade can finish behind one that plods at 8 percent. The rate must also be real after fees, tax and inflation, each of which compounds against you with the same arithmetic: a 1 percent annual fee on a 7 percent return turns the 7,612 after 30 years into 5,743, a cost of about 1,900 on 1,000 invested. And debts compound exactly as assets do. Technical debt, unanswered email and deferred maintenance all grow on their own base. The question is never whether compounding is happening, only which direction it is running in.
Example
The 2024 Berkshire Hathaway shareholder letter, dated 22 February 2025, opens with a table of Berkshire's per-share market value against the S&P 500 with dividends, year by year from 1965. Berkshire's compounded annual gain over 1965 to 2024 was 19.9 percent and the index's 10.4 percent. The overall gains over the sixty years were 5,502,284 percent and 39,054 percent, so 1,000 dollars would have become about 55 million dollars in Berkshire shares and about 391,000 in the index. The table also shows the rate was anything but steady: Berkshire fell 48.7 percent in 1974 and 31.8 percent in 2008. Compounding carried it through because no single year destroyed the base.
Where it applies
Pick the few things in your life that compound (a skill you use daily, savings, a codebase's test coverage, a handful of relationships) and protect their base above everything else. In engineering, treat every quick hack as a loan with a rate: write down when it will be repaid, because the interest is paid by whoever touches that code next.
Think about it
What are you currently paying interest on, and what are you earning it on?
Applied it to something real? Log it.
Test yourself1,000 grows at 7 percent a year for 30 years. Roughly what does it become, and how much of that is interest on interest?
About 7,612 (1,000 × 1.07^30). Simple interest would give 3,100, so about 4,500 is return earned on earlier returns.
Related models
- The rule of 72Divide 72 by the annual growth rate in percent to get the number of years a quantity takes to double.
- Exponential versus linear growthLinear growth adds the same amount each step; exponential growth multiplies by the same factor, and the two look alike only at the start.
- The Lindy effectFor things that do not wear out, such as ideas, books and technologies, every year they survive raises how long they can be expected to last.
- Margin of safetyLeave a wide gap between what you expect to need and what you build or pay, so that your errors and bad luck do not sink you.