The History of Compound Interest: From Babylonian Tablets to APR Disclosure

Ancient Mesopotamia: Interest Before Algebra

Interest is older than coinage. Cuneiform tablets from Mesopotamia record loans of silver and grain carrying stated rates, and the Code of Hammurabi, dating to roughly 1750 BCE, set legal maximum rates that differed for the two commodities. More strikingly, Old Babylonian mathematical tablets contain problems that amount to compound interest calculations — including one that asks how long a sum lent at 20 percent will take to double, which the scribe solves to just under four years. That problem is a doubling-time question of exactly the kind the Rule of 72 later approximated. The mathematics was worked in a sexagesimal system without algebraic notation, which makes it a remarkable piece of computation and shows that the underlying idea of interest accumulating on interest was understood long before it had a name.

1494: Pacioli States the Rule of 72

Luca Pacioli's Summa de Arithmetica (1494), the same volume that carried double-entry bookkeeping into print, also contains an early statement of what is now called the Rule of 72: dividing 72 by the interest rate expressed as a percentage gives an approximation of the number of periods required for a sum to double. Pacioli gave the rule without a derivation. It works because the exact doubling time is the natural logarithm of 2, roughly 0.693, divided by the rate, and 72 is a conveniently divisible number close to 69.3 that also corrects slightly for the fact that most real interest is compounded discretely rather than continuously. The approximation is closest for rates in the range of about 6 to 10 percent.

1613-1683: Tables, Continuous Compounding, and the Number e

Richard Witt, a London mathematical practitioner, published Arithmeticall Questions, touching the Buying or Exchanging of Annuities in 1613 — generally regarded as the first English book devoted to compound interest, with worked tables that let a user look up a result rather than recompute it. Seventy years later the Swiss mathematician Jacob Bernoulli examined a question that arises directly from compounding frequency: if interest is compounded ever more often over a year, does the total grow without limit? He showed in 1683 that it does not, but converges to a value between 2 and 3. That limit is the number later denoted e by Leonhard Euler, roughly 2.71828 — a constant fundamental to mathematics that was first encountered in a question about interest.

1790-1990: The Franklin Bequests

Benjamin Franklin died in 1790 leaving a codicil to his will that is one of the few long-horizon compounding experiments actually carried out. He directed 1,000 pounds sterling each to the cities of Boston and Philadelphia, to be lent in small sums at interest to young married tradesmen, with the funds to remain in trust for 200 years: a partial distribution was permitted at 100 years and the remainder at 200. Both trusts ran their full course and were wound up in 1990. Their end values are commonly reported at roughly $5 million for Boston and about $2 million for Philadelphia. The gap between the two is instructive in itself — the funds were managed differently, and over two centuries small differences in realised return produce very large differences in outcome.

1968 and 1991: Making the Rate Comparable

Compounding is only useful to a consumer if the disclosed rate reflects it, and for most of history it did not. The US Truth in Lending Act, enacted in 1968 as part of the Consumer Credit Protection Act, required creditors to disclose the cost of credit on a standardised basis, including the annual percentage rate (APR), so that offers with different fee structures and compounding conventions could be compared. The Truth in Savings Act, enacted in 1991, did the parallel job for deposits, requiring institutions to disclose the annual percentage yield (APY), which reflects compounding within the year. Both statutes are administered today with the Consumer Financial Protection Bureau, created under the Dodd-Frank Act in 2010, holding principal rule-writing authority.