Compound interest
Grow $100 for 30 years, find Bernoulli's number, and run Benjamin Franklin's 200-year bet.
Interest on interest
$100 earns 10% a year. Each year’s interest joins the pile, and then it earns interest too. Drag the years.
$1,745, against $400 with simple interest. The bright part is interest earned on interest, and by year 30 it’s most of the pile.
Simple interest pays on the original amount only: A = P(1 + rt). Compound interest pays on everything so far: A = P(1 + r)ᵗ.
simple: 100 × (1 + 0.1 × 30) = $400
compound: 100 × 1.1³⁰ ≈ $1,745
About the famous quote. “Compound interest is the eighth wonder of the world” is usually credited to Albert Einstein, but there’s no evidence he said it. The earliest close match is an unsigned 1925 savings-and-loan advert. The first Einstein attribution found so far is from 1988. The Ultimate Quotable Einstein (Princeton University Press) files it under “Probably Not By Einstein”.
All rates on this page are examples, not forecasts or financial advice.
It doubles, and doubles
Compound growth doubles in equal steps. Find the yearly rate that doubles your money every 10 years.
About 7.2%. Then ×2 by year 10, ×4 by 20, ×8 by 30, ×16 by 40. That’s why 72 is a handy number: divide it by the rate to get the doubling time.
The exact doubling time is ln 2 ÷ ln(1 + r). For everyday rates that’s close to 72 ÷ (rate in %).
7.2%: exact 9.97 years · rule 10
6%: exact 11.9 years · rule 12
The earliest known written version is in Luca Pacioli’s Summa de arithmetica (Venice, 1494). His example: at 6% a year, 72 ÷ 6 = 12 years. He doesn’t explain where it comes from, so it’s probably older still.
Bernoulli's question
1683: Jacob Bernoulli asks what happens if a bank pays 100% a year, but in smaller, more frequent slices. Try it.
Daily gives $2.7146. Hourly barely adds anything. The total closes in on a limit, 2.71828…, the number we now call e.
Paid n times a year, each slice is 1/n of 100%, so $1 becomes (1 + 1/n)ⁿ.
n = 1: 2.00 · 2: 2.25 · 4: 2.4414 · 12: 2.6130
52: 2.6926 · 365: 2.7146 · ∞: 2.71828…
Bernoulli showed the limit had to lie between 2 and 3, but didn’t compute it. Euler later gave it the name e. It turns up all over mathematics, from growth and decay to probability.
Franklin's 200-year bet
In 1789 Benjamin Franklin left £1,000 each to Boston and Philadelphia, to be lent out at 5%, with instructions for the next two centuries.
His plan: £131,000 by 1890, spend £100,000, then £4,061,000 by 1990. The arithmetic holds up. Real life didn’t follow it exactly. Boston’s fund reached about $4.5 million.
Franklin’s codicil is dated 23 June 1789. The money was to be lent to young married tradesmen under 25, each repaying a tenth of the principal a year.
1.05¹⁰⁰ ≈ 131.5, so £1,000 → ≈ £131,500
£31,000 × 131.5 ≈ £4.08 million
A lawsuit by his heirs in 1891 held up the 100-year split. In 1894, $329,300 was paid out towards building the Franklin Union, a technical school. By 1990, the 200-year mark, Boston’s fund was worth about $4.5 million and Philadelphia’s about $2 million.
The chart is a log scale: each gridline is ten times the one below, so steady compounding draws a straight line.
Early or late?
Early saves $1,000 a year from 25 to 34, then stops. Late saves from 35 to 64: three times as much. At 5%, Late wins. Change the rate.
Above about 6.3%, Early ends up with more, despite paying in a third as much. Those first ten years had the longest to compound.
Each saver pays $1,000 at the start of each year, and the balance is measured at 65. Growth is the same every year, which real investments never are.
at 7%: Early ≈ $112,500 (paid $10,000) · Late ≈ $101,100 (paid $30,000)
at 6%: Early ≈ $80,200 · Late ≈ $83,800
These rates are examples, not forecasts, and they ignore inflation and tax.
It works against you too
$10,000 growing 7% a year for 40 years. Now take a small yearly fee out of that growth.
A 1% fee leaves you with about 31% less. The fee compounds too, because every dollar it takes stops growing.
no fee: 10,000 × 1.07⁴⁰ ≈ $149,700
1% fee: 10,000 × 1.06⁴⁰ ≈ $102,900 (≈ 31% less)
Debt compounds the same way. US credit cards that charge interest average about 22% a year (Federal Reserve, 2026). At that rate an unpaid balance doubles in about 3.5 years, or a little faster with monthly compounding.
Your numbers
Everything’s unlocked. Set your own numbers, or try the example: $820 a month from 25, at 7%, is about $1M by 55.
- Constant growth: real returns vary from year to year and can be negative.
- No inflation or tax: the totals are in today’s numbers, before tax.
- Monthly compounding: deposits land at the end of each month.
- The $1M example is before inflation: a million dollars in 30 years will buy less than a million does today.
This is a calculator for intuition, not financial advice.