Investing
How To Calculate The Rule Of 72
The rule of 72 is a mental shortcut for doubling time: divide 72 by the annual return and you get roughly how many years it takes for money to double. It is fast, memorable and close enough for nearly every rate you will meet.
Quick Answer
Doubling time = 72 / annual return percent
- rate
- Annual return as a percentage, such as 8 for 8%
- 72
- The magic numerator that makes the estimate work
- years
- Approximate years needed to double
Divide 72 by the percentage return. At 8% a year, 72 divided by 8 gives 9 years to double. At 6% it is 12 years, and at 12% it is 6 years. The exact answers using the compound formula are close, differing by a few months at typical rates.
What Is The Rule Of 72?
The rule of 72 is a shortcut that estimates how many years it takes for an investment to double at a fixed compound return. Divide 72 by the annual percentage return, and the result is the doubling time in years.
It works because the logarithm of two, the mathematical basis of doubling, is approximately 0.693, and dividing 72 by the rate is a close rational approximation to the exact calculation of ln(2) divided by ln(1 + rate). The choice of 72 rather than 69.3 makes the arithmetic easier and compensates for the way reasonable rates behave.
The estimate is most accurate in the 6% to 10% range. At 8% the rule gives 9 years and the exact figure is about 9.01 years, an error of a few days. At very low or very high rates the gap widens, and at rates above about 20% you notice it.
The rule also works in reverse. To find what return is needed to double in a target number of years, divide 72 by the years. If you want to double in 10 years you need roughly 7.2% a year.
It applies to any exponentially growing quantity, not just money. A population, a price index or a debt balance all double at a time given by the same rule, which is why it turns up across economics and finance.
Compounding frequency matters slightly. The rule assumes annual compounding but is close enough for monthly compounding at typical rates. The more frequent the compounding, the shorter the true doubling time, though the difference is small at ordinary rates.
Inflation is a useful application. At 3% inflation, prices double in about 72 divided by 3, or 24 years. That single figure makes the slow erosion of purchasing power vivid in a way that an annual percentage does not.
Fees and taxes are the rule's practical limitation. A fund returning 8% before fees and 7.2% after a 0.8% expense ratio takes about 10 years to double rather than 9, and after-tax returns stretch it further. Always apply the rule to the net return you actually keep.
The rule is an estimate, not a substitute for the compounding formula. For precision, use the exact doubling time of ln(2) divided by ln(1 + rate), which is what a spreadsheet will compute. Use the rule of 72 when you need a quick answer in your head.
A related variant is the rule of 70, which uses 70 instead of 72 and is slightly more accurate at low rates. The rule of 69.3 is the mathematically derived version. All three are close enough for conversation, and 72 is the one that is easiest to divide mentally.
The rule breaks down at high rates because the approximation assumes the growth rate stays small. At 30% a year the rule gives 2.4 years, while the exact answer is about 2.64 years, a noticeable gap that grows worse as the rate rises.
The most powerful use is comparative. Doubling at 4% takes about 18 years; doubling at 8% takes 9. Halving the return doubles the wait, which is why even a modest improvement in return rate, sustained over decades, transforms a long-term outcome.
Formula
Years = 72 / rate percent
Divide 72 by the annual percentage return to estimate the doubling time.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| r | Annual return | rate | Return as a percentage, such as 8 for 8%. |
Years = ln(2) / ln(1 + rate)
The precise time to double from the compound growth equation.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| ln2 | Natural log of two | number | Constant approximately 0.693. |
How To Calculate The Rule Of 72
- 1
Identify the annual return
Use the net return after fees and, if relevant, after tax. An 8% gross return with a 1% fee is 7% for this purpose.
- 2
Divide 72 by the rate
72 divided by 8 is 9, so money doubles in about 9 years at 8%.
- 3
Check the range
The estimate is tightest between 6% and 10%. Outside that range, treat it as a rough guide and compute the exact figure if precision matters.
- 4
Compare with the exact answer
For confirmation, divide the natural log of two by the natural log of one plus the rate. The two should agree closely at ordinary returns.
- 5
Apply it to the outcome you care about
Use the rule on inflation, debt or savings to see how long the quantity takes to double, which is often more intuitive than an annual percentage.
Examples
Example 1: Doubling time at 8% a year
- Annual return
- 8%
| Step | Calculation | Result |
|---|---|---|
| Rule of 72 | 72 / 8 | 9 |
| Exact doubling time | ln(2) / ln(1.08) | 9.0065 |
Result: At 8% the rule of 72 gives 9 years, and the exact doubling time is 9.0065 years, so 10,000 grows to 20,000 in roughly 9 years.
Example 2: Inflation doubling prices at 3%
- Annual return
- 3%
| Step | Calculation | Result |
|---|---|---|
| Rule of 72 | 72 / 3 | 24 |
| Exact doubling time | ln(2) / ln(1.03) | 23.45 |
Result: At 3% inflation the rule of 72 gives 24 years to double prices, while the exact figure is about 23.45 years.
Calculator
Doubling time (rule of 72)
9
- Exact doubling time
- 9.0065
- Value of 1,000 after that time
- $2,000.00
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Rule Of 72 calculator page.
Common Mistakes
Using the gross return instead of the net
Fees and taxes cut the rate you actually compound at. A 1% fee turns 9 years into about 10, and tax stretches it further.
Trusting the rule at extreme rates
Above roughly 20% the approximation drifts noticeably. At 30% the rule says 2.4 years but the exact answer is about 2.64.
Confusing doubling with a fixed gain
The rule assumes the return compounds. A simple 8% of the original sum each year does not double at the same speed as compound growth.
Ignoring the effect of contributions
Adding money along the way changes the picture entirely. The rule applies to a lump sum left to grow, not to a stream of deposits.
Applying it to a fluctuating return
The rule needs a steady rate. A volatile return compounds to a different result than its average suggests, and the doubling time should be computed from the actual sequence.
Forgetting that it is an estimate
The rule is a mental shortcut, not the accounting. For a contract or a projection, use the exact logarithm formula or a spreadsheet.
Overlooking that half the return doubles the wait
Doubling time is inversely proportional to the rate. Cutting the return from 8% to 4% doubles the years, which is why fees matter so much over long horizons.
FAQ
How does the rule of 72 work?
Divide 72 by the annual percentage return to estimate the number of years needed to double. At 8% it is 72 divided by 8, or 9 years. The 72 comes from a close approximation to the logarithm of two.
Why 72 and not 70 or 69?
The mathematically derived constant is about 69.3, but 72 is easier to divide mentally and compensates for compounding effects across the usual range of rates, making it more accurate between 6% and 10%.
What return doubles money in 10 years?
Divide 72 by 10 to get 7.2. A return of roughly 7.2% a year doubles an investment in about ten years.
Is the rule of 72 accurate?
It is close within a few months for most returns between 6% and 10%. Outside that range the error grows, so use the exact logarithm formula when precision matters.
Does compounding frequency matter?
Slightly. The rule assumes annual compounding but stays close for monthly compounding at ordinary rates. More frequent compounding shortens the true doubling time a little.
Can I use it for debt?
Yes. A balance growing at a given interest rate doubles in about 72 divided by that rate, which is a vivid way to see how quickly unpaid interest compounds against you.
References
- [1]Investor.gov, U.S. Securities and Exchange Commission, Compound interest — https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator
- [2]Investopedia, Rule of 72 — https://www.investopedia.com/terms/r/ruleof72.asp
- [3]Consumer Financial Protection Bureau, Saving and investing — https://www.consumerfinance.gov/consumer-tools/saving/