Head-to-head
Rule of 72 vs Compound Interest
The rule of 72 is a shortcut: divide 72 by the annual percentage rate and you get the number of years it takes to double. It works because the natural logarithm of two is 0.693, and for small rates the logarithm of 1 + r is close to r, so the doubling time is roughly 69.3 divided by the rate. Seventy-two is used instead of 69.3 because it divides neatly by 2, 3, 4, 6, 8, 9 and 12.
Compound interest is the actual calculation: the balance grows by a factor of 1 + r each period, raised to the number of periods. There is no approximation in it. It answers any target multiple, not just doubling, at any rate — and it costs a logarithm.
The gap is small where the rule is normally used and large where it is not. Between 6% and 10% the two agree to within a few days. At 4% the rule runs slightly long; above 12% it runs short, and the error grows faster than the rate does. Over a long horizon a small error in doubling time compounds into a large error in the final balance, which is the part people miss.
Rule of 72
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.
Exact compounding
How To Calculate Compound Interest
Compound interest is interest that earns interest. Once interest is added to your balance, that added amount starts earning too — which is why long horizons grow faster than people expect.
How They Differ
| Attribute | Rule of 72 | Exact compounding |
|---|---|---|
| What it is | A mental approximation | The actual formula |
| Formula | 72 ÷ rate | A = P(1 + r)^t |
| At 4% a year | 18.00 years | 17.67 years |
| At 8% a year | 9.00 years | 9.01 years |
| At 12% a year | 6.00 years | 6.12 years |
| At 24% a year | 3.00 years | 3.22 years |
| At 36% a year | 2.00 years | 2.25 years |
| Accuracy | Best between 6% and 10% | Exact at any rate |
| Answers | Doubling only | Any target multiple |
| Effort | One division | Needs a logarithm |
The Same Numbers, Both Ways
$10,000 left to compound at 24% a year for 30 years, with the rule of 72 predicting how many doublings happen.
- Starting balance
- $10,000
- Annual rate
- 24%
- Horizon
- 30 years
| Line | Rule of 72 | Exact compounding |
|---|---|---|
| Doubling time | 3.00 years | 3.2223 years |
| Doublings in 30 years | 10 | 9.31 |
| Value after 30 years | $10,240,000 | $6,348,199 |
| Overstatement | $3,891,801 | $0 |
| As a share of the true value | 61.3% | 0.0% |
The rule says 24% doubles money every 3.00 years, so $10,000 becomes $10,240,000 in 30 years. Exact compounding says the doubling takes 3.2223 years, which is 9.31 doublings, and the true balance is $6,348,199. The shortcut overstates the result by $3,891,801, or 61.3%, and every dollar of that gap comes from a 0.2223-year error repeated ten times.
When To Use Each
Rule of 72
- You are in a meeting and need a doubling estimate in your head.
- You are sanity-checking a quoted return before reaching for a calculator.
- The rate is between 6% and 10%, where the approximation is closest.
Exact compounding
- You are producing a projection that someone will make a decision from.
- The rate is high, the horizon is long, or you need something other than a doubling.
- You are comparing scenarios and need the figures to reconcile exactly.
Common Questions
Why 72 and not 69.3?
Because 72 has more divisors. It divides by 2, 3, 4, 6, 8, 9, 12 and 18, so the mental arithmetic works for the rates people actually quote. The cost is a small bias: 72 is slightly high for low rates and slightly low for high ones.
How accurate is the rule of 72?
Within a few days between 6% and 10%. At 4% it overstates the doubling time by about four months; at 24% it understates it by about 81 days. The error grows roughly with the square of the rate, which is why it is invisible at ordinary rates and obvious at credit-card rates.
Does the rule work for inflation?
Yes, and it is one of its best uses. At 3% inflation, prices double in about 24 years, so a 30-year retirement projection at that rate should assume roughly two and a half times today's cost of living.
What if I want tripling rather than doubling?
Use 114 instead of 72, since the logarithm of three is 1.099. For quadrupling, use 144. The same logic applies: multiply the natural logarithm of the target multiple by 100 and round to a number with convenient divisors.
Why is the rule short at high rates?
Because the logarithm of 1 + r grows more slowly than r itself as r rises. Approximating the logarithm by r overstates growth per period, so it understates the time needed to double. The bias is second-order in the rate, which is exactly why 8% looks perfect and 24% does not.
Can I use it for continuous compounding?
Roughly, yes. Continuous compounding doubles in 69.3 divided by the rate, so the rule of 72 comes out about 4% long in that setting. The difference is small enough to ignore for a mental estimate.
More Comparisons
- Simple Interest vs Compound Interest
Simple interest charges the principal only; compound interest charges the balance. On $10,000 at 5% over 10 years that gap is $1,288.95 — here is where it comes from.
- ROI vs CAGR
ROI measures the whole return; CAGR turns it into a compound annual rate. A 100% return over five years is 20% a year by division but 14.87% a year in reality.
- Nominal vs Real Return
A 10% return with 3% inflation is a 6.80% real return, not 7%. $10,000 grows to $16,105 in five years, but only $13,892 of that is in today's money — $2,213 of the gain is the price level moving, not you.
Sources
- Rule of 72 — derivation and accuracy — Wikipedia
- Compound interest — formula and frequency — Wikipedia
- Compound interest — worked examples — Corporate Finance Institute
Both methods sit in Finance.