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Rule of 72 Doubling Time Calculator

Puts the rule of 72 beside the exact doubling time for your rate and compounding frequency, and reports how far the rule is off in years and as a share, the numerator that would be exact at your rate, and the range of rates over which the rule stays within a quarter of a year. It also gives the exact time to triple and to grow tenfold, and tabulates the error across a grid of rates and numerators so the safe region can be seen rather than assumed.

Last updated
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Calculator overview

Inputs and outputs

This summary comes from the calculator's published input and output contract.

Inputs

Rule Numerator (required)
About this input

The number the rule divides by the rate in percent. 72 is the classic choice because it has many divisors and is nearly exact near eight percent; 70 is closer for low rates and 69.3 is exact for continuous compounding. Its unit is a rate in percent multiplied by a number of years, which is why it is 72 and not 0.72.

Unit percent-years Default 72 Range 60 to 80
Compounding Frequency (required)
About this input

How often growth is added to the balance within a year. Yearly compounding is what the rule of 72 was tuned for; monthly, daily and continuous compounding raise the effective rate and shorten the exact doubling time, so the rule drifts further from the truth the more often the money compounds.

Default Annually Allowed Annually, Semi-annually, Quarterly, Monthly, Daily, Continuously
Annual Rate (required)
About this input

The rate the money grows at each year, before any compounding within the year is applied. Enter it as a percentage of the balance, so eight percent is 8.00%. It has to be above zero: money earning nothing never doubles.

Unit fraction Default 0.08 Range 0 to 0.25

Outputs

Safe Rate To
About this output

The highest rate on the same sweep at which the rule is still within a quarter of a year. Between this and the lowest safe rate the rule is safe to use in your head; outside that range, use the exact figure.

Unit fraction
Table1 Rule Column Axis
About this output

The values across the top of the grid: the rule's numerator, in steps of one either side of yours. Read a column to hold the numerator fixed and vary the rate. The middle entry is your own numerator.

No unit declared
Rule Error Years
About this output

The rule less the exact answer. Positive means the rule says doubling takes longer than it really does; negative means it says sooner. The rule overstates at low rates and understates at high ones, and the crossing is where the numerator happens to be exact.

Unit years
Safe Rate From
About this output

The lowest rate between 1 and 25 percent, in half-point steps, at which your numerator keeps the rule within a quarter of a year of the exact answer at this compounding. Zero means the rule is never that close anywhere on that sweep.

Unit fraction
Table1 Rule Column Input
About this output

Machinery, and NOT an input. The same for the column axis: Excel substitutes into it while filling the grid, and the rest of the time it mirrors Rule_Numerator.

Unit percent-years
Table1 Rule Row Input
About this output

Machinery, and NOT an input. Excel substitutes each value from the row axis into this cell while it fills the grid; the rest of the time it mirrors Annual_Rate. Change Annual_Rate above, never this: the axis is derived from it, so editing the mirror moves the axis while the table walks and the grid comes out meaningless.

Unit fraction
Table1 Rule Values
About this output

The body of the grid: the rule's error in years, sign ignored, recomputed for every combination of the two axes. One hundred and twenty-one cells, each one the whole calculation run again. It is a component of the table rather than a result on its own.

No unit declared
Table1 Rule Corner
About this output

Machinery. Excel requires the formula being tabulated to sit in the grid's top-left corner, where it means nothing to a reader, so it is formatted away. It holds Absolute_Error_Years.

Unit years
Table1 Rule Row Axis
About this output

The values down the left of the grid: the annual rate of return, in half-point steps either side of yours and never below a tenth of a percent. Read a row to hold the rate fixed and vary the numerator. The middle entry is your own rate.

No unit declared
Exact Doubling Time
About this output

How long a single sum takes to double at this rate and compounding, solved exactly from the growth relation rather than approximated. This is the figure to rely on.

Unit years
Exact Numerator
About this output

The numerator that would make the rule exact at your rate and compounding: the exact doubling time multiplied by the rate in percent. It is close to 72 near eight percent yearly, closer to 70 at low rates, and exactly 69.31 for continuous compounding at any rate.

Unit percent-years
Absolute Error Years
About this output

The error with its sign removed. This is what the grid at the foot tabulates and what the status line compares against its quarter-year tolerance.

Unit years
Effective Annual Rate
About this output

What the money actually grows by in a year once compounding within the year is counted. Equal to the annual rate for yearly compounding and higher for every other choice; the exact doubling time is computed from this rate, which is why the compounding choice moves it.

Unit fraction
Exact Tenfold Time
About this output

How long the same sum takes to grow tenfold, exactly: the doubling time scaled by the natural log of 10 over the natural log of 2, about 3.32 times as long.

Unit years
Rule Doubling Time
About this output

The numerator divided by the rate in percent. Quick to do in your head, and right to within a few days near eight percent compounded yearly, which is where it was tuned.

Unit years
Rule Error Percent
About this output

The same error as a share of the exact doubling time, so a low rate with a long doubling time is not judged in years alone.

Unit fraction
Exact Tripling Time
About this output

How long the same sum takes to triple, exactly. It is the doubling time scaled by the natural log of 3 over the natural log of 2, about 1.585 times as long.

Unit years
Model Status
About this output

Reads OK, or explains why the inputs are not valid or why the rule deserves a second look at this rate.

No unit declared
This page is provided by LogicCommons for informational purposes only. Results are model outputs computed from the inputs you supply and are not financial, investment, tax, accounting, or legal advice, and no advisory relationship is created. Verify all inputs and results independently before relying on them in any decision.

LogicCommons is in beta. If a result, label, or reference looks wrong, tell us here; we read every message.

Methodology

Purpose and model boundary

Use this to see how good the rule of 72 actually is at the rate you care about. The rule is a mental shortcut; this puts the exact doubling time next to it and names the error in years rather than leaving it as a feeling.

This is an arithmetic comparison at a constant rate. A real return that varies does not have a single doubling time, and the rule is not more accurate for being memorable.

Inputs and units

Input Unit Accepted range What it means
Annual Rate % 0% through 25% The rate the money grows at each year, before any compounding within the year is applied. Enter it as a percentage of the balance, so eight percent is 8.00%. It has to be above zero: money earning nothing never doubles.
Compounding Frequency Choose from the list How often growth is added to the balance within a year. Yearly compounding is what the rule of 72 was tuned for; monthly, daily and continuous compounding raise the effective rate and shorten the exact doubling time, so the rule drifts further from the truth the more often the money compounds.
Rule Numerator percent-years 60 through 80 The number the rule divides by the rate in percent. 72 is the classic choice because it has many divisors and is nearly exact near eight percent; 70 is closer for low rates and 69.3 is exact for continuous compounding. Its unit is a rate in percent multiplied by a number of years, which is why it is 72 and not 0.72.

Governing relationships

The rule divides a numerator by the rate expressed in percent. That numerator is 72 by convention, though it is an input here. The exact doubling time comes from the compounding the account actually uses, which is why the frequency is an input. The calculator reports the difference between them, and the numerator that would have been exact at your rate.

Calculation sequence

  1. Convert the annual rate to the effective annual rate implied by the chosen compounding frequency.
  2. Solve exactly for the time a single sum takes to double at that rate.
  3. Divide your numerator by the rate in percent for the rule's answer.
  4. Difference the two, both in years and as a share of the exact time.
  5. Multiply the exact doubling time by the rate in percent to get the numerator that would have been exact here.
  6. Scale the exact doubling time by the natural log of 3 over the natural log of 2, and by the natural log of 10 over the natural log of 2, for the tripling and tenfold times.
  7. Sweep rates from 1 to 25 percent in half-point steps to find the range over which your numerator keeps the rule within a quarter of a year.

Outputs and interpretation

Output Role Unit What it means
Rule Error Years primary years The rule less the exact answer. Positive means the rule says doubling takes longer than it really does; negative means it says sooner. The rule overstates at low rates and understates at high ones, and the crossing is where the numerator happens to be exact.
Exact Doubling Time primary years How long a single sum takes to double at this rate and compounding, solved exactly from the growth relation rather than approximated. This is the figure to rely on.
Rule Doubling Time primary years The numerator divided by the rate in percent. Quick to do in your head, and right to within a few days near eight percent compounded yearly, which is where it was tuned.
Safe Rate To detail % The highest rate on the same sweep at which the rule is still within a quarter of a year. Between this and the lowest safe rate the rule is safe to use in your head; outside that range, use the exact figure.
Safe Rate From detail % The lowest rate between 1 and 25 percent, in half-point steps, at which your numerator keeps the rule within a quarter of a year of the exact answer at this compounding. Zero means the rule is never that close anywhere on that sweep.
Exact Numerator detail percent-years The numerator that would make the rule exact at your rate and compounding: the exact doubling time multiplied by the rate in percent. It is close to 72 near eight percent yearly, closer to 70 at low rates, and exactly 69.31 for continuous compounding at any rate.
Absolute Error Years detail years The error with its sign removed. This is what the grid at the foot tabulates and what the status line compares against its quarter-year tolerance.
Effective Annual Rate detail % What the money actually grows by in a year once compounding within the year is counted. Equal to the annual rate for yearly compounding and higher for every other choice; the exact doubling time is computed from this rate, which is why the compounding choice moves it.
Exact Tenfold Time detail years How long the same sum takes to grow tenfold, exactly: the doubling time scaled by the natural log of 10 over the natural log of 2, about 3.32 times as long.
Rule Error Percent detail % The same error as a share of the exact doubling time, so a low rate with a long doubling time is not judged in years alone.
Exact Tripling Time detail years How long the same sum takes to triple, exactly. It is the doubling time scaled by the natural log of 3 over the natural log of 2, about 1.585 times as long.

Model Status reads OK, or explains why the inputs are not valid or why the rule deserves a second look at this rate. It is shown alongside the results rather than in place of them.

The calculator also returns a grid that reruns the calculation across two varying assumptions at once. Its axes, corner and body arrive as separate outputs and are the grid's parts rather than results to read on their own; the page assembles them into the table.

Every figure above is returned by the workbook. The page arranges and formats them; it computes none of them.

Validation and status logic

The workbook returns one status alongside the figures. These are the states its delivered test cases exercise, so the list records what it has been observed to return rather than every branch it could take; a figure that moves with the inputs is shown as ….

Outcome Returned status
Refuses to answer NOT VALID: the annual rate has to be above zero; money earning nothing never doubles
Answers, and flags it CHECK: at this rate the rule is … years too long, past the …-year tolerance; use the exact figure
Answers, and flags it CHECK: at this rate the rule is … years too short, past the …-year tolerance; use the exact figure
Answers plainly OK

Assumptions and limitations

  • The rate is constant. A return that varies has no single doubling time, and the exact figure here would not describe it.
  • Inflation, tax and fees are not modelled; the doubling is of the nominal amount.
  • The safe range is found by sweeping rates from 1 to 25 percent in half-point steps, so it reports the tolerance over that sweep and not beyond it.
  • The quarter-year tolerance is this workbook's convention for "close enough", not a standard.

Restrictions and non-computing states

The declared bounds are enforced before the calculation runs, so a value outside them is refused rather than answered:

  • Annual Rate: 0% through 25%.
  • Rule Numerator: 60 through 80.

Errors and warnings

NOT VALID means the inputs do not describe a question this calculator can answer, and the figures beside it should not be relied on. CHECK is not an error: the arithmetic is sound and the figures stand. Something about the combination is still worth knowing before the answer is used, such as an assumption at the edge of its range, a comparison that has collapsed to a single case, or a result whose sign is the opposite of what the page's framing suggests. The status is shown with the results rather than in place of them, so a flagged answer is still a readable one.

References

The rule of 72 as a mental approximation is described in the U.S. Securities and Exchange Commission's Investor.gov material on compound interest: divide 72 by the expected rate of return and you have roughly how long the money takes to double, so 72 over 9 gives about eight years at nine percent. Investor.gov offers the rule as an estimate and does not say at which rates it stops being a close one, which is the gap this calculator measures.

For the effect of compounding frequency on how quickly a balance grows, see the Consumer Financial Protection Bureau on how compound interest works, which names the compounding frequency as an input in its own right and lists increasing it among the ways to make savings grow faster.

These sources provide background; they do not supply the calculator's assumptions or certify its result. This calculator is informational and is not financial, investment, or tax advice. Results follow directly from the rates and amounts you enter, which are assumptions rather than forecasts.

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