What is the doubling time formula?

For periodic compounding: T = ln(2) / (n * ln(1 + r/n)), where r is the annual rate as a decimal, n is the number of compounding periods per year, and ln is the natural logarithm. For continuous compounding the formula simplifies to T = ln(2) / r, because as n approaches infinity n * ln(1 + r/n) converges to r. ln(2) is the mathematical constant 0.693147.

What is the Rule of 72 and how accurate is it?

The Rule of 72 is a mental shortcut: divide 72 by your annual percentage rate to estimate the doubling time in years. At 6% that gives 72/6 = 12 years. It is most accurate for rates between roughly 6% and 10%; outside that range the error grows. The exact formula is always more accurate, but the Rule of 72 is easy to do in your head.

Why does compounding frequency affect doubling time?

More frequent compounding means interest is calculated on a slightly larger balance each period, so the effective annual rate is higher than the stated nominal rate. A 7% rate compounded monthly grows slightly faster than 7% compounded annually, which means your money doubles in a little less time. Continuous compounding is the theoretical limit where interest is applied instantaneously at every moment.

Does this work for population growth, inflation, or any exponential process?

Yes. The formula is identical for any quantity that grows at a constant percentage rate per period, whether that is a savings account, an investment portfolio, a bacterial colony, a population, GDP, or inflation. Just enter the annual growth rate and the compounding frequency that matches your scenario (use "Annual" for most population or GDP figures, "Continuous" for biological growth models).

What is the Rule of 69.3 and when should I use it?

The Rule of 69.3 comes from the fact that ln(2) = 0.693147, so dividing 69.3 by the rate gives the exact doubling time for continuous compounding. It is more accurate than 72 for very low rates (below 2%) and for continuously compounding processes. The Rule of 72 tends to be more accurate at higher rates because of offsetting errors in the approximation.

How do I find the rate needed to double my money in a specific time?

Switch to "Find required rate" mode, enter your target time in years and choose a compounding frequency. The calculator inverts the formula: for periodic compounding r = n * (exp(ln(2) / (n * T)) - 1) and for continuous compounding r = ln(2) / T. The result is the annual nominal rate you need.

Doubling Time Calculator

The doubling time calculator tells you how long any exponentially growing quantity — savings, an investment portfolio, a business, a population — takes to double in size at a constant growth rate. It also works in reverse: enter your target doubling period and it calculates the annual rate you need to hit it. Both directions use the exact mathematical formula based on the natural logarithm, with support for annual, quarterly, monthly, daily, and continuous compounding.

How it works

For any quantity growing at nominal annual rate r (as a decimal), compounded n times per year, the exact doubling time is:

T = ln(2) / (n × ln(1 + r/n))

where ln is the natural logarithm and ln(2) = 0.693147. For continuous compounding (the limit as n → ∞) the formula simplifies to:

T = ln(2) / r

The reverse calculation (finding the required rate to double in T years) inverts the same formula:

r = n × (e^(ln(2) / (n × T)) − 1) for periodic compounding

r = ln(2) / T for continuous compounding

The calculator also reports three classic rules of thumb alongside the exact result: the Rule of 72 (72 ÷ rate%), the Rule of 70 (70 ÷ rate%), and the Rule of 69.3 (69.3 ÷ rate%), showing the error of each approximation in years so you can see which shortcut is most accurate at your chosen rate.

Worked example

Suppose you invest in an index fund that historically averages 7% per year, compounded annually:

Exact doubling time: ln(2) / ln(1.07) ≈ 10.24 years (about 10 years 3 months)
Rule of 72: 72 / 7 = 10.29 years (off by 0.04 yr — very close)
Rule of 70: 70 / 7 = 10.00 years (off by 0.24 yr)

Now switch to monthly compounding at the same 7% nominal rate:

Monthly rate: 7% / 12 = 0.5833%
Exact: ln(2) / (12 × ln(1.005833)) ≈ 9.93 years (about 9 years 11 months)

The difference between annual and monthly compounding at 7% is just 0.31 years — meaningful over decades but small for a quick estimate.

Rate	Compounding	Doubling time	Rule of 72 error
2%	Annual	35.00 yr	1.00 yr
5%	Annual	14.21 yr	0.19 yr
7%	Annual	10.24 yr	0.05 yr
10%	Annual	7.27 yr	0.07 yr
7%	Monthly	9.93 yr	0.36 yr
7%	Continuous	9.90 yr	0.39 yr

The Rule of 72 is remarkably accurate near 7–8%, which is why it became the canonical mental shortcut for investors — at typical stock-market return rates the error is under 0.1 years.

Formula note

ln(2) is the natural logarithm of 2, equal to 0.693147… This constant appears because doubling means multiplying by 2, and the inverse of the exponential function is the natural log. All rules of thumb (69.3, 70, 72) are approximations derived by rounding or slightly biasing this constant to numbers that divide evenly by common interest rates. The Rule of 72 uses 72 because it is divisible by 1, 2, 3, 4, 6, 8, 9, and 12 — covering all common rate values — even though 72 is slightly larger than 100 × ln(2) ≈ 69.3.

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