acalculator

What does the rule of 72 say?

Type a yearly rate of return to see roughly how many years your money takes to double, or type the years to see the rate you need. The rule of 72 calculator also shows the exact answer, so you can see how close the rule is.

Your numbers

Find
Rule
Rule estimate
8

By the rule of 72, the answer is about 8 years; exactly 8.04 with yearly compounding.

Exact, compounded yearly
8.04
Exact, compounded continuously
7.7
Rule minus exact
-0.04
Answer in
years

Rule estimate: 8. By the rule of 72, the answer is about 8 years; exactly 8.04 with yearly compounding.

The results are estimates for information only. They are not financial, tax, or legal advice. Check the numbers with your lender or a qualified professional before you decide. Terms of use

How to calculate

Uses the rule of 72 to estimate how many years money takes to double at a yearly return, or the return that doubles it in a given time, with the exact answer and the rules of 70 and 69.3.

Example with the default inputs (Find Years to double, Yearly rate of return 9%, Rule 72): By the rule of 72, the answer is about 8 years; exactly 8.04 with yearly compounding.

Method: Years ≈ 72 ÷ r and rate ≈ 72 ÷ t (or 70, 69.3). Exact, yearly: t = ln 2 ÷ ln(1 + r/100), r = (2^(1/t) − 1) × 100. Continuous: t = ln 2 ÷ (r/100), r = ln 2 ÷ t × 100.

  • The rate stays the same every year and all growth is reinvested.
  • Taxes, fees and inflation are not taken off.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Find Years to double, Yearly rate of return 9%, Rule 72 gives Rule estimate 8, Exact, compounded yearly 8.043232, Exact, compounded continuously 7.701635, Answer in years.Source: SEC, Tips for Teaching Students About Saving and Investing: divide 72 by the expected rate of return; at 9% money doubles about every 8 years, https://www.sec.gov/investor/students/tips.htm
  2. Find Years to double, Yearly rate of return 6%, Rule 72 gives Rule estimate 12, Exact, compounded yearly 11.895661, Rule minus exact 0.104339.Source: SEC, Tips for Teaching Students About Saving and Investing: divide 72 by the expected rate of return; at 9% money doubles about every 8 years, https://www.sec.gov/investor/students/tips.htm
  3. Find Rate to double, Years to double 10, Rule 72 gives Rule estimate 7.2, Exact, compounded yearly 7.177346, Exact, compounded continuously 6.931472, Answer in percent a year.Source: SEC, Tips for Teaching Students About Saving and Investing: divide 72 by the expected rate of return; at 9% money doubles about every 8 years, https://www.sec.gov/investor/students/tips.htm
  4. Find Years to double, Yearly rate of return 7%, Rule 70 gives Rule estimate 10, Exact, compounded yearly 10.244768.Source: SEC, Tips for Teaching Students About Saving and Investing: divide 72 by the expected rate of return; at 9% money doubles about every 8 years, https://www.sec.gov/investor/students/tips.htm (the same rule with 70)

How it works

Pick what to find and the rule (72, 70 or 69.3). With r the yearly rate in percent, t the years and N the rule:

Years to double (you type r):

  • Rule estimate = N ÷ r
  • Exact, compounded yearly = ln 2 ÷ ln(1 + r ÷ 100)
  • Exact, compounded continuously = ln 2 ÷ (r ÷ 100)

Rate to double (you type t):

  • Rule estimate = N ÷ t
  • Exact, compounded yearly = (2^(1/t) − 1) × 100 = (e^(ln 2 ÷ t) − 1) × 100
  • Exact, compounded continuously = ln 2 ÷ t × 100

Rule minus exact = the rule estimate − the exact yearly-compounding answer. "Answer in" says "years" or "percent a year". Every number shows to 2 decimal places; the maths runs in floating point and rounds for display only.

Assumptions: the rate is the same every year and all growth stays invested; taxes, fees and inflation are not taken off.

Rules

  • The rate is from 0.01% to 100% a year. The years are from 0.1 to 1,000. A value outside that shows a message on its field.

Worked examples by hand

9% a year (the SEC example). 72 ÷ 9 = 8 years. Exact: ln 2 ÷ ln 1.09 = 0.693147 ÷ 0.086178 = 8.04 years. Continuous: 0.693147 ÷ 0.09 = 7.70 years.

6% a year. 72 ÷ 6 = 12 years. Exact: ln 2 ÷ ln 1.06 = 11.90 years, so the rule is 0.10 years long.

Double in 10 years. 72 ÷ 10 = 7.2%. Exact: (2^0.1 − 1) × 100 = 7.18%. Continuous: 69.31 ÷ 10 = 6.93%.

Rule of 70 at 7%. 70 ÷ 7 = 10 years; exact 10.24 years.

Other questions people ask

What is the rule of 72?

A quick way to estimate doubling time: divide 72 by the yearly rate of return in percent. At 9% a year, money doubles in about 72 ÷ 9 = 8 years (the SEC uses this example).

How accurate is the rule of 72?

Close for rates from about 5% to 12%. At 9% the exact doubling time with yearly compounding is 8.04 years against the rule’s 8. At 2% the rule gives 36 years and the exact answer is 35.0; at 30% the rule gives 2.4 years and the exact answer is 2.64.

How do I find the rate that doubles my money in a set time?

Divide 72 by the years. To double in 10 years you need about 72 ÷ 10 = 7.2% a year; the exact rate with yearly compounding is (2^(1/10) − 1) × 100 = 7.18%.

What are the rules of 70 and 69.3?

The same idea with a different number. ln 2 × 100 = 69.3, so 69.3 ÷ r is almost exact for continuous compounding, and 70 is a round number close to it. 72 is popular because it divides evenly by 2, 3, 4, 6, 8, 9 and 12, and it is closer for yearly compounding at common rates.

Does the rule of 72 work for inflation or debt?

Yes, for anything that grows at a steady percent. At 3% inflation prices double in about 24 years. A credit card balance at 18% that is not paid doubles in about 4 years.

Why is the exact answer ln 2 ÷ ln(1 + r)?

Money growing at r a year is multiplied by (1 + r)^t after t years. It has doubled when (1 + r)^t = 2, so t = ln 2 ÷ ln(1 + r).