Use the Rule of 72

The Rule of 72 is a handy tool used in finance to estimate the number of years it would take to double a sum of money through interest payments, given a particular interest rate. The rule can also estimate the annual interest rate required to double a sum of money in a specified number of years. The rule states that the interest rate multiplied by the time period required to double an amount of money is approximately equal to 72.

The Rule of 72 is applicable in exponential growth, (as in compound interest) or in exponential "decay," as in the loss of purchasing power caused by monetary inflation.

Steps

Exponential growth

Estimating "doubling" time

  1. Let R x T = 72, where R = the rate of growth (for example, the interest rate), T = doubling time (in years) (for example, the time it takes to double an amount of money).
  2. Insert a value for R. For example, how long does it take to turn $100 into $200 at a yearly interest rate of 5%? Letting R = 5, we get 5 x T = 72.
  3. Solve for the Use Equations. In this example, divide both sides of the above equation by R (that is, 5) to get T = 72 ÷ 5 = 14.4. So it takes 14.4 years to double $100 to $200 at an interest rate of 5% per annum. (The initial amount of money doesn't matter. It will take the same amount of time to double no matter what the beginning amount is.)
  4. Study these additional examples:
    • How long does it take to double an amount of money at a rate of 10% per annum? Let 10 x T = 72, so that T = 7.2 years.
    • How long does it take to turn $100 into $1600 at a rate of 7.2% per annum? Recognize that 100 must double four times to reach 1600 ($100 → $200, $200 → $400, $400 → $800, $800 → $1600). For each doubling, 7.2 x T = 72, so T = 10. So as each doubling takes ten years, the total time required (to multiply $100 by 16) is 40 years.

Estimating growth rate

  1. Let R x T = 72, where R is the rate of growth (in these examples it's the interest rate), and T is the time it takes to double any amount of money.
  2. Enter the value of T. For example, let's say you want to double your money in ten years. What Determine Home Equity Loan Rates would you need in order to do that? Enter 10 for T in the equation. R x 10 = 72.
  3. Solve for R. Divide both sides by 10 to get R = 72 ÷ 10 = 7.2. So you will need an annual interest rate of 7.2% in order to double your money in ten years.

Estimating exponential "decay" (loss)

  1. Estimate the time it would take to lose half of your money (or its purchasing power in the wake of inflation). Solve T = 72 ÷ R. This is the same equation as above, just slightly rearranged. You simply enter a value for R. For example:
    • How long will it take for $100 to assume the purchasing power of $50, given an inflation rate of 5% per year?
      • Let 5 x T = 72, so that T = 72 ÷ 5, meaning T = 14.4. That's how many years it would take for money to lose half its buying power in a period of 5% inflation.
  2. Estimate the rate of decay (R) over a given time span: R = 72 ÷ T. Enter a value for T, and solve for R. For example:
    • If the buying power of $100 becomes worth only $50 in ten years, what is the inflation rate during that time?
      • R x 10 = 72, where T = 10. Then R = 72 ÷ 10 = 7.2%.
  3. Ignore any unusual data. If you detect general trends, don't worry about numbers that are wildly out of range. Drop them from consideration.

Doubling Time Chart

Doc:Doubling Time Chart



Derivation

Periodic compounding

  1. For periodic compounding, FV = PV (1 + r)^T, where FV = future value, PV = present value, r = growth rate, T = time.
  2. If money has doubled, FV = 2*PV, so 2PV = PV (1 + r)^T, or 2 = (1 + r)^T, assuming the present value is not zero.
  3. Solve for T by taking the natural logs on both sides, and rearranging, to get T = ln(2) / ln(1 + r).
  4. The Taylor series for ln(1 + r) around 0 is r - r2/2 + r3/3 - ... For low values of r, the contributions from the higher power terms are small, and the expression approximates r, so that t = ln(2) / r.
  5. Note that ln(2) ~ 0.693, so that T ~ 0.693 / r (or T = 69.3 / R, expressing the interest rate as a percentage R from 0-100%), which is the rule of 69.3. Other numbers such as 69, 70, and 72 are used for easier calculations.

Continuous compounding

  1. For periodic compounding with multiple compounding per year, the future value is given by FV = PV (1 + r/n)^nT, where FV = future value, PV = present value, r = growth rate, T = time, and n = number of compounding periods per year. For continuous compounding, n approaches infinity. Using the definition of e = lim (1 + 1/n)^n as n approaches infinity, the expression becomes FV = PV e^(rT).
  2. If money has doubled, FV = 2*PV, so 2PV = PV e^(rT), or 2 = e^(rT), assuming the present value is not zero.
  3. Solve for T by taking natural logs on both sides, and rearranging, to get T = ln(2)/r = 69.3/R (where R = 100r to express the growth rate as a percentage). This is the rule of 69.3.
  • For continuous compounding, 69.3 (or approximately 69) gives more accurate results, since ln(2) is approximately 69.3%, and R * T = ln(2), where R = growth (or decay) rate, T = the doubling (or halving) time, and ln(2) is the natural log of 2. 70 may also be used as an approximation for continuous or daily (which is close to continuous) compounding, for ease of calculation. These variations are known as rule of 69.3, rule of 69, or rule of 70.
    • A similar accuracy adjustment for the rule of 69.3 is used for high rates with daily compounding: T = (69.3 + R/3) / R.
  • The Eckart-McHale second order rule, or E-M rule, gives a multiplicative correction to the Rule of 69.3 or 70 (but not 72), for better accuracy for higher interest rate ranges. To compute the E-M approximation, multiply the Rule of 69.3 (or 70) result by 200/(200-R), i.e., T = (69.3/R) * (200/(200-R)). For example, if the interest rate is 18%, the Rule of 69.3 says t = 3.85 years. The E-M Rule multiplies this by 200/(200-18), giving a doubling time of 4.23 years, which better approximates the actual doubling time 4.19 years at this rate.
    • The third-order Padé approximant gives even better approximation, using the correction factor (600 + 4R) / (600 + R), i.e., T = (69.3/R) * ((600 + 4R) / (600 + R)). If the interest rate is 18%, the third-order Padé approximant gives T = 4.19 years.
  • To estimate doubling time for higher rates, adjust 72 by adding 1 for every 3 percentages greater than 8%. That is, T = [72 + (R - 8%)/3] / R. For example, if the interest rate is 32%, the time it takes to double a given amount of money is T = [72 + (32 - 8)/3] / 32 = 2.5 years. Note that 80 is used here instead of 72, which would have given 2.25 years for the doubling time.
  • Here is a table giving the number of years it takes to double any given amount of money at various interest rates, and comparing the approximation with various rules:
Rate Actual
Years 		Rule
of 72 		Rule
of 70 		Rule of
69.3 		E-M
rule
0.25% 277.605 288.000 280.000 277.200 277.547
0.5% 138.976 144.000 140.000 138.600 138.947
1% 69.661 72.000 70.000 69.300 69.648
2% 35.003 36.000 35.000 34.650 35.000
3% 23.450 24.000 23.333 23.100 23.452
4% 17.673 18.000 17.500 17.325 17.679
5% 14.207 14.400 14.000 13.860 14.215
6% 11.896 12.000 11.667 11.550 11.907
7% 10.245 10.286 10.000 9.900 10.259
8% 9.006 9.000 8.750 8.663 9.023
9% 8.043 8.000 7.778 7.700 8.062
10% 7.273 7.200 7.000 6.930 7.295
11% 6.642 6.545 6.364 6.300 6.667
12% 6.116 6.000 5.833 5.775 6.144
15% 4.959 4.800 4.667 4.620 4.995
18% 4.188 4.000 3.889 3.850 4.231
20% 3.802 3.600 3.500 3.465 3.850
25% 3.106 2.880 2.800 2.772 3.168
30% 2.642 2.400 2.333 2.310 2.718
40% 2.060 1.800 1.750 1.733 2.166
50% 1.710 1.440 1.400 1.386 1.848
60% 1.475 1.200 1.167 1.155 1.650
70% 1.306 1.029 1.000 0.990 1.523

Tips

  • The value of 72 was chosen as a convenient numerator in the above equation. 72 is easily divisible by several small numbers: 1,2,3,4,6,8,9, and 12. It provides a good approximation for annual compounding at typical rates (from 6% to 10%. The approximations are less exact at higher interest rates.
  • Let the Rule of 72 work for you, by starting to save now. At a growth rate of 8% a year (the approximate rate of return in the stock market), you would double your money in nine years (72 ÷ 8 = 9), quadruple your money in 18 years, and have 16 times your money in 36 years.
  • You can use Felix's Corollary to the Rule of 72 to calculate the "future value" of an annuity (that is, what the annuity's face value will be at a specified future time). You can read about the Corollary on various financial and investing websites.

Warnings

  • Don't let the rule of 72 work against you by taking on high-interest debt (as is typical with credit cards). At an average interest rate of 18%, credit card debt doubles in just four years (72 ÷ 18 = 4), quadruples in eight years, and never stops growing over time.

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