How to Use the 72: 10 Step Rule (with Pictures)

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How to Use the 72: 10 Step Rule (with Pictures)
How to Use the 72: 10 Step Rule (with Pictures)
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The "rule of 72" is a rule of thumb used in finance to quickly estimate the number of years it takes to double a sum of principal, with a given annual interest rate, or to estimate the annual interest rate it takes to double a sum of money over a given number of years. The rule states that the interest rate multiplied by the number of years required to double the capital lot is approximately 72.

The rule of 72 is applicable in the hypothesis of exponential growth (such as compound interest) or exponential decrease (such as inflation).

Steps

Method 1 of 2: Exponential Growth

Estimation of the doubling time

Use the Rule of 72 Step 1
Use the Rule of 72 Step 1

Step 1. Let's say R * T = 72, where R = growth rate (for example, the interest rate), T = doubling time (for example, the time it takes to double an amount of money)

Use the Rule of 72 Step 2
Use the Rule of 72 Step 2

Step 2. Enter the value for R = growth rate

For example, how long does it take to double $ 100 at an annual interest rate of 5%? Putting R = 5, we get 5 * T = 72.

Use the Rule of 72 Step 3
Use the Rule of 72 Step 3

Step 3. Solve the equation

In the example given, divide both sides by R = 5, to get T = 72/5 = 14.4. So it takes 14.4 years to double $ 100 at an annual interest rate of 5%.

Use the Rule of 72 Step 4
Use the Rule of 72 Step 4

Step 4. Study these additional examples:

  • How long does it take to double a given amount of money at an annual interest rate of 10%? Let's say 10 * T = 72, so T = 7, 2 years.
  • How long does it take to transform 100 euros into 1600 euros at an annual interest rate of 7.2%? It takes 4 double to get 1600 euros from 100 euros (double of 100 is 200, double of 200 is 400, double of 400 is 800, double of 800 is 1600). For each doubling, 7, 2 * T = 72, so T = 10. Multiply by 4, and the result is 40 years.

Estimation of the Growth Rate

Use the Rule of 72 Step 5
Use the Rule of 72 Step 5

Step 1. Let's say R * T = 72, where R = growth rate (for example, the interest rate), T = doubling time (for example, the time it takes to double an amount of money)

Use the Rule of 72 Step 6
Use the Rule of 72 Step 6

Step 2. Enter the value for T = doubling time

For example, if you want to double your money in ten years, what interest rate do you need to calculate? Substituting T = 10, we get R * 10 = 72.

Use the Rule of 72 Step 7
Use the Rule of 72 Step 7

Step 3. Solve the equation

In the example given, divide both sides by T = 10, to get R = 72/10 = 7.2. So you'll need an annual interest rate of 7.2% to double your money in ten years.

Method 2 of 2: Estimating Exponential Degrowth

Use the Rule of 72 Step 8
Use the Rule of 72 Step 8

Step 1. Estimate the time to lose half of your capital, as in the case of inflation

Solve T = 72 / R ', after entering the value for R, similar to the doubling time for exponential growth (this is the same formula as doubling, but think of the result as decrease rather than growth), for example:

  • How long will it take € 100 to depreciate to € 50 with an inflation rate of 5%?

    Let's say 5 * T = 72, so 72/5 = T, so T = 14, 4 years to halve purchasing power at an inflation rate of 5%

Use the Rule of 72 Step 9
Use the Rule of 72 Step 9

Step 2. Estimate the rate of degrowth over a period of time:

Solve R = 72 / T, after entering the value of T, similarly to the estimate of the exponential growth rate for example:

  • If the purchasing power of 100 euros becomes only 50 euros in ten years, what is the annual inflation rate?

    We put R * 10 = 72, where T = 10 so we find R = 72/10 = 7, 2% in this case

Use the Rule of 72 Step 10
Use the Rule of 72 Step 10

Step 3. Attention

a general (or average) trend of inflation - and "out of bounds" or strange examples are simply ignored and not considered.

Advice

  • Felix's corollary of the Rule of 72 it is used to estimate the future value of an annuity (a series of regular payments). It states that the future value of an annuity whose annual interest rate and the number of payments multiplied together give 72, can be roughly determined by multiplying the sum of the payments by 1, 5. For example, 12 periodic payments of 1000 euros with a growth of 6% per period, they will be worth around 18,000 euros after the last period. This is an application of Felix's corollary since 6 (the annual interest rate) multiplied by 12 (the number of payments) is 72, so the value of the annuity is about 1.5 times 12 times 1000 euros.
  • The value 72 is chosen as a convenient numerator, because it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It gives a good approximation for annual compounding at a typical interest rate (6% to 10%). The approximations are less accurate with higher interest rates.
  • Let the rule of 72 work for you, starting to save immediately. At a growth rate of 8% per year (the approximate rate of return of the stock market), you can double your money in 9 years (8 * 9 = 72), quadruple it in 18 years, and have 16 times your money in 36 years old.

Demonstration

Periodic Capitalization

  1. For periodic compounding, FV = PV (1 + r) ^ T, where FV = future value, PV = present value, r = growth rate, T = time.
  2. If the 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 extracting the natural logarithms of both sides, and rearrange 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 of the higher terms are small, and the expression estimates r, so that t = ln (2) / r.

  5. Note that ln (2) ~ 0.693, hence T ~ 0.693 / r (or T = 69.3 / R, expressing the interest rate as a percentage of R from 0 to 100%), which is the rule of 69, 3. Others numbers like 69, 70 and 72 are used for convenience only, to make calculations easier.

    Continuous capitalization

    1. For periodic capitalizations with multiple capitalizations during the 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, en = number of compounding periods per year. For continuous compounding, n tends to infinity. Using the definition of e = lim (1 + 1 / n) ^ n with n tending towards infinity, the expression becomes FV = PV e ^ (rT).
    2. If the 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 extracting the natural logarithms of both sides, and rearrange 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 capitalizations, 69, 3 (or approximately 69) yields better results, since ln (2) is about 69.3%, and R * T = ln (2), where R = rate of growth (or decrease), T = the doubling (or half-life) time and ln (2) is the natural logarithm of 2. You can also use 70 as an approximation for continuous or daily capitalizations, to facilitate calculations. These variations are known as the rule of 69, 3 ', rule of 69 or rule of 70.

        A similar fine adjustment for the rule of 69, 3 is used for high rates with daily compounding: T = (69.3 + R / 3) / R.

      • To estimate doubling for high rates, adjust the rule of 72 by adding one unit for each percentage point 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 we used 80 instead of 72, which would have given a period of 2.25 years for the doubling time
      • Here is a table with the number of years it takes to double any amount of money at various interest rates, and compare the approximation by various rules.

      Effective

      of 72

      of 70

      69.3

      E-M

      Badger Years Rule Rule Rule of 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

      • The Eckart-McHale Second Order Rule, or the E-M rule, gives a multiplicative correction to the rule of 69, 3, or 70 (but not 72), for better accuracy for high interest rates. To calculate the E-M approximation, multiply the result of the rule of 69, 3 (or 70) by 200 / (200-R), i.e. T = (69.3 / R) * (200 / (200-R)). For example, if the interest rate is 18%, the 69.3 rule says that t = 3.85 years. The E-M Rule multiplies this by 200 / (200-18), giving a doubling time of 4.23 years, which best estimates the effective doubling time of 4.19 years at this rate.

        Padé's third-order rule gives an 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%, Padé's third-order rule estimates T = 4.19 years

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