The rule of 72 is a handy rule used in finance to estimate quickly the number of years it takes to double a sum of capital given an 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 interest percentage times the number of years it takes to double a principal 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. Here's Tips On How to Use the Rule of 72 :
- The rule 72 is chosen as a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less exact at higher interest rates.
- To estimate doubling time for higher rates, adjust 72 rule 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.
- 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.
- 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 calculator of 70 Rule of 69.3 E-M rule.
- Felix's Corollary to the Rule of 72 is used to approximate the future value of an annuity (a series of regular payments). It states that the future value of an annuity whose percentage interest rate and number of payments multiply to be 72 can be approximated by multiplying the sum of the payments times 1.5. For example, 12 periodic payments of $1000 growing at 6% per period will be worth approximately $18,000 after the last period. This is an application of Felix's Corollary to the Rule of 72 since 6 (the percentage interest rate) times 12 (the number of payments) equals 72, so the value of the annuity approximates 1.5 times 12 times $1000.
- Start reading and learning. Understand about that you are starting. Let the rule of 72 work for you, by starting saving now. At a growth rate of 8% per annum (the approximate rate of return in the stock market), you would double your money in 9 years (8 * 9 = 72), quadruple your money in 18 years, and have 16 times your money in 36 years.
- Read also about Forward Contracts.
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