# Rule of 72 Exact doubling times of an investment (dashed lines) and approximations with the rule of 72 (short lines with numbers) for different interest rates

The 72 rule is a rule of thumb from the interest calculation . The rule approximates the doubling time , i.e. the time after which an interest-bearing capital investment doubles in nominal value (through the effect of compound interest ). To do this, divide 72 by the percentage of the interest rate on the invested amount, hence the name of the rule. Variants of the 72 rule are the 70 rule and the 69 rule .

## formula

The time (in years) in which an investment with interest rate doubles (in percent per year) is, according to the 72 rule: ${\ displaystyle t}$ ${\ displaystyle p}$ ${\ displaystyle t \ approx {\ frac {72} {p}} ~ {\ text {years}}}$ .

The same formula can be used to estimate what interest rate is required to double a capital in a given time : ${\ displaystyle p}$ ${\ displaystyle t}$ ${\ displaystyle p \ approx {\ frac {72 ~ {\ text {years}}} {t}}}$ .

## Examples

In what time will an amount invested at an interest rate of (percent per year) double? ${\ displaystyle t}$ ${\ displaystyle p = 8}$ ${\ displaystyle t = {\ frac {72} {p}} ~ {\ text {years}} = {\ frac {72} {8}} ~ {\ text {years}} = 9 ~ {\ text {years }}}$ What interest rate (in percent per year) do you need to double a capital in the period ? ${\ displaystyle p}$ ${\ displaystyle t = 12 ~ {\ text {years}}}$ ${\ displaystyle p = {\ frac {72 ~ {\ text {years}}} {t}} = {\ frac {72 ~ {\ text {years}}} {12 ~ {\ text {years}}}} = 6}$ Of course, the rule of 72 can be applied not only to the calculation of interest, but to any kind of exponential growth . For example, the generation time , ie the time until a population doubles, with an annual population growth of about years. ${\ displaystyle 4 \, \%}$ ${\ displaystyle {\ tfrac {72} {4}} = 18}$ ## Derivation

According to the compound interest formula , the final capital of a fixed-income investment with initial capital at an interest rate of (in percent) after a term of years with annual interest ${\ displaystyle K_ {t}}$ ${\ displaystyle K_ {0}}$ ${\ displaystyle p}$ ${\ displaystyle t}$ ${\ displaystyle K_ {t} = K_ {0} \ left (1 + {\ frac {p} {100}} \ right) ^ {t}}$ .

If you now set , apply the logarithm to both sides of the equation and solve for , the number of years until doubling results as ${\ displaystyle K_ {t} = 2K_ {0}}$ ${\ displaystyle t}$ ${\ displaystyle t = {\ frac {\ ln (2)} {\ ln \ left (1 + {\ frac {p} {100}} \ right)}}}$ .

After converging for small amounts against (see Taylor series ) and with results as an approximation formula ${\ displaystyle \ ln (1 + x)}$ ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle \ ln (2) \ approx 0 {,} 6931}$ ${\ displaystyle t \ approx {\ frac {0 {,} 6931} {\ frac {p} {100}}} = {\ frac {69 {,} 31} {p}}}$ .

If one approaches through or , it is called the 69-rule or the 70-rule. As a rule of thumb, however, the approximation has proven itself, among other things because the number has many small divisors . There is also a modification of the form for the rule of 69 in the literature ${\ displaystyle \ ln (2)}$ ${\ displaystyle 0 {,} 69}$ ${\ displaystyle 0 {,} 70}$ ${\ displaystyle 0 {,} 72}$ ${\ displaystyle 72}$ ${\ displaystyle (72 = 2 ^ {3} \ cdot 3 ^ {2})}$ ${\ displaystyle t \ approx {\ frac {69} {p}} + 0 {,} 35}$ ,

obtained by the Taylor expansion of the logarithmic function up to the second order.

## accuracy

The following table compares the estimates according to the rule of 72, rule of 70, rule of 69 and the modified rule of 69 with the actual values ​​for typical interest rates.

interest rate ${\ displaystyle p}$ Doubling
time${\ displaystyle t}$ Rule of 72 70s rule 69 rule 69 rule
(modified)
0.25% 277,605 288,000 280,000 276,000 276,350
0.5% 138.976 144,000 140,000 138,000 138,350
1 % 69.661 72,000 70,000 69,000 69,350
2% 35.003 36,000 35,000 34,500 34,850
3% 23,450 24,000 23,333 23,000 23,350
4% 17.673 18,000 17,500 17.250 17,600
5% 14.207 14,400 14,000 13,800 14.150
6% 11,896 12,000 11.667 11,500 11,850
7% 10.245 10.286 10,000 9,857 10.207
8th % 9.006 9,000 8,750 8.625 8.975
9% 8.043 8,000 7.778 7.667 8.017
10% 7.273 7,200 7,000 6,900 7.250
11% 6.642 6.545 6.364 6.273 6.623
12% 6.116 6,000 5.833 5.750 6,100
15% 4,959 4,800 4,667 4,600 4,950
18% 4,188 4,000 3.889 3.833 4.183
20% 3.802 3,600 3,500 3,450 3,800
25% 3.106 2,880 2,800 2.760 3.110
30% 2,642 2,400 2.333 2,300 2,650
40% 2.060 1,800 1,750 1.725 2.075
50% 1.710 1,440 1,400 1,380 1.730