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: ${\displaystyle t}$ ${\displaystyle p}$

${\displaystyle t\approx {\frac {72}{p}}~{\text{Jahre}}}$.

The same formula can be used to estimate what interest rate is required to double a capital in a given time : ${\displaystyle p}$ ${\displaystyle t}$

${\displaystyle p\approx {\frac {72~{\text{Jahre}}}{t}}}$.

Examples

In what time will an amount invested at an interest rate of (percent per year) double? ${\displaystyle t}$ ${\displaystyle p=8}$

${\displaystyle t={\frac {72}{p}}~{\text{Jahre}}={\frac {72}{8}}~{\text{Jahre}}=9~{\text{Jahre}}}$

What interest rate (in percent per year) do you need to double a capital in the period ? ${\displaystyle p}$ ${\displaystyle t=12~{\text{Jahre}}}$

${\displaystyle p={\frac {72~{\text{Jahre}}}{t}}={\frac {72~{\text{Jahre}}}{12~{\text{Jahre}}}}=6}$

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

See also

• Time value of money

Web links

• Eric W. Weisstein : Rule of 72 . In: MathWorld (English).
• Rule of 72. Online calculator from moneychimp.com (English)

Individual evidence

1. ^ Pamela Peterson Drake, Frank J. Fabozzi: Foundations and Applications of the Time Value of Money . John Wiley & Sons, 2009, p. 89 .
2. ^ MC Lovell: Economics With Calculus . World Scientific, 2004, pp. 361 .
3. ^ RL Finney, GB Thomas: Calculus . Addison-Wesley, 1990, pp. 360 .
4. ^ JP Gould, RL Weil: The Rule of 69 . In: Journal of Business . tape 39 , 1974, p. 397-398 .
5. ↑ Richard P. Brief: A note on “rediscovery” and the rule of 69 . In: The Accounting Review . tape 52 , no. 4 , 1977, pp. 810-812 .