The rule of 72 gives quick approximate answers to compound interest and compound growth problems. The rule tells us how many periods it takes for wealth to double with a specified rate of return, and is exact for a rate of 7.85 percent. For smaller rates, doubling is a little quicker than what the rule calculates; for greater rates, it takes a little longer. The table compares the rule in column 2 with the exact value in column 3. The “exact rule” column shows the number that should replace 72 to calculate each rate of return. For an 8 percent return, the number, rounded to two decimal places, is 72.05, which shows how close the rule of 72 is. Notice that the number in column 4 for the exact rule should equal the column 1 return per period multiplied by the corresponding values in column 3 (actual number of periods to double), but that the column 4 figures don’t quite agree with this. That’s because the numbers in columns 3 and 4 are rounded off from the exact figures, correct to two decimal places.
The mental calculator may notice that the exact rule changes by about one-third for each 1 percent change in the return per period; so an easy approximation to the exact rule is 72 + (R−8%)/3. For 1 percent this gives 69.67 compared with the exact 69.66, and for 20 percent we get 76.00 compared with the exact 76.04. The formula fits well for the rest of the table, too.
Number of Periods to Double |
|||
Return Per Period |
By Rule of 72 |
Actual |
Exact Rule |
|
1% |
72 |
69.66 |
69.66 |
|
2% |
36 |
35.00 |
70.01 |
|
3% |
24 |
23.45 |
70.35 |
|
4% |
18 |
17.67 |
70.69 |
|
5% |
14.4 |
14.21 |
71.03 |
|
6% |
12 |
11.90 |
71.37 |
|
7% |
10.29 |
10.24 |
71.71 |
|
8% |
9 |
9.01 |
72.05 |
|
9% |
8 |
8.04 |
72.39 |
|
10% |
7.2 |
7.27 |
72.73 |
|
12% |
6 |
6.12 |
73.40 |
|
15% |
4.8 |
4.96 |
74.39 |
|
20% |
3.6 |
3.80 |
76.04 |
|
24% |
3 |
3.22 |
77.33 |
|
30% |
2.4 |
2.64 |
79.26 |
|
36% |
2.0 |
2.25 |
81.15 |
The idea behind the rule works for other wealth multiples. For instance, to get a rule for multiplying by 10, divide all the numbers in the table by 0.30103 (which is log10 2). Thus for 8 percent we get approximately 240, so we have a “rule of 240” for multiples of 10. We conclude that a return of 8 percent multiplies wealth by 10 in about 240 ÷ 8 = 30 years.
When Berkshire Hathaway offered to buy Shaw Industries for about $2 billion in cash, one manager mentioned that their earnings were up ten times from sixteen years before. By the rule of 240, we quickly find an approximate growth rate of 240 ÷ 16 = 15%. The actual figure is 15.48 percent.