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The Rule of 72 (Posted on 2005-01-25) Difficulty: 3 of 5
The Rule of 72 is a rule of thumb that states that the number of time periods (such as years) that it takes for a sum to double at compound interest is very nearly 72 divided by the percentage interest rate per period. Thus, for example, it takes (almost exactly) 9 years for a sum to double at 8% interest compounded yearly.

1. Using the rule, find the annual rate of increase for an investment that has quadrupled in 24 years. Compare to the exact value.

2. At what interest rate is the rule exact?

3. Justify the rule using mathematical analysis and a few numerical calculations.

  Submitted by Richard    
Rating: 3.6667 (3 votes)
Solution: (Hide)
1. Quadrupling in 24 years is doubling in 12 years, so the rate is 6%. The exact value is 5.9463094%.

2. The rule is exact at (slightly less than) 8%, but no whole number n corresponds to that rate of interest.

3. Let n be the number of periods at interest rate i (not in percent) per period. Since there is a doubling, (1+i)^n=2, or taking natural logs, n*ln(1+i)=ln(2). Replace ln(1+i) by (ln(1.08)/.08)*i, the secant line from (0,0) to (.08,ln(1.08)). This yields n*i=.720517467, so that rounding off and multiplying both sides by 100, we get n*(100*i)=72. Denoting by n' the approximate value of n gotten using the rule, we have n'/n ~= 1.04*(ln(1+i)/i), and for i=.2 (20% interest) this ratio is .95, while for i=.01 (1% interest) it is 1.03.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionNo SubjectCharlie2005-01-25 20:00:23
SolutionFurther justificationTristan2005-01-25 19:45:13
re(2): Continuous compounding caseJer2005-01-25 18:53:51
re: Continuous compounding caseJohn2005-01-25 18:37:39
Continuous compounding caseJer2005-01-25 18:08:11
Whole number of yearsJer2005-01-25 17:46:57
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