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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.

See The Solution Submitted by Richard    
Rating: 3.6667 (3 votes)

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Solution No Subject Comment 6 of 6 |

1. Using the rule: Since the investment must double in 12 years in order to quadruple in 24, the rule gives 72/12 = 6 percent interest.  The exact value can be found from (1+r)^12 = 2, or 12*log(1+r) = log(2), so that r=.059463, or 5.9463%.

2. As the interest rate at which the formula is exact will probably involve a fractional year doubling time, we have to know how the interest is calculated for a partial year.  But disregarding this, a naive method would be to solve log(2)/log(1+r) = .72/r, which yields .07846871 or 7.846871%.

But, actually, since interest is compounded annually, the actual payout after 9.17563 (the predicted doubling time) depends on how the extra .17563 years is handled.  Most likely the correct compounding rate is applied up to the exact 9-year point, and then simple interest is counted for the fraction of a year.  When this happens, the answer comes out to 7.921216%, for a doubling time of 9.089513 years.  Here's how that works:

After 9 years exactly, the investment has grown to (1+.0792121611748568)^9 = 1.98591874498216 times its original value.  The simple interest for the remainder of the time is then .089513394422314 * .0792121611748568 * 1.98591874498216  = .01408125501789, which when added to the  1.98591874498216 comes out to 2.

3. The goal is to solve (1+r)^n = 2. By logarithms, this is n * log(1+r) = log(2).  If the logarithms are natural logarithms, then r constitutes an approximation for log(1+r) -- meaning ln(1+r). The natural log of 2 is approximately .69 and the r is the percentage divided by 100.  However, using r as an approximation for ln(1+r) is only valid as r approaches zero.  The logarithmic function then starts to slope less upward than the number itself.  As shown above, by the time about r=.09 is reached, its close to 69/72 of r.

The following table shows, for various interest rates, the doubling time based on the rule of 72, the doubling time based on the naive formula (1+r)^n = 2, and the doubling time taking into consideration that the fractional year will be at simple interest.

 1 72.00 69.66 69.66
 2 36.00 35.00 35.00
 3 24.00 23.45 23.45
 4 18.00 17.67 17.67
 5 14.40 14.21 14.20
 6 12.00 11.90 11.89
 7 10.29 10.24 10.24
 8  9.00  9.01  9.01
 9  8.00  8.04  8.04
10  7.20  7.27  7.26
11  6.55  6.64  6.63
12  6.00  6.12  6.11
13  5.54  5.67  5.66
14  5.14  5.29  5.28
15  4.80  4.96  4.96
16  4.50  4.67  4.65
17  4.24  4.41  4.40
18  4.00  4.19  4.18
19  3.79  3.98  3.98
20  3.60  3.80  3.79
21  3.43  3.64  3.61
22  3.27  3.49  3.46
23  3.13  3.35  3.33
24  3.00  3.22  3.20
25  2.88  3.11  3.10
26  2.77  3.00  3.00
27  2.67  2.90  2.89
28  2.57  2.81  2.79
29  2.48  2.72  2.70
30  2.40  2.64  2.61
31  2.32  2.57  2.53
32  2.25  2.50  2.46
33  2.18  2.43  2.40
34  2.12  2.37  2.33
35  2.06  2.31  2.28
36  2.00  2.25  2.23
37  1.95  2.20  2.18
38  1.89  2.15  2.13
39  1.85  2.10  2.09
40  1.80  2.06  2.05

In fact, based on this, we can work out a table of what the "rule of x" should be at various interest rates.  The table below shows the rate r, ln(1+1), 1/ln(1+r) and the corresponding numerator that should be used:

0.01  0.009950331  1.00499 69.34443
0.02  0.019802627  1.00997 69.68772
0.03  0.029558802  1.01493 70.02990
0.04  0.039220713  1.01987 70.37098
0.05  0.048790164  1.02480 70.71097
0.06  0.058268908  1.02971 71.04990
0.07  0.067658648  1.03461 71.38777
0.08  0.076961041  1.03949 71.72460
0.09  0.086177696  1.04435 72.06041
0.10  0.095310180  1.04921 72.39520
0.11  0.104360015  1.05404 72.72900
0.12  0.113328685  1.05887 73.06182
0.13  0.122217633  1.06368 73.39367
0.14  0.131028262  1.06847 73.72455
0.15  0.139761942  1.07325 74.05449

Based on this, a rule of thumb might be to make it a rule of 69+I/3, so that the doubling time would be (69+I/3)/I.  Maybe easier still would be to call it a rule of 69, and always add 1/3 year to the total.

Since it is indeed a rule of thumb, it of course does not take into consideration the fractional year at simple interest.

 

Edited on January 25, 2005, 8:03 pm
  Posted by Charlie on 2005-01-25 20:00:23

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