Choice of Rule
The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less accurate at higher interest rates.
For continuous compounding, 69 gives accurate results for any rate. This is because ln(2) is about 69.3%; see derivation below. Since daily compounding is close enough to continuous compounding, for most purposes 69, 69.3 or 70 are better than 72 for daily compounding. For lower annual rates than those above, 69.3 would also be more accurate than 72.
| Rate | Actual Years | Rule of 72 | Rule of 70 | Rule of 69.3 | 72 adjusted | E-M rule |
|---|---|---|---|---|---|---|
| 0.25% | 277.605 | 288.000 | 280.000 | 277.200 | 277.667 | 277.547 |
| 0.5% | 138.976 | 144.000 | 140.000 | 138.600 | 139.000 | 138.947 |
| 1% | 69.661 | 72.000 | 70.000 | 69.300 | 69.667 | 69.648 |
| 2% | 35.003 | 36.000 | 35.000 | 34.650 | 35.000 | 35.000 |
| 3% | 23.450 | 24.000 | 23.333 | 23.100 | 23.444 | 23.452 |
| 4% | 17.673 | 18.000 | 17.500 | 17.325 | 17.667 | 17.679 |
| 5% | 14.207 | 14.400 | 14.000 | 13.860 | 14.200 | 14.215 |
| 6% | 11.896 | 12.000 | 11.667 | 11.550 | 11.889 | 11.907 |
| 7% | 10.245 | 10.286 | 10.000 | 9.900 | 10.238 | 10.259 |
| 8% | 9.006 | 9.000 | 8.750 | 8.663 | 9.000 | 9.023 |
| 9% | 8.043 | 8.000 | 7.778 | 7.700 | 8.037 | 8.062 |
| 10% | 7.273 | 7.200 | 7.000 | 6.930 | 7.267 | 7.295 |
| 11% | 6.642 | 6.545 | 6.364 | 6.300 | 6.636 | 6.667 |
| 12% | 6.116 | 6.000 | 5.833 | 5.775 | 6.111 | 6.144 |
| 15% | 4.959 | 4.800 | 4.667 | 4.620 | 4.956 | 4.995 |
| 18% | 4.188 | 4.000 | 3.889 | 3.850 | 4.185 | 4.231 |
| 20% | 3.802 | 3.600 | 3.500 | 3.465 | 3.800 | 3.850 |
| 25% | 3.106 | 2.880 | 2.800 | 2.772 | 3.107 | 3.168 |
| 30% | 2.642 | 2.400 | 2.333 | 2.310 | 2.644 | 2.718 |
| 40% | 2.060 | 1.800 | 1.750 | 1.733 | 2.067 | 2.166 |
| 50% | 1.710 | 1.440 | 1.400 | 1.386 | 1.720 | 1.848 |
| 60% | 1.475 | 1.200 | 1.167 | 1.155 | 1.489 | 1.650 |
| 70% | 1.306 | 1.029 | 1.000 | 0.990 | 1.324 | 1.523 |
Read more about this topic: Rule Of 72
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