Financial
Rule of 72 vs exact doubling & tripling time formula (LN(2) / LN(1 + r))
Calculate exact compound doubling time LN(2)/LN(1+r), continuous doubling LN(2)/r, and N-fold multiplication time—and see where Rule of 72 drifts.
Compare an approximation with the compound-growth equation
For a constant positive growth rate per period, solving (1+r)^n=2 gives n=LN(2)/LN(1+r). At 8%, the result is 9.0065 periods, versus 9 from the rule of 72. If periods are years, the difference is about 2.4 days.
At 24%, the logarithmic result is 3.2223 years, versus 3 from the approximation. This is a hypothetical constant-growth calculation; it does not forecast investment returns or business growth.
The logarithmic result interpolates a fractional period. If growth is credited only at whole-period ends, round the unrounded result up to the next integer to find the first credited balance that reaches the target. LN(2)/r instead assumes a continuously compounded rate; using the same numeric r is not comparing equivalent effective rates.
| Annual Growth Rate r | Rule of 72 (72 / 100r) | Exact Discrete LN(2)/LN(1+r) | Continuous LN(2)/r |
|---|---|---|---|
| 8.0% (Stocks) | 9.0000 yrs | 9.0065 yrs (Spot-on!) | 8.6643 yrs |
| 24.0% (High Growth) | 3.0000 yrs | 3.2223 yrs (+2.7 mos longer!) | 2.8881 yrs |
| 100.0% (Doubling/yr) | 0.7200 yrs (Wrong!) | 1.0000 yr exact! | 0.6931 yrs |
Exact discrete doubling time vs Rule of 72 and general K-fold multiple time
Assumes positive growth and reports fractional periods rounded to four decimals.
=IF(OR(NOT(ISNUMBER(A2)),A2<=0),"Invalid growth inputs",ROUND(LN(2) / LN(1 + A2), 4))General K-fold multiplication time (e.g., B2 = 3 for tripling, B2 = 10 for 10×)
For positive growth, targets of at least one produce nonnegative time. A target of one takes zero periods.
=IF(OR(NOT(ISNUMBER(A2)),A2<=0,NOT(ISNUMBER(B2)),B2<1),"Invalid growth inputs",ROUND(LN(B2) / LN(1 + A2), 4))Try the example
Use the original values or change them to test your understanding.