Doubling time and compound growth
Doubling time of a growth (exact and rule of 70), rate from an observed doubling, or value after n periods.
Result
10.245 periods
- Calculation
- T = ln 2 / ln(1 + 7/100) = 10.245 periods
- Rule of 70
- 10 periods
The quantity doubles in 10.245 periods; the rule of 70 estimates 10 periods.
Note
Rule of 70: accurate to about 4% below a 10% rate; at 100%, it predicts 0.7 of a period for a doubling that takes 1.
Calculation, assumptions and sources
OverviewNumber of periods for a quantity growing at r% per period to double, exactly (ln 2 / ln(1 + r)) and by the rule of 70.
Number of periods for a quantity growing at r% per period to double, exactly (ln 2 / ln(1 + r)) and by the rule of 70. The reverse calculation gives the rate from an observed doubling, the projection compounds the rate over n periods, and a negative rate gives a halving time.
Method & formulasT = ln 2 / ln(1 + r/100)
The exact doubling is the power that brings (1 + r) to 2. The rule of 70 replaces ln(1 + r) with r, an approximation valid at small rates.
- Compound growth
- each period applies to the previous period’s result, hence the exponential.
- Doubling time
- the number of periods to multiply by two, constant at a constant rate.
- Rule of 70
- 70 divided by the rate in %, an approximation accurate below 10%. Some use 72, which has more divisors.
- Halving time
- the counterpart of the doubling time for a negative rate, for example the erosion of purchasing power by inflation.
Validity domainThe model assumes a constant rate, rarely held for long: an epidemic saturates, a population runs out of resources, an interest rate changes.
The model assumes a constant rate, rarely held for long: an epidemic saturates, a population runs out of resources, an interest rate changes. Compounding is over whole, identical periods; for continuous compounding, replacing ln(1 + r) with r becomes exact and the rule of 70 becomes the rule of 69.3.
Compound growthAt 1% per day, the quantity is multiplied by 1.01 each day, that is by 37.8 over the year; adding the rates would give 365%.
At 1% per day, the quantity is multiplied by 1.01 each day, that is by 37.8 over the year; adding the rates would give 365%. Half of the total growth arrives in the last doubling time, and thirty doublings multiply by a billion.