Compound growth rates

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When a quantity is increasing in time by the same fraction every period, it is said to exhibit compound growth - every time period it increases by a larger amount.

It is possible to show that the behaviour of the quantity over time follows the law

x(t) = x(0) * exp(r * t)


  • x(t) is the value of the quantity now
  • x(0) is the value of the quantity at time t = 0
  • t is time in units of [time_period]
  • r is the growth rate in units of 1 / [time_period]

Calculating growth rates

From the equation above, to calculate a growth rate you need to know:

  • The value at the start of a period
  • The value at the end of a period
  • The duration of the period

Then, following some algebra, the growth rate is given by

r = ln(x(t) / x(0)) / t


For a concrete example, assume a table with columns:

  • num_this_month - this is x(t)
  • num_last_month - this is x(0)

In the following then, growth_rate is the equivalent yearly growth rate for that month: