compound-growth-rates.md

Compound growth rates

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Description

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)

where

• 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)
• this_month & last_month - t (in years) = datetime_diff(this_month, last_month, month) / 12

Inferred growth rates

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

-- with_growth_rate
select
  *,
  ln(num_this_month / num_last_month) * 12 / datetime_diff(this_month, last_month, month) growth_rate
from with_previous_month

Using growth rates

To better depict the effects of a given growth rate, it can be converted to a year-on-year growth factor by inserting into the exponential formula above with t = 1 (one year duration). Or mathematically,

yoy_growth = x(1) / x(0) = exp(r)

It is always sensible to spot-check what a growth rate actually represents to decide whether or not it is a useful metric.

Extrapolated year-on-year growth

-- with_yoy_growth
select
  *,
  exp(growth_rate) as yoy_growth
from with_growth_rate

num_this_month	num_last_year	growth_rate	yoy_growth
343386934	361476484	-0.6160690738834281	0.5400632190921264
1151727105	343386934	14.521920316267764	2026701.8316519475