Growth rates multiply. Everyone knows this about interest and forgets it immediately about everything else, because twelve times ten is such an available number.
ending = starting × (1 + monthly rate)ⁿ
Start at 10,000 and grow ten per cent a month for twelve months and you finish at 31,384. The compound monthly growth rate calculator confirms the rate from the two endpoints: 10.00 per cent a month, a total growth of 213.8 per cent over the year.
Not 120 per cent. The extra 94 points are growth on growth, and they are most of the result.
Small monthly numbers are large annual ones
Six per cent a month sounds modest and the month-over-month calculator puts its annualised rate at 101.2 per cent: a doubling, from a figure most people would describe as steady rather than fast.
This cuts both ways and the downside is less discussed. A six per cent monthly decline compounds too, and a metric losing that much each month has lost more than half its value inside a year while never having a dramatic month. Slow compounding decline is the thing that does not trigger an alarm.
Run rate assumes the opposite of growth
Annualised run rate takes a period and multiplies it up. Forty-two thousand dollars in a month becomes a $504,000 run rate, and the calculation is one multiplication.
What it quietly assumes is that nothing changes. For a growing company the run rate understates the next twelve months, sometimes badly. For a seasonal one it is worse than useless: a retailer annualising December is describing a year that will never happen, and the same retailer annualising February is describing a different year that will also never happen.
Run rate is a snapshot of the current pace, and that is all it is. Pairing it with a growth rate is the minimum needed to make it a forecast, which is why the two numbers belong together rather than alone.
Which period you compare also decides the story
The same underlying business produces different headline numbers depending on the comparison chosen. Month-over-month is noisy and responds to a single good week. Year-over-year is quiet and removes seasonality by comparing like with like, at the cost of taking a year to notice anything.
Quarter-over-quarter sits in between and is where most reporting ends up. None of these is the honest one in general: they answer different questions, and the usual tell that a number is being presented rather than reported is that the period changes between slides.
The Rule of 40 exists because of exactly this, by refusing to look at growth alone. Twenty-eight per cent growth with a 15 per cent margin scores 43 and clears the benchmark. Forty per cent growth at break-even lands exactly on 40, and so does 10 per cent growth at a 30 per cent margin. It is a crude measure and its value is that it cannot be improved by choosing a different comparison period.
For the same compounding applied to money rather than metrics, see why small differences in rate matter so much.