## Saturday, December 17, 2011

### Defining Churn Rate--is instantaneous better?

I enjoyed this post on defining "churn rate" at the Shopify tech blog, though I don't remember at all how it ended up in my reading list.

The author goes through ways you might define define "churn rate", showing that each of them could be misleading.

Like a good ex-calculus teacher (from one year in grad school), I wondered: Isn't any "rate" more naturally understood as an instantaneous quantity? Doesn't the problem come from the fact that he's trying to understand churn over an interval first?

To illustrate, here's my oversimplified model:

• You start with some number of customers (initial_customers) at time t=1
• Each day you gain new customers at some number of new customers, drawn from a Poisson distribution with expected value expected new customer rate.
• Each day you lose some proportion of your customers, drawn from a binomial distribution where the expected value of the proportion you lose is your daily churn rate.
I suspect that everything I say will apply also to more complicated/realistic models.

Here's a silly simulation (here's code for the simulation) of this model, where I've determined the expected churn rate (flat, then slopes up) and  expected new customer rate (flat).

Churn rates:

Daily number of customers:

Assuming your number of customers doesn't change too much in a day, the daily churn rate is almost like an instantaneous rate.

So, I have two notions of daily churn rate:

The definition he ended up with is an appropriate weighted average of daily churn rate and he must have been thinking of this.

Why take an average over a period? Because actual churn rates are noisy, so averaging is one way to smooth out that noise.

But once we're thinking in these terms, aren't there all sorts of standard time series methods for helping us model (and even project) churn rates?

In other words: Averaging isn't the only way to separate the "noise (randomness in churn) from the "signal" (expected value of churn).