On the surface, churn rate may seem like a natural proxy for changes in customer lifetimes..In fact, as we will see, having a constant churn probability over time essentially implies that you’ll never have long term customers.If a user has a constant churn probability over time, this implies that customer lifetimes come from an Exponential distribution.If you have a constant churn of c per month, the probability that a customer stays subscribed for n months is (1-c)^n..In this case, this implies that customer lifetimes comes from a Lomax distribution..This is equivalent to a Pareto distribution shifted to start at 0.The Lomax distribution can express churn probabilities that get lower with time.What your churn is actually like, with help from Waloddi WeibullIf you suspect that churn probability per day may increase the longer a user has been subscribed, the Lomax distribution won’t work for you..Let’s say customer lifetimes come from a Lomax distribution..We can simulate it and find out.Keep in mind, in each of the examples below we simulate lifetimes from the same customer lifetime distribution, and this distribution does not change over time.With a shrinking business, churn appears to improve because subscriptions are getting fewer and older.This is clearly a dying business, yet the churn rate graph is looking great!.The churn rate per day is falling steadily, even if we know that there is no change in customer lifetimes in our model.So what’s going on?.We’ll keep the customer lifetimes exactly the same, but change it so that the number of new sign ups per day is growing superlinearly.With a growing business, churn rate appears to not change, only because most subscriptions are new.Even if the customer lifetimes are unchanged from before, the churn rate graph here is flat..For a more thorough derivation of this estimator under censored data, see this PDF from Daowen Zhang at NCSU.Now that we have an estimate of the distribution parameters, we can look at the mean and medians of the distribution.This outputs:weibull shape=1.0935179324818296, scale=122.3601743694174 weibull mean 118.29884582009447 weibull median 87.51413316428012 lomax shape=101.65488165157542, scale=12575.372370928875 lomax mean 124.93554375692996 lomax median 86.03983407302042The mean tells us the mean customer lifetime, in days..The existence of these rarer, very long term customers matter a lot to the outlook of your business, as we will see.Tail effects: The rare long-term customers make you rich slowlyLet’s look at customer survival for two business..Each will have customer lifetimes coming from a Lomax distribution..Keep in mind that the typical customers (found by the median) stick around equally long in either company, but it’s the rare long term customers that shift the Lifetime Customer Value massively in favor of the orange company.Let’s simulate the Monthly Recurring Revenue (MRR) for these two businesses..Monthly Recurring Revenue (MRR) for the first 2000 days of operation, for two companies with equal median customer lifetimes, and different mean customer lifetimes.The two companies have similar trajectories in the start, but eventually the effect of the rarer, long term customers show themselves.. More details