Converting Churn Rate in Customer Lifetime

Question

A service has an 80% Annual Customer Churn Rate. It means that the service loses 80% of its customers every year.

The -80% Annual Churn Rate can be converted to a -12.55% Monthly Churn Rate with the formula:

(1-0.8)^(1/12)-1

The image below shows the churns period after period. We can see that the 80% Annual Rate leads to the same number of customers of the 12.55% Monthly Rate after 1, 2 and 3 years.

When talking about Customer Churn Rate, it is also common to talk about Average Customer Lifetime, that is the average time that a customer uses the service before churning. This site says that we can use the following formula to find the average Customer Lifetime:

Customer Lifetime = 1 / Customer Churn Rate

The same site also says:

Note that if the Customer Churn rate is a monthly % or annual %, then the Customer Lifetime will be for the same time period. Here is a monthly and annual example to illustrate the point:

a) If the Monthly customer churn rate is 3%, then the Customer Lifetime will be 1/0.03 which is 33 months.

b) if the Annual customer churn rate is 20%, then the Customer Lifetime will be 1/0.20 which is 5 years.

According to that we can say that:

a) If the Monthly customer churn rate is 12.55%, then the Customer Lifetime will be 1/0.1255 which is 7.97 months.

b) if the Annual customer churn rate is 80%, then the Customer Lifetime will be 1/0.80 which is 1.25 years (15 months).

But we have calculated in the beginning that an 80% annual churn rate corresponds to an 12.55% monthly churn rate. How can these equivalent rates lead to two different lifetimes?

In other words, how can an 80% annual Churn Rate be equivalent to a 12.55% Monthly Churn Rate, but the former leads to a lifetime of 15 months, while the latter leads to a lifetime of 7.97 months?

---

Edit:

After David's answer, I've got new questions related to the topic. I am editing this question to address them and also to confirm if I have understood correctly what David have kindly explained.

From what I could understand, the formula Average Customer Lifetime = 1 / Churn Rate assumes that cancellations occur in a linear fashion during the cohort period. For example: a Cohort with an annual Churn Rate of 80% and that starts with 240,000 customers, would have to lose 16,000 each month. And, within each month, it would have to lose 533 per day.

However, in an exponential decay model, the above formula is not perfect. I understood that if the Churn Rate is small, it becomes acceptable to use it, because the result would be close to reality. But what to do when the Churn Rate is high?

I thought the following ...

By David's response, I understood that it is correct to use the formula (1-0.8)^(1/12)-1 to convert the annual rate of -80% into a monthly rate of -12.55%, when we are in an exponential decay model. So, why not to convert the high annual Churn Rate of 80% to a low daily Churn Rate, and then use the formula that calculates Lifetime (suitable for low Churn Rates)?

For example:

Suppose that I have enough data to know that my service has an annual Churn Rate of 80% and that the decay is exponential. I want to calculate an Average Customer Lifetime that is close to reality. So I do the following:

1. Convert the -80% annual rate to a daily rate: (1-0.8)^(1/365)-1 = -0.44%
2. Calculate the Average Lifetime (in days): 1 / 0.0044 = 227 days = 7.57 months.

If I had calculated the Average Lifetime using the 80% rate directly, I would have come up with a result of 15 months (1 / 0.8 = 1.25 years = 15 months), which I believe to be much further from reality in comparison to the 7.57 months resulted from using the daily rate.

Does this make any sense?

Answer 1

Suppose a service maintains $240,000$ subscribers with an annual churn rate of $80\%.$ This service has $192,000$ cancellations per year (offset by $192,000$ new subscriptions if the part about "maintains" is true).

That averages to $16,000$ cancellations per month, which works out to a monthly churn rate of $(16,000/240,000)\times100\%\approx6.67\%.$

Not $12.55\%.$

A nice feature of this kind of churn calculation is that the average duration of a subscription is the reciprocal of the churn rate.

If you want to calculate churn by cohorts instead, then you need a model of how the cohort decays. For example, if you start with a cohort of $240,000$ and have $16,000$ cancellations from that cohort every month, you'll have an $80\%$ churn rate in the first year, and nobody from that cohort will remain after $15$ months. The monthly churn rate starts at just $6.67\%$ but rapidly increases (in the $15$th month it's $100\%.$)

The reciprocal relationship doesn't work out so neatly, however. If the cohort of $240,000$ were all new subscribers, their average duration is only $7.5$ months.

Another model is exponential decay. In this model the monthly attrition rate is constant, and the annual attrition rate is related to the monthly rate by a compounding formula as shown in the question. But the average duration of a subscription in that model depends on the instantaneous rate of attrition, which is related to both the monthly and annual rates by a formula of continuous compounding, and it predicts a shorter average duration than either the reciprocal of the monthly attrition or of the annual attrition.

For an exponential model with a small churn rate, however, the reciprocal formula for a short enough time period predicts about the same average subscription duration as the instantaneous rate does. For example, suppose the model is that the size of the cohort is $ne^{-0.03t}$ where $t$ is the number of months elapsed. This mathematical model has an average subscription duration of exactly $1/0.03=33\frac13$ months. After one month, this model predicts that you will lose $2.9554\%$ of your customers. The reciprocal formula then gives you $33.836$ months--not the same exact duration, but the difference may be a lot less than the inaccuracy of the model.

If we project the same exponential model to one year, it predicts you lose $30.23\%$ of customers. The reciprocal formula then gives you $3.308$ years, which is $39.69$ months. That's $6$ months longer than the mathematically precise answer for this model--but if the kind of answer you want is "about three years" then it's still good enough. Or you can choose your time unit so that the churn is a much smaller percentage and thereby avoid this source of disagreement with the model.

People who track these things for a living may be able to ignore a lot of these finer points because they are working with lower churn rates and/or they only need a rough approximation.