Subscriber Lifetime Value Modeling

While working on a marketing campaign dashboard that reports Return on Ad Spend (ROAS) at the campaign level, I was asked to include predicted lifetime value (LTV) alongside current 'ROAS to date'. This would make a case for increased spend on ads by justifying higher bids at the margin.

There are multiple ways to calculate subscriber survival and expected lifetime value. In this example I calculated survival curves, then integrated the best fitting model to get expected value.

Example Data

One well-known churn dataset is the IBM Telco Customer Churn Dataset, also hosted on Kaggle.

Spreadsheet

I downloaded the Telco Churn data into Google Sheets here.

For each tenure, I took the count of churned accounts as the numerator and, for the denominator, accounts with as much or more tenure i.e., if an account is only 3 months old, it is not included in the denominator for survival of tenures of 4 or more months. Here's the resulting survival curve:

This curve provides the probability that an account remains active after a given number of months, e.g., 10, 20, 30 months gives 85%, 80%, and 77%.

Integrating a survival curve provides the mean expected lifetime value, in this case of 54 months.

A drawback of this non-parametric method is that the LTV is determined by how much historic data you have. The table below, from the same linked spreadsheet above, calculates the estimated value, but having restricted the available history to each corresponding bin.

The curve above from the spreadsheet approach is known as Kaplan-Meier, and it's better for understanding survival probability at a given timepoint where history exists within the data rather than providing an overall expected survival value.

Kaplan-Meier is often a first look at survival analysis for a business and is just churned accounts / accounts that could have churned for each time period.

Parametric Models

Unlike Kaplan-Meier, parametric models can extrapolate to predict survival for future, unseen time periods.

To demonstrate this, pretend there's only data for 24 months whereas the goal might be to understand survival by 36 or even 48 months.

Use cases:

• New business with less historic data trying to estimate future retention
• Product AB testing, where you can model expected survival vs. a test group
• Adjust campaign ad spend based on expected lifetime ROAS of a cohort

Workflow

Code used for this analysis is here.

Since there are 24 months total history in this scenario:

• Train a model on the first 12 months
• Test the model against actual between 13 and 24 months
• Refit the chosen best fit model on the full 24 months of available data and then Predict out to 36 months

The plots below show actual survival in dark blue, while the lighter blue line is the predicted survival for each parametric model I tried.

The models were only trained on 12 months of data, so everything after 12 months on the light blue curves is extrapolated and can be compared to the actual dark blue line.

In this case, just eyeballing the plots, the mixture model combining Weibull and Exponential Decay fitted actual data out to 24 months a little better than Weibull or LogLogistic by themselves.

Now that a best fit model has been identified, retrain it on all available 24 months of data, and then predict into the future for time periods we don't yet have history for, such as 36 months.

Expected Survival Time (LTV)

Integrating a survival curve gives the mean expected survival time. Multiply monthly or annual revenue by this value to estimate LTV for a new subscriber.

Since parametric curves level off, a hard cutoff, such as 3, 4, or 5 years, must be defined before integration.

With only 24 months of history, integrating the Kaplan-Meier curve estimates survival at 20.5 months (see the table in the "Spreadsheet" section above).

In contrast, provided there is a good fit, the parametric model provides a more reliable estimate by extending beyond the initial 24 months.

Integrating the model’s 36-month survival curve yields an average survival of 29.6 months.

These LTV multipliers can be applied to new subscriptions to estimate expected lifetime value.