CLV Episode 1: Time Series

What can the passage of time tell us?

Working under the dual assumptions of “history repeats itself” and “we are creature of habits” time series approaches leverage the pattern of historic sales or transaction data to estimate the future.

With these methods you are driving down the road looking out the rear view mirror.  They are based on simple assumptions and make no attempt to explain why anyone is or is not profitable.  That said these techniques might out perform more sophisticated models. 

Let’s imagine a scenario with sufficient historic information to estimate the profit of each customer for each month in the past two years.  That is we know what he bought and what our cost to market, sell and service has been.

NPV
Given either average or most recent profitability we can create the present value of that stream of cash over the foreseeable future.     It looks like this:

where

• N is the number of time periods
• R is the profit at time t
• i is the discount rate that addresses uncertainty about the future.

An interesting twist is to vary both time and risk on a per customer basis.

Since customer lifetime value is predicated on transacting in the future we can adjust the current value accordingly.  In certain categories a person’s age is a surrogate answer to the question:  How long will she be a customer?   And if there are insights in terms of attrition, then the risk factor can be raised or lowered accordingly.

This approach works well when average or current profit itself isn't expected to change that much.  It also fits expected ideas about customers:  Young, active customers will be more profitable over the long run. 

Trend
Given sufficient history of profit, one could apply time series techniques to predict the trend going forward.  There are two approaches possible here using regression:

• The simplest way would be to estimate profit as simply the passage of time in order to isolate the trend component.  

Yt = a + bt + et

where the value at time t is some multiple of the monthly change in a customer's profit (b) plus a baseline figure (a).    Future profit is simply the number of months since the start of the measurement window times the rate of change.

• A slight extension would be to estimate profit based on the previous profit levels themselves.  It looks like:

where the value at time t is a weighted value of previous profits.   Future profit is based on estimating t+1, t+2, etc. up to the expected end of the relationship.   

As with NPV, both methods can create the desired spread by varying the length of time for each customer.

This approach fits well over the short-to-medium term when customer behavior, and thus profit, trends one-way or another over time.  However, at some point there is a bend in the road that we won’t see and this type of trending can produce surprises if used too far out into the future.

Depth of Repeat
“Once a customer, always a customer” is a big fat myth.   Consumers defect and do so at a surprisingly predictable rate.   We can take advantage of this fact and develop a depth of repeat model that estimates the number of purchases over a given time span.

The date requirements are slightly different than above and focus on using recency and frequency type metrics as the first.   For each customer we need to answer the following questions:

• How many repeat transactions did they make?
• How long did they have to make those transactions?
• How long since their last transaction?

Rather than the regression or econometric approach typical used by marketing analysts, this method uses the distribution of past behavior to estimate the probability of a customer doing something.   To provide a basis of discussion, we’ll start with a picture of the results first.   The following comes from a published analysis of repeat purchases for CDNow and it shows good fit with the number of future transactions. 

And since this is a more technical piece, here’s how it looks to the analyst (and yes it can be implemented in Excel).

The above simply turns things marketers believe into useful planning information.  The probability that a customer will make a certain number of transactions given her history is based on two key assumptions.

• The number of transactions a customer makes varies around their historic average.
• Customers may stop being customers after a transaction and thus drop out never to be seen again.

The result is a series of four parameters that generate the best fitting depth-of-repeat curve.   Thus we can now estimate the number of transactions each customer will have in the future.  And given the profitability of a transaction, we have future lifetime value.

This approach (BG/NBD) and its variants have been used for years in estimating the success or failure of new products from survey data so has a lot of validity.  But as with other time series approaches it is a bit of a black box from a tactical point of view.

Next up:  Segment Migration
Then: Predictive Models