Apparently, his publishers told Stephen Hawking that for every formula he put into his book, A Brief History of Time, he would halve his sales. I sense our small readership may be decimated by this article. We’ll try to keep it brief, and you can skip to the final line in the final formula.

In Return on Customer, Peppers & Rogers propose a metric, called ROC as in the book title. They follow the form of the return on a traditional financial security which is the dividends you received plus or minus the change in the value of the stock, divided by the value of the stock.

\[ ROC = \frac{FC_{i} + ∆CE_{i}}{CE_{i-1}} \]

Here ROC is the free cash flow FC in this period plus the corresponding change in customer equity ∆CE, divided by the customer equity at the beginning of he period.

For our purposes we will follow our previous analysis and equate free cash flow with the ARPU less the subscriber acquisition and retention costs SARC.

\[ FC = ARPU - SARC \]

Customer equity is the same as lifetime customer value. All mobile companies publish churn rates, which are the inverse of the average length of a customer stay with the operator. Multiply this length of stay with the free cash flow from that customer, and you have his customer equity. (You should probably discount future cash flows but it doesn’t make any real difference to the numbers and just complicates the maths.) Hence:

\[ CE = \frac{ARPU-SARC}{Churn} = \frac{FC}{Churn} \]

This is a steady-state model of the business, and obviously a simple one, though one that turns out to be useful and predictive, as we have seen.

Now we can easily simplify the ROC formula if we remember our differential calculus and substitute \(FC/Churn\) for \(CE\):

\[\begin{align} ROC &= \frac{FC + ∆CE}{CE} = Churn + \frac{∆CE}{CE} \\\\ &= Churn + \frac{Churn}{FC} ∆\left(\frac{FC}{Churn}\right) \\\\ &= Churn + \frac{Churn}{FC} \left(\frac{∆FC}{Churn} - \frac{FC \times ∆Churn}{Churn^2}\right) \\\\ ROC &= Churn - \frac{∆Churn}{Churn} + \frac{∆FC}{FC} \end{align}\]

In other words, ROC is the churn plus the percentage points decrease in churn plus the percentage improvement in free cash flow (which essentially means ARPU).

There is no doubt that ROC brings together key metrics for the industry, but does the combination make sense? In an unchanging business, ROC is simply the churn rate. Normally, less churn is better while you want more return on an asset. The relationship seems to be the wrong way.

The change elements are better. In our model, it doesn’t matter if you double the ARPU or halve the churn, and that is correctly represented in the formula.

We are not convinced. Stay tuned.

“ROC” and “Return on Customer” may be a service mark, trademark, or registered trademark of Peppers & Rogers Group, Inc. in certain jurisdictions.