# Math behind Customer Lifetime Value

## Deriving the formula

Published in

--

When you conduct a bottom-up analysis of a business with customers there is one metric you definitely need to know — Customer Lifetime Value (LTV), i.e. how much profit a customer brings in over a whole period of economic relationship.

LTV is a very popular topic of business discussion and you will find a lot of different formulas to calculate it if you google it, but I will dive into plain mathematical aspect of this concept, which is not so well represented over the internet.

Consider a subscription based service (e.g. Netflix) where a customer pays at the end of every period, and we get profit of amount \$M from that. Customer can cancel the subscription in any given period T with probability c.

In this setting discrete random variable cancelation time T has a geometric distribution with probability of cancelation in each period c. Total Profit(T) is therefore a function of random variable, which in this case is just a sum of all profits for T periods. Then LTV is the expected value of Total profit(T):

The last sum was folded due to the trick with derivative:

When customer pays at the beginning of each period we just have to add the corresponding profit to the LTV formula:

We’ve found formulas for LTV, great! But what assumptions did we make?

1. Regular profits \$M from a customer are constant and don’t depend on time of using service. No cross-sell, upsell. No scale economy savings. No support costs reduction.
2. Value of money today is the same as value of money tomorrow. No discounting of cash flows \$M.
3. Customer is equally likely to leave at any time — probability of cancelation today is independent from probability of cancelation any day before. No accounting of stickiness to the product that grows with time in reality. This assumption was inherited from geometric distribution of cancelation period T.
4. Probability of cancelation c is constant, i.e. the same for every period. It’s inherited from geometric distribution nature of T as well.
5. Customer can be subscribed infinitely long.

# General case of LTV

In the setting above we’ve made many assumptions. Now let’s look at LTV without assumptions.

In this general setting, where cancelation / churn period T is a discrete random variable, LTV is the expected present value of future net profits.

Last double sum looks scary. Can we simply things here? Indeed we can. We can calculate expectation of random variable via integration of its complementary cumulative distribution function (aka survival function).

There is a continuous generalization of this formula. You can find more discussion about this fact at [1] and [2]. I just provide illustrative proof for discrete case.

Having formula (7) in mind, where function f(t) in our case is PresentValue(TotalProfit(t)), one period increment f(t+1)-f(t) equals to PresentValue(Profit(t)). Therefore we can transform our LTV formula to a more friendly form:

# LTV under relaxed assumptions for quick estimation

Now consider our first LTV setting (1) with T having geometric distribution, but with slightly relaxed assumptions which are more realistic:

1. Instead of constant regular profit for each period \$M we’ll use profit with a constant linear growth component \$G
2. Take into account time value of money (DCF with a constant discount rate d)

Using our compressed formula (9) for general case derive LTV in a closed form expression for the setting above.

# Example

Check our results with a synthetic numerical example for setting (10). Take

• M = \$23 — constant component of profit from one customer for one period
• G = \$3 — growth component of profit from one customer for one period
• c = 13%— constant probability of cancelation/churn rate in any given period
• d=7%— discount rate

Let’s calculate it in three ways by substituting into WolframAlpha: use brute force expression for expectation (6), short expression for expectation (9) and closed form formula (11).

# LTV infinite sum approximation issues

When we have to compute LTV using general formula (9) there is a desire to upper bound number of periods from infinity to finite N.

Some sources over the internet recommend to take N as an average lifespan.

But such proxy might significantly underestimate LTV because it ignores the future cash flows from the the customers with a longer than average lifetime [3].

To illustrate what portion of customers we don’t take into consideration, take our setting (10) with geometrically distributed cancelation period T.

Then the share of customers with longer than average lifespan is a function of probability of cancelation c. The share is quite significant when c is small. So underestimation of LTV with that approximation approach could be significant too.

# Compromise between realistic assumptions and simplicity of formulas

So far we’ve seen that either we have a simple formula (2) under a lot of oversimplified assumptions or a complex formula with an unfolded sum (9) without any assumptions. Is there a compromise?

Yes. Most of our assumptions are driven by customer behavior, which is represented by distribution of random variable T — period of relationship cancelation. The simpler assumptions we make about T the simpler probability distribution we obtain and therefore the simpler LTV formula we get. Then realistic assumptions can be incorporated with an accurate estimation of P(T = t).

Probability distribution approximation is the area of statistics and machine learning and we’ll not explore it here. Instead, below I provide some references of application different probabilistic models to LTV analysis.

• In [4] authors provide overview of sophisticated probabilistic models applied to different settings
• In [5] the same authors apply geometric distribution for the likelihood and beta distribution for the prior of churn probability

# Conclusion

We started with elaboration of assumptions that we make calculating LTV in a simplified setting (1) where cancelation period T has geometric distribution and profits in each period are constant. Then, in order to get full perspective of LTV, we wrote LTV as the expected present value of future net profits from customer (6) using plain definition of expected value. It looked cumbersome and we transformed it into a more convenient form (9) using alternative representation of expectation as an integral of survival function (7). After that we considered a common simplified setting (10) with geometrically distributed cancelation period T, linearly growing profit and discount rate, for which we expressed LTV in a closed form formula (11). We also validated our formulas with the example (12). Then we discussed potential underestimation of LTV (16) using approximation of infinite sum in LTV formula with a finite sum upper bounded by average lifespan of a customer. And finally we discovered the way of incorporation more realistic assumptions about customer behaviour into LTV and more precisely into probability distribution of random variable T.

P.S. We didn’t dive into a business aspects of LTV because it wasn’t the goal. For example, we didn’t discuss what particular costs should be taken into account computing net profits. Let’s just note that for upper bound of LTV, only costs associated with serving and supporting customer are included. But some authors also subtract customer acquisition costs (CAC) [6].

# References

[2] A. Lo, Demystifying the Integrated Tail Probability Expectation Formula (2019), The American Statistician, Volume 73, Issue 4

[3] P. Fader, B. Hardie, What’s Wrong With This CLV Formula? (2014), www.brucehardie.com

[4] P. Fader, B. Hardie, Probability Models for Customer-Base Analysis (2009), 20th Annual Advanced Research Techniques Forum 2009

[5] P. Fader, B. Hardie, How to project customer retention (2007), Journal of Interactive Marketing, Volume 21, Issue 1