# Understanding Customer Lifetime Value (LTV)

- 7 mins

There are plenty of articles about LTV to get a general overview of this term. However, if you want to go deeper, it could be quite difficult to understand what’s behind these fancy formulas. That’s why after reading 10+ articles about LTV I decided to try to aggregate all acquired knowledge in this paper.

# What is LTV?

LTV is a sum of all returns a company expects to get from a customer during their current and future relationships. The easiest way to understand this concept in theory is to look at customers’ cohorts. This term originates from a demographic study and generally means people who made some action during some time period (for example, people who married in 2015). We also could define a cohort like all customers who made their first orders within a month. So, there are cohorts of December 2017, January 2018, etc. Now if you want to get a cohorts lifetime value you need to sum all expected returns from them:

\begin{equation} \text{Cohort LTV} = \text{Cohort Gross Margin in Month 0} + … + \text{Cohort Gross Margin in Month N + …} \end{equation}

Sure, you could sum revenue flows, but it’s more appropriate to use Gross Margin:

\begin{equation} \text{ Gross Margin = Revenue } * \text { Gross Margin % = Revenue } * \frac{\text{Revenue - COGS}}{\text{Revenue}} \end{equation}

You could see the formula for Gross Margin above, but what’s COGS? This is a cost of goods sold. For example, your Gross Margin could be just the sum of the commission your company gets from every order.

If you look at Cohorts LTV formula above, the whole point is that we couldn’t get accurate LTV but only try to predict it varying our aggregation assumptions. We could simplify it (and lose some accuracy) in the following way:

\begin{equation} \text{ Cohort LTV = Cohort Size } * \text{R}_0 * \text{AGMPU}_0 + \text{Cohort Size} * \text{R}_1 * \text{AGMPU}_1 + \text{ …} \\ \text{where} \\ \text{Cohort Size – initial cohort size (e.g. 100 people who made their first order in some month, constant) } \\ \text{R}_n \text{ – cohort retention rate in month n (e.g 100 % in the 0th month, 35 % in the 1st month, etc.) } \\ \text{ AGMPU}_n \text{ – average gross margin per user in month n (e.g. 5 bucks in the 0th month, 11 bucks in 1st month, etc.) } \end{equation}

Sometimes people discount it by rate to get the present value of future revenue (yes, 100 dollars next year ≠ 100 dollars today):

\begin{equation} \text{ Cohort LTV = Cohort Size } * \text{R}_0 * \text{AGMPU}_0 + \text{Cohort Size} * \text{R}_1 * \frac{\text{AGMPU}_1}{\text{ (1 + d)}} + \text{ …} \\ \text{where} \\ \text{d - discount rate, i.e. the interest rate you can get if you put your money in the bank } \\ \text{ (by the way we don’t discount cash flow of 0th month) } \end{equation}

The final step is just to get rid of this cohort terminology and talk about one customer. Just remove Cohort Size from the formula and we get Customer Lifetime Value:

\begin{equation} \text{ Customer LTV = } \text{R}_0 * \text{AGMPU}_0 + \text{R}_1 * \frac{\text{AGMPU}_1}{\text{ (1 + d)}^1} + \text{ …} + \text{R}_n * \frac{\text{AGMPU}_n}{\text{ (1 + d)}^n} + \text{ … } \qquad \text{ (1) } \end{equation}

Understanding $R_0, R_1$ could be not clear without cohorts’ context, but just think about them as some kind of probability that a user will bring revenue in the n-th period of their customer life cycle.

So, cool! We got it! Let’s move to a more practical side. Technically I see two ways of calculating LTV in practice depending of the accuracy we want to achieve and the data we have:

1. A simple but less accurate approach (more aggregations): assume that R and AGMPU are constant over customer’s lifetime
2. A less simple but more accurate approach (less aggregations): assume that R and AGMPU are not constant over customer’s lifetime

## A simple but less accurate approach

Okay, let’s make the following assumptions:

• AGMPU is a constant: every month a customer brings us the same amount of revenue (we can calculate it on historical data or just assume some amount)
• R is a constant: each month R % of the last month customers continue using a service or simply saying each month the constant percent (1 - R) of customers churn (we can calculate it on historical data or assume)

Assuming this, we get the following:

\begin{equation} \text{ Customer LTV = } \text{AGMPU} + \text{R}^1 * \frac{\text{AGMPU}}{\text{(1 + d)}^1} + \text{ …} + \text{R}^n * \frac{\text{AGMPU}}{\text{ (1 + d)}^n} + \text{ … = } \sum_{i=0}^∞ \text{R}^i * \frac{\text{AGMPU}}{\text{ (1 + d)}^i} \qquad \text{ (2) } \end{equation}

It means:

• In month 0 a customer brings us just AGMPU
• In month 1 a customer brings us discounted AGMPU with a probabiliy R
• In month 2 a customer brings us two times discounted AGMPU with a probabiliy R $*$ R (e.g. it was 95 % retained users in the month 1, then in month it will be 90.25 % of them)
• In month 3 …

Now look at the final formula (2) again. Wait, wait, wait. Something very familiar… Damn, this is geometric series! Common ratio is $\frac{\text{R}}{\text{ (1 + d)}}$, while AGMPU is the first term of the series. It means that we can use the following formula to calculate the total sum of (3):

\begin{equation} \text{ Customer LTV = } \frac{\text{AGMPU}}{1 -\frac{\text{R}}{\text{1 + d}}} \text{ = } \frac{\text{AGMPU} * \text{(1+d)}}{\text{1 + d} - \text{R}} \qquad \text{ (3) } \end{equation}

If we didn’t discount, we would get the following very common formula:

\begin{equation} \text{ Customer LTV = } \frac{AGMPU}{1-R} = \frac{AGMPU}{\text{churn rate}} \end{equation}

Manipulations above could be also explained using an exponential decay constant, but, for me, geometric series is the clearest way. However, if you start reading some articles about LTV, you could find that some authors just mention it without any explanation.

Let’s look at some example. Imagine that:

• the discount rate equals 2 % (US dicount rate currently)
• every month your company loses 5 % of your old customers, then a retention rate equals 95 %
• your average monthly gross margin per user equals $50 Then your customer LTV will be the following: \begin{equation} \text{ Customer LTV = } \frac{$50 * (1+0.02)}{(1 + 0.02) - 0.95} = \$728.57 \end{equation}

Other useful recommendations regarding this approach (3):

• Some people also multiply the calculated LTV by a factor (ex. 0.75), because a churn rate could be higher in reality.
• If you try to calculate LTV for SaaS, then look here. The logic is the same, you just need to make more assumptions.
• Use the above approach (3) if you have a lack of time or data.

## A less simple but more accurate approach

The point of more complicated approach is just to take the formula (1) as it is. Don’t assume that a retention rate and AGMPU are constants. The algorithm will be the following:

1. Calculate historical retention rates and AGMPU for cohorts (if you have historical data, otherwise, use the first approach above).
2. Calculate average historical retention rates and AGMPU weighted on cohorts’ sizes.
3. Fit statistical models for retention rates and AGMPU versus cohorts’ lifespan.
4. Predict retention rates and AGMPU for the future using created models (ideally, you will find such an exponential function for a retention rate that it will go to zero after some lifetime).
5. Calculate LTV using the formula (1)

Yes, it sounds a bit complicated, but really it’s not. I will try to explain this approach more precisely in my next post. Tschüss!

comments powered by Disqus
Griefberg © 2018