How to read lifetime value: review of methods

The issue of calculating lifetime value (also known as LTV, customer lifetime value, CLV) sooner or later rises before the developers of mobile (by the way, not only) applications. There are a lot of calculation methods, and there are as many opinions about how to calculate LTV. In this article, I decided to describe the most common methods, identify their pros and cons. These methods are suitable primarily for the description of the f2p model.


1. After the fact
This method stands out against all subsequent ones, since it does not model LTV and does not predict it, but considers the actual LTV.
For this method, you need to take a cohort of users who have already left the project, see how much money the whole cohort has brought, then divide this amount by the size of the cohort. It is desirable that users are registered at about the same time (in one month, and better - in one day).
In practice, this method is poorly applicable, since there will necessarily be at least one person from the cohort who is still active, no matter how long the cohort has been registered. Therefore, in practice, LTVs are precisely the ones that are being modeled, but not calculated in fact. And all subsequent methods will exactly model the future LTV, and not evaluate the past.
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2. Take everything and divide, or Sharikov method

The quickest, but roughest method. We take the entire income of the application for the period and divide by the total number of users for the same period.


Plus , this method has only one: it is considered quite quickly, literally in one action.

The minus is the obvious inaccuracy of the method, which may be due, for example, to the following reasons:

1. the income from those users who have already managed to become active (hit the denominator), but have not yet had time to bring revenue (which would have entered the numerator) is not taken into account;
2. values ​​of application metrics are included in the calculation from the very beginning of its life; Do not forget that applications have their life cycle, and as a rule, at the beginning of their life cycle, indicators are better than some time after (read about this excellent study from GameAnalytics ). In the same method, all stages of the application's life are combined.
3. Also in this method it is difficult to calculate LTV separately for each user segment, for this you need to know in advance the size of the segment and the amount of money brought by users of this segment.

3. Lifetime is simple
If we know how many days the user lives on average in the application, and how much money he brings on average per day of his life, then we can estimate how much money he will bring in his entire life in the application. And this is our LTV. The formula for this method is:

Then the question arises how to count the lifetime. There are two methods, and the first is the calculation in a simple way (as you might have already noticed from the title):
1) We define a certain period of inactivity, that is, the time after which the user most likely will not return to the application. This is determined either on the basis of retention values, or, more often, expertly. Usually expertly this value is set equal to one or two weeks.
2) Every day we look at users who have a period of inactivity on this particular day.
3) For each user, we calculate the number of days from his first visit to the current day.
4) Calculate the average value for all users. This is the lifetime.

Well, ARPU (in this case, ARPU = ARPDAU) is calculated as daily Revenue divided by DAU. Multiply the lifetime on ARPU and get LTV.

Advantages of the method:

1. Ease of calculations. Calculating the lifetime in this way is not difficult; it is even easier to calculate the ARPU. And any student can multiply one another.
2. You can expect LTV at least every day.
3. LTV can be calculated for each user segment separately.

The disadvantages are again inaccuracies, which in this case is due to the following reasons:

1. The value strongly depends on the period of inactivity, which is usually set expertly.
2. We multiply the average lifetime value by the average ARPU value, we get the accumulated error.
3. When calculating lifetime, we look at those users who have already left the application. When calculating the same ARPU, we look at the users of the current day. It turns out that the sets of users that form lifetime and ARPU do not overlap: a lifetime is considered according to the data of past days, ARPU is considered according to the current day.
4. Strong assumption of unchanged ARPU. We take ARPU in just one day and, based on it, we forecast LTV for many days ahead.

4. Lifetime is hard, or Bottoms Up
The second name of this method is taken from Wooga material , and this, you see, is the source to which you should listen. The formula of the method is exactly the same:

But the lifetime here is considered a bit more complicated and it turns out much more accurate. Recall what the retention schedule looks like:

The fact is that lifetime is the area of ​​the figure under the retention schedule, in other words, the integral of retention over time.
But before considering the integral, it is necessary to construct the function itself. How it's done:
1) As a rule, you have values ​​of retention rates for several days (for example, for 1 day, 7 days, 28 days). If there are other days, and even better - for longer periods of time - this is fine, it will make the calculations only more precisely.
2) Based on the known values ​​(say, for 1, 7, and 28 days), we need to build a retention curve. We will look for the equation of the curve of the form:

where t is the number of days from the first visit, F (t) is the future retention equation, and A, B and C are the model coefficients.
3) We substitute the known values ​​of retention, no matter how many, into the equation, and we obtain the system of equations for the coefficients A, B and C.
4) Calculate the sum of the squared differences in the deviations between the actual and simulated values ​​of F (t).
5) Find values ​​A, B and C that minimize the total deviation. This can be done perfectly, for example, using the Solver tool (Search for a solution) in MS Excel.
6) We substitute the found values ​​of A, B, C into the equation and get a function with which you can estimate retention for as many days as you like.
This is not all, but we are close. Then you can still choose a complex or simple method.
The difficult method is to find the integral of the retention function.
Recall that


The simple method is to divide the retention curve into segments depending on the lifetime value. For example, for users who have gone through the day, have lived in the application from 2 to 7 days, from 8 to 30 days, from 1 to 3 months, over 3 months. The more segments, the better. For each segment, calculate according to the retention table the percentage of users (segment weight) related to it, and then calculate the weighted average lifetime over all segments.

But whichever method you choose, you will be confronted with the question to what point is LTV considered (in the case of an integral, this will be the right edge of the integration area, in the case of a sum, the number of days in the most recent segment). And here again there are two methods of solution: simple and complex.
A simple method is that the right edge is set expertly. This usually happens like this:
- and let's take half a year!
- why?
- why not?
- well, let's six months.

The difficult method is to use discounting and find the WACC discount rate (admit you didn't expect to see financial math in this material?). The fact is that a thousand dollars now and a thousand dollars tomorrow are different amounts of money. Tomorrow's thousand dollars today will be equal to nine hundred dollars or so, depending on the choice of the discount rate.
The formula is as follows:

Here PV (present value) is the present value of future money,
CFi - the money you receive in i time periods,
WACC (weighted average cost of capital) is the discount rate.
How to find her? Usually, WACCs are made equal to the actual return on equity of the average for the firm. You can also equate it to the desired return on equity, or to the return on equity of alternative projects. If you do not understand this paragraph, ask your financiers, they probably know your company's WACC.
So, knowing the WACC, you will be able to discount future time flows, and therefore, choose at least infinity as the right edge of integration. The fact is that adding a WACC makes of your sum (or of your integral) an infinitely decreasing sequence, in which you can find a sum.
We assume that we have considered the lifetime. Now we consider ARPU (Revenue / DAU), multiply ARPU by the lifetime and get LTV.

Advantages of the method:

1. Accuracy. Lifetime is calculated very accurately, the error in it is minimal.
2. A side effect of the calculation of this method is that you also receive a bonus retention forecast for as many days.
3. Ability to calculate LTV for each segment separately.

Cons of the method:

1. Difficult to count (although an experienced analyst with all the data will consider you an LTV in five minutes).
2. Once again, the assumption that ARPU is unchanged over time. You can be a little safe and take into account not ARPU for one day, but average daily ARPU for a lifetime, this will increase accuracy.

5. Accumulative ARPU, or Top Down
The second name of the method is again taken from Wooga material , which gives +10 to trust in this method. The picture is taken from the same material:

We will explain. Suppose a group of new players came to you in the project, and you began to follow it. You measure how much money an average of one player from this group brought to you in 7 days, for 14, for 28, and so on. That is, in fact, you go from a regular ARPU to a cumulative in N days.
Well, knowing Cumulative ARPU for 7, 14, 28, etc. days, we will again be able to build a mathematical model of a curve that will predict the Cumulative ARPU values ​​for as many days. We will look for the equation of the curve of the form:

where t is the number of days from the first visit of the user, F (t) is the future equation, A and B are the coefficients of the model.
Again, we calculate the sum of squared deviations and minimize it by selecting the optimal values ​​of the coefficients A and B.
If you have more Cumulative ARPU values ​​(say, 60 and 90 days), then you can add additional terms like C * t or D / t to the equation, this can increase accuracy. Well, in general - there is no one equation, which is guaranteed to give the minimum deviation. Experiment with the look of the equation!
Through several iterations, you still get an equation that suits you. Now, substituting the value of t you need into this equation, you get a Cumulative ARPU (t), which in essence will be equal to LTV.
How to choose the value of t for calculating LTV?

1. First, you can take a lifetime.
2. Secondly, you can re-set it t expertly.
3. Third, you can go back to discounting and add the denominator to the resulting equation. In this case, sooner or later, an asymptotic value will appear on the graph (as in the picture above - approximately $ 3.7, above which LTV cannot be. Take this value.

So, we looked at five methods for calculating LTV, which, as you can see, are ranked from least accurate to most accurate. Choose the method you like, calculate your LTV and make the right decisions. And now the main rule of LTV: divide users into segments, and count LTV of each segment separately. This will give you both higher accuracy and more reasons to make the right decisions for your product.