MODEL OVERVIEW

Estimating User Lifetime Value from Retention Data

A cohort-based approach to early lifecycle forecasting

MODEL EXPLAINED

One curve. Two planning questions.

Start with a cohort’s daily retention, then choose a finite planning horizon.

One retention curve, two planning questions Choose a finite planning horizon T, then sum daily retention through T to estimate active days. Multiply by value per active day for LTV, or by equal daily acquisition for expected DAU. HORIZON T Daily retention R₁, R₂, …, Rₜ Expected active days LTₜ = R₁ + R₂ + … + Rₜ Value per acquired user LTVₜ = LTₜ × daily value Active-user forecast DAUₜ = LTₜ × daily new users One retention curve, two planning questions Choose a finite planning horizon T, then sum daily retention through T to estimate active days. Multiply by value per active day for LTV, or by equal daily acquisition for expected DAU. HORIZON T Daily retention Who is active on each day? Expected active days Add retention through day T. Value per acquired user Multiply by value per active day. Active-user forecast Multiply by daily new users.
The value branch assumes a constant value per active day. The growth branch assumes equal daily acquisition, a shared retention curve and no pre-existing users.
UNIT ECONOMICS

Core Business Model for Digital Products

Connecting user value, acquisition cost, and acquisition volume

Simplified contribution model

$$\text{Net contribution}_T = N\,(LTV_T - CAC)$$
01

Revenue and acquisition spend

$$\text{Revenue}_T = N \cdot LTV_T$$
$$\text{Acquisition spend} = N \cdot CAC$$
02

Digital product unit economics

$$\text{Contribution per user}_T = LTV_T - CAC$$
$$LTV_T = LT_T \cdot V$$
LTV: Revenue per acquired user CAC: Acquisition Cost

Units and scope. Here, LTVₜ is undiscounted revenue per acquired user through day T; V is constant revenue per active user-day; N is acquired users. Subtracting CAC gives contribution after acquisition spend, before operating costs. This is a currency amount, not an ROI percentage or final profit. If daily value varies, use $LTV_T = \sum_{t=1}^{T} R_t V_t$.

Planning question: How can LTV be estimated before a full year of data is available?

A full-year observation period delays acquisition and budget decisions. Early retention data provides a basis for estimating the remaining lifecycle, subject to the model assumptions described below.

RETENTION AND LIFETIME

Estimating lifetime from retention

Expected active days derived from a cohort retention curve

Retention rate

The retention rate is the share of the original cohort that is active on day $t$, including users who return after an inactive day.

$$t\text{-day retention} = \frac{\text{Active Users on Day } t}{\text{Initial Acquired Cohort}}$$
MODEL EXPLAINED

Retention adds up to active days.

Each probability contributes a fraction of one expected active day.

Seven days of retention sum to 2.62 active days Daily retention is 100, 39, 30, 26, 24, 22 and 21 percent. Their sum is 2.62 expected active days per acquired user during the first seven days. Daily activity probability 0% 25% 50% 75% 100% 100% Day 1 39% Day 2 30% Day 3 26% Day 4 24% Day 5 22% Day 6 21% Day 7 Same cohort · first seven days · Day 1 is the acquisition day Seven days of retention sum to 2.62 active days Daily retention is 100, 39, 30, 26, 24, 22 and 21 percent. Their sum is 2.62 expected active days per acquired user during the first seven days. Daily activity probability 0% 25% 50% 75% 100% Day 1 100% Day 2 39% Day 3 30% Day 4 26% Day 5 24% Day 6 22% Day 7 21% Same cohort · first seven days
1 + 0.39 + 0.30 + 0.26 + 0.24 + 0.22 + 0.21 = 2.62 active days.
That is activity within seven calendar days, not a prediction that a user permanently leaves after 2.62 days.
$$LT_T = \sum_{t=1}^{T} R_t$$

Why summing works. For one acquired user, let Iₜ be 1 when they are active on day t and 0 otherwise. Since E[Iₜ] = Rₜ, linearity of expectation gives E[∑ Iₜ] = ∑ Rₜ. Activity on different days does not need to be independent. Day 1 is the acquisition day, with R₁ = 1 in this example.

From days to value. At USD 0.05 in revenue per active user-day, the seven-day revenue LTV is $LTV_7 = 2.62 \times 0.05 = 0.131$ dollars per acquired user. A 365-day estimate requires retention through day 365, including an explicitly modeled tail.

When is an infinite-horizon lifetime finite?

An infinite sum is valid only when the modeled tail is summable. For a positive power-law tail $R_t = a t^b$, this requires $b < -1$. The example fit below has $b = -0.3526$, so it does not imply a finite infinite-horizon lifetime or a finite steady-state DAU under constant acquisition. Use a finite horizon, or validate a different tail model. Daily activity retention is not necessarily a survival probability: users may return after an inactive day.

MODEL SELECTION

Comparing candidate retention curves

Fit and extrapolation behavior of five candidate functions

Five models on one shared scale

All five functions are fitted to the same six observations from cohort Days 2–7, then shown from Day 2 through Day 30 on a shared 0–100% retention scale.

  • Observed Days 2–7
  • Linear
  • Logarithmic
  • Polynomial
  • Exponential
  • Power · focal
Shared domain: cohort Day 2–30 and retention 0–100%. Lines outside the valid retention range are clipped at the plot boundary; the cards below explain where each tail becomes unsuitable.

Linear Function

Not suitable
$$R(t) = -0.0339t + 0.3917$$

Days 2–7 fit: $R^2 = 0.8688$

Limitation: Linear decay eventually predicts negative retention ($R(t) < 0$), outside the valid range for a retention rate.

Logarithmic Function

Not suitable
$$R(t) = -0.102\ln(t) + 0.3849$$

Days 2–7 fit: $R^2 = 0.9856$

Limitation: The fitted curve eventually crosses zero as $t \to \infty$, so it cannot support long-term extrapolation.

Polynomial Function

Poor extrapolation
$$R(t) = 0.0083t^2 - 0.0921t + 0.4693$$

Days 2–7 fit: $R^2 = 0.9802$

Limitation: The quadratic curve turns upward beyond Day 7, which conflicts with the assumed monotonic decline in retention.

Exponential Function

Underfit
$$R(t) = 0.4046e^{-0.1191t}$$

Days 2–7 fit: $R^2 = 0.9126$

Limitation: The six-point fit decays too quickly, predicting about 1.3% retention on Day 30 against an observed value near 12% in the comparison sample.

Power Function

Best sample fit
$$R(t) = 0.3925t^{-0.3526}$$

Days 2–7 fit: $R^2 = 0.9988$

Result: The power function has the highest in-sample fit ($R^2 = 0.9988$) and preserves a positive, slowly decaying tail.

Reading the fitted curves. The chart uses actual cohort-day labels. Internally, the equations use elapsed time since acquisition, so cohort Day 2 corresponds to $t = 1$. The six observed points cover Days 2–7; the acquisition-day value $R_1 = 1$ is separate. The shared axes support direct comparison, but a strong in-sample fit alone does not validate the long-term tail.

Cross-product comparison

Observed and power-function estimates of 30-day retention

Platform Predicted Actual Error Rate
WeChat 81.37% 84.96% -4.23%
YouTube 51.70% 51.43% +0.52%
Facebook 47.71% 47.66% +0.10%
Taobao 28.91% 28.50% +1.44%
Momo 12.82% 12.65% +1.34%

Error profile

Four of the five products have an absolute error below 1.5%; WeChat is the exception at 4.23%.

Product coverage

The comparison includes social, video, ecommerce, and dating products. Broader use requires validation on the target product's own cohorts.

ACQUISITION SCENARIOS

Comparing three acquisition schedules

The schedules differ in timing, spend concentration, and retained DAU

Model 01

Target-day acquisition

Acquire the full cohort on the target date

Metric Day 1 Day 2 Day 3
DNU 0 0 100
DAU 0 0 100
Advantage

Simple to execute and reaches the target on the specified day.

Disadvantage

Concentrates all spend on one day, while DAU declines immediately afterward.

Model 02

Front-loaded acquisition

Acquire users early to offset subsequent retention decay

Metric Day 1 Day 2 Day 3
DNU 333 0 0
DAU 333 133 100
Advantage

Reaches the Day 3 target under the stated retention assumptions.

Disadvantage

Requires 3.3 times the target volume in Day 1 acquisition.

Model 03 (planning baseline)

Equal daily acquisition

Distribute acquisition evenly across the planning period

Metric Day 1 Day 2 Day 3
DNU 59 59 59
DAU 59 83 100
Advantage

Produces a predictable daily budget and a gradual build in DAU.

Assumption

Assumes consistent daily execution and a retention curve that remains suitable for forecasting.

COHORT ACCUMULATION

DAU under equal daily acquisition

Each daily cohort contributes retained users to total DAU

MODEL EXPLAINED

Read across for retention. Down for DAU.

Acquire 100 users each day. Every row is a new cohort; every column is a calendar day.

Three cohorts contribute 169 active users on day three With 100 new users each day and age-specific retention of 100, 39 and 30 percent, day three has 30 active users from cohort one, 39 from cohort two and 100 from cohort three: 169 in total. Calendar day Day 1 Day 2 Day 3 Cohort 1 100 39 30 Cohort 2 — 100 39 Cohort 3 — — 100 Total DAU 100 139 169 — Not acquired yet · darker = more active users Three cohorts contribute 169 active users on day three With 100 new users each day and age-specific retention of 100, 39 and 30 percent, day three has 30 active users from cohort one, 39 from cohort two and 100 from cohort three: 169 in total. Calendar day Day 1 Day 2 Day 3 Cohort 1 100 39 30 Cohort 2 — 100 39 Cohort 3 — — 100 Total DAU 100 139 169 — Not acquired yet · darker = more active users
On Day 3: 30 + 39 + 100 = 169 expected active users.
Equivalently: 100 × (0.30 + 0.39 + 1) = 100 × LT₃. The same retention curve is shifted one day for each new cohort.
Expand the full 365-day relationship

365-Day Cohort Accumulation Matrix

Unit DNU Inflow
Cohort Day 1 Day 2 Day 3 Day 4 Day 5 ... Day 364 Day 365 DAU
Cohort 1 R₁ R₂ R₃ R₄ R₅ ... R₃₆₄ R₃₆₅
Cohort 2 - R₁ R₂ R₃ R₄ ... R₃₆₃ R₃₆₄
Cohort 3 - - R₁ R₂ R₃ ... R₃₆₂ R₃₆₃
Cohort 365 - - - - - ... - R₁
$$DAU_T = DNU \cdot \sum_{t=1}^{T} R_t = DNU \cdot LT_T$$

Assumptions. Acquisition starts on Day 1 with no existing users. DNU is constant, and every cohort follows the same retention curve by age. DAUₜ is the expected active-user count on calendar day T; LTₜ is expected active days per user through age T. This finite-horizon identity does not require a steady state.

What if daily acquisition changes?

Add each cohort’s contribution separately. If Aₛ users arrive on calendar day s, then:

$$DAU_T = \sum_{s=1}^{T} A_s R_{T-s+1}$$

If cohorts have different retention, use each cohort’s own age-specific curve as well. Existing users need an additional contribution term.

WORKED EXAMPLE

Acquisition volume for a target DAU

A worked example using a 365-day planning period

Scenario and planning target

Given: A mobile game with a 365-day expected active-day total ($LT_{365}$) of 28 days per acquired user, and a target to reach 1,000,000 DAU on Day 365 under equal daily acquisition.

Question: How many total users must be acquired ($Total$), and what is the required daily acquisition rate ($DNU$)?

Algebraic steps

Step 1: Fundamental identities
$$DAU_M = LT_M \cdot DNU \quad \text{and} \quad Total = M \cdot DNU$$
Step 2: Ratio substitution
$$\frac{DAU_M}{Total} = \frac{LT_M \cdot DNU}{M \cdot DNU} = \frac{LT_M}{M}$$
Step 3: Final acquisition formula
$$Total = \frac{DAU_M \cdot M}{LT_M}$$

Calculation and result

Substitute values ($DAU = 1M, M = 365, LT = 28$):
$$Total = \frac{1,000,000 \times 365}{28} \approx 13,035,714$$
Required acquisition rate
DNU ≈ 35,714 / day

Under the model assumptions, acquiring about 35,714 new users per day, or 13.04 million over the year, yields approximately 1 million DAU on Day 365. For a constant whole-user daily target that meets or exceeds the modeled goal, round up to 35,715. Acquisition spend equals total acquired users × CAC; CAC is not specified here.

DECISION SUPPORT

How the model supports planning

Using early retention observations to estimate lifecycle and acquisition needs

Lifecycle forecasting

Fit an early retention curve and use it to estimate active days beyond the observation window.

Budget planning

Translate a target DAU into daily and total acquisition requirements, then compare alternative spend schedules.

DAU forecasting

Estimate how new cohorts and retained users combine to produce DAU over the planning period.

Practical use and limitations

The power-function model produces an early lifecycle estimate from seven days of retention data. In this five-product comparison, four estimates are within 1.5% of observed 30-day retention and one differs by 4.23%. Results should be recalibrated as more cohort data becomes available.