A Mathematical Model for Calculating User Lifetime Value

Based on Retention Rates

Universal Business Problem Framework

How to quantify internet product ROI? How to determine budget allocation? How to predict user growth? How to optimize acquisition strategies? These seemingly unrelated questions share a common mathematical foundation.

ROI Quantification
Budget Planning
Growth Prediction
Strategy Optimization

Mathematical Foundation

Power function retention models with proven accuracy

Real-World Validation

Tested across WeChat, YouTube, Facebook, and more

Practical Implementation

From 365-day wait to 7-day prediction

Core Business Model for Internet Products

Mathematical foundations of digital business models

Mathematical Business Framework

$$\text{Profit} = \text{ROI} \times \text{Quantity}$$

ROI Decomposition

$$\text{ROI} = \frac{\text{Return}}{\text{Investment}}$$
$$\text{ROI} = \text{Return} - \text{Investment}$$

Internet Product ROI

$$\text{ROI} = \text{LTV} - \text{CAC}$$
$$\text{LTV} = \text{Lifetime Value}$$
$$\text{CAC} = \text{Customer Acquisition Cost}$$
$$\text{LTV} = \text{LT} \times \text{V}$$

Critical Question: How to calculate LTV without waiting 365 days?

This fundamental challenge drives the need for predictive mathematical models based on early retention data.

LT Calculation Begins with "Retention"

Lifetime estimation methodology: retention-based modeling

Retention Rate Definition

Retention rate: ratio of active users at time t to initial cohort size.

$$t\text{-day retention} = \frac{\text{Users still active on day t}}{\text{Users acquired on day 1}}$$
Next-day retention = Day 2 active users / Day 1 acquired users

Empirical Retention Dataset

Day 1
100%
Day 2
39%
Day 3
30%
Day 4
26%
Day 5
24%
Day 6
22%
Day 7
21%

Mathematical Derivation

Step 1: User Behavior Model

Unit cohort assumption: retention rate $$R_t$$ represents day t activity probability

Step 2: Expected Value Calculation

Expected lifetime across user lifecycle:

$$\sum[\text{Active Days}] = R_1 + R_2 + R_3 + ... + R_t + ...$$

Step 3: LT Derivation

Lifetime defined as expected activity duration:

$$LT = \sum[\text{Active Days}] = \sum_{t=1}^{\infty}{R_t}$$

Core Formula

$$LT_{\infty} = \sum_{t=1}^{\infty}{R_t}$$
Where $$R_t$$ represents day t retention rate

Numerical Example

7-day calculation:

LT₇ = 100% + 39% + 30% + 26% + 24% + 22% + 21% = 2.62 days

Function Fitting

Function Selection via Comparative Analysis

Linear Function

$$R(t) = a + bt$$

Problems:

  • • Inadequate in-sample fit
  • • Critical flaw: Predicted values may be negative
  • • Retention rate domain constraint: R(t) ≥ 0

Logarithmic Function

$$R(t) = a + b \ln(t)$$

Problems:

  • • Acceptable in-sample performance
  • • Extrapolation issues: Asymptotic negativity
  • • Model excluded from consideration

Polynomial Function

$$R(t) = at^2 + bt + c$$

Problems:

  • • Overfitting: Positive curvature beyond day 7
  • • Monotonic decay constraint violated
  • • Pattern recognition unreliable

Exponential Function

$$R(t) = ae^{bt}$$

Problems:

  • • In-sample underfitting
  • • Excessive decay rate: Day 15: 6.7% retention
  • • Empirical day 30: ~12% retention

Power Function

$$R(t) = at^b$$

Optimal characteristics:

  • • Perfect match predicted vs actual data
  • • Cross-platform universality
  • • Minimal prediction error

Power Function: Empirical Validation

Cross-platform prediction accuracy assessment

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%

Prediction Accuracy

  • • Error rate < 2% across platforms
  • • Facebook: 0.10% error
  • • YouTube, Momo: <1.5% error
  • • High overall prediction accuracy

Universality Verification

  • • Consistent across different product types
  • • Social, Video, E-commerce, Dating
  • • Universal mathematical pattern
  • • Robust model validation

Three User Acquisition Models

User Acquisition Strategy Analysis

Method 1: "Ignore the Future"

　 Day 1 Day 2 Day 3
Daily New Users 0 0 100
Daily Active Users 0 0 100
Advantages
  • • Minimal complexity
  • • Immediate scale achievement
  • • Emergency deployment compatible
Disadvantages
  • • user scale highly unstable，sharp decline starting from day 4
  • • Concentrated cost burden
  • • Unsustainable growth model

Method 2: "Ignore Cost"

　 Day 1 Day 2 Day 3
Daily New Users 333 0 0
Daily Active Users 333 133 100
Advantages
  • • Model-based precision
  • • Accurate target achievement
  • • Stable retention profile
Disadvantages
  • • Excessive acquisition cost
  • • Poor resource efficiency
  • • High capital concentration risk

Method 3: "Equal Acquisition"

　 Day 1 Day 2 Day 3
Daily New Users 59 59 59
Daily Active Users 59 83 100
Advantages
  • • Scale stability maintained
  • • Optimized cost efficiency
  • • Distributed investment model
  • • Sustainable growth framework
Considerations
  • • Data dependency requirement
  • • Implementation complexity
  • • Continuous execution required

Retention, LT and User Scale Growth

Equal Acquisition Model: Mathematical Analysis

Daily Equal Acquisition Model

Unit daily acquisition with retention Rt over 365 days:

Day 1 Day 2 Day 3 Day 4 Day 5 ... Day 363 Day 364 Day 365
Day 1 R1 R2 R3 R4 R5 ... R363 R364 R365
Day 2 - R1 R2 R3 R4 ... R362 R363 R364
Day 3 - - R1 R2 R3 ... R361 R362 R363
Day 4 - - - R1 R2 ... R360 R361 R362
... ... ... ... ... ... ... ... ... ...
Day 363 - - - - - ... R1 R2 R3
Day 364 - - - - - ... - R1 R2
Day 365 - - - - - ... - - R1

DAU-Lifecycle Relationship

$$DAU_{\infty} = LT_{\infty} \cdot DNU = \sum_{t=1}^{\infty} R_t \cdot DNU$$
$$DAU_{\infty}$$

Steady-state DAU

$$LT_{\infty}$$

Mean user lifetime

$$DNU$$

Daily acquisition volume

Calculation

Model Application: Business Problem Solving

Business Problem Statement

Problem Context: Given a product with a 365-day user lifecycle of 28 days, and a target of 1 million DAU after 365 days, what acquisition strategy should be used? What is the total acquisition cost?

The acquisition method is "Equal Daily Acquisition". The "acquisition cost" essentially asks how many new users need to be acquired to ensure that under equal daily acquisition, the DAU scale reaches 1 million on the final day.

Data Preparation

Retention Rate Data

Day 1:100%
Day 2:39%
Day 3:30%
Day 4:26%

Revenue Data

Average Daily Revenue:$0.05
User Segment:Mobile Games
Observation Period:7 Days

Mathematical Derivation Process

Key reasoning process and mathematical steps

Variable Definition

Total: Total acquisition count

M: Acquisition period

DNU: Daily new users

DAU_M: DAU on day M

LT_M: User lifecycle on day M

Core Formula Derivation

$$\because DAU_M = LT_M \cdot DNU$$
$$Total = M \cdot DNU$$

Basic relationship establishment

$$\therefore \frac{DAU_M}{Total} = \frac{LT_M \cdot DNU}{M \cdot DNU}$$
$$\therefore \frac{DAU_M}{Total} = \frac{LT_M}{M}$$

Substitution and simplification

$$\therefore Total = \frac{DAU_M \cdot M}{LT_M}$$

Final solution formula

Numerical Substitution and Results

Final numerical calculation results

DAU_M = 1,000,000

Acquisition Period M = 365 days

LT_M = 28 days

$$Total = \frac{1,000,000 \times 365}{28} = 13,035,714$$

Conclusion: Need to acquire 13,035,714 new users

13,035,714 ÷ 365 = 35,714, meaning acquiring 35,714 new users daily, with a user lifecycle of 28 days, DAU will reach 1 million after one year.

Business Applications

Practical Implementation of Mathematical Models

Predictive Modeling

Transform short-term retention data into long-term user lifecycle predictions, enabling proactive business planning and strategic decision-making.

Cost Optimization

Optimize Customer Acquisition Cost (CAC) and Lifetime Value (LTV) ratios through data-driven mathematical models and strategic resource allocation.

Growth Modeling

Build sustainable growth frameworks using mathematical retention models to forecast user acquisition needs and scale planning strategies.

Core Value Proposition

Mathematical retention modeling transforms retrospective lifecycle data into predictive metrics, reducing observation time from 365 to 7 days.

Time Efficiency

Accelerate decision-making with early insights from mathematical models

Strategic Precision

Make data-driven decisions with quantified user lifecycle predictions