Subscription Churn

Solution & Explanation

Problem Breakdown:

• Initial Churn Rate: 10% of customers who purchased subscriptions in January canceled before February.
• Churn Reduction: The churn rate decreases by 20% each subsequent month.
• Objective: Calculate the expected churn rate for March for all customers who subscribed since January 1st.

Step-by-Step Solution:

1. January to February:

   • Assume x customers subscribe in January.
   • 10% churn by February 1st:
     • Customers remaining by February 1st = $(1 - 0.1) \times x = 0.9x$.

2. February to March:

   • Another x customers subscribe in February.
   • Churn rate for January subscribers decreases by 20%, so new churn rate = $10\% \times (1 - 0.2) = 8\%$.
   • February churn:
     • For January subscribers: $0.9x \times 0.08 = 0.072x$ churned.
     • For February subscribers: $x \times 0.1 = 0.1x$ churned.

3. Total Customers by March 1st:

   • January subscribers remaining by March 1st = $0.9x - 0.072x = 0.828x$.
   • February subscribers remaining by March 1st = $x - 0.1x = 0.9x$.
   • Total customers = $0.828x + 0.9x = 1.728x$.

4. Churn Rate Calculation for March 1st:

   • Total churned customers by March 1st = $x - 0.828x + x - 0.9x = 0.172x$.
   • Expected churn rate for March 1st = $\frac{0.172x}{2x} = 0.086$ or 8.6%.

Explanation:

• Churn Reduction Impact: The 20% reduction in churn rate is applied to the original churn rate for January subscribers, resulting in an 8% churn rate applied to those who remained by February.
• Inclusion of New Subscribers: The calculation includes both the January and February cohorts, reflecting the churn dynamics for all customers who subscribed since January.
• Churn Rate Interpretation: The final churn rate of 8.6% reflects the proportion of customers lost relative to the total customer base by March 1st.

This solution methodically accounts for the dynamics of churn reduction and new customer acquisition, providing a comprehensive view of the expected churn rate by March 1st.