Compute unit economics and insurance break-even

Part A (15-month term): A subscription service charges $40 per month, but the first three months are free; variable service cost is $25 per month; installation cost is $35 per customer, paid once at the start; marketing plus overhead is $120 per customer, all treated as variable in this part. The contract lasts exactly 15 months with no churn. Compute the net value per new customer, defined as contribution margin minus acquisition and setup costs.

Part B (18-month term): Everything is the same except the contract term is 18 months. Recompute the net value and explain exactly why it changes.

Part C (21-month term with churn and mixed costs): The contract term is 21 months; the first three months are free; 10% of customers break the contract uniformly over the paid months and pay a $100 penalty immediately upon breaking; marketing becomes variable at $20 per customer; overhead becomes fixed at $1,000,000 per year; installation stays at $35 per customer; variable service cost stays at $25 per month. You acquire $N$ customers this year. Derive a formula for the minimum $N$ required to break even company-wide, and solve for $N$.

Part D (curves): Qualitatively sketch or describe (1) demand versus price and (2) total profit versus price for this service. Identify where revenue is maximized versus where profit is maximized, and explain why those points can differ.

Part E (weather insurance break-even rate): An insurer sells a 12-month policy with the premium paid upfront at $30 per month; servicing cost is $3 per month; if a covered failure occurs, the benefit paid is $8,000; regulatory expense is $4 per quarter plus $300 conditional on paying a benefit. Ignore investment income and time value. Let $p$ be the probability of at least one claim in 12 months. Write and solve the equation for $p$ such that expected profit is zero.

Part F (MLE and interval): If 400 policies are observed for a year and 18 claims occur, compute the MLE for $p$ and a 95% Wald confidence interval; discuss when the Wald interval is unreliable and propose a better interval method.

Overview: This question tests a candidate's ability in unit economics, insurance break-even calculations, expected-value modeling, churn effects, pricing versus demand trade-offs, and basic statistical inference including MLE and confidence intervals.