Compute unit economics and insurance break-even

Q: Compute unit economics and insurance break-even

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Question

Subscription unit economics and insurance math. Assume no discounting and ignore taxes unless stated. Part A (15-month term): A subscription service charges revenue = $40/month with the first 3 months free; variable service cost = $25/month; installation cost = $35/customer (one-time at start); marketing + overhead = $120/customer (all variable in this part). Contract term is fixed at 15 months, no churn. Compute net value per new customer (contribution margin minus acquisition/setup). Part B (18-month term): Same parameters except contract term is 18 months. Recompute net value and explain precisely why it changes. Part C (21-month term with churn and mixed costs): Contract term 21 months; first 3 months free; 10% of customers break the contract uniformly during the paid months and pay a $100 penalty immediately upon break; marketing becomes variable $20/customer; overhead becomes fixed at $1,000,000 per year; installation remains $35/customer; variable service cost remains $25/month. You acquire N customers this year. Derive a formula for the minimum N needed to break even company-wide and solve for N. Part D (curves): Qualitatively sketch or describe (1) demand vs. price and (2) total profit vs. price for this service. Identify where revenue is maximized vs. where profit is maximized and explain why they can differ. Part E (weather insurance break-even rate): An insurer sells a 12-month policy with premium paid upfront at $30/month; servicing cost = $3/month; if a covered failure occurs, the benefit paid is $8,000; regulatory expense = $4/quarter plus $300 conditional on paying a benefit. Ignore investment income/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.

Compute unit economics and insurance break-even

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