A bank is evaluating a new credit card. Segments: A: 30,000 customers, capture rate 20%, avg annual spend per captured customer = $100,000. B: any population, capture rate 0% (ignore for revenue). C: 150,000 customers, capture rate 5%, avg annual spend per captured customer = $20,000. Assumptions: revenue = 3% of cardholder spend; default/charge-off cost = 1% of spend; acquisition cost = $200 per active card (one-time, paid at activation); fixed annual cost = $10,000,000. a) Compute total annual spend captured, gross revenue, costs, and profit. b) Name two additional costs you would include (e.g., rewards/cashback, servicing) and explain directionally how they change profit. c) A partnership adds 5,000 new customers, each spending $40,000/year; recompute annual profit. d) First year only, each new customer gets a $500 signup bonus paid at activation. Assume spend accrues uniformly, retention is indefinite, no discounting, and the $200 acquisition cost and $500 bonus are both paid immediately. At what time (years, 2 decimals) does the incremental $500 break even? Sketch the cumulative net profit curve over time and label breakeven. e) Derive the default rate r (as a function of parameters) that makes profit exactly zero.