Assess card rewards profitability and break-even spend

A bank is considering launching a new credit card that pays 1% rewards on all card spending. Per-customer assumptions: interchange is 2% of purchase volume; baseline average monthly spend is $S = 500$ dollars (treat $S$ as a variable when solving for break-even); average revolving balance is 2,000 dollars; net interest margin (APR minus cost of funds) is 10% per year on the revolving balance; operating expense is 50 dollars per year; loss rate is 3% per year on the average balance. Answer: 1) Calculate annual unit economics per customer—revenue, costs, and profit—at a 1% rewards rate when $S = 500$; should the bank issue the card? 2) Solve for the minimum monthly spend $S^*$ that makes expected annual profit equal zero under two cases: (a) average balance fixed at 2,000 dollars; (b) 20% of annual purchase volume becomes revolving balance for one year in addition to the 2,000-dollar baseline (that is, average balance equals $2{,}000 + 0.20 \times 12 \times S$). 3) Stress test: if interchange falls to 1.6% and the loss rate rises to 5%, recompute $S^*$ for cases (a) and (b). 4) In incremental analysis for a rewards change on an existing customer, when is it valid to ignore the outstanding balance term, and what shortcut does that imply for estimating break-even new spend? Overview: This question tests a data scientist's ability to perform unit-economics and break-even analysis, including modeling interchange revenue, rewards costs, interest income on revolving balances, credit losses, operating expenses, sensitivity/stress testing, and incremental impact assessment.