A bank considers issuing a new credit card with a 1% rewards rate on all card spend. Assumptions per customer: interchange = 2% of purchase volume; average monthly spend S = $500 baseline (you may treat S as a variable when solving for break‑even); average revolving balance = $2,000; net interest margin (APR minus cost of funds) = 10% annually on the revolving balance; operating expense = $50 per year; loss rate = 3% annually on the average balance. Answer: 1) Compute annual unit economics (revenue, costs, profit) per customer with rewards at 1% when S = $500; should the bank issue the card? 2) Solve for the minimum monthly spend S* that makes expected annual profit = 0 under two cases: (a) average balance fixed at $2,000; (b) 20% of annual purchase volume becomes revolving balance for one year in addition to the $2,000 baseline (i.e., avg balance = $2,000 + 0.20 × 12 × 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?