Calculate break-even new customers for a 30% Rent-a-Home discount

Question

Capital One (C1) is assessing a promotional offer for its Rent-a-Home (RH) product and asks you to work through the unit economics. Base assumptions

• C1's active cardholder count is 2,000,000.
• At present, 5% use Rent-a-Home (RH) annually.
• Annual RH spend averages $500 per RH user.
• The proposed promotion gives 30% off RH spend, with C1 covering the full cost.
• First-year net revenue per new cardholder, after non-RH rewards, servicing, and expected credit losses, averages $550.
• For simplicity, ignore interchange on RH transactions. Let x be the number of new customers acquired because of the promotion.

1. Base break-even. Set up the break-even equation in which incremental revenue from the new customers equals the promo cost across existing RH users plus new RH users, then solve for x. Show your work and final numbers.
2. Sizing and realism. State x as a share of the 2,000,000 base, and discuss whether that target is realistic given typical acquisition funnels and approval rates.
3. Spend-uplift extension. Extend the scenario now and assume that during the promotion: (i) existing RH users lift RH spend by 20% (from $500 to $600); (ii) new customers who join because of the promotion spend twice as much on RH as existing users during the promotion period (i.e., $1,000). The discount remains 30% on RH spend, and average first-year net revenue per new cardholder remains $550. Write and solve the break-even equation: $550x - 0.30\left(1000x + 2,000,000 \cdot 5\% \cdot 600\right) = 0$
4. Compare and explain. Compare the break-even x from part 3 with the original break-even from part 1, and explain intuitively why the required new-customer count changes even though spend is higher. Overview: A Capital One data-scientist technical screen covering break-even and unit economics for a 30%-off Rent-a-Home promotion. You build and solve the break-even equation for incremental new customers, express it relative to the 2M cardholder base, then re-solve with spend-uplift assumptions and explain why the required count nearly doubles.