Katkat
Back

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

Capital OneData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

Last updated

A Capital One data-scientist technical screen on break-even and unit economics for a 30%-off Rent-a-Home promotion. You set up and solve the break-even equation for incremental new customers, size it against the 2M cardholder base, then re-solve under spend-uplift assumptions and explain why the required count nearly doubles.

Question

Capital One (C1) is evaluating a promotional offer on its Rent-a-Home (RH) product and wants you to size the unit economics.

Base assumptions

  • C1 has 2,000,000 active cardholders.
  • Currently 5% use Rent-a-Home (RH) annually.
  • Average RH spend per RH user per year is $500.
  • A proposed promo gives 30% off RH spend, fully funded by C1.
  • Average first-year net revenue per new cardholder (after non-RH rewards, servicing, and expected credit loss) is $550.
  • Ignore interchange on RH transactions for simplicity.

Let x = the number of new customers acquired because of the promo.

  1. Base break-even. Write the break-even equation where 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. Express x as a percentage of the 2,000,000 base, and discuss whether this target is realistic given typical acquisition funnels and approval rates.
  3. Spend-uplift extension. Now extend the scenario and assume that during the promo: (i) existing RH users increase RH spend by 20% (from $500 to $600); (ii) new customers who join because of the promo spend twice as much on RH as existing users during the promo 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:

550·x − 0.30·(1,000·x + 2,000,000·5%·600) = 0

  1. Compare and explain. Compare the new break-even x from part 3 to the original break-even from part 1, and explain intuitively why the required new-customer count changes despite the higher spend.