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Compute profit and break-even for memberships

Capital OneData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This problem tests your ability to model unit economics for a subscription-based loyalty program and perform sensitivity analysis to determine the break-even point for a business intervention.

The Problem: Loyalty Membership Profitability

You are working for a bank that is launching a premium credit card loyalty membership. The program has an annual fee of $F = $150. A member costs the bank $C = $40 per year in administrative overhead and rewards processing. Additionally, the bank earns an average transaction-based profit (interchange fees) of $p = $12 for every $500 spent by a member.

Task

  1. Define a function profit(n_spend) that calculates the annual net profit per member based on their total annual spending n_spend (in units of $500).
  2. Calculate the 'break-even' annual spend: the minimum dollar amount a customer must spend per year for the bank to avoid losing money on that member.
  3. The marketing department proposes a 'Gold Tier' status that increases the administrative cost $C$ to $70, but increases the expected transaction profit $p$ by 20%. Calculate the new break-even point and determine if this tier is more or less profitable for a customer who spends $5,000 annually.

Constraints

  • n_spend represents the number of $500 blocks spent (e.g., $2,500 spend = 5 units).
  • Assume n_spend is a continuous variable.

Examples

  • If n_spend is 1 ($500), profit(1) = ($150 - $40) + ($12 * 1) = $122.
  • If a customer spends $0, the bank loses the difference between the fee and the overhead.

Edge Cases

  • What happens if the bank changes the membership fee to $0 (freemium model)?
  • Consider the scenario where the transaction profit $p$ is negative (e.g., extremely high rewards payout).

Hints

Hint 1

Express the total annual profit as a linear equation: Profit = (Fee - Cost) + (Profit per unit * number of units).

Hint 2

To find the break-even point, set the profit equation to zero and solve for the total dollar amount spent, noting that the variable 'n_spend' corresponds to (Total Spend / 500).

Hint 3

For the sensitivity analysis, recalculate the intercept (Fixed Cost) and the slope (Variable Profit per $500) with the new values provided for the Gold Tier.

Follow-up questions

  • How would you modify this model if the rewards cost was a percentage of the total spend rather than a fixed administrative cost?
  • If customer spend follows a log-normal distribution, how would you estimate the aggregate annual profit for a cohort of 10,000 members?
  • What metrics other than 'annual profit per member' would you track to determine if this loyalty program is successful?