Credit Card Product Break-Even Analysis

Context

You are tasked with evaluating the financial viability of a new credit card product. The product's financial model includes both revenue streams and cost components. All values are assumed to be annual. Each active cardholder is expected to generate an average annual purchase volume, denoted as S.

Revenue Streams

• Annual Fee (A): A fixed fee charged per active cardholder.
• Interchange Revenue (i): A percentage i of the total cardholder purchase volume (S).

Cost Components

• Cashback (c): A percentage c paid out on the total cardholder purchase volume (S).
• Per-User Variable Cost (v): An optional variable cost incurred per active cardholder.
• Fixed Cost (F): A one-time or recurring fixed cost associated with launching and operating the product, which must be fully recovered.

Tasks

1. Derive the break-even equation for this credit card product and solve for N, the number of active cardholders required to cover the fixed costs F.
2. Explain how the break-even point (N) changes when either the cashback percentage (c) or the annual fee (A) is adjusted. Specify both the direction and magnitude of the change.
3. Given specific numerical inputs for A, i, c, S, v, and F, calculate the exact value of N.