Break-Even for a Credit Card with Annual Fee, Interchange, and Cashback

Context

You are evaluating a new credit-card product. Revenue comes from:

• An annual fee per active cardholder (A), and
• Interchange revenue at rate i on cardholder purchase volume (S).

Costs include:

• Cashback paid at rate c on purchase volume, and
• Any other per-user variable cost (v, optional),
• Plus a fixed cost to launch/operate the product (F) that must be recovered.

Assume values are annual and that each active cardholder generates average annual purchase volume S.

Tasks

1. Derive the break-even equation and solve for the number of active cardholders N needed to cover fixed costs F.
2. State how the break-even point changes when the cashback percentage c or the annual fee A changes (direction and magnitude).
3. If specific inputs for A, i, c, S, v, and F are provided, compute N.

Hint: Set Total revenue − Total cost = 0 and solve for N.