The Stamp Card Is a Tiny Loan: Nine Coffees, One Rebate

Your wallet probably holds at least one loyalty card. Maybe it's the cardboard one from the café near the station: nine little circles, then a tenth that says free coffee. It feels like a small gift. It is not a gift. It is a tiny loan you make to the café, repaid in caffeine.

That isn't a complaint. Loyalty cards can be genuinely useful. But if you want to know whether to keep walking to the cheaper place, or switch to the cashback app, you need to turn the card into algebra. The good news: it's mostly one number—the effective price per coffee.

The card as a finite series

Let p be the price of a coffee. On a buy-nine-get-one-free card, you pay 9p and receive 10 coffees. The value is 10p, the spend is 9p, so the saving is p. As a percentage of what you would have spent without the card, that's p / 10p = 10%. Your average cost is 9p / 10 = 0.9p.

The free coffee is not a separate miracle. It is funded by the nine paid ones. Each paid coffee carries an extra p/9 of 'stamp value'. Nine of those make p. In series form: p/9 + p/9 + ... nine times = p. That's a finite series, not a very exciting one, but it tells the truth. You have already bought the tenth coffee in instalments.

Most stamp cards exclude the free drink from earning another stamp. Check the small print. If it does earn one, the maths becomes an actual geometric series: 9 paid coffees produce 9 stamps, which produce 1 free coffee; that free coffee produces 1 stamp, worth 1/9 of a coffee; that produces 1/9 of a stamp, and so on. Total coffees = 9 + 1 + 1/9 + 1/81 + ... = 9 + 1/(1 - 1/9) = 10.125. Effective price = 9p / 10.125 ≈ 0.889p, a discount of about 11.1%. Nice, but rare. Don't bank on it unless the card says so.

Cashback isn't always what it says either

Cashback apps often advertise '5% back'. If you simply get 5% of your spend returned as cash, the effective price is 0.95p—a 5% discount. But if the app lets you spend cashback and earn cashback on that spending, the rebate becomes a geometric series: 0.05p + 0.05²p + 0.05³p + ... = p × 0.05/(1 - 0.05) ≈ 0.0526p. That's 5.26%, not 5%. Small difference, but it's the same idea as compound interest, pointed at your flat white.

Straight percentage discounts are simpler. A 15% off deal makes the effective price 0.85p. No stamps, no app balance, no expiry date to track.

Put three offers on one line

Imagine three coffees on your high street. Café A charges £3.40 and gives a stamp card. Café B charges £3.55 but has a 15% off app deal. Café C charges £3.20 and gives 5% cashback, simple, no compounding. Which is cheapest per coffee?

Use effective prices:

• Café A: 0.9 × £3.40 = £3.06
• Café B: 0.85 × £3.55 = £3.0175
• Café C: 0.95 × £3.20 = £3.04

Café B wins by a whisker, then C, then A. The headline price at B is the highest, but the discount does the work. This is why comparing sticker prices is a trap.

The break-even formula is short. Suppose Café A has price p_A and a 10% effective stamp-card discount. Café B has price p_B and discount d. A is cheaper when 0.9p_A < (1 - d)p_B. Rearranged, B's price can rise to p_B = 0.9p_A / (1 - d) before A wins. With p_A = £3.40 and d = 15%, that's £3.06 / 0.85 = £3.60. So B can charge up to £3.60 and still match A. Above that, the stamp card pulls ahead.

The break-even you can't put on a card

There's a second break-even: the one in your head. A stamp card only pays 10% if you finish it. If you get to seven stamps and move house, lose the card, or switch to a new job, you've paid extra for nothing. The expected discount falls with every abandoned card. That's not algebra's fault; it's a behaviour tax.

So the practical rule is dull but sound. Use a stamp card only at a café you already visit often enough to complete cards. Compare its effective price with the alternatives. If the cheaper option is more than 10% cheaper after discounts, the stamp card is not worth the loyalty. If it's within 10%, convenience, queue length and whether the coffee is any good can decide.

Your tenth coffee isn't free. It's a rebate, paid in nine instalments, with a small risk of never arriving. Once you see it as a series, the high street gets a little clearer—and your coffee budget gets a little more honest.