What will this instalment plan really cost?

What a deposit and a monthly instalment come to against the cash price of the same thing, and the annual rate that difference is equivalent to. It reads one offer alone, and the price you type is what the plan is measured against, and it counts no late fee, admin charge or insurance, each of which raises the real rate.

2,000 down and 900 a month for 24 months comes to 23,600, which is 3,600 above the cash price of 20,000 and costs the same as borrowing it at 18.16% a year, against the cash price you entered.

Two figures, and they answer different questions. What the plan adds is a subtraction anybody can check: the deposit plus every instalment, less the cash price of the same thing. It is in money, it is the number that leaves your account, and it is the one at the top of the card.

The rate is the same difference expressed the way a loan would express it. The plan lends you what you did not pay up front, and you repay that amount in equal monthly instalments; the rate is the annual figure at which those instalments repay exactly that amount and no more. There is no closed form for it, so it is found by bisection — narrowing an interval until the two sides agree to within a billionth — which is the same solver the murabaha calculator uses on the same shape of problem.

The rate exists so that two offers can be read against one another when their lengths differ. The same money added to a price over one year and over four are not the same charge, and the money figure alone cannot say so: a longer plan almost always adds more in total while charging less per year. Neither figure is the right one on its own, which is why both are on the card.

Where the deposit covers the whole price there is nothing financed, so there is no rate to find, and the card prints nothing rather than a zero that would read as free credit. The same is true where the instalments never come to the amount financed at any rate at all.

what it adds = deposit + instalment × months − cash price

The rate is the annual figure at which those instalments repay the amount financed exactly, found by bisection.

A worked example

A cash price of 20,000, with 2,000 down and 900 a month for two years, comes to 23,600. That is 3,600 above the price, and it costs the same as borrowing the 18,000 financed at 18.16% a year.

What it assumes

• The cash price is one you could actually pay today for the same thing. Where a shop quotes one price for cash and another for the plan, the gap between them is part of what the plan costs and belongs in this figure.
• Every instalment is equal and paid on the day it is due. A plan with a larger final payment, or one carrying a late charge, is not the schedule the rate is solved against.
• The plan runs its full length. Settling early usually costs less than this and occasionally costs the same, and it is the agreement rather than the arithmetic that says which.
• The deposit is paid at the start and only what is left of the price is treated as financed. A plan whose deposit is really its first instalment is a different schedule and gives a higher rate.