Cash or installments?
A store offers a discount for paying now, or the same purchase in installments "with no interest". Enter both prices and see the interest the installments hide, and how the two compare once your money earns something in the meantime.
The purchase
R$
R$
R$
Your money in the meantime
%
- The installment plan (10x) embeds 1.96% interest a month over the cash price.
- Together the installments add up to R$ 2,000.00: R$ 200.00 more than the cash price, which is 11.1% more.
How the cash or installments comparison works
What the page compares
When a store offers one price to pay now and a higher one split into installments "with no interest", the difference between the two is interest that the price has hidden. The page does two calculations with that pair of prices. The first finds the rate the installments embed, as if the difference were a loan the store makes you. The second asks what would happen to your money if, instead of paying in cash, you kept it invested and paid each installment on its due date. Together, the two calculations say which way costs less, in today's money.
What you enter
- The cash price, discount included, and the down payment, if there is one. The down payment is what is paid today on top of the installments: if the first installment is paid at the purchase, enter it as the down payment and count only the rest.
- The number of installments and the amount of each. The calculation assumes they fall due month by month, the first one a month from now.
- How your money would earn in the meantime: a fixed rate, a percentage of an index (such as 100% of the CDI) or an index plus interest, the type of product when the country has a list, and whether the yield is taxed or exempt. If the index has no quote, the field stays empty and the page says there is no comparison, rather than comparing with a number nobody calculated.
How the calculation works
The embedded rate is the one on the contract check page, read backwards: the amount financed is the cash price minus the down payment, and the rate is the one that makes equal installments pay exactly that amount, month by month, with compound interest. Dividing the difference by the price and by the number of installments would give a smaller number, because with each installment paid the balance falls and interest is charged on an ever smaller amount.
For the yield, the page works out how much you need to set aside today to pay each installment on its due date. The investment earns, the tax on the yield comes from the country's table, the same one as on the investments page, and what is left has to be worth the installment. Because the tax falls with time, each installment has its own: in Brazil it is 22.5% up to 180 days and 20% from 181 to 360 days, so the first installments pay more tax than the last ones, and the page does not use an average term for all of them. The sum of those amounts, plus the down payment, is the cost of the installments in today's money, and that is what is compared with the cash price.
An example with numbers
A product costs R$ 1,800.00 in cash or 10 installments of R$ 200.00, which add up to R$ 2,000.00: a 10% cash discount. The money is in a fixed 12% a year deposit, taxed.
| Measure | Result |
|---|---|
| Total of the installments | R$ 2,000.00 |
| Difference from the cash price | R$ 200.00 |
| Rate embedded in the installments | 1.96% a month (26.27% a year) |
| Net return of the money over 10 months | 9.58% a year |
| Cost of the installments in today's money | R$ 1,919.22 |
| How much less paying in cash costs | R$ 119.22 |
| Cash price that ties | R$ 1,919.22 (a 4.04% discount) |
Paying in cash costs R$ 119.22 less because the installments embed 1.96% a month and the money, after tax, earns far less than that. With a smaller discount the answer changes: at R$ 1,950.00 in cash, a 2.5% discount, the embedded rate falls to 0.46% a month, below what the money earns, and the installments cost R$ 30.78 less. The turning point is the price of R$ 1,919.22, a 4.04% discount on the total of the installments: above it paying in cash costs less, below it the installments do. If the same money were in a tax-exempt investment at 12% a year, the cost of the installments would fall to R$ 1,899.47 and the turning point would rise to 5.03%: the more the money earns, the larger the cash discount has to be.
Mistakes to avoid
- Looking only at the discount as a percentage. Ten percent off seems small next to a 12% a year investment, but it is paid over ten months, and the embedded rate is the one worth comparing with the yield.
- Counting the gross yield. Tax reduces what the money earns, and it is the net yield that pays the installments.
- Entering a yield your money does not have. The calculation holds for the rate you typed: with the money sitting idle the rate is zero and the installments cost the total of the installments.
- Forgetting the down payment or a first installment paid at the purchase. They earn nothing and enter the cost of the installments as they are.
- Comparing with money you do not have. If paying in cash would take a loan, the comparison is between the embedded rate and the rate of that loan.
What the page does not cover
The page does not model the risk of the investment or the redemption period, the card limit, points, cashback, a price that changes with the payment method or late-payment interest. The yield is today's rate and the calculation keeps it until the last installment: a CDI that moves in the middle of the plan changes the answer. Outside Brazil the page has no table of tax on yield, so the rate you type is taken as it is and should be entered already net. It is a calculation with the assumptions it states, not a recommendation.
Where to go next
The guide on cash or installments explains why "no interest" almost always has interest hidden in it, and the guide on the total effective cost shows how the rate of one offer compares with another's. If the money is meant to decide between a debt and an investment, use the pay off or invest page; the methodology page describes how the calculations are made.