Cash or installments "with no interest": how to really compare

A store offers R$ 1,800.00 in cash or 10 installments of R$ 200.00 "with no interest". Which way costs less? The answer is not in the size of the discount but in a comparison: the rate the installments hide against what your money would earn, after tax, while it waits to pay each installment. The tax rules cited here are Brazil's.

"No interest" almost never means no interest

When the cash price is lower than the sum of the installments, the difference is the price of paying later, that is, interest, even if the store does not use the word. In the example the ten installments add up to R$ 2,000.00 and the cash price is R$ 1,800.00: whoever pays in installments pays R$ 200.00 more for the same product. The opposite case exists, where the price is the same either way. Then the installments really are interest-free, and the money that stays invested until the last installment only helps whoever pays in installments.

How to find the hidden rate

The calculation treats the difference as a loan. The amount financed is the cash price minus the down payment, in the example R$ 1,800.00, and the rate is the one that makes ten installments of R$ 200.00 pay exactly that amount, with compound interest. The result is 1.96% a month, or 26.27% a year, for a discount that looks modest: R$ 200.00 out of R$ 2,000.00, the 10% cash discount.

Dividing the difference by the price and by the number of installments would give a much smaller number, and it is a common mistake. With each installment paid the outstanding balance falls, and interest is charged on an ever smaller amount, so the true rate is higher than that simple division. The contract check page does the same calculation for an ordinary loan, and the guide on the total effective cost explains why the effective rate is the right number to compare offers.

What your money would do while it waits

Paying in cash means giving up the money today. Paying in installments means leaving it invested until the day of each installment. So the comparison is not between R$ 1,800.00 and R$ 2,000.00 but between R$ 1,800.00 and the amount you would need to set aside today to pay the ten installments on their due dates. That amount is less than R$ 2,000.00, because the money earns along the way, but how much less depends on what it earns.

With the money in a fixed 12% a year deposit, taxed, setting aside R$ 1,919.22 today pays the ten installments of R$ 200.00. That is the cost of the installments in today's money, and the comparison with the cash price looks like this:

The turning point is R$ 1,919.22, a 4.04% discount on the total of the installments. Above that discount paying in cash costs less; below it, the installments do. What decides it is the relation between the two rates: when the rate the installments embed is higher than the money's net yield, paying in cash costs less, and when it is lower, the installments cost less.

Tax moves the turning point

What pays the installments is the yield after tax, not the gross yield. On taxed fixed income, Brazil charges income tax by a regressive table, with rates that fall with the length of the investment, and so each installment has its own: the one due in a month is paid with money that earned little and paid the highest rate, the one due in ten months with money that earned more and paid less. The guide on paying off a debt or investing has the table as it stood when it was updated.

In the example, the 12% gross becomes a net yield of 9.58% a year over the ten months. If the same money were in a tax-exempt investment earning 12% a year, the cost of the installments would fall to R$ 1,899.47 and the turning point would rise to 5.03%. In other words, the more the money earns, the larger the cash discount has to be to pay off. Tax rules change: confirm the ones in force with the tax authority or with the institution that offers the product.

What the calculation does not see

A comparison of rates leaves out what is not a rate. Before deciding, also weigh:

• Liquidity and risk. The money set aside to pay the installments has to be available on their due dates, with no redemption penalty or loss of value. The calculation assumes the yield is today's until the last installment, and an index such as the CDI moves.
• Money that does not exist. If paying in cash would take a loan, the comparison becomes the rate embedded in the installments against the rate of that loan, and revolving credit card debt is usually much higher.
• Advantages the calculation ignores. Points, cashback, a price that changes with the payment method and the chance of negotiating the price when paying in cash are left out.
• The emergency reserve. Money that would serve as a reserve should not enter the calculation as if it were free.
• Inflation. Fixed installments lose real value over time, but the nominal yield of the money usually embeds the expected inflation. To separate the effect, see the guide on the real interest rate.

How Finasst compares

On the Cash or installments page, enter the cash price, the down payment if there is one, the number and amount of the installments, and say how your money earns: a fixed rate, a percentage of an index or an index plus interest, taxed or exempt. The result appears at once, with no account needed: the rate the installments embed, their cost in today's money, how much more or less paying in cash costs and the cash price that ties. The page only describes what the calculations show, with the assumptions it states, and does not recommend a way to pay.