[tfehring]

It’s still useful to convert to an annualized interest rate even if your “investment” period is less than a year, because annualized returns are the standard unit of measurement for investment returns.

For example, if you’re debating whether to use a 2% rewards credit card or to use BNPL and keep the difference in a savings or money market account for 90 days, the latter strategy only wins if that account is yielding more than ~8%. Or if you’re deciding whether to use BNPL to let you pay down other debt a little faster, it’s only worth it if the interest rate on that debt is above ~8%.

[RC_ITR]

Yes. But let’s break this into first principles. You get $200 rewards on a $10k purchase. In no world do you get $800 on $10k. You only get $800 on $40k because the 2% isn’t a rate in principle so 8% literally never happens.

[tfehring]

The opportunity cost of 2% credit card rewards isn't a rate, but you can convert it to a rate by dividing by the time period. If you forego $200 in rewards to keep $10k for an extra 3 months, you're paying $67 or 0.67% per month for that money. If you instead forego that same $200 in rewards but get to keep your money for 6 months, you're only paying $33 or 0.33% per month. Converting those monthly rates to annualized rates of ~8% and ~4% respectively is just a matter of convenience, the math still works out the same if you keep everything on a monthly basis instead.

[RC_ITR]

But I’m telling you annualizing something like that is not convenience, it’s misleading.

Does every cash purchase also have an infinite APR? Because there I’m paying $200 for nothing. If that’s not logical then where does it start being logical?

[pandaman]

It's pretty simple really. Consider two scenarios of buying a $100 item:

a) pay with a 2% reward credit card

b) pay with BNPL scheme with a 3 month term.

In the scenario a) you effectively paid $98 right away and in the scenario b) you pay $100 but after 90 days (let's say you don't have monthly payments for simplicity). For b) to be no more expensive than a) you need a way to make at least $2 from a $100 investment in 90 days. This means that whatever investment you make with the $100 has to yield 8% annually. Even though you withdraw money after 90 days.

If you have monthly payments you need even higher yield to compete with the 2% reward. E.g. if you pay $33.33 every month then you need to make 1% yield per month to collect $2 off your initial $100 in 3 months: you get 3 months of yield from the final $33.33 payment, 2 months of the previous payment and 1 month of the first payment for total 6 months yield from $33.33. That's 12% annualized yield.

[RC_ITR]

You’re not answering my question about cash.

[pandaman]

What is your question about cash? There is no APR on credit card purchase in this scenario. There is also no APR on cash purchases. If you pay cash you pay $100 and without delay so you just lose 2% discount from the credit card or any chance to make any interest from your money on BNPL.