The credit card APR: why this number is misleading

I have two principles in credit card management, one “hard” and one “soft.”  The hard principle, never to be violated under any circumstances, is:

“Never miss a payment”

Failing to follow the hard principle would damage my credit profile to an unimaginable extent and therefore is not allowed to happen.

The soft principle goes:

“Avoid carrying a balance except under special circumstances”

The soft principle, serves the purpose of avoiding wasting money on interest payments. From time to time there will be a credit card promotion with an introductory 0% period offer, and in such a situation carrying a balance until the end of the promotional period doesn’t do any harm unless your balance is so high that it implies bankruptcy risk.

Anyhoo, you should never have to pay credit card interest, for the simple reason that it is too expensive. The lowest credit card annual interest rate, or APR (Annual Percentage Rate), that I have seen, is 9.99%. The highest I’ve seen? Around 30%. And worse yet, these rates do not reflect the actual interest you get charged. This blog post explains why, by showing you the math used to compute the APR.

First of all, the APR on your card is often variable. Most commonly it is the prime rate plus some percentage, as explained in the terms coming in the same envelope as your card. Since the prime rate varies according to market conditions, so does the APR. It is therefore impossible to compute an exact annual rate. The APR quoted is actually just an estimate of the annual rate. It is the daily interest rate on a certain date multiplied by the number of days in the year, usually 365. How is the result of this calculation different from the actual annual rate, the Effective Annual Rate (EAR)?

The Effective Annual Rate, let’s call this R, is the annual interest rate effective as of the date picked for APR computation.

Further, let’s call the daily rate effective as of that day r. We assume that the daily interest rate for the next 364 days is invariable.

For simplicity, let’s assume you have a $1,000 balance on your credit card.

The way credit card interest accumulation works is, for each passing day your daily balance gets charged an amount equal to the balance multiply by the daily interest rate. So after the first day, your new balance will become: 1,000+1,000*r=1,000*(1+r)

This is your daily balance for the second. After the second, your new balance will be: 1,000*(1+r)*(1+r)=1,000*(1+r)^2

After these recursive calculations, after 365 days you end up with: 1,000*(1+r)^365

Remember we called the Effective Annual Rate R? Another way to calculate the balance after 365 days of interest accumulation is:  1,000+1,000*R=1,000*(1+R)

From the two calculations above, we have: 1,000*(1+r)^365=1,000*(1+R)

It then follows that: 1+R=(1+r)^365 , or R=(1+r)^365-1. This is the EAR.

The APR, on the other hand, is simply calculated as: APR=r*365 . That’s why the EAR is different from the APR.

To illustrate the magnitude of the difference, let’s assume the APR is quoted as 20%. Follow our calculations, the effective daily rate, r=20%/365=0.0548%.

The effective annual interest rate, R=(1+0.0548%)^365-1=22.1% – that’s more than 2% higher than the APR! If you carry the original $1,000 balance through the year, you’ll end up having to pay $221 in interest, not just $200 as the APR would imply! The bad gets worse.

The bottom line is, APR is ridiculously high, but the actual interest rate you pay is even higher than that. Therefore you should avoid paying credit card interest, by paying off the balance every month. This is the best way to control your spending and not fall into the credit debt spiral that has tainted the credit card market in the last few years.

As long as you follow both the “hard” principle and the “soft” principle, you only have things to gain from credit cards.