# APR and EAR Differences and Calculation (Intermediate Accounting I #7)

So how many of you guys out there own credit cards? I’m guessing many of you do and you’ve probably received a credit card statement recently and like always the interest rate is reported usually hovers around the rate of 20% which is pretty much industry standard and my question to you is if we have a rate of 20% does this mean that we’re paying 20% of interest a year? And surprisingly the answer is not always. and it has to do with the difference between APR and the term EAR. So APR is actually the standardized term reported on every credit card statement and it stands for the annual percentage rate And the annual percentage rate reflects the amount of simple interest that we owe. And the APR can be calculated by writing the percentage rate per period multipled by the number of periods. So for example If we had a rate of 4% a month Of course we have 12 months then our APR would be 48% If we had a 1% rate/day then we’ll have an APR rate of 365% year. Just simple multiplication. Now the difference between these two terms besides what they actually stand for, which is Effective Annual Rate is that EAR reflects compounding or the amount of compound interest So EAR can be calculated as the interest rate per period but instead of multiplying it by the number of periods it’s going to be to the power of the number of periods. and that way we will get an exponential rate because compound interest is exponential in nature Now, APR is what is always displayed on credit card statements where it gets kind of stick is that if we have interest on interest or compound interest and then we’re trying to calculate it as APR if we have interest on interest and the APR is calulated Then it’s actually going to understate the interest rate that we owe. And that’s because it’s calculating it as simple interest when compound interest is definitely much more. so quick example This is a timeline and lets say that we borrowed \$1000. This is an example of simple interest So if we have the APR rate of 20% and making interest payments monthly, then our interest rate per month would be 1.67% *writing 12 marks for the months of the year* So we’re going to be paying \$16.70 of interest A month. If the APR was reported and it was 20% then that would be an accurate depiction of the interest we owe. Now if we have compound interest It changes using a time line, lets say that interest is accrued on a daily basis. Because for most credit cards it is. Then we’re going to take the APR, 20%, and divide it by 365 days. And that will give us a rate of 0.054794% and this can also be expressed as a decimal as 0.00054794 And we’re going to have 365 periods on our timeline So the amount of interest that we would owe along with interest.. Day 1 would be \$1000 multiplied by 1.00054794 And I’m adding a 1 because we have to pay back the principal of \$1000 but we also need to pay the interest. So the decimal 0.00054794 represents the interest part. So that would be the amount we owe after day 1 Day 2 would look similar instead of multiplying it by just 1.00054794, we’ll multiply it by 1.00054794 twice. So that it’s simpler, you can just say 1.00054794 to the power of 2 And you can see it would be the same up until day 365. so the rate would actually be 1.00054794 to the power of 365 days and this would give us an answer of 1.2213 what the 1.2213 stands for is that we owe principal \$1000 back which is represented by the 1. and the 2213 represents 22.13%, the EAR or effective annual rate. So then if we were to look at the APR in our contract The 20% will be much lower than the real 22.13% effective interest rate SO even if the rate was stated at 20% We would still be effectively paying 22.13% of interest and that’s if it is compound interest charged So sometimes APR can be misleading like we’ve seen. And now, why would they state the APR? Well of course 20% is much less than 22.13% And it’s more appealing to consumers so it’s a way to let lenders to hide the true rate if you’re having interest charged on a compounded basis. and the last thing I wanted to throw up a chart that gives you an idea of the differences between APR rates and their EAR rate whether they’re compounded daily, monthly, quarterly, or semi annually. The bolded amounts are APR rates, so if we had a rate of 1% APR if it was compounded semi annually it’d be 1.003% if it was compounded on a daily basis it would be 1.005% and you see it makes a larger difference if the interest rate is higher so if it’s 50% APR…compounded semi-annually it would be 56.25% as the EAR. which is considerably different if you have a large balance outstanding I’ll see you guys in the next tutorial where we’ll discuss notes receivable! 