# Annual Percentage Rate vs Annual Percentage Yield

##### This video explains how to determine the annual percentage rate (APR) charged by a payday loan company and how to calculate the annual percentage yield charged by a credit card company.

In this video we’re going to talk about how to calculate the annual percentage rate the apr and the apy the annual percentage yield so let’s do it through a math problem a credit card company charges 0.5 per month for any unpaid balance calculate the apr so the apr or the annual percentage rate that’s equal to the periodic rate which we’ll call capital r times

The number of periods per year which is n so for this particular credit card company they charge 0.5 percent per month so that is the periodic rate now in one year how many months are there we know that there’s 12 months per calendar year and so the annual percentage rate or the interest rate that they’re charging per year is going to be 0.5 percent times 12.

Half of 12 is six percent so the apr is six percent per year so that’s how you can calculate the apr in this particular example now let’s move on to part b what is the annual percentage yield to calculate the apy from the apr you could use this formula it’s going to be 1 plus r r is the interest rate the annual interest rate which is the annual percentage rate

Or the apr n is the number of times the interest is credited per year which in this example is 12 and then -1 now the difference between the apy and the apr the apy takes into account the effect of compounded interest whereas the apr it doesn’t take that into account the apr is more of a simple interest calculation so the annual percentage rate is six percent

As a decimal that’s point zero six he needs to divide this by a hundred and it’s compounded monthly so n is twelve we know that point six divided by twelve is this number 0.5 so this becomes 1 plus 0.05 0.6 divided by 12 is point five percent or it’s actually point zero zero five so one point zero zero five raised to the twelfth power minus one gives you this

Number point zero six one six seven eight now that’s the annual percentage yield as a decimal so what we need to do is we need to multiply this by a hundred to convert it into a percentage so the apy for part b is going to be if you round it approximately 6.168 percent so that’s how you can calculate the apy if you know the apr but looking at these two values

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It’s not that much different and the reason being is the interest rate is relatively low when the interest rate is high the apy can be significantly different than the apr when it’s low they’re approximately the same now let’s move on to part c what would be the apy if the credit card company charges 20 per year compounded monthly and let’s compare that to the

Answer part d if it’s compounded daily so the amount that they charge per year that’s the annual percentage rate is 20 so now let’s calculate the apy using this formula one plus r over n raised to the n minus one so first let’s convert twenty percent to a decimal if we divide that by a hundred that’ll give us an r value of point twenty now compounded monthly so

The interest is going to be charge each month and it’s 12 months in a year so n is 12. one plus point 20 divided by 12 that’s 1.016 repeating and then if you raise that to the 12th power and then subtract it by one you’re going to get .2194 so the apy is approximately 21.94 so notice that this difference is more significant than the last example now that

The interest is higher the apr and the apy they’re not very close to each other the apy is significantly higher than the apr almost 22 percent compared to twenty percent now let’s see what happens if interest is compounded daily so we’re going to use the same formula r is going to be .20 but if it’s compounded daily what is the value of n well how many days are

In a year we know that there’s 365 days per year so that’s going to be our end value one plus point 20 divided by 365 that’s 1.000 five four seven nine four five two if you raise that to the three sixty fifth power and then subtract it by one you’re going to get point two two one three three five eight five so now let’s take that value let’s multiply by a

Hundred percent so this is going to give us an annual percentage yield of 22.13 so this value is not that much different than 21.94 but nevertheless it’s slightly higher thus as the n value increases the annual percentage yield goes up due to the effect of compounding so the interest that’s being charged it’s being charged not only on the principal but on the

Previous accumulated interest and so if a credit card company is charging you interest daily you’re going to pay more than if they were to charge you interest monthly that’s what you want to take from the less than part c versus part d as you can see the annual percentage yield will be 21.94 if the interest is charged on a monthly basis but if it’s charged in a

Daily basis it’s going to be slightly higher 22.13 but in both cases the apy is still significantly higher than the apr and that’s we want to take into account if the apr is relatively high the apy will be even higher if the apr is relatively low the apy will be slightly higher but not that much different but for the most part the apy is typically higher than

The apr now let’s move on to number two payday loan company charges 10 for a 1 000 loan plus a 20 processing fee which must be repaid in a 14 day period calculate the apr for this type of loan the annual percentage rate is going to be equal to the periodic rate times the number of periods per year well how do we calculate the periodic rate in this case you can

Think of it as basically the earning that this company is making for their principal so they’re making money out of the interest that they’re charging which we’ll call i plus the fees that they’re charging which we’ll call f divided by the principal which we’ll call p so that’s going to make up the periodic rate generated by this company and then we’re going

To multiply that by the number of periods per year well each period represents 14 days so how many 14-day periods are in one year to calculate that it’s going to be 365 days divided by in this case we’ll call n where n is the number of days in a period which is 14. so capital n in this example is 14. and then finally we need to multiply this by 100 to turn

The answer from a percentage i mean from a decimal to a percent but we’ll do that later but for now know that capital n is 14. lowercase n is 365 over 14. 14 days is approximately two weeks and we know that there’s 52 weeks in a year so there’s approximately 26 periods per year in this example if you take 365 divided by 14 you get 26.071 because there’s really 52

Weeks and one day in a year 52 times 7 that’s 364. and so that’s why the answer is not exactly 26. it’s a little bit higher because of that extra one day in the year but now let’s go ahead and plug this information in so this particular company charges a ten percent interest on a one thousand dollar loan so the principal is a thousand ten percent of a thousand

Is a hundred so they’re charging a hundred dollars in interest for a one thousand dollar loan plus a twenty dollar processing fee and there’s 14 days four so this 10 percent will be charged in the 14 day period so they’re making 120 out of every 1 000 that they lend so that is a periodic rate of 0.12 or 12 percent and if they do this every two weeks they can

Do this 26. point zero seven one times in a year so for a thousand dollars they can collect 120 dollars every two weeks you times that by 26 can collect 3 120 in a year just by using a thousand dollars so that’s more than a 300 return let’s get the exact figure so if we multiply 0.12 by 26.071 the apr is 3.1285 now if we multiply that by 100 the apr in this

Example is 312.85 percent which is pretty huge but when dealing with payday loans not everyone is going to pay back the loan and so this huge interest rate makes up for the risk of not getting that money back because some people may take this loan and just not pay back and so that’s the risk that this company has to take when offering payday loans so to make up

For that huge risk they charge a huge annual percentage rate

Transcribed from video
Annual Percentage Rate vs Annual Percentage Yield By The Organic Chemistry Tutor