What is the difference between APY and APR? How is interest actually computed in my savings account or on my loan?
posted by Presh | 16 August 2007
Whether you are financing a loan or getting a savings account, you need to know about interest rates. Banks talk about interest rates using the acronyms APR and APY, but what exactly do they mean? I’ll explain why banks conveniently quote one figure or the other. But first, I will discuss simple interest and compound interest which are what APR and APY really boil down to.
Simple Interest (no compounding)
The nominal interest rate is an annual rate quoted in percentages. The simple interest method does not consider the effects of compounding. This method calculates interest as the product of the original balance, the nominal interest rate, and the time period (in years).
Here’s an example. Consider a $5,000 savings account balance with a 12% nominal interest rate, using the simple interest method. In one year, your account would equal the interest payment of $5,000 x 12% = $600 plus your original balance of $5,000, for a grand total of $5,600.
Let’s suppose you could request to be paid in half of a year. In this case, your balance would be $5,000 x (1 + 12% x 0.5) = $5,300. As you can see, when you use the simple interest method, the yearly interest of $600 is exactly double the semi-annual interest of $300.
To generalize, simple interest can be computed once you know:
- the balance of the loan or savings account (P)
- the annual nominal interest rate (r)
- the time in years (t)
The general formula is:
Compounding Interest
Compound interest is calculated very much like simple interest, but it takes into account that the balance changes after each time interest is paid out.
Consider a $5,000 savings account with a 12% simple interest rate, interest is paid semi-annually, but this time use the compound interest method.
What is the account balance in one year? For the first half of the year, no interest has been paid, so the compound method is the same as the simple method. The balance grows to $5,300 = $5,000 x (1+12%/2) in half of a year.
Now comes the interesting part. For the second half of the year, interest is computed on top of the interest that’s already accumulated. Interest is therefore computed using a $5,300 balance (instead of a $5,000 balance in the simple interest case). This is what differentiates compound interest from simple interest. After one year, the ending balance is $5,618 = $5,300 x (1+12%/2)^2, which is slightly higher than the $5,600 from the simple interest method.
Compound interest uses the same variables as simple interest, but we also need to know the frequency of compounding:
- the balance of the loan or savings account (P)
- the annual nominal interest rate (r)
- the time in years (t)
- the frequency of compounding in one year (m)
The compound interest formula is:
The formula differs from simple interest in a few ways: (1) the nominal interest rate is expressed as an interest rate per m periods (so a 12% nominal rate is a 6% semi-annual rate) (2) interest is compounded on top of what’s already paid: the exponent takes care of compounding m times each year.
Annual Percentage Yield (APY)
The annual percentage yield (APY) is the interest yield you would get on a balance held for one year in a financial product, taking compounding into account.
In the previous example, a $5,000 savings account balance becomes $5,618 in one-year, so the APY is equal to $618 / $5,000 = 12.36%. Another way of saying this, is that a 12% nominal interest rate, compounded semi-annually, has an APY of 12.36%.
As you can see, the APY is a simplification of compound interest formula, where we consider an investment of one year (t=1) and an investment of one dollar (P=1).
Here is the formula for the APY of an investment where r is the annual interest rate and m is the number of compounding periods:
Example: ING Electric Orange Checking
ING advertises interest rates and APYs for its various checking accounts (see table below).
Let’s check that ING has made a correct conversion between APY and interest rates. Based on the August 16, 2007 rates, ING shows a 5.25% APY for a 5.13% interest rate (compounded monthly). Using the APY formula, we have:
Wonderful–ING did the math correctly.
Annual Percentage Rate (APR)
The annual percentage rate (APR) is the same as what I’ve been calling the nominal interest rate. It is the rate of interest in one year, without taking compounding into account. In the ING example, the 5.25% APY is equal to a 5.13% APR.
You can convert APR to APY once you know the frequency of compounding per year (m). Many credit cards, loans, and savings accounts are monthly (m=12):
Because of how they are defined, the APY is higher than the APR. This mathematical property is why banks like to quote both figures:
- For savings accounts, companies advertise the APY to inflate how high their rates are.
- For loans and credit cards, companies advertise the APR to conceal the true cost of the loan.
The APY is more realistic because it takes compounding into effect. For this reason, I typically ignore APRs and only consider APYs. The law says that the APY must be quoted by banks, though it usually appears in small print. I find it is easier just to calculate it myself.
Update: I’ve made a spreadsheet that converts APR and APY. Find the file in my Financial Tools.
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31 Responses to “What is the difference between APY and APR? How is interest actually computed in my savings account or on my loan?”
I don’t understand why in your simple interest example you calculated the interest earned using 12% in you equation for a nominal interest rate of 6%. Could you please explain this apparent discrepancy?
By Gerald J. Gaestel on Nov 30, 2007
@Gerarld J. Gaestel: Ack. The discrepancy is a typo–12% is the nominal rate I meant and I corrected the article.
I am learning how hard it is to write flawlessly. I’m suddenly more forgiving of error prone college textbooks I had to use.
Thank you for pointing this out to me.
By Presh Talwalkar on Dec 1, 2007
I read that you look for the APY over the APR. So if a bank is offering 1.00%APR on a traditional savings account and another bank is offering 3.55%APY on an online savings account, which would you turn to to invest your money. I was thinking the 3.55%APY, but I am new at this so I would like a more educated decision. I found the APY and APR convert spreadsheet to be helpful.
Thanks
By Malissa on Feb 5, 2008
@Malissa: Glad you found the spreadsheet helpful.
To answer your question, yes, the online savings account will give you a higher return to money. Most young people I know have online accounts like ING or Emigrant Direct instead of a traditional savings account.
If you are interested in an ING account, let me know and I’ll send you a link so you can get $25 when you open (if you open with $250).
By Presh Talwalkar on Feb 5, 2008
It’s like my grandfather used to say to me: “Sweetheart the secret to getting rich is in just 2 words: compound interest.”
Words of wisdome to live by.
By Amy Simpson on Mar 19, 2008
Please explain the steps and the equation to convert APY to APR. I’ve forgotten most of my algebra but I’ll give it a try
Thanks for your help
JB
By Jerry Browne on Apr 9, 2008
Jerry Browne: Good question. The answer is found by rearranging terms in the formula for APY I derived in the article.
Here is the derivation as an image; I hope it helps:
By Presh Talwalkar on Apr 10, 2008
Many thanks for the explanation.
I either never learned or have forgotten how to use such a small decimal exponent. I can’t make it work.
I can understand common exponents such as 2 or 3, etc.
Jerry B.
By Jerry Browne on Apr 15, 2008
Jerry Browne: Hmm…If you are primarily interested in computations, you can download my spreadsheet that converts between the two in an intuitive manner (enter APY and m to get an APR):
http://mindyourdecisions.com/blog/excel/APY_and_APR_Converter.xls
Otherwise, feel free to email me (presh@mindyourdecisions.com) the specific problem or more details, and I am more than willing to help out. I love this stuff
By Presh Talwalkar on Apr 15, 2008
At 83 lts of things are forgotten.
I want to write an algorithm in Visual Basic
so I can convert to APR.
I was successful in doing that for APR to APY.
I’m missing something but I don’t know what.
Thank you
Jerry B.
By Jerry Browne on Apr 16, 2008
At 83 lots of things are forgotten.
I want to write an algorithm in Visual Basic
so I can convert to APR.
I was successful in doing that for APR to APY.
I’m missing something but I don’t know what.
Thank you
Jerry B.
By Jerry Browne on Apr 16, 2008
Jerry Browne: Here’s are my two guesses:
1. Check the parentheses in your code. There are several so it’s easy to miss one.
2. Make sure APY is a decimal (0.05) or a percentage (5 percent) instead of a whole number (like 5)
3. Especially check the parentheses for the exponent. If you write ^1/m you are just dividing by m. What you want is ^(1/m) which raises to a fractional exponent.
Here is the formula again:
APR = m [(1+APY)^(1/m)-1]
Hope this fixes it.
By Presh Talwalkar on Apr 16, 2008
Sorry to be a pest.
I’m having trouble using the 1/m exponent
I divide 1 by 365 to get .002739
I subtract 1 from .002739 and get -.99726
I can’t figure out how to use the -.99726
Any way I try, I don’t get a usable figure form the resulting amount.
I’m using .0613 as the APY 1 + 5.13 yield on 5%
By Jerry Browne on Apr 17, 2008
Jerry Browne: I enjoy the questions–this is educational for all
I think I see the problem–it’s in your second step. The exponent is (1/m) not (1/m - 1).
I think you have APY = 0.0613 and m=365 (daily compounding). Here is how the formula would work:
APR = 365 [(1+0.0613)^(1/365)-1]
= 365 [(1.0613)^(.002739)-1]
= 365 [1.000163012-1]
= 365 [0.000163012]
= 0.0595
Let me know if you have more questions.
By Presh Talwalkar on Apr 17, 2008
Hi Presh,
First of Thanks for the explanation of APR and APY. I was looking for this exact question. I had one more query. Is AER ,which is advertised by Banks in UK like HSBC, Llyods TSB, same as APR? I am confused by their statement.
For example consider HSBC’s Regular Saver Savings Account (Bank Account Plus).It says “10% AER/gross fixed for 12 months”?
Source: http://www.hsbc.co.uk
By Nirav on Aug 13, 2008
Nirav:
Thanks. Per your question, AER stands for annual effective rate and it is the same thing as APY–annual percentage yield. Here’s my source on that:
http://moneyterms.co.uk/aer/
The fact the banks are advertising the AER further confirms it’s the APY. They want to advertise the larger number so savings appear greater. If you held 100 pounds in that 10% AER HSBC account for one year, you would get 110 pounds at the year end. This number takes monthly compounding into effect, hence it is a yield or an effective percentage.
By Presh Talwalkar on Aug 13, 2008
Hi,
This blog is a great one for people like me who are zero in this field.
Can you help me understand following case:
A bank is offering 6.01% Annual Percentage Yield (rate of 5.84%).
It says interest rate is calculated using Daily balance method, compounded daily and paid monthly.
In this case how do I find my earnings.
Can you please explain me this case,
Thank you,
Chintan
By Chintan on Aug 29, 2008
Thank you for your blog. I have found it extremely helpful. My question is: if I already have the APY, and the frequency of compounding in one year (m), and I want to calculate the annual nominal interest rate (r), could you please solve then equation of the APY for (r). In other words, I have the 5.25% APY and I would like to find out, if the interest is computed monthly, what is my interest rate?
Thanks again,
Milton Sambolín
By Milton E. Sambolín on Sep 1, 2008
Chintan:
If you simply deposit money at the start of a year and hold, then you’ll get a return of 6.01 percent–that’s what the APY is. In any given month you’ll earn a little less than half of a percent (interest compounds over the year).
But most of us regularly deposit or withdraw money. So the question is: what is the “base” for which interest is calculated? Your bank uses the “average daily balance” which is a weighted average of money. If you had $1,000 at the start of a month but withdrew it all halfway through, you would have an “average” balance of $500.
Here’s a good article on computing interest using the average daily balance:
http://www.streetauthority.com/terms/a/average-daily-balance-method.asp
By Presh Talwalkar on Sep 4, 2008
Milton E Sambolin:
The formula is
APR = m [(1+APY)^(1/m)-1]
Read my reply to Jerry Browne in the comments above for the derivation.
By Presh Talwalkar on Sep 4, 2008
Hi,
For customer deposits in US bank, its interest basis is 365 days.
For inter-bank money market, its interest basis is 360 days for US$.
How APY conversion rule to show out the days basis ?
Best Regards
Daisy
By Daisy Ares on Nov 10, 2008
Daisy:
This is a very good question. I *think* the APY will be the same, or at least very, very close regardless of the day count convention. My guess is that both terms are over 1 year, so it doesn’t matter if interest is based on a 365 day year or 360 day year. The difference would come if you withdraw money earlier than one year, in which case the day count would affect how much interest accumulated.
Here’s a good article on the pesky day count convention:
http://www.margill.com/Interest-calculation-White-paper.htm
But if you find a better answer, please do let us all know. Thanks.
By Presh Talwalkar on Nov 12, 2008
hey, thanks for the explanations and such, I have a question though, what exactly does this mean: “8.3% per annum compounding biweekly”, I was wondering is biweekly meant twice a week or once every two weeks.., this is for a CD account, so I’m not sure which it meant, many thanks ahead of time.
By Eric on Nov 30, 2008
This is a great question. I think people use biweekly to mean both every two weeks and twice a week! More often I hear it to mean every two weeks, but there are exceptions.
In either case, the relevant rate is the 8.3% per annum. On $1,000 held for a year, you would end up with $1,083 by the year end.
The only difference would be if you withdrew money earlier (but in a CD most people don’t). In that case, the difference in twice a week versus every two weeks would be reflected in how often interest is credited to the account.
If you really need to know, I would suggest calling the bank and asking.
By Presh Talwalkar on Dec 3, 2008
This site has been quite helpful, as I am considering a CD, and I made an Excel file using the formula to convert APR to APY. But, I am trying to actually understand the formula. Specifically, why is the 1 added and subtracted? Thanks.
By Robert on Dec 22, 2008
Good question Robert.
The 1 is there to account for the original money we invested. Let me go through an example to make this clear.
If we invested $100 and it grew by 5 percent, then we end up with:
($100 * 0.05) + $100 = $100 (1 + 0.05)
The 1 represents the $100 original investment. The factor 1.05 is the interest rate plus a factor to account for the original investment. To get back to the interest rate, we have to subtract the 1.
That’s the same idea for why the 1 appears in the APR and APY formula–it’s a way to isolate the interest portion.
By Presh Talwalkar on Dec 27, 2008
Thanks for the explanation. Of course, now it seems perfectly obvious!
By Robert on Jan 2, 2009
Hi,
How do we calculate the APR. Is there any standard formula to calculate APR. Please advise me a Official Website that explain the actual calculation for an APR.
Many Thanks
Shaun S
By Shaun on Jan 20, 2009
The APR is the nominal cost of a loan or investment product, and the method to calculate it is to use a discounted cash flow method or do an IRR calculation. I wrote an article about how to do this in Excel:
http://mindyourdecisions.com/blog/2008/11/13/calculating-the-rate-of-return-on-investments-roi-versus-irr/
Once you get the periodical IRR (which can be daily, monthly, etc.), you scale it to find the APR. For example, a 1 percent monthly IRR is a 12 percent APR (or a 12.7 percent APY).
I don’t know of any official resources on the web, so I suggest you borrow a financial management textbook…hope that helps get you in the right direction.
By Presh Talwalkar on Jan 22, 2009
some banks pay interest on certificate of deposits monthly, quarterly and semi-annually.If you let you interest add-on.Do you earn anymore interest by monthly rather than semi-annual?
By Janell Kimmel on May 11, 2009
Janell: The best way to find out is to calculate the annualized APY for the various compounding schedules.
In practice, I would guess deposits that roll-over for longer periods pay the most, as it entails the largest risk to the consumer and allows the bank to hold the money the longest.
By Presh Talwalkar on Jun 12, 2009