Talk:Annual percentage rate

Statutory EU formula - minor point
Would it not be better to use a different letter to represent the powers on the two side of this equation? For example, if the first advance occurs on a different date from the first repayment, then t1 takes on a different meaning on the lhs than on the rhs. Should one or other be changed to T or t', or something? —62.77.181.1 (talk) 12:41, 7 May 2008 (UTC)

Also, I suggest removing the assertion at the end of the list of terms below the formula that t1=0, which is often not the case. —62.77.181.1 (talk) 12:44, 7 May 2008 (UTC)


 * No, it should not have a different letter. tn must be identical on both sides for any instances where tk  and tl represent the same date.  If the dates of advance and repayment are different, you are merely saying that at that point, k &ne; l. ''t1 always equals zero by definition.  Interest does not start until some cash flow occurs.  Rossami (talk) 14:58, 7 May 2008 (UTC)
 * But your assertion does not make sense in the normal meaning of subscripted terms in mathematics. t2 on the left hand side refers to the date of the second advance; t2 on the right hand side refers to the date of the second repayment. Therefore it is clearly possible for t2 to be representing two different numbers in the same equation.  Take the example of a regular loan consisting of a single advance being repayed with three equal annual instalments, the first repayment being made one year after the advance.  In this case, you have t1 = 0 on the left hand side, and on the right hand side you have t1=1, t2=2 and t3=3. You're clearly then using t1 to represent two different things. On my second point, the EU directive defines the zero date for calculations as the date of the first advance. And, since this is not necessarily the same date as the date of the first repayment,  t1 as used on the left hand side is always 0 by definition, but t1 as used on the right hand side is not. If one were to use, say, Tk instead of tk on the right hand side, then the ambiguities would disappear, and one could then legitimately retain the assertion that tl=0.  —62.77.181.1 (talk) 14:22, 8 May 2008 (UTC)


 * That's the point. k=2 and l=2 may represent different dates but they're both on the same timeline, t.  The normal mathematical convention is to use the same variable with the two different subscripts. On your second point, you are missing the fact that this calculation can be run in either direction.  The APR calculation works the same whether you are calculating a standard loan (where the first advance would be before the first payment) or calculating in annuity with balloon payments (where you might well make payments before receiving a payout).  Rossami (talk) 14:55, 8 May 2008 (UTC)
 * I understand the calculations completely. I am unable to accept your assertion that a notation than leads to, for example, t2 taking on two different values in the same equation is "The normal mathematical convention" and am unable to find examples of this in a reputable source in any other context than in this formula. I stand by my position on this matter.  I also stand by my position that, if t1 is allowed to represent the date of the first repayment, then it is incorrect to assert that t1 always equals 0. I understand the different contexts in which the formula can be used, and perfectly comfortable with the possibity of negative values for tk on the right-hand side. I merely pointed out that the EU directive defined time=0 to be the date of the first advance. Anyway, I leave the matter to rest and will not attempt to edit the main page. You are clearly knowledgable about this topic; if I have failed to convince you and the other major contributors to this article of the merits of the points I raised, then I'm happy to just let it go. (I believe this to be a technical inaccuracy in any case and don't believe that it will lead anyone who actually uses the formula astray, since they will know what was intended anyway.) [Comment from 2008]


 * The formula used by the EU standard was actually amended to fix this issue in 2008/48/EC. (Strangely enough, the document is dated just two weeks before the comments above were added, now 11 years ago!) The updated formula now correctly uses different symbols for each side of the equation so that, for example, t3 represents the time of the 3rd drawdown, and  s3 represents the time of the 3rd repayment. I agree with the original comment regarding the ambiguity and confusion from the old formula using t3 to refer both to the time of the third drawdown and the time of the third repayment. It is even more curious when you consider index one: the old formula used t1 to refer both to the first drawdown time (by definition, zero) and the first repayment time (possibly non-zero, either positive or negative). Using a symbol for multiple meanings, especially within a single formula, is certainly not typical in typesetting conventions used in most fields of pure or applied mathematics, engineering or physics, and I presume not accounting/economics. Even in, for example, tensor notion used in relativity the use of different subscripts on a common symbol might represent different bases or coordinate systems, but the parent symbol (t) would still represent a single quantity (e.g. the coordinate independent quantity). Yet in the original EU formula from 1999 the symbol t was being used both for the drawdown time series (a vector, roughly speaking) as well as the repayment time series (a vector of different length). 82.22.66.201 (talk) 00:07, 31 October 2019 (UTC)

Tags
I've tagged this article with refimprove and confusing. The former tag is because the entire article is basically OR; there's only one reference given. All of the calculations, formulas, and so on are entirely unsourced. The confusing tag was added because I find the text of the page rather jumbled. To someone with little or no experience working with finance math, this article is obtuse. &mdash;  Hello Annyong  (say whaaat?!) 23:12, 3 June 2008 (UTC)

I agree. I've tackled the second section. I haven't added new references yet; I just reorganized what was there. Mebden (talk) 11:36, 12 July 2009 (UTC)

Effective interest rate, APR and APY
What is the relationship between effective interest rate, annual percentage rate and annual percentage yield? Finnancier (talk) 11:48, 23 August 2008 (UTC)


 * http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics#Annual_Percentage_Rate Sentriclecub (talk) 17:12, 15 September 2008 (UTC)


 * APR = (Per-period rate) X (Periods per year)
 * EAR = $$e^{APR}-1\,$$
 * APR = $$ln ( 1 + EPR ) \,$$
 * I'm going to look more into all this for you, and will have a full report back in an hour--it looks as though neither the APR nor EAR are independent of the number of compounding periods. The best way to interpret interest rates is with calculus.  It appears that the APR and EAR are dependent on the number of periods per year, which bothers me.  I'll re-read everything, and will also consult my HP-10Bii user manual, 2 feet away from me at all times.  While you are waiting, check out Khan academy on youtube, this guy is a bona fide expert.  The finance videos he has produced are wonderful. Sentriclecub (talk) 13:35, 15 September 2008 (UTC)

Disputed section
The following paragraph was added at the top of the Does not represent the total cost of borrowing subsection back in July. I do not think that it is appropriate in the encyclopedia for two reasons. I am removing the paragraph from the page pending discussion here. Rossami (talk) 23:31, 27 August 2008 (UTC) "APR used for mortgages lacks several factors such as length of time the borrower intends to have the loan and lender rebates that a lender may receive and not pass on to a borrower by selling a borrower a higher interest rate than they qualify for. Lender Police Effective Annual Rate accounts for these factors and gives the borrower a true account of the rate of interest the mortgage borrower will be paying during the time the borrower plans to have the loan."
 * 1) The first sentence is factually incorrect.  Mortgages do account for the length of the loan.  And while an advertised APR can not account for an individual borrower's intent to prepay a loan, the basic calculations can be used by any borrower based on his/her own intent.  That is not a failing of the concept of APR.  Furthermore, lender rebates which are not passed on to the borrower have no bearing on the borrower's rates.  Money out is money out.  The rate they "qualify for" is irrelevant to the calculation of what they will pay.
 * 2) The second sentence comes across more as a spam link than as proper encyclopedic content.  The Lender Police article was properly speedy-deleted back in July for a failure to meet even the minimum standards of an assertion of notability.  All sources about this organization that I can find are self-published (their own website or republication of press releases, for example).  The algorithm described is also not independently sourcable that I can find.  It certainly doesn't seem to have attracted any attention beyond the company itself.


 * I was planning on removing that section too. See below... Sentriclecub (talk) 15:44, 18 September 2008 (UTC)

Content Issues
I'm doing a clean-thru and over the next few weeks, will tackle a couple paragraphs at a time, and will delete unsourced stuff which is bad for the article (the stuff which I believe warrants the two tags at the top of the article that say this article may be confusing or unclear).

I have already tackled and finished everything above the Contents_Box. I plan to use a different formula than this one... $$p = \cfrac{P_0*r^{(n+1)}}{\sum_{l=1}^n r^l} $$ because its not the best one for APR, as it is not 100% relevant. There is a better equation which I plan to substitute.

The "Failings" dominates the article and is 50% of the article on my screen resolution. I'll condense it to 3 well written, clear paragraphs (I too believe strongly in the content, i'm not defending APR, I actually have an issue that APR depends on the number of periods e.g. daily, monthly, biyearly). Also i'll keep that table, as it makes some very good points.

I plan on making a section purely devoted to UK style APR, instead of merging the two articles (as they are now) and trying to make them explained every other sentence. They are so core different, that it's easier to explain the US version one time, then make a section on the UK version. (may sound unglobal, but you'll see it's one of the reasons why the article is bad, its like talking about the Tampa Bay bucaneers and the Tampa Bay Devil rays, all in the same article, one section at a time, and trying to make it "flow" well in style, which is impossible. Just because the UK APR and US APR differ by only one letter, they are about as dissimilar as the two different sports teams.

I plan on cutting down all stuff about other types of rates. Let them be explained on their own page (main article... template).

Style issues
Seems like many different editors took turns submitting paragraphs. Look at the top two paragraphs, how I idealized the style, and texture of the should-be article.

Length I think ideally the article should be much smaller than its current size.


 * "In the U.S., the calculation and disclosure of APR is governed by the Truth in Lending Act (also known as Regulation Z). In general, APR in the United States is expressed as the periodic interest rate times the number of compounding periods in a year (also known as the nominal interest rate); since the APR must include certain non-interest charges and fees, however, it requires more detailed calculation."

This section of the article does not belong at the bottom of the article, but rather towards the top or maybe I'll incorporate it into the intro. These are just after a 10 minute brainstorming. Ideally, I'm going to print the article, and use red-ink like marking up a poorly written essay by a classmate who procrastinated an assignment to write a wikipedia article and did it all the night before. Ideally guys and gals, I'm going to simulate that I'm on the tv show The Apprentice and I dont want to get fired, and my assignment is to fix this article. Collaborators, please drop me a note on my talk page (I work very slow, if left to do this alone) Sentriclecub (talk) 16:05, 18 September 2008 (UTC)

Retrofit topic year headers/subpages
14-Nov-2008: I have added subheaders above as "Topics from 2004" (etc.) to emphasize the dates of topics in the talk-page. Older topics might still apply, but using the year headers helps to focus on more current issues as well. The topic-year boundaries were located by searching from bottom for the prior year#. Afterward, I dated/named unsigned comments & cut auto-signature comments. Then I added "Talk-page subpages" beside the TOC. -Wikid77 (talk) 13:44, 14 November 2008 (UTC)

Archived 2004 to mid-2006
14-Nov-2008: I have archived 7 older topics (49%) into new archive page "Talk:Annual percentage rate/Archive_prior_talk" listed under "Talk-page subpages" at top (near TOC). -Wikid77 (talk) 15:49, 14 November 2008 (UTC)

Added nominal APR & effective APR
14-Nov-2008: I am adding the main legal terms "nominal APR" and "effective APR" (used over 20 years in USA) into the article. I was very confused reading the article, due to lack of those 2 terms, which clearly separate nominal (simple-interest rate) from effective (compound-interest+fees), according to laws in the U.S. Without those terms, the article can seem very confusing for readers using USA meanings. -Wikid77 (talk) 15:48, 14 Nov 2008

Money factor
Due to its AfD resulting in a not-delete, I've merged money factor into this article. It's a very rudimentary merge (essentially a cut-and-paste, and some minor changes) so any help on cleaning up this section would be amazing. flaminglawyerc 01:51, 15 January 2009 (UTC)

Doesn't include negative points
My experience has been that if you offset costs with negative points, taking a higher interest rate, the APR still includes the standard closing costs even though those costs are being paid by the lender. One would expect that if I had no closing costs at all, the APR would be equal to the interest rate (absent prepaid interest for partial months, or the like), but I noticed that the difference between APR and interest rate was the same as between an interest rate at zero points and its APR. Ex. 3% rate, $2200 cost, 3.125 APR | 3.5% rate, $300 cost, 3.625 APR. Is this just another instance of how APR fails to compare rates correctly, and should that be included in the article (or is it already and I'm just not seeing it?) 99.186.225.51 (talk) 18:08, 9 January 2012 (UTC)

Math in Nominal APR Does not...
Just reading over this I am not sure if the math is accurate, unless it is an issue with windows calculator and trailing decimals. Using the 'common credit card quotes at 12.99% APR compounded monthly, the one year EAR is ((1 + 0.129949/12)^12) - 1, or 14.7975%. I get .1379750 or 13.7975%. Then for a 42.99% APR compounded daily, using the above formula replacing the changed values of ((1 + .429949/365)^365)-1 I get .53679 or 53.67%, a far cry from the 13.87% listed there and counter to it's original purpose of showing an increased APR % with different compounded period of having a higher 'true cost'. I don't believe the issue is with my calculator as I do come up with the same values for the 29.99% compounded monthly. ~Edit forgot to sign 65.51.196.34 (talk) 19:01, 15 February 2012 (UTC)2/15/2012 David — Preceding unsigned comment added by 65.51.196.34 (talk) 18:57, 15 February 2012 (UTC)
 * It should all tie out now. About 2 weeks ago, anonymous user:87.41.193.88 vandalized the page, changing many of the numbers on this page.  The vandalism was overlooked until you asked your question.  Thank you for helping to keep Wikipedia honest.  Rossami (talk) 23:19, 15 February 2012 (UTC)

Dependance on loan period
I am having trouble wrapping my head around the statement:
 * If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated.

This seems like the inverse. If I have a loan for N periods, and I pay it off after only a single period, I only pay interest for one period. For example if we use the example in the picture at the top of the page; if I have a loan for $100, with $10 in fees and pay it off in a single month, you will pay the $100 loan amount, the $10 fee, and $5.50 in interest (5% of the balance). This leads to a payment of $115.50 or $15.50 more than the loan amount. Using the equation from the image, Effective APR = (interest + fees) / loanAmt ==> Effective APR = 15.50/100 = 0.155 = 15.5% which is much less then the 49% shown in the image. If the formula in the box is correct, than this make the statement about rate being higher being incorrect.

Overall this page feels poorly written, and contradictory and could use the help of an expert, or at least someone with a decent text book. — Preceding unsigned comment added by 140.252.14.13 (talk) 22:46, 27 February 2012 (UTC)
 * The 15.5% interest you calculated above is a monthly interest (since you are paying off the loan in a single month). To annualize that rate, you must multiply by 12 for an effective annual rate of 186%.  Rossami (talk) 02:27, 28 February 2012 (UTC)

Quality of this Article
I find the overall quality of this article on APR to be very low. The main issues are (1) it is predominately written from the perspective of US residential real estate lending, and this unnecessarily complicates the whole article and (2) it is just very porrly written.

It is possible, and it should be done, to define APR ignoring fees as just the annual simle interest rate. This is the way it is normally defined in basic finance textbooks. Then, if you want to include fees into the caculation then this should be done specifically in a discussion of how APR is caclualted for residential mortgages in the US, which also involves a discussion of the regulation. — Preceding unsigned comment added by 68.51.78.9 (talk) 16:55, 23 March 2012 (UTC)
 * No, it is not possible because APR is not the same as the annual simple interest rate. US residential real estate loans are one example but the concept applies to any loan with non-unitary duration and fees.  Having said that, if you think you can improve it, be bold and make some edits.  Rossami (talk) 20:54, 23 March 2012 (UTC)


 * You are right, 68.51.78.9. More work is needed. I renamed the "Failings" section which seems to speak about USA only (only that country is mentioned and none of that applies to EU) and moved it to the bottom together with the section on rate format which also seems to apply to some country only but who knows which. I also rearranged or removed some other confusing content and slightly updated the EU section. --Nemo 10:03, 28 September 2013 (UTC)

Separation of APR and APY
There is an article for annual percentage yield - equivalently, effective annual rate - which should be linked to in this article. Maybe, the two concepts APR and APY should be separated. The into is not clear. — Preceding unsigned comment added by 137.132.3.10 (talk) 02:23, 30 September 2012 (UTC)

The Loan Figure Payment is Incorrect
In the section "Multiple definitions of effective APR" there is the following example:

"For example, consider a $100 loan which must be repaid after one month, plus 5%, plus a $10 fee. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). Hence there are at least two possible "effective APRs": 80% and 435%. Laws vary as to whether fees must be included in APR calculations."

The math is not totally correct. If the $10 fee were considered as well, it would be 100 ( 1 + .05) ^ 12 which is about $179.59, minus the principal ($100), plus the $10 fee. So the amount paid would be about 89 dollars on top of the 100 borrowed. Therefore it would be 89% APR.

The statement "If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase)." is totally bogus and doesn't make any sense.

The figure is incorrect because it follows the example, so instead of 39 dollars of compound interest it should be 79 dollars + the 10 dollar loan fee. — Preceding unsigned comment added by 2601:9:2A80:1BE:A409:F133:E0E1:F6BB (talk) 22:37, 17 April 2014 (UTC)


 * In the figure, the value of the loan starts decreasing after the first month, i.e. it is not $100 for the whole year. --Petteri Aimonen (talk) 08:07, 19 April 2014 (UTC)

Misleading Confusion
It is clear that this term - the Wikipedia article and the comments outrageously evidence it itself even more - is in fact more confusing, misleading and deceitful than minimally helpful, as it is in general everything related to finance and the financial services industry. This indeed should be object of study (and of attention to Wikipedia) as well as all the legal loopholes, which always benefit (and lag behind) the corporations doing lucrative business from it. — Preceding unsigned comment added by 90.244.15.8 (talk) 13:50, 3 November 2014 (UTC)

Best APR on TV
Which is best for me as a borrower; APR 35.6%  APR 603%  APR 1264%  Papa.kirana@hotmail.com   82.43.161.235 (talk) 10:09, 21 April 2015 (UTC)  (82.43.161.235 (talk) 10:09, 21 April 2015 (UTC))

Wikidata
This article is linked to a Wikidata element with interwikis. Why does that have no effect on the article's page (no link to Wikidata, no interwikis)? --AVRS (talk) 08:37, 4 February 2016 (UTC)

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Alternate source for cited info?
In the section "Nominal APR dies not reflect...", the parenthetical note "(see ~credit card interest~ for the .000049 addition to the 12.99% APR)." doesn't actually explain anything, and the credit card interest page does not explain this percentage point discrepancy at all, as of today's date. Does anyone have an alternate, steady, source for this math? Trumblej1986 (talk) 19:01, 29 May 2019 (UTC)

Project loone
@Monica (my virtual assistant) 2600:6C40:6F0:A260:CDCE:7419:BB10:93FD (talk) 14:09, 11 July 2023 (UTC)