# Talk:Compound interest

WikiProject Mathematics (Rated B-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 Mid Importance
Field: Basics
One of the 500 most frequently viewed mathematics articles.
B  This article has been rated as B-Class on the project's quality scale.
Top  This article has been rated as Top-importance on the project's importance scale.
WikiProject Economics (Rated B-class, High-importance)
This article is within the scope of WikiProject Economics, a collaborative effort to improve the coverage of Economics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
B  This article has been rated as B-Class on the project's quality scale.
High  This article has been rated as High-importance on the project's importance scale.

## Einstein : greatest mathematical discovery?

Even this (supposed) quote seems unlikely / unverified. See Urban Legends Reference Pages. FMalan 07:12, 7 December 2006 (UTC)

## Canadian mortgages do not compound semi-annually

As documented (on my user talk page in response to a question), Canadian mortgages use semi-annual compounding. This term is widely used in the industry, on the net, by Canadian banks, by the Crown corporation responsible for govt mortgage policy, etc. It seems preferable to use this terminology rather than saying "In Canada the interest charged (each month) is only slightly different but more complicated." Perhaps there is some coherent objection? --Gregalton 12:31, 20 January 2007 (UTC)

The statement that Canadian mortgages compound semi-annually, regardless of whether they are paid monthly or more frequently is wrong. After a discussion at [[1]], I understood now how he was making the error. The term "semi-annual compounding" is used to describe how the interest rate applied is calculated. But the mortgage itself is compounded monthly (weekly, etc). This is an important distinction. The mortgages compound monthly, and the calculation of the interest rate used in that monthly compounding is derived from an bi-annual rate.
If you look at two of his own references [2] and [3] you will in fact find the term "semi-annual compounding". But if read carefully it is clear they do NOT mean that the mortgage itself compounds semi-annually. The references are very clear that the mortgages compound with the same frequency as the payment structure. In one it says so directly. In another the parameters of the equations given show unequivocally that the compounding periods equal the payment periods. Retail Investor 00:14, 22 January 2007 (UTC)

Please do not delete the text in the discussion page to which you are responding. The issue still seems to boil down to some objection to the term semi-annual compounding. The term is widely used , widely documented and understood. The current text with constant reversion still reads terribly: "slightly different but more complicated". Why not simply use the standard terminology?--Gregalton 05:28, 22 January 2007 (UTC)

I deleted your comments to save you face after I reposted the explanation that shows that your references support me, not you. The issue never was "preferable to use....". Nor was the issue ever "some objection to the term ...". The issue is the same as this section heading (which you erased) "Canadian mortgages do NOT compound semi-annually". Your own references prove it. Retail Investor 22:02, 22 January 2007 (UTC)

The saving face support is not needed, thanks. I've left the original order for coherence. If you wish to insist on changing the heading from neutral to the current reading, so be it, although saying I erased a heading that you changed is a misrepresentation. I believe I now understand your objection, and the difference of opinion, but will have to respond later. --Gregalton 14:50, 23 January 2007 (UTC)

I have documented extensively that this term – semi-annual compounding – is used in Canada as the convention in mortgage lending. Since there has been no attempt to deny that the term is in broad use and widely accepted, it clearly meets the standards for use.

The objection appears to be that semi-annual compounding does not compound every six months when the payment schedule does not coincide. But the phrase is in use by virtually everyone: are they all wrong? No, because compounding periods can be converted easily.

The text that is referred to was evidently read selectively: it reads “With the exception of variable rate mortgages, all mortgages are compounded semi-annually, by law.” It then goes on to say “However, you make your interest payments monthly, so your mortgage lender needs to use a monthly rate based on an annual rate that is less than 6%. Why? Because this rate will get compounded monthly.” If interpreted literally, these two sentences would appear to contradict one another; to choose one of them as "proof" is disingenuous.

There is a semantic issue that can be debated: when does compounding occur? But this is irrelevant: as noted, any periodic compounding rate can be converted into any other. When payment periods do not coincide with the compounding period, it is easiest to calculate by converting one into the other. This is what the text referred to was intended to show (in response to questions about how to do the calculations).

The phrase “semi-annual compounding”, like an APR, is used to specify the rate exactly. Any actual compounding period can be used when administering a loan, as long as the effective rate is equivalent to the stated rate. If payments are monthly, it will likely be converted to a monthly compounding rate, if weekly, weekly, etc.

What appeared to be the position proposed was that semi-annual compounding is the same as monthly compounding. They are not, but the difference is not in the compounding (they can always be converted between them), but in the rate. This is how the term is used, it is standard, and it is widely understood.

As a compromise, I have specified this in a way that should be acceptable. "Canadian mortgages use interest rates based on semi-annual compounding; the actual compounding may be calculated in accordance with the payment schedule at an equivalent effective rate."

I have also modified the text to correct errors: as written now, it states that mortgage interest is quoted as simple interest. This is incorrect: simple interest is not compound interest. See dictionary.com for example: "interest payable only on the principal; interest that is not compounded." Mortgages are never quoted this way as far as I am aware (and indeed it would probably be illegal in many jurisdictions).--Gregalton 12:21, 26 January 2007 (UTC)

Current text seems a reasonable compromise. If the part about simple interest is not clear, let me know - I will show further references. As for the math, ran out of space in the tagline, but all I meant was that the formula as written is not in "formula form" and perhaps not clear to others - I know what is meant but did not edit it. Also apologise for an inadvertent revert that was there for a few hours, somehow system did not take; now fixed.--Gregalton 13:21, 30 January 2007 (UTC)

You have now reverted the text with the tagline comment "If you don't understand simple interest do not edit". I was trying to be polite; it seems you are making no effort to do so. I have documented that simple interest and compound interest are different. While you may disagree with me, please keep in mind that I have shown references to support my edit and tried to explain the difference. If you wish to respond substantively, I look forward to hearing your point of view rather than instructions not to edit based on your opinion that I do not understand. For background, please consider doing a simple google search on simple interest, and consult the article on annual percentage rate (the usual standard for disclosing interest rates in the US, for example, as well as other jurisdictions). The commonly understood difference between simple interest and compound interest may then be clear, and probably explained in those places better than I can do.--Gregalton 20:56, 30 January 2007 (UTC)

## Simple interest, nominal and other

There is now sufficient documentation and references provided to distinguish between simple interest, nominal and effective interest. Since no documentation supporting the other position has been provided, I will edit to reflect what credible sources say. Grateful provision of documentation when editing.--Gregalton 20:08, 8 February 2007 (UTC)

This article rambles a lot. Compound interest is basically interest on interest. Could you please just say this, give some examples, give some history, then exit the topic?—Preceding unsigned comment added by 65.78.214.134 (talkcontribs)

You can do this yourself! Wikipedia is a wiki — a reader-edited encyclopedia. See WP:WELCOME for information on how you can contribute. — Feezo (Talk) 01:41, 11 February 2007 (UTC)

## Question

Hope this is allowed. In the first section entitled Compound Interest, 2nd Par say's "Compound interest rates can be called variously Annual Percentage Yield, Annual Equivalent Rate, Effective Annual Rate, Effective Annual Interest, Effective Compound Interest." Is it saying that they are all the same thing?

Yes. People use all kinds of terms. But because interest is so misunderstood it becomes incumbent on YOU to clarify in each situation, exactly what measurement is being used. The best course of action is to ALWAYS get the future cash flows in writing, or get the calculation used in writing. Then, using the cash flows determine the interest rate yourself, so you know what calculation is used. Retail Investor 18:59, 13 February 2007 (UTC)

I've been struggling to build a spread sheet that compares APR & AER. So I can see how much money I'm not getting from the bank. Does it already exist? Thank you, Peter. PS Wikipedia is a wonderful thing.

I did not include the Annual Percentage Yield (APR) in the list of equivalents because I don't know enough about it. I think it is different. Certainly the Wiki page for APR makes no attempt to start from the annual compounding rate and make adjustments to it. The page talks about all kinds of other things. Retail Investor 18:59, 13 February 2007 (UTC)
Annual Equivalent Rate, Effective Annual Rate, and Effective Annual Interest are all the same (although there may be local specifics and translations of similar foreign terms may cause confusion): they are the rate re-stated to annual compounding, e.g. a bank loan that compounds daily will be higher when re-stated as effective rate.
APR is a term used in many places and differs in two ways: 1) Usually, "non-interest fees" (such as "points" in the U.S., sometimes known as front-end fees and other obligatory payments like use of the lender's appraiser) have to be included, but the specifics of which fees must be included can differ from place to place, and depending on how the contract is written; 2) the specifics of whether it has to be re-stated as annual compounding (EU generally) or if other compounding conventions can be used vary by jurisdiction (such as nominal interest rate, i.e. monthly or other periodic compounding, as in the U.S.). Hence APR may only be comparable within a country, and even then there may be ways to exclude certain fees in a way that may make comparison impossible. APR usually has some legal or regulatory definition specifying how it is calculated. So, an APR for a loan where there is a "qualifying fee" and the borrower has to pay legal costs will be higher than the advertised note rate. It may be impossible to know the APR before the final contract is signed.
Annual Effective Yield and Annual Percentage Yield correspond to the terms above respectively, but for deposits and investments. In other words, "rate" for loans to customers, "yield" for loans to banks/financial institutions (from customers). As noted, however, there is a lack of consistency from place to place, and context matters.
For example, a banker may talk about the yield on his mortgage portfolio - meaning he's thinking about his income from loans to customers. So in a more general sense, yield refers to the return you get, rate to the return you're giving someone else, but consistency between institutions and individuals for use of the terms is even worse.--Gregalton 20:32, 13 February 2007 (UTC)

## compound

Question: What does compound mean? The article says it's compounded by *something* but what is it? —The preceding unsigned comment was added by 67.173.12.250 (talk) 02:33, 27 April 2007 (UTC).

Skand swarup (talk) 13:20, 9 May 2008 (UTC)Compound interest is basically interest on interest.

## $i%$

There are several occurences of "$i%$", which surely isn't correct but should be just "$i$", right? —Bromskloss 12:59, 5 July 2007 (UTC)

## Exceptions Subsection

I cleaned up the wording of "continuously compounded" to say that it's the limit as the compounding period approaches zero. The previous wording was "where the maximum (bordering on infinite) frequency of compounding is used", which is technically imprecise and mysterious to the layman.

I also replaced "Continuous compounding allows for the use of certain mathematical approaches that are more elegant and easier to compute." with "Continuous compounding in pricing these instruments is a natural consequence of Ito Calculus, where derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time." Unfortunately, I don't think this really makes sense to the lay reader, but I'm doubtful that concepts like derivatives valuation can be easily explained. They require some investigation to really understand. The former statement, however, is not technically true; derivatives valuation is particularly unlike mathematics, where elegance is a goal. In fact, most exotic options don't have "closed form" solutions, and are modelled using algorithms instead (monte carlos for non-american options and lattice approaches for american options) being examples. Of course, Black Scholes works for european options, but even Black-Scholes can't be used once we stop believing in geometric brownian motion and move to GARCH models.

Jason 07:37, 7 September 2007 (UTC)

## The example just above "The Rule of 72"

The solution to this example is, to be blunt, utterly wrong. The only correct value in the answer is the "-1." I have no idea where the author of the solution obtained the figures, as they seem to have been plucked from the ether seemingly at random. Before I change the solution, I would like someone else to verify that the answer is indeed incorrect. Thanks a lot. Permarbor0 13:55, 12 September 2007 (UTC)

Just corrected it. If anyone has complaints with the edit, please post to my usertalk. Permarbor0 14:01, 12 September 2007 (UTC)

## Merger proposal

Should this page be merged into the general interest article? I am writing a new draft for the latter and it makes sense to me to include compound interest (along with simple interest) in with a discussion of the meaning of interest generally. If accepted, this article for compound interest should then be converted to a redirect to interest. It is already the case that simple interest redirects to interest.

JJMcVey 08:57, 13 September 2007 (UTC)

• Keep The subject of interest and the derivatives thereof is a vast topic and it can't possibly be reflected on one page. In my view the subject has so much depth that a category list should be created linking the relevant interest-related articles together. RichyBoy 22:37, 15 September 2007 (UTC)

## missing math

In example B: "The mathematics to find the 0.9853% is discussed at Time value of money,...." The problem is that this is not -- or at least not clearly -- discussed in time value of money. I think these formulae need to be included in this article. I prefer to use the euler number version of this formulae:

$e ^{ \frac {ln(1+i)} {n} } - 1$

So in this example, the formula would be: $e ^{ \frac {ln(1+.04)} {4} } - 1 = .00985341$

The other version of this formula is: $\sqrt[4]{1+.04}-1 = .00985341$

or genericly, $\sqrt[n]{1+i}-1$

Do you agree that this formula (either or both forms) needs to be in the article? 66.94.95.194 21:08, 18 September 2007 (UTC)

## worst kind of usury

Anyone know when the thoughts changed from "that's bad," to "this is a good thing"? When did the change in attitue shift? 192.44.136.113 (talk) 17:19, 5 December 2007 (UTC)

See usury. That article puts it at 15th century.--Gregalton (talk) 20:55, 5 December 2007 (UTC)

I question the guilder to Euro conversion. As of 2008 I get the Indians would have made only 579 billion Euros —Preceding unsigned comment added by 129.7.88.82 (talk) 02:04, 19 February 2008 (UTC)

THIS WEB PAGE WAS VERY HELPFUL TO ME!!!! —Preceding unsigned comment added by 75.73.206.76 (talk) 22:20, 25 March 2008 (UTC)

## Force of interest section

The definition seems to be poor, it does not give any reason why the accumulation $A(t, t+h)$ converges to e as $h \rarr 0^+$, e.g. compounding yearly, monthly, daily tends toward the force of interest.

## External Compound Interest Calculator with Addition

External Compound Interest Calculator with Addition —Preceding unsigned comment added by 216.119.215.193 (talk) 04:15, 11 October 2008 (UTC)

p=(1+r/100)n —Preceding unsigned comment added by 117.199.163.237 (talk) 11:00, 11 February 2009 (UTC)

## Interest rate vs APY as used by US banks

If you look at the interest rate disclosures in the US, you see the interest rate and APY, and these two numbers are usually almost identical but differ by 0.01. For example, an interest rate might be advertised as 1.99% while the APY is 2.00%. What is the reason for the slight discrepancy? How is the 1.99% figure actually used? 76.254.13.85 (talk) 06:20, 29 January 2010 (UTC)

There are generally two reasons why the rates differ by small amounts (and depends if it is a loan or a deposit). First, and particularly in this case likely reason is because they are compounding more frequently. For a deposit account, they may compound daily ('daily interest'). Second, various fees and others. In this case, and purely guessing, it sounds like a deposit, likely compounding daily, and they have to state the yield (rate) in monthly or annual compounding terms (I don't know the statutory requirements in the US). At low interest rates, the difference is not large. Hope this helps.--Gregalton (talk) 18:05, 8 March 2010 (UTC)

## why does the '1' appear in the formula for compound interest?

I pulled this question from the article page. I would answer it but I don't know the answer: "This is not a formal 'edit', it is to enquire if there is an explanation as to why the '1' appears in the formula for compound interest? (thank you in advance)" --auk (talk) 06:21, 3 February 2010 (UTC)

The savings formula describes periodic contributions to a compounding situation. Could someone please add a derivation of this:

--68.195.44.36 (talk) 02:55, 6 May 2010 (UTC)

## Compound interest taking the world

Trust Issues mentions Hartwick College, Benjamin Franklin's will, Peter Thellusson, When the Sleeper Wakes and Jonathan Holden. You may want to work them into the article. --Error (talk) 00:58, 15 December 2011 (UTC)

## Example of compound interest

The evangelical parable how one poor widow at the time of Jesus Christ sacrificed the last to the temple that it had - two smallest coins, contributions is known. From here, by the way, expression "also went to bring the contribution". If to imagine, what at that time there were banks, and it would bring one coin in bank, what sum would collect on the bank account by today, considering, what the bank provides capitalization of percent in the sum, say, five annual interest rates?

The subsequent calculations illustrate application of difficult percent. It will be easier to us to speak, not about a contribution, but about cent. After the first year of storage the capital would make one cent plus of 5% from it, i.e. math> time would increase in $(1+0,05)$. For the second year of 5% would calculate any more from one cent, and from the size of bigger it by (1+0,05) times. And, in turn, this size would increase too in a year by (1+0,05) times. Means, in comparison with primary sum the contribution in two years would increase in $(1 + 0,05) ^2$ time. In three years - in $(1 + 0,05) ^3$ time.

By 2012 primary contribution would grow up to the size in $(1 + 0,05) ^ {2012}$ times more the initial. The size $(1 + 0,05) ^ {2012}$ makes $4,29\cdot10^ {42}$. At an initial contribution to one cent by 2012 the sum will make $4,29\cdot10^ {40}$ dollars.

The initial idea of application to an ancient parable of estimates in difficult percent belongs to the Polish mathematician Stanislav Kovalya and is published by it in the early seventies in the book "500 zagadek matematycznych".Vladimir Baykov (talk) 13:18, 30 January 2013 (UTC)

## template

I moved the {{E (mathematical constant)}} template to the top. Don't revert OR ELSE! (Just kidding.) 68.173.113.106 (talk) 22:46, 6 March 2012 (UTC)

## Reverting new section pending discussion

I'm deleting the new section "Comparison interest rate" so we can discuss it here.

My primary objection is that it is unsourced, appears to be original research, appears to invent a neologism (Comparison interest rate), and is entirely based on the equations

$S = n * P$
$= \frac{nLi}{1-\frac{1}{(1+i)^n}}$

and

$S_2 = S + F$
$= n * P + F$

where P is the periodic payment and n is the number of such payments. Here the first equation adds together the non-comparable payments from different time periods to get a sum that is meaningless because payments made at different times actually have different values (see time value of money), and where the second equation adds to that the total amount of fees F, again without reference to when the fees are paid. I doubt there is any reputable source that does this. If you have such a source, please mention it here for further discussion. Duoduoduo (talk) 14:12, 24 July 2013 (UTC)

## History - quotes from Bible and Qur'an

I was bold and removed the two biblical quotes - they have nothing to do specifically with compound interest. They are more relevant to, covered more thoroughly in, and formatted more nicely in the article on Usury, which is linked from the History section. Whikie (talk) 08:02, 13 July 2014 (UTC)