Talk:Interest
|
|
| Contents |
POV for neoclassical economics
I have to debate the definition of interest given in this topic. Interest is not necessitated by inflation, but inflation instead necessitated by interest. Inflation is caused by allowing the central banks to print currency for the government to use to pay off its debt _to the central banks_. (A surplus of currency makes your money worth less). Since the interest on this debt is exponentially larger than the principal of the debt all payments go directly fighting off the evil curse of interest.
- I agree completely. The article in its present form is propaganda for neoclassical economics. Your explanation is basically valid, as far as I know from reading John Kenneith Galbraith and others who know the process of money supply and central banking. Islamic economics and green economics have plenty of alternatives to interest as solutions to inflation. There is no excuse for this bias in the current article. EofT
- Well, printing money is one cause of inflation, but there are others, no? Martin
- I have edited this page for accuracy and taken the NPOV alert off. I agree there are alternative financial systems other than capitalism and they should be written about, but as an introduction to the role of interest in capitalism, this article, as it presently stands, is a good start. mydogategodshat 22:39, 27 Sep 2003 (UTC)
- "Interest is not necessitated by inflation"? This is nonsense; an economy which had inflation but a zero interest rate is simply inconceivable/impossible. "..but inflation instead necessitated by interest"? Again this is simply untrue; there have been deflationary economies with positive interest rates. As for calling the article propagands, that's bull; the article simply reflects orthodox (not just neoclassical) economic thinking. Islamic economics and green economics may have plenty of alternatives but few or none are taken all that seriously in the field of economics. User:jimg
Inconsistent results?
Would someone care to explain in what sense using simple interest when the interest remains in the account would "produce mathematically inconsistent results"? Although I can see that this situation might not make the lender happy, what is "inconsistent" about it? Either a fuller discussion or removal of this claim seems in order. --Ryguasu 01:20, 1 Oct 2003 (UTC)
- I agree that this claim is not at all clear. I think what the author was trying to say was, the interest received would not be consistent with what they should receive because they are not getting interest on their past interest earnings. That is, the past interest portion of their account would yield a zero % return even though the financial institution was claiming to be providing a positive rate of return.mydogategodshat 10:28, 1 Oct 2003 (UTC)
Simple and compound interest for the layman
It would be nice to have the formulas for simple and compound interest included and explained nicely. - Omegatron 15:56, Apr 9, 2004 (UTC)
Sure!
See Future value
First the nomenclature.
I - The stated interest rate, for example, 5%/year. This is not the APR (annualized percentage rate).
m - The number of periods in the time frame of I. I is usually based on a year but it could be based on any amount of time.
i - The interest rate for the compounding period which is needed for the calculation. For example, a real property mortgage is usually based on a monthly period. In this case i=I*1/12 where I is based on the normal yearly period. In general i=I/m. Also I needs to be a decimal not a percent thus it also needs to be divided by 100.
n - The total number of periods or payments. Things like mortgages usually cover multiple years.
B - The balance, for example, the balance remaining on a mortgage or an interest baring check book or savings (pass) book balance.
Simple Interest: <math>B_0 (1 + in) \,<math>
Inside the parentheses the first term, namely 1, gives back the original investment and the second term, namely in , generates the period interest and multiplies it by the number of periods.
Compound Interest: <math>B_0 (1 + i)^n \,<math>
In the compound case we have a binomial expansion where the first two terms are the same as the simple interest and the remaining terms calculate the interest on interest. Actually all interest calculations can be carried out using simple interest. Compound interest is simply a special case when the calculations can be simplified by the use of the binominal expansion.
Lets take <math>B_0 = 1\,<math>, I = .06 and n = m and consider the case where m = 1, 12, 365 and infinity, compounding namely, yearly, monthly, daily and instantaneously. For the first three cases we can use the binomial expansion <math>(1 + I/m)^m \,<math>. In the last case we need to modify the limit equation in the main article getting
- <math>\lim_{m\to\infty} \left(1+\frac{I}{m}\right)^m = e^I,<math>
Running the calculations gives:
for yearly (m = 1) 1.06
for monthly (m = 12) 1.061677812
for daily (m = 365) 1.061831287
for instantaneously (m = infinity) 1.061836547
Subtracting one and multiplying by 100 to get the percentage interest rate gives: 6, 6.1677812, 6.1831287 and 6.1836547.
The first number is simple interest since there is only a single period. The remaining numbers give the simple interest required to provide the same value as that given compounding at .06. Thus they are the APR the annual percentage rate. In the 1960s banks were attempting to lure customers by compounding instantaneously rather than daily. As one can see there is not a lot of difference, less than a hundredth of a percent.
Mortgage Calculations:
Let B0 be the original mortgage or opening bank balance.
Let B1, B2, B3 etc. be the balance after the first, second, third period respectively.
Obviously, one can think of B0 as the balance after the zeroth period namely the beginning balance.
P - The payment in the case of a mortgage or a deposit or withdrawal (a negative deposit) in the case of a bank account.
Now lets write down the balances. First the initial balance, the amount of the mortgage.
B0
Now lets calculate the balance after one period or payment.
- <math>B_1 = B_0 (1 + i) - P \,<math>
During the first period the initial balance has grown by the period interest and has been decreased by the first payment. Similarly
- <math>B_2 = B_1 (1 + i) - P = B_0 (1 + i)^2 - P (1 + i) - P\,<math>
Again
- <math>B_3 = B_2 (1 + i) - P = B_0 (1 + i)^3 - P (1 + i)^2 - P (1 + i) - P\,<math>
After n periods or payments we have
- <math>B_n = B_0 (1 + i)^n - P (1 + i)^{n-1} ..... - P (1 + i)^2 - P (1 + i) - P\,<math>
Bn is set equal to zero. When the mortgage is paid off the balance is zero. Now one can solve for P the payment. Rearranging gives:
- <math>B_0 (1 + i)^n = P [1 + (1 + i) + (1 + i)^2 + .... + (1 + i)^{n-1}]\,<math>
The righthand side is a geometric series where each term is equal to the preceding term multiplied by (1 + i) which is known as the ratio.
Multiplying the righthand side by [1 - (i + 1)]/(-i) gives:
- <math>B_0 (1 + i)^n = P [1 - (1 + i)^n]/(-i) = P [(1 + i)^n - 1]/i\,<math>
Note: What one is doing is multiplying and dividing by -i and in the numerator adding and subtracting 1. The reason for this is that multiplying a geometric series by one minus the ratio leaves simply the first term minus the last term with the exponent incremented by one since all the other terms cancel in pairs.
Solving for P gives:
- <math>P = B_0 [i(1 + i)^n]/[(1 + i)^n - 1]\,<math>
The payment can be readily calculated to the penny with a scientific calculator. Does a spread sheet have enough accuracy?
Note: B0 is just a simple multiplier. Therefore one can do the calculation for a unit of currency such as a dollar and then multiply the result by the amount of the loan. In essence B0 is just a scale factor. For example think of the loan amount as my dollar where my dollar is just a currency whose exchange rate is just the loan amount difference.
Now lets do some calculations. Mortgages are usually for 15, 20 or 30 years. Interest rates use to be around 9%/year and today around 6%/year. For all calculations B0 = 1
years, n, (1 + i)^n, P, nP for i = .09/12 = .0075
15 180 3.838043267 .010142665 1.8256797
20 240 6.009151524 .008997259559 2.15934216
30 360 14.73057612 .00804622617 2.89664136
years, n, (1 + i)^n, P, nP for i = .06/12 = .005
15 180 2.454093562 .008438568281 1.51894224
20 240 3.310204476 .007164310585 1.7194344
30 360 6.022575212 .005995505252 2.158381891
First calculate (1 + i)^n since it occurs in both the numerator and the denominator. Then complete the calculation for the payment P. In the first case, for each dollar of loan the payment is a little over a penny per month. Multiplying the amount of the payment P by the number of payments n gives the total amount paid. In the first case, for each dollar of loan the repayment is a little over a dollar and 82 cents. The 1.82 is also the ratio of the repayment amount to the amount of the loan.
Again this is the best I can do with the tables, etc. Also someone may want to work this into the main article. Next chance I get I will go into bank accounts, US Treasury Bills and whatever else I can think of. Also I will bring out the where, what, when, why of simple and compound interest.
Sorry be back as soon as I figure out how to make the math show up right. Well a little bit more. I'll keep working on it. Thanks for the help.
- Go here: meta:MediaWiki_User's_Guide:_Editing_mathematical_formulae - Omegatron 20:31, Jun 29, 2004 (UTC)
look here for more material: http://mathforum.org/dr.math/faq/faq.interest.html
Um
"This formula is usually written:
- <math>I = Pe^{rt}<math>"
So if i have $1000 at 10% interest,
- <math>I = $1000 \cdot e^{0.1 \cdot 1} = $1105.17<math>
I is not the interest. It is the principal plus interest after 1 year. - Omegatron 03:04, Aug 6, 2004 (UTC)
- I fixed it sort of. Not too happy with the nomenclature. Guess, I should get around to improving
things.
- Problem is, I've heard of P=ert as "PERT" before, so it is usually called P. But it needs to be explained concisely - Omegatron 01:03, Aug 12, 2004 (UTC)
- Sorry for the edit i did awhile ago, i didn't read this before hand. The notations are inconsistant throughout the different fields that use TVM. I took the notations of my notes from "interest theory" because it actually defines "accumulation" a(t) and "amount" A(t) in function format. (feels more mathematical) --Voidvector 19:37, Nov 8, 2004 (UTC)
You did a nice job improving this article. Nomenclature is always a problem. I believe being consistant is the top priority. Also, reducing interest and compounding periods to single variables makes for improved clarity.
Suggestions: Perhaps there should be a note providing additional details on interest and compounding periods. For example see Mortgage 5 Fixed rate mortgage calculations under Contents. Everyone reads the encyclopedia, so I feel that we should be careful not to assume that the reader is familiar with the subject. Also, concerning the sentence, "Since the principle k is simply a coefficient, it is often dropped for simplicity.", I feel there is something deeper than just simplicity involved. I would try something like: Without loss of generality, the principle k can be taken as unity since it is simply a coefficient or scale factor. For example see Geometric progression.
I just noticed that in Continuous Compounding the meaning of t has changed.
with interest rate article
This article covers material similar to interest rate article. We should either consider differentiating or merging the two articles. --Voidvector 19:37, Nov 8, 2004 (UTC)
I agree. It seems to me that the current content of the interest rate article should be moved and/or incorporated into the article on interest and that the interest rate article should be about interest rates. Namely, how interest rates are set, what variables affect interest rates, historical charts, etc. There are many links to interest rate which would need to be shifted to interest.
AKK EKK HEEEELLLLPPPP MMMEEEEE
Uh....**gasp**....*pant pant** **Ok, calm down, no need to panic...) Whew,...Ok, let's say I have a starting balance and it is compounded monthly AND I want to put an extra $5.00 a month into the account how do I figure out my ending balance after so many years??? What equation should I use to figure it out??? Comprehensive article you math guys have here... Jaberwocky6669 21:03, Mar 28, 2005 (UTC)