Interest

In finance, interest has three general definitions. *Interest is a surcharge on the repayment of debt (borrowed money). *Interest is the return (finance) derived from an investment. *Interest is the right to one's claim in a corporation, such as that of an owner or creditor. This article covers the first definition listed above. Economics sometimes refer to interest as economic rent on money. As with any rental, the market price (or rate) is subject to change to reflect market conditions. The fraction by which the balances grow is called the interest rate. The original balance is called the principal. Interest rates are very closely watched market indicators, and have a dramatic effect on finance and economics. The fact that lenders demand interest for loans in capitalist countries can be attributed to the following reasons: * Time value of money or time preference ** (TVM: Having money now is more valuable than having it at some future time because interest is earnt) ** (TP: Interest is the value borrowers place on having money now) * Opportunity cost ** (OC: The cost in terms of options no longer available once one particular option is chosen) ==History== Historical documents dating back to the Sumerian civilization, circa 3000 B.C., reveal that the ancient world had developed a formalized system of credit based on two major commodities, grain and silver. Before there were coins, metal loans were based on weight. Archaeologists have uncovered pieces of metal that were used in trade in Troy, Minoan civilization and Mycenaean civilizations, Babylonia, Assyria, Egypt and Persian Empire. Before money loans came into existence, loans of grain and silver served to facilitate trade. Silver was used in town economies, while grain was used in the country. The collection of interest was restricted by Jewish, Christian and other religions under laws of usury. This is still the case with Islam, which results in a special type of Islamic banking. Silvio Gesell researched the destabilizing effect of interest (an asset will increase beyond any limit over time) in his Freiwirtschaft theory, which includes negative interest rates. Sometimes income tax has to do with interest rates. Depending on the source, Albert Einstein referred to compound interest as the eighth wonder of the world, the human race's greatest invention, or the most powerful force of the universe. ==Types of compounding== The method by which interest accumulates generally falls in one of the following two categories: ===Simple interest=== Simple interest is the method in which outstanding balances grow linearly with time. In each period, the total balance grows by some fraction ''of the principal'' (that is, of the original investment). Simple interest is seldom used in practice, mostly for estimating compound interest in short durations. In most cases, this is because the interest earned in previous periods is assumed to remain in the account. Only when the interest earned is immediately withdrawn from the account should simple interest be used. When interest remains in the account with the principal, the interest increases the amount of money subject to interest. In this case, simple interest would not reflect the opportunity cost that the lender experiences.

Balance = Principal \cdot (1 + Periods \cdot Interest\ rate)\,

===Compound interest=== Compound interest is the method in which outstanding balances grow exponential growth with time. In each period, the total balance grows by some fraction ''of the sum of the principal and the interest paid on all previous periods''. With compound interest, the frequency of compounding influences the total amount of interest paid over the life of the loan. The accumulation function for compound interest is an exponential function in terms of time.

Balance = Principal \cdot (1 + Interest Rate) ^ {Periods}

===Variables=== :Balance - :Principal - :Interest rate - :Periods - ==Types of interest rate== Two general types of interest rate exist for debt instruments: * The most common type of interest rate is fixed-rate. Fixed-rate instruments contain a fixed denomination throughout the life of the instrument. Most bonds exhibit this type of interest rate. * Another type of interest rate is variable-rate. Variable rate instruments are usually attached to an index that floats based on the economics condition such as Prime rate or CPI. The inflation-indexed instrument is a type of variable-rate instrument that is created to combat inflation. It is very common for firms to create contracts that swap between the two types of interest rate. These kind of contractual agreements are called interest rate swaps. ==Analysis of interest-rate risks== Interest involves the future, which is uncertain. Some interest bearing investments are riskier than others are. The greater the risk of the security, the more interest the investors will expect to receive. The fundamental determinants of interest rate of a debt instrument are these risks. The following is a list of risks commonly associated with interest rates: * Nonsystematic risks ** Credit risk - the risk of default on the loan due to bankruptcy ** Maturity/Term risk - the risk involved in a long-term investment ** Liquidity risk - the need of compensating the illiquidity of the debt * Systematic risks ** inflation risk - macroeconomic price changes ** exchange rate risk - currency fluctuation Interest rate has been analyzed in almost every way possible. All the above listed risks have been scrutinized to test their effects on the interest rate. ===Credit risk=== The credit risk is the most commonly associated risk. It determines the different amount individuals or firms pay based on their credit-worthiness. Different parties will be offered different rates on debt obligations (such as loans). The measure of credit worthiness of an individual is called a credit rating or credit score. Other entities (such as governments and companies) will acquire a bond rating if they are active in bond markets. The credit spread between an instrument and its risk-free equivalent is called the risk premium. ===Maturity/Term risk=== See term structure of interest rates. ===Liquidity risk=== Liquidity risk is the risk that the lender might not be able to liquidate the debt on short notice. The difference in interest rate due to liquidity risk is called liquidity spread. Instruments such as bonds have an active secondary markets. Other instruments such as savings deposits are easily transferable to cash. On the other hand 30-year US Government Savings Bond is nontransferable. It can only be redeemed at half price before maturity. The savings bond will obviously offer a higher return. Another interesting phenomenon observed from liquidity spread is that on-the-run securities (primary market) have lower interest rates compare to the off-the-run securities (secondary market). This implies that there is a higher demand for on-the-run securities. ===Inflation and exchange-rate risks=== Majority of the inflation and exchange rate risk come from loans to developing countries. Therefore, loans offered by banks in developed countries usually denominate the loan contract in stable currencies such as the US Dollar, Pound Sterling, or Euro. This has led to unfavorable consequences for the borrowers of developing countries because the economies of developing countries often have high inflation and unstable exchange rate. ==Mathematics of interest== The Amount functions for simple and compound interest are defined as the following: :

A(t)=k(1+t i)\,

A(t)=k(1+i)^t\,

''A''(''t'') = amount at time ''t'' ''k'' = principal ''t'' = compounding periods ''i'' = interest To use these functions, simply substitute the values into the appropriate variable and solve. Since the principal ''k'' is simply a coefficient, it is often dropped for simplicity. The accumulation function is the resulting function. Accumulation functions for simple and compound interest are listed below: :

a(t)=1+t i\,

a(t)=(1+i)^t\,

Note: ''A''(''t'') is the amount function and ''a''(''t'') is the accumulation function. ===Force of interest=== In mathematics, the accumulation function are often expressed in terms of ''E (mathematical constant)'', the base of the natural logarithm. This facilitates the use of calculus methods in manipulation of interest formulas. This is called the force of interest. The force of interest is defined as the following: :

\delta_{t}=\frac{a'(t)}{a(t)}\,

a(n)=e^{\int_0^n \delta_t\, dt}\,

When the above formula is written in differential equation format, the force of interest is simply the coefficient of amount of change. :

da(t)=\delta_{t}a(t)\,dt\,

The force of interest for compound interest is a constant for a given ''i'', and the accumulation function of compounding interest in terms of force of interest is a simple power of e: :

\delta=\ln(1+i)\,

a(t)=e^{t\delta}\,

===Continuous compounding=== For interest compounded a certain number of times, ''n'', per year, such as monthly or quarterly, the formula is: :

a(t)=\left(1+\frac{i}{n}\right)^{n \cdot t}\,

Continuous compounding can be thought as making the compounding period infinitely small; therefore achieved by taking the Limit (mathematics) of ''n'' to infinity. One should consult definitions of the exponential function for the mathematical proof of this limit. :

a(t)=\lim_{n\to\infty}\left(1+\frac{i}{n}\right)^{n \cdot t}

a(t)=e^{i \cdot t}

The amount function is simply :

A(t)=k e^{i \cdot t}

Interest

==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. User:EntmootsOfTrolls :: Well, printing money is ''one'' cause of inflation, but there are others, no? User:MyRedDice :::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. User: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. --User: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.User: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. - User: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:

B_0 (1 + in) \,

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:

B_0 (1 + i)^n \,

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

B_0 = 1\,

, 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

(1 + I/m)^m \,

. In the last case we need to modify the limit equation in the main article getting :

\lim_{m\to\infty} \left(1+\frac{I}{m}\right)^m = e^I,

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. :

B_1 = B_0 (1 + i) - P \,

During the first period the initial balance has grown by the period interest and has been decreased by the first payment. Similarly :

B_2 = B_1 (1 + i) - P = B_0 (1 + i)^2 - P (1 + i) - P\,

Again :

B_3 = B_2 (1 + i) - P = B_0 (1 + i)^3 - P (1 + i)^2 - P (1 + i) - P\,

After n periods or payments we have :

B_n = B_0 (1 + i)^n - P (1 + i)^{n-1} ..... - P (1 + i)^2 - P (1 + i) - P\,

B_n 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: :

B_0 (1 + i)^n = P [1 + (1 + i) + (1 + i)^2 + .... + (1 + i)^{n-1}]\,

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: :

B_0 (1 + i)^n = P [1 - (1 + i)^n]/(-i) = P [(1 + i)^n - 1]/i\,

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: :

P = B_0 [i(1 + i)^n]/[(1 + i)^n - 1]\,

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%27s_Guide:_Editing_mathematical_formulae - User: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: :

I = Pe^{rt}

" So if i have $1000 at 10% interest, :

I = $1000 \cdot e^{0.1 \cdot 1} = $1105.17

I is not the interest. It is the principal ''plus'' interest after 1 year. - User: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 - User: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) --User: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. --User: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... User:Jaberwocky6669 21:03, Mar 28, 2005 (UTC)

See other meanings of words starting from letter:

I

IA | IB | IC | ID | IE | IF | IG | IH | IJ | IK | IL | IM | IN | IO | IP | IR | IS | IT | IU | IW | IX | IY | IZ |

Words begining with Interest:

Interest
Interest
Interest-only_loan
Interest-only_mortgage
Interest-rate_swaps
Interesting
Interesting
Interesting_and_uninteresting_numbers
Interesting_and_uninteresting_numbers
Interesting_number_paradox
Interesting_number_paradox
Interesting_Times
Interests_in_securities
Interest_group
Interest_group
Interest_groups
Interest_in_securities
Interest_only_mortgage
Interest_on_lawyer_trust_accounts
Interest_parity_condition
Interest_rate
Interest_rate
Interest_rates
Interest_rates
Interest_rate_basis
Interest_Rate_Cap
Interest_rate_cap
Interest_rate_cap/floor
Interest_rate_cap/floor
Interest_rate_caps/floors
Interest_rate_cap_and_floor
Interest_rate_cap_and_floor
Interest_rate_derivative
Interest_rate_derivatives
Interest_Rate_Floor
Interest_rate_floor
Interest_Rate_Future
Interest_rate_option
Interest_rate_options
Interest_rate_risk
Interest_rate_swap
Interest_rate_swap
Interest_rate_swaps