Interest is a
fee paid on
borrowed assets. It is the price paid for the use of borrowed
money, or, money earned by deposited funds.
Assets that are sometimes lent with interest include
money,
shares,
consumer goods through
hire purchase, major assets such as
aircraft, and even entire factories in
finance lease arrangements. The
interest is calculated upon the value of the assets in the same
manner as upon money. Interest can be thought of as "
rent of money". When money is deposited in a bank,
interest is typically paid to the depositor as a percentage of the
amount deposited; when money is borrowed, interest is typically
paid to the lender as a percentage of the amount owed. The
percentage of the principal that is paid as a fee over a certain
period of time (typically one month or year), is called the
interest rate.
Interest is compensation to the lender for the risk of not being
paid back, and for forgoing other useful
investments that could have been made with the
loaned asset. These forgone investments are known as the
opportunity cost. Instead of the lender
using the assets directly, they are advanced to the borrower. The
borrower then enjoys the benefit of using the assets ahead of the
effort required to obtain them, while the lender enjoys the benefit
of the fee paid by the borrower for the privilege. The amount lent,
or the value of the assets lent, is called the principal. This
principal value is held by the borrower on
credit. Interest is therefore the price of
credit, not the price of money as it is commonly believed to
be.
History of interest
Interest is the price paid for the use of savings over a given
period of time. In ancient biblical Israel, it was against the Law
of Moses to charge interest on private loans.During the
Middle Ages, time was considered to be property
of
God. Therefore, to charge
interest was considered to be commerce with God's property. Also,
St.
Thomas Aquinas, the leading
theologian of the
Catholic Church,
argued that the charging of interest is wrong because it amounts to
"
double
charging", charging for both the thing and the use of the
thing. The church regarded this as a sin of
usury; nevertheless, this rule was never strictly
obeyed and eroded gradually until it disappeared during the
industrial revolution.
Usury has always been viewed negatively by the
Roman Catholic Church. The
Second
Lateran Council condemned any repayment of a debt with more
money than was originally loaned, the
Council of Vienna explicitly prohibited
usury and declared any legislation tolerant of usury to be
heretical, and the first scholastics reproved the charging of
interest. In the
medieval economy,
loans were entirely a consequence of necessity (bad harvests, fire
in a workplace) and, under those conditions, it was considered
morally reproachable to charge interest. It was also considered
morally dubious, since no goods were produced through the lending
of money, and thus it should not be compensated, unlike other
activities with direct physical output such as blacksmithing or
farming.
Interest has often been looked down upon in
Islamic civilization as well for the
same reason for which usury was forbidden by the Catholic Church,
with most scholars agreeing that the Qur'an explicitly forbids
charging interest. Medieval jurists therefore developed several
financial instruments to encourage responsible lending.
In the
Renaissance era, greater mobility
of people facilitated an increase in commerce and the appearance of
appropriate conditions for
entrepreneurs to start new, lucrative
businesses. Given that borrowed money was no longer strictly for
consumption but for production as well, interest was no longer
viewed in the same manner. The
School of Salamanca elaborated on
various reasons that justified the charging of interest: the person
who received a loan benefited, and one could consider interest as a
premium paid for the risk taken by the loaning party. There was
also the question of
opportunity
cost, in that the loaning party lost other possibilities of
using the loaned money. Finally and perhaps most originally was the
consideration of money itself as merchandise, and the use of one's
money as something for which one should receive a benefit in the
form of interest.
Martín de
Azpilcueta also considered the effect of time. Other things
being equal, one would prefer to receive a given good now rather
than in the future. This
preference
indicates greater value. Interest, under this theory, is the
payment for the time the loaning individual is deprived of the
money.
Economically, the
interest rate is the
cost of capital and is subject to the laws of
supply and demand of the
money supply.
The first attempt to control interest
rates through manipulation of the money supply was made by the
French Central
Bank in 1847.
The first formal studies of interest rates and their impact on
society were conducted by
Adam Smith,
Jeremy Bentham and
Mirabeau during the birth of classic economic
thought. In the early
20th century,
Irving Fisher made a major
breakthrough in the economic analysis of interest rates by
distinguishing nominal interest from real interest. Several
perspectives on the nature and impact of interest rates have arisen
since then. Among academics, the more modern views of
John Maynard Keynes and
Milton Friedman are widely accepted.
The latter half of the 20th century saw the rise of interestfree
Islamic banking and finance, a movement which attempts to apply
religious law developed in the medieval period to the modern
economy. Some entire countries, including Iran, Sudan, and
Pakistan, have taken steps to eradicate interest from their
financial systems entirely.Rather than charging interest, the
interestfree lender charges a "fee" for the service of lending. As
any such fee can be shown to be mathematically identical to an
interest charge, the distinction between "interestfree" banking
and "forinterest" banking is merely one of semantics
Types of interest
Simple interest
Simple interest is calculated only on the principal amount, or on
that portion of the principal amount which remains unpaid.
The amount of simple interest is calculated according to the
following formula:
 I_{simp} = (r \cdot B_0) \cdot m
where
r is the period interest rate (I/m),
B_{0} the initial balance and
m the
number of time periods elapsed.
To calculate the period interest rate
r, one divides the
interest rate
I by the number of periods
m.
For example, imagine that a credit card holder has an outstanding
balance of $2500 and that the simple
interest rate is 12.99% per annum. The
interest added at the end of 3 months would be,
 I_{simp} = \bigg(\frac{0.1299}{12}\cdot $2500\bigg) \cdot
3=$81.19
and he would have to pay $2581.19 to pay off the balance at this
point.
If instead he makes interestonly payments for each of those 3
months at the period rate
r, the amount of interest paid
would be,
 I = \bigg(\frac{0.1299}{12}\cdot $2500\bigg) \cdot 3=
($27.0625/month) \cdot 3=$81.19
His balance at the end of 3 months would still be $2500.
In this case, the
time value of
money is not factored in. The steady payments have an
additional cost that needs to be considered when comparing loans.
For example, given a $100 principal:
 Credit card debt where $1/day is charged: 1/100 = 1%/day =
7%/week = 365%/year.
 Corporate bond where the first $3 are due after six months, and
the second $3 are due at the year's end: (3+3)/100 = 6%/year.
 Certificate of deposit (GIC) where $6 is paid at
the year's end: 6/100 = 6%/year.
There are two complications involved when comparing different
simple interest bearing offers.
 When rates are the same but the periods are different a direct
comparison is inaccurate because of the time value of money. Paying $3 every six
months costs more than $6 paid at year end so, the 6% bond cannot
be 'equated' to the 6% GIC.
 When interest is due, but not paid, does it remain 'interest
payable', like the bond's $3 payment after six months or, will it
be added to the balance due? In the latter case it is no longer
simple interest, but compound interest.
A bank account offering only simple interest and from which money
can freely be withdrawn is unlikely, since withdrawing money and
immediately depositing it again would be advantageous.
Compound interest
Compound interest is very similar to simple interest; however, with
time, the difference becomes considerably larger. This difference
is because unpaid interest is added to the balance due. Put another
way, the borrower is charged interest on previous interest.
Assuming that no part of the principal or
subsequent interest has been paid, the debt is
calculated by the following formulas:
\begin{align}&I_{comp}=B_0\cdot\big[\left(1+r\right)^m1\big]\\&B_m=B_0+I_{comp}\end{align}
where
I_{comp} is the compound interest,
B_{0} the initial balance,
B_{m}
the balance after m periods (where
m is not necessarily an
integer) and
r the period rate.
For example, if the credit card holder above chose not to make any
payments, the interest would accumulate
\begin{align}&\mbox{Calculation for Compound
Interest}:\\I_{comp}&=$2500\cdot\bigg[\bigg(1+\frac{0.1299}{12}\bigg)^31\bigg]\\&=$2500\cdot\left(1.010825^31\right)\\&=$82.07\\\end{align}
\begin{align}B_m&=B_0+I_{comp}\\&=$2500+$82.07\\&=$2582.07\end{align}
So, at the end of 3 months the credit card holder's balance would
be $2582.07 and he would now have to pay $82.07 to get it down to
the initial balance. Simple interest is approximately the same as
compound interest over short periods of time, so frequent payments
are the best (least expensive) payment strategy.
A problem with compound interest is that the resulting obligation
can be difficult to interpret. To simplify this problem, a common
convention in economics is to disclose the interest rate as though
the term were one year, with annual compounding, yielding the
effective interest rate.
However, interest rates in
lending are often
quoted as
nominal interest
rates (
i.e., compounding interest uncorrected for the
frequency of compounding).
Loans often include various noninterest charges and fees. One
example are
points on a
mortgage loan in the United States. When such
fees are present, lenders are regularly required to provide
information on the 'true' cost of finance, often expressed as an
annual percentage rate (APR).
The APR attempts to express the total cost of a loan as an interest
rate
after including the additional fees and expenses,
although details may vary by jurisdiction.
In economics,
continuous
compounding is often used due to its particular
mathematical properties.
Fixed and floating rates
Commercial loans generally use
simple interest, but they may not
always have a single interest rate over the life of the loan. Loans
for which the interest rate does not change are referred to as
fixed rate loans. Loans may also have
a changeable rate over the life of the loan based on some
reference rate (such as
LIBOR and
EURIBOR), usually
plus (or minus) a fixed margin. These are known as
floating rate, variable rate or
adjustable rate loans.
Combinations of fixedrate and floatingrate loans are possible and
frequently used. Loans may also have different interest rates
applied over the life of the loan, where the changes to the
interest rate are governed by specific criteria other than an
underlying interest rate. An example would be a loan that uses
specific periods of time to dictate specific changes in the rate,
such as a rate of 5% in the first year, 6% in the second, and 7% in
the third.
Composition of interest rates
In economics, interest is considered the price of credit,
therefore, it is also subject to distortions due to
inflation. The nominal interest rate, which refers
to the price before adjustment to inflation, is the one visible to
the consumer (
i.e., the interest tagged in a loan
contract, credit card statement, etc). Nominal interest is composed
of the
real interest rate plus
inflation, among other factors. A simple formula for the nominal
interest is:
i= r + \pi
Where
i is the nominal interest,
r is the real
interest and
\pi is inflation.And the following block
diagram represents the HH value
This formula attempts to measure the value of the interest in units
of stable purchasing power. However, if this statement were true,
it would imply at least two misconceptions. First, that all
interest rates within an area that shares the same inflation (that
is, the same country) should be the same. Second, that the lenders
know the inflation for the period of time that they are going to
lend the money.
One reason behind the difference between the interest that yields a
treasury bond and
the interest that yields a
mortgage
loan is the risk that the lender takes from lending money to an
economic agent. In this particular case, a government is more
likely to pay than a private citizen. Therefore, the interest rate
charged to a private citizen is larger than the rate charged to the
government.
To take into account the
information asymmetry aforementioned,
both the value of inflation and the real price of money are changed
to their
expected values resulting in the
following equation:
i_t = r_{(t+1)} + \pi_{(t+1)} + \sigma
Here, i_t is the nominal interest at the time of the loan,
r_{(t+1)} is the real interest expected over the period of the
loan, \pi_{(t+1)} is the inflation expected over the period of the
loan and \sigma is the representative value for the risk engaged in
the operation.
Cumulative interest or return
The calculation for cumulative interest is (FV/PV)1. It ignores
the 'per year' convention and assumes compounding at every payment
date. It is usually used to compare two long term
opportunities.
Other conventions and uses
Exceptions:
 US and Canadian TBills (short term Government debt) have a
different calculation for interest. Their interest is calculated as
(100P)/P where 'P' is the price paid. Instead of normalizing it to
a year, the interest is prorated by the number of days 't':
(365/t)*100. (See also: Day count
convention). The total calculation is
((100P)/P)*((365/t)*100). This is equivalent to calculating the
price by a process called discounting at a simple interest
rate.
 Corporate Bonds are most frequently payable twice yearly. The
amount of interest paid is the simple interest
disclosed divided by two (multiplied by the face value of
debt).
Flat Rate Loans and the Rule of 78s: Some consumer
loans have been structured as flat rate loans, with the loan
outstanding determined by allocating the total interest across the
term of the loan by using the "
Rule of
78s" or "Sum of digits" method. Seventyeight is the sum of the
numbers 1 through 12, inclusive. The practice enabled quick
calculations of interest in the precomputer days. In a loan with
interest calculated per the Rule of 78s, the total interest over
the life of the loan is calculated as either simple or compound
interest and amounts to the same as either of the above methods.
Payments remain constant over the life of the loan; however,
payments are allocated to interest in progressively smaller
amounts. In a oneyear loan, in the first month, 12/78 of all
interest owed over the life of the loan is due; in the second
month, 11/78; progressing to the twelfth month where only 1/78 of
all interest is due. The practical effect of the Rule of 78s is to
make early payoffs of term loans more expensive. For a one year
loan, approximately 3/4 of all interest due is collected by the
sixth month, and payoff of the principal then will cause the
effective interest rate to be much higher than the APY used to
calculate the payments.
In 1992,
the United
States outlawed the use of "Rule of 78s" interest in
connection with mortgage refinancing and other consumer loans over
five years in term. Certain other jurisdictions have
outlawed application of the Rule of 78s in certain types of loans,
particularly consumer loans.
Rule of 72: The "
Rule of
72" is a "quick and dirty" method for finding out how fast
money doubles for a given interest rate. For example, if you have
an interest rate of 6%, it will take 72/6 or 12 years for your
money to double, compounding at 6%. This is an approximation that
starts to break down above 10%.
Market interest rates
There are markets for investments (which include the money market,
bond market, as well as retail financial institutions like banks)
set
interest rates. Each specific debt
takes into account the following factors in determining its
interest rate:
Opportunity cost
This encompasses any other use to
which the money could be put, including lending to others,
investing elsewhere, holding cash (for safety, for example), and
simply spending the funds.
Inflation
Since the lender is deferring his consumption, he will at a bare
minimum, want to recover enough to pay the increased cost of goods
due to
inflation. Because future inflation
is unknown, there are three tactics.
 Charge X% interest 'plus inflation'. Many governments issue
'realreturn' or 'inflation indexed' bonds. The principal amount or
the interest payments are continually increased by the rate of
inflation. See the discussion at real
interest rate.
 Decide on the 'expected' inflation rate. This still leaves both
parties exposed to the risk of 'unexpected' inflation.
 Allow the interest rate to be periodically changed. While a
'fixed interest rate' remains the same throughout the life of the
debt, 'variable' or 'floating' rates can be reset. There are
derivative products that allow for hedging and swaps between the
two.
Default
There is always the risk the borrower will become
bankrupt,
abscond or
otherwise
default on the loan. The
risk premium attempts to measure the integrity
of the borrower, the risk of his enterprise succeeding and the
security of any collateral pledged. For example, loans to
developing countries have higher risk premiums than those to the US
government due to the difference in creditworthiness. An operating
line of credit to a business will have a higher rate than a
mortgage.
The
creditworthiness of businesses
is measured by
bond rating
services and individual's
credit
scores by
credit bureaus. The
risks of an individual debt may have a large standard deviation of
possibilities. The lender may want to cover his maximum risk, but
lenders with portfolios of debt can lower the risk premium to cover
just the most probable outcome.
Deferred consumption
Charging interest equal only to inflation will leave the lender
with the same purchasing power, but he would prefer his own
consumption sooner rather than later. There will be an interest
premium of the delay. He may not want to consume, but instead would
invest in another product. The possible return he could realize in
competing investments will determine what interest he
charges.
Length of time
Shorter terms have less risk of default and inflation because the
near future is easier to predict. Broadly speaking, if interest
rates increase, then investment decreases due to the higher cost of
borrowing (all else being equal).
Interest rates are generally determined by the market, but
government intervention  usually by a
central bank may strongly influence shortterm
interest rates, and is used as the main tool of
monetary policy. The central bank offers to
buy or sell money at the desired rate and, due to their control of
certain tools (such as, in many countries, the ability to print
money) they are able to influence overall market interest
rates.
Investment can change rapidly in response to changes in interest
rates, affecting national income, and, through
Okun's Law, changes in output affect
unemployment.
Open market operations in the United States
The effective federal funds rate
charted over fifty years.
The
Federal Reserve (Fed) implements
monetary policy largely by targeting the
federal funds rate. This is the rate that
banks charge each other for overnight loans of
federal funds. Federal funds are the reserves
held by banks at the Fed.
Open market operations are
one tool within monetary policy implemented by the Federal Reserve
to steer shortterm interest rates.
Using the power to buy and sell treasury
securities, the Open Market Desk at the
Federal
Reserve Bank of New York can supply the market with dollars by purchasing
Tnotes, hence increasing the nation's money
supply. By increasing the money supply or
Aggregate Supply of Funding
(ASF), interest rates will fall due to the excess of dollars banks
will end up with in their reserves.
Excess reserves may be lent in the
Fed funds market to other banks, thus driving
down rates.
Interest rates and credit risk
It is increasingly recognized that the business cycle,
interest rates and
credit risk are tightly interrelated. The
JarrowTurnbull model was the
first model of credit risk which explicitly had random interest
rates at its core. Lando (2004),
Darrell
Duffie and Singleton (2003), and van Deventer and Imai (2003)
discuss interest rates when the issuer of the interestbearing
instrument can default.
Money and inflation
Loans, bonds, and shares have some of the characteristics of money
and are included in the broad money supply.
By setting
i*n, the government institution can
affect the markets to alter the total of loans, bonds and shares
issued. Generally speaking, a higher real interest rate reduces the
broad money supply.
Through the
quantity theory of
money, increases in the money supply lead to inflation. This
means that interest rates can affect inflation in the future.
Interest in mathematics
It is thought that
Jacob Bernoulli
discovered the mathematical constant
e by studying a question about
compound interest.
He realized that if an account that starts with $1.00 and pays 100%
interest per year, at the end of the year, the value is $2.00; but
if the interest is computed and added twice in the year, the $1 is
multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25.
Compounding quarterly yields
$1.00×1.25
^{4} = $2.4414…, and so on.
Bernoulli noticed that this sequence can be modeled as
follows:
 \lim_{n\rightarrow\infty} \left(1+\dfrac{1}{n}\right)^n=e,
where
n is the number of times the interest is to be
compounded in a year.
Formulae
The
balance of a loan with
regular monthly payments is augmented by the monthly interest
charge and decreased by the payment so,
 B_{k+1}=\big(1+r\big)B_kp.
where,
 i = loan rate/100 = annual rate in decimal form (e.g. 10% =
0.10 The loan rate is the rate used to compute payments and
balances.)
 r = period rate = i/12 for monthly payments (customary usage
for convenience)[22526]
 B_{0} = initial balance (loan principal)
 B_{k} = balance after k payments
 k = balance index
 p = period (monthly) payment
By repeated substitution one obtains expressions for B
_{k}
which are linearly proportional to B
_{0} and p and use of
the formula for the partial sum of a
geometric series results in,
 B_k=(1+r)^k B_0  \frac {r}\ p
A solution of this expression for p in terms of B
_{0} and
B
_{n} reduces to,
 p=r\Bigg[\frac{B_0B_n}{({1+r})^n1}+B_0\Bigg]
To find the payment if the loan is to be paid off in n payments one
sets B
_{n} = 0.
The PMT function found in
spreadsheet
programs can be used to calculate the monthly payment of a
loan:
 p=PMT(rate,num,PV,FV,) = PMT(r,n,B_0,B_n,)\;
An interestonly payment on the current balance would be,
 p_I=r B\;
The total interest, I
_{T}, paid on the loan is,
 I_T=npB_0\;
The formulas for a regular savings program are similar but the
payments are added to the balances instead of being subtracted and
the formula for the payment is the negative of the one above. These
formulas are only approximate since actual loan balances are
affected by rounding. In order to avoid an underpayment at the end
of the loan the payment needs to be rounded up to the next cent.
The final payment would then be (1+r)B
_{n1}.
Consider a similar loan but with a new period equal to k periods of
the problem above. If r
_{k} and p
_{k} are the new
rate and payment, we now have,
 B_k=B'_0=(1+r_k)B_0p_k\;
Comparing this with the expression for B
_{k} above we note
that,
 r_k=(1+r)^k1\;
 p_k=\frac{p}{r} r_k
The last equation allows us to define a constant which is the same
for both problems,
 B^*=\frac{p}{r}=\frac{p_k}{r_k}
and B
_{k} can be written,
 B_k=(1+r_k)B_0r_k B^*\;
Solving for r
_{k} we find a formula for r
_{k}
involving known quantities and B
_{k}, the balance after k
periods,
 r_k=\frac{B_0B_k}{B^*B_0}
Since B
_{0} could be any balance in the loan, the formula
works for any two balances separate by k periods and can be used to
compute a value for the annual interest rate.
B* is a
scale invariant since it
does not change with changes in the length of the period.
Rearranging the equation for B
^{*} one gets a
transformation coefficient (
scale
factor),

\lambda_k=\frac{p_k}{p}=\frac{r_k}{r}=\frac{(1+r)^k1}{r}=k[1+\frac{(k1)r}{2}+\cdots]
(see binomial theorem)
and we see that r and p transform in the same manner,
 r_k=\lambda_k r\;
 p_k=\lambda_k p\;
The change in the balance transforms likewise,
 \Delta B_k=B'B=(\lambda_k rB\lambda_k p)=\lambda_k \Delta B
\;
which gives an insight into the meaning of some of the coefficients
found in the formulas above. The annual rate, r
_{12},
assumes only one payment per year and is not an "effective" rate
for monthly payments. With monthly payments the monthly interest is
paid out of each payment and so should not be compounded and an
annual rate of 12·r would make more sense. If one just made
interestonly payments the amount paid for the year would be
12·r·B
_{0}.
Substituting p
_{k} = r
_{k} B* into the equation for
the B
_{k} we get,
 B_k=B_0r_k(B^*B_0)\;
Since B
_{n} = 0 we can solve for B*,
 B^*=B_0\bigg(\frac{1}{r_n}+1\bigg)
Substituting back into the formula for the B
_{k} shows that
they are a linear function of the r
_{k} and therefore the
λ
_{k},

B_k=B_0\bigg(1\frac{r_k}{r_n}\bigg)=B_0\bigg(1\frac{\lambda_k}{\lambda_n}\bigg)
This is the easiest way of estimating the balances if the
λ
_{k} are known. Substituting into the first formula for
B
_{k} above and solving for λ
_{k+1} we get,
 \lambda_{k+1}=1+(1+r)\lambda_k\;
λ
_{0} and λ
_{n} can be found using the formula for
λ
_{k} above or computing the λ
_{k} recursively from
λ
_{0} = 0 to λ
_{n}.
Since p=rB* the formula for the payment reduces to,
 p=\bigg(r+\frac{1}{\lambda_n}\bigg)B_0
and the average interest rate over the period of the loan is,

r_{loan}=\frac{I_T}{nB_0}=r+\frac{1}{\lambda_n}\frac{1}{n}
which is less than r if n>1.
See also
References
Specific references
 http://scripturetext.com/deuteronomy/2319.htm
 http://www.uh.edu/engines/epi2547.htm
 [1]
General references
External links