Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding (i.e. the interest is compounded). A loan, for example, may have its interest compounded every month: in this case, a loan with $100 initial principal and 1% interest per month would have a balance of $101 at the end of the first month, $102.01 at the end of the second month, and so on.
In order to define an interest rate fully, and enable one to compare it with other interest rates, the interest rate and the compounding frequency must be disclosed. Since most people prefer to think of rates as a yearly percentage, many governments require financial institutions to disclose the equivalent yearly compounded interest rate on deposits or advances. For instance the yearly rate for the loan in the above example is approximately 12.68%. This equivalent yearly rate may be referred to as annual percentage rate (APR), annual equivalent rate (AER), annual percentage yield , effective interest rate , effective annual rate , and by other terms. When a fee is charged up front to obtain a loan, APR usually counts that cost as well as the compound interest in converting to the equivalent rate. These government requirements assist consumers to compare the actual costs of borrowing more easily.
For any given interest rate and compounding frequency, an "equivalent" rate for any different compounding frequency exists.
Compound interest may be contrasted with simple interest, where interest is not added to the principal (there is no compounding). Compound interest is standard in finance and economics, and simple interest is used infrequently (although certain financial products may contain elements of simple interest).
Terminology
The effect of compounding depends on the frequency with which interest is compounded and the periodic interest rate which is applied. Therefore, in order to define accurately the amount to be paid under a legal contract with interest, the frequency of compounding (yearly, half-yearly, quarterly, monthly, daily, etc.) and the interest rate must be specified. Different conventions may be used from country to country, but in finance and economics the following usages are common:
Periodic rate: the interest that is charged (and subsequently compounded) for each period, divided by the amount of the principal. The periodic rate is used primarily for calculations, and is rarely used for comparison. The nominal annual rate or nominal interest rate is defined as the periodic rate multiplied by the number of compounding periods per year. For example, a monthly rate of 1% is equivalent to an annual nominal interest of 12%.
Effective annual rate: this reflects the effective rate as if annual compounding were applied: in other words it is the total accumulated interest that would be payable up to the end of one year.
Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may however be the one quoted instead. When quoted together with the compounding frequency, a loan with a given nominal annual rate is fully specified (the effect of interest for a given loan scenario can be precisely determined), but the nominal rate cannot be directly compared with loans that have a different compounding frequency.
Loans and finance may have other "non-interest" charges, and the terms above do not attempt to capture these differences. Other terms such as annual percentage rate and annual percentage yield may have specific legal definitions and may or may not be comparable, depending on the jurisdiction.
The use of the terms above (and other similar terms) may be inconsistent, and vary according to local custom or marketing demands, for simplicity or for other reasons.
Exceptions
- US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated as (100 − P )/ Pbnm , 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 day count convention).
- The interest on corporate bonds and government bonds is usually payable twice yearly. The amount of interest paid (each six months) is the disclosed interest rate divided by two (multiplied by the principal). The yearly compounded rate is higher than the disclosed rate.
- Canadian mortgage loans are generally semi-annual compounding with monthly (or more frequent) payments.
- U.S. mortgages generally use monthly compounding (with corresponding payment periods).
- It is sometimes mathematically simpler, e.g. in the valuation of derivatives to use continuous compounding , which is the limit as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Itō calculus, where derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time.
Mathematics of interest rates
Simplified calculation
Formulae are presented in greater detail at time value of money.
In the formula below, i is the effective interest rate per period. FV and PV represent the future and present value of a sum. n represents the number of periods.
These are the most basic formulae:
The above calculates the future value ( FV ) of an investment's present value ( PV ) accruing at a fixed interest rate ( i ) for n periods.
The above calculates what present value ( PV ) would be needed to produce a certain future value ( FV ) if interest ( i ) accrues for n periods.
The above calculates the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods.
The above formula calculates the number of periods required to get FV given the PV and the interest rate ( i ). The log function can be in any base, e.g. natural log (ln)
Compound
Formula for calculating compound interest:
Where,
- P = principal amount (initial investment)
- r = annual nominal interest rate (as a decimal)
- n = number of times the interest is compounded per year
- t = number of years
- A = amount after time t
Example usage: An amount of $1500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years.
A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6:
So, the balance after 6 years is approximately $1,938.84.
Periodic compounding
The amount function for compound interest is an exponential function in terms of time.
- t = Total time in years
- n = Number of compounding periods per year (note that the total number of compounding periods is
)
- r = Nominal annual interest rate expressed as a decimal. e.g.: 6% = 0.06
As n increases, the rate approaches an upper limit of e r . This rate is called continuous compounding , see below.
Since the principal A ( 0 ) is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used in interest theory instead. Accumulation functions for simple and compound interest are listed below:
Note: A ( t ) is the amount function and a ( t ) is the accumulation function.
Continuous compounding
Continuous compounding can be thought as making the compounding period infinitesimally small; therefore achieved by taking the limit of n to infinity. One should consult definitions of the exponential function for the mathematical proof of this limit.
The amount function is simply
The interest rate expressed as a continuously compounded rate is called the force of interest. The annual force of interest is simply 12 times the monthly force of interest.
The effective interest rate per year is
Using this i the amount function can be written as:
or
See also logarithmic or continuously compounded return.
Force of interest
In mathematics, the accumulation functions are often expressed in terms of e , the base of the natural logarithm. This facilitates the use of calculus methods in manipulation of interest formulae.
For any continuously differentiable accumulation function a(t) the force of interest, or more generally the logarithmic or continuously compounded return is a function of time defined as follows:
which is the rate of change with time of the natural logarithm of the accumulation function.
Conversely:
When the above formula is written in differential equation format, the force of interest is simply the coefficient of amoun
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