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Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (i.e. interest is compounded).
Interest rates must be comparable in order to be useful, and in order to be comparable, the interest rate
and the compounding frequency must be disclosed. Since most people think of rates as a yearly percentage, many governments require financial institutions to disclose a (notionally) comparable yearly interest rate on deposits or advances. Compound interest rates may be referred to as
Annual percentage rate,
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 more easily compare the actual cost of borrowing.
Compound interest rates may be converted to allow for comparison: for any given interest rate and compounding frequency, an "equivalent" rate for a 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 predominates 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. The periodic rate is used primarily for calculations, and is rarely used for comparison. The periodic rate is defined as the annual nominal rate divided by the number of compounding periods per year.
Nominal interest rate or nominal annual rate: the annual rate, unadjusted for compounding. For example, 12% annual nominal interest compounded monthly has a periodic (monthly) rate of 1%.
Effective annual rate: the nominal annual rate "adjusted" to allow comparisons; the nominal rate is restated to reflect the effective rate as if annual compounding were applied.
Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may be the most frequently used. When quoted 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 cannot be compared to loans with 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, marketing demands, 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)/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 day count convention).
- Corporate Bonds are most frequently 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.http://laws.justice.gc.ca/en/showdoc/cs/I-15/bo-ga:s_6//en#anchorbo-ga:s_6 Interest Act (Canada), Department of Justice. The Interest Act specifies that interest is not recoverable unless the mortgage loan contains a statement showing the rate of interest chargeable, "calculated yearly or half-yearly, not in advance." In practice, banks use the half-yearly rate.
- U.S. mortgages generally use monthly compounding (with corresponding payment periods).
- Certain techniques for, e.g., valuation of derivatives may use continuous compounding, which is the Limit of a function as the compounding period approaches zero. 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.
Mathematics of interest rates
Simple Formulae
Formulae are presented in greater detail at
time value of money.
In the formulae below,
i or
r are the interest rate, expressed as a true percentage (i.e. 10% = 10/100 = 0.10). FV and PV represent the future and present value of a sum.
These are the most basic formulae required by a new student:
FV = PV ( 1+i )^n\,
The above calculates the future value, (FV), of an investment, (PV), accruing at a fixed interest rate of i for n periods. Also a=p(1+r/t)^nt can be used.
PV = \frac {FV} {\left( 1+i \right)^n}\,
The above calculates what present value (PV) would be needed to produce a certain future value (FV) if interest of i accrues for n periods.
i = \left(\sqrt{\left( \frac {FV} {PV} \right)} -1\right) \, or i = \left(\left( \frac {FV} {PV} \right)^\left(\frac {1} {n} \right)- 1\right),
The above two formulae are the same and calculate the compound interest rate i achieved if an initial investment of PV returns a value of FV after n accrual periods.
Translating different compounding periods
Each time unpaid interest is compounded and added to the principal, the resulting principal is grossed up to equal P(1+i%).
A) You are told the interest is 8% a year, with 2%(=8/4) interest charged every quarter. What is the equivalent annual rate?. Start with $100. At the end of one year it will be:$100 (1+ .02) (1+ .02) (1+ .02) (1+ .02) = $108.24 We know that $100 invested at 8.24% will give you $108.24 at year end. So the equivalent rate is 8.24%. Using a financial calculator or a table is simpler still. Using the Future Value of a currency function, input
- PV = 100
- n = 4
- i = .02
- solve for FV = 108.24
B) You know the equivalent annual interest rate is 4%, but it will be compounded quarterly. You need to find the interest rate that will be applied each quarter.
\sqrt{1+.04}-1 = .00985341
$100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104 The mathematics to find the 0.9853% is discussed at Time value of money, but using a financial calculator or table is easier. Input
- PV = 100
- n = 4
- FV = 104
- solve for interest = 0.9853%
C) You sold your house for a 60% profit. What was the annual return? You owned the house for 4 years, paid $100,000 originally, and sold it for $160,000. $100,000 (1+ .1247) (1+ .1247) (1+ .1247) (1+ .1247) = $160,000 Find the 12.47% annual rate the same way as B.) above, using a financial calculator or table. Input
- PV = 100,000
- n = 4
- FV = 160,000
- solve for interest = 12.47%
Example question:
In January 1970 the
S&P 500 index stood at 92.06 and in January 2006 the index stood at 1248.29.What has been the annual rate of return achieved? (ignoring dividends).
PV = 92.06 \,
FV = 1248.29 \,
n = 36 (years) \,
Answer:
i = \sqrt{\left( \frac {1248.29} {92.06} \right)} -1 = 7.51% \,
The Rule of 72
The Rule of 72 is a very simple way of illustrating the growth potential of compound interest. The rule says simply this:
\frac {72} {i} \approx n, where \ i is the interest rate in percentage ( i.e, \ 100 \ i ) and \ n is the number of time periods needed to double the principal.
For example, say a mutual fund grows at 12% average interest rate. According to the rule of 72, if money were invested in this mutual fund, then it would double every 6 \left(\frac {72} {12}=6\right) years. This calculation deals only with the gross amount, taxes must be factored into growth if taxable vehicles (such as CDs, mutual funds, etc) are used.
However, the above
Rule of 72 merely gives an approximation of the time needed to retain an investment before it doubles in value. The accurate calculation is as follows:
n = \frac{\ln 2}{\ln(1+i)}
Periodic compounding
The amount function for compound interest is an exponential function in terms of time.
A(t) = A_0 \left(1 + \frac {r} {n}\right) ^ {n \cdot t}
- n = Number of compounding periods per each t (time in years) (note that the total number of compounding periods is n \cdot t )
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 interest and compound interest are listed below:
a(t)=1+t r\,
a(t) = \left(1 + \frac {r} {n}\right) ^ {n \cdot t}
Note:
A(
t) is the amount function and
a(
t) is the accumulation function.
Force of interest
In mathematics, the accumulation functions 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 formulae. 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}\,. Note that this equation contains an ERROR given the previous equation. The below is a deemed correction.
a(n)=a(0)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
r, and the accumulation function of compounding interest in terms of force of interest is a simple power of e:
\delta=\ln(1+r)\,
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{r}{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{r}{n}\right)^{n \cdot t}
a(t)=e^{r \cdot t}
The amount function is simply
A(t)=A_0 e^{r \cdot t}
A common mnemonic device considers the equation in the form
A = P e^{r \cdot t}
called 'PERT' where P is the principal amount, E (mathematical constant) is the base of the natural log, R is the rate per period, and T is the time (in the same units as the rate's period), and A is the final amount.
Compounding bases
See Day count conventionTo convert an interest rate from one compounding basis to another compounding basis, the following formula applies:
r_2=\left{\times}n_2
where
r1 is the stated interest rate with compounding frequency
n1 and
r2 is the stated interest rate with compounding frequency
n2.
When interest is
#Continuous compounding:
R=n{\times}\ln{\left(1+\frac{r}{n}\right)}
where
R is the interest rate on a continuous compounding basis and
r is the stated interest rate with a compounding frequency
n.
History
If the
Native Americans in the United States tribe that accepted goods worth 60
guilders for the sale of Manhattan in
1626 had invested the money in a
Netherlands bank at 6.5% interest, compounded annually, then in 2005 their investment would be worth over €700 billion (around US$820 billion), more than the assessed value of the real estate in all five boroughs of
New York City.
Compound interest was once regarded as the worst kind of
usury, and was severely condemned by
Roman law, as well as the
common laws of many other countries.
Richard Witt's book
Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called
anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.
See also
References
Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (i.e. interest is compounded).
Interest rates must be comparable in order to be useful, and in order to be comparable, the interest rate
and the compounding frequency must be disclosed. Since most people think of rates as a yearly percentage, many governments require financial institutions to disclose a (notionally) comparable yearly interest rate on deposits or advances. Compound interest rates may be referred to as
Annual percentage rate,
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 more easily compare the actual cost of borrowing.
Compound interest rates may be converted to allow for comparison: for any given interest rate and compounding frequency, an "equivalent" rate for a 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 predominates 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. The periodic rate is used primarily for calculations, and is rarely used for comparison. The periodic rate is defined as the annual nominal rate divided by the number of compounding periods per year.
Nominal interest rate or nominal annual rate: the annual rate, unadjusted for compounding. For example, 12% annual nominal interest compounded monthly has a periodic (monthly) rate of 1%.
Effective annual rate: the nominal annual rate "adjusted" to allow comparisons; the nominal rate is restated to reflect the effective rate as if annual compounding were applied.
Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may be the most frequently used. When quoted 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 cannot be compared to loans with 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, marketing demands, 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)/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 day count convention).
- Corporate Bonds are most frequently 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.http://laws.justice.gc.ca/en/showdoc/cs/I-15/bo-ga:s_6//en#anchorbo-ga:s_6 Interest Act (Canada), Department of Justice. The Interest Act specifies that interest is not recoverable unless the mortgage loan contains a statement showing the rate of interest chargeable, "calculated yearly or half-yearly, not in advance." In practice, banks use the half-yearly rate.
- U.S. mortgages generally use monthly compounding (with corresponding payment periods).
- Certain techniques for, e.g., valuation of derivatives may use continuous compounding, which is the Limit of a function as the compounding period approaches zero. 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.
Mathematics of interest rates
Simple Formulae
Formulae are presented in greater detail at time value of money.
In the formulae below,
i or
r are the interest rate, expressed as a true percentage (i.e. 10% = 10/100 = 0.10). FV and PV represent the future and present value of a sum.
These are the most basic formulae required by a new student:
FV = PV ( 1+i )^n\,
The above calculates the future value, (FV), of an investment, (PV), accruing at a fixed interest rate of i for n periods. Also a=p(1+r/t)^nt can be used.
PV = \frac {FV} {\left( 1+i \right)^n}\,
The above calculates what present value (PV) would be needed to produce a certain future value (FV) if interest of i accrues for n periods.
i = \left(\sqrt{\left( \frac {FV} {PV} \right)} -1\right) \, or i = \left(\left( \frac {FV} {PV} \right)^\left(\frac {1} {n} \right)- 1\right),
The above two formulae are the same and calculate the compound interest rate i achieved if an initial investment of PV returns a value of FV after n accrual periods.
Translating different compounding periods
Each time unpaid interest is compounded and added to the principal, the resulting principal is grossed up to equal P(1+i%).
A) You are told the interest is 8% a year, with 2%(=8/4) interest charged every quarter. What is the equivalent annual rate?. Start with $100. At the end of one year it will be:$100 (1+ .02) (1+ .02) (1+ .02) (1+ .02) = $108.24 We know that $100 invested at 8.24% will give you $108.24 at year end. So the equivalent rate is 8.24%. Using a financial calculator or a table is simpler still. Using the Future Value of a currency function, input
- PV = 100
- n = 4
- i = .02
- solve for FV = 108.24
B) You know the equivalent annual interest rate is 4%, but it will be compounded quarterly. You need to find the interest rate that will be applied each quarter.
\sqrt{1+.04}-1 = .00985341
$100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104 The mathematics to find the 0.9853% is discussed at Time value of money, but using a financial calculator or table is easier. Input
- PV = 100
- n = 4
- FV = 104
- solve for interest = 0.9853%
C) You sold your house for a 60% profit. What was the annual return? You owned the house for 4 years, paid $100,000 originally, and sold it for $160,000. $100,000 (1+ .1247) (1+ .1247) (1+ .1247) (1+ .1247) = $160,000 Find the 12.47% annual rate the same way as B.) above, using a financial calculator or table. Input
- PV = 100,000
- n = 4
- FV = 160,000
- solve for interest = 12.47%
Example question:
In January 1970 the
S&P 500 index stood at 92.06 and in January 2006 the index stood at 1248.29.What has been the annual rate of return achieved? (ignoring dividends).
PV = 92.06 \,
FV = 1248.29 \,
n = 36 (years) \,
Answer:
i = \sqrt{\left( \frac {1248.29} {92.06} \right)} -1 = 7.51% \,
The Rule of 72
The
Rule of 72 is a very simple way of illustrating the growth potential of compound interest. The rule says simply this:
\frac {72} {i} \approx n, where \ i is the interest rate in percentage ( i.e, \ 100 \ i ) and \ n is the number of time periods needed to double the principal.
For example, say a mutual fund grows at 12% average interest rate. According to the rule of 72, if money were invested in this mutual fund, then it would double every 6 \left(\frac {72} {12}=6\right) years. This calculation deals only with the gross amount, taxes must be factored into growth if taxable vehicles (such as CDs, mutual funds, etc) are used.
However, the above
Rule of 72 merely gives an approximation of the time needed to retain an investment before it doubles in value. The accurate calculation is as follows:
n = \frac{\ln 2}{\ln(1+i)}
Periodic compounding
The amount function for compound interest is an exponential function in terms of time.
A(t) = A_0 \left(1 + \frac {r} {n}\right) ^ {n \cdot t}
- n = Number of compounding periods per each t (time in years) (note that the total number of compounding periods is n \cdot t )
- r = Nominal 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 interest and compound interest are listed below:
a(t)=1+t r\,
a(t) = \left(1 + \frac {r} {n}\right) ^ {n \cdot t}
Note:
A(
t) is the amount function and
a(
t) is the accumulation function.
Force of interest
In mathematics, the accumulation functions 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 formulae. 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}\,. Note that this equation contains an ERROR given the previous equation. The below is a deemed correction.
a(n)=a(0)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
r, and the accumulation function of compounding interest in terms of force of interest is a simple power of e:
\delta=\ln(1+r)\,
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{r}{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{r}{n}\right)^{n \cdot t}
a(t)=e^{r \cdot t}
The amount function is simply
A(t)=A_0 e^{r \cdot t}
A common mnemonic device considers the equation in the form
A = P e^{r \cdot t}
called 'PERT' where P is the principal amount,
E (mathematical constant) is the base of the natural log, R is the rate per period, and T is the time (in the same units as the rate's period), and A is the final amount.
Compounding bases
See Day count conventionTo convert an interest rate from one compounding basis to another compounding basis, the following formula applies:
r_2=\left{\times}n_2
where
r1 is the stated interest rate with compounding frequency
n1 and
r2 is the stated interest rate with compounding frequency
n2.
When interest is
#Continuous compounding:
R=n{\times}\ln{\left(1+\frac{r}{n}\right)}
where
R is the interest rate on a continuous compounding basis and
r is the stated interest rate with a compounding frequency
n.
History
If the
Native Americans in the United States tribe that accepted goods worth 60 guilders for the sale of Manhattan in
1626 had invested the money in a
Netherlands bank at 6.5% interest, compounded annually, then in 2005 their investment would be worth over €700 billion (around US$820 billion), more than the assessed value of the real estate in all five boroughs of New York City.
Compound interest was once regarded as the worst kind of usury, and was severely condemned by Roman law, as well as the common laws of many other countries.
Richard Witt's book
Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called
anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.
See also
References
