Canadian mortgage loans are generally semi-annual compounding with monthly (or more frequent) payments.[1]
- 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 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 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. n represents the number of periods.
These are the most basic formulae:
- Failed to parse (Missing texvc executable; please see math/README to configure.): FV = PV ( 1+i )^n\,
The above calculates the future value of FV of an investment's present value of PV accruing at a fixed interest rate of i for n periods.
- Failed to parse (Missing texvc executable; please see math/README to configure.): PV = \frac {FV} {\left( 1+i \right)^n}\,
The above calculates what present value of PV would be needed to produce a certain future value of FV if interest of i accrues for n periods.
- Failed to parse (Missing texvc executable; please see math/README to configure.): i = \sqrt[n]{\left( \frac {FV} {PV} \right)} -1 \,
- or
- Failed to parse (Missing texvc executable; please see math/README to configure.): i = \left( \frac {FV} {PV} \right)^\left(\frac {1} {n} \right)- 1
The above two formulae are the same and calculate the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods.
- Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac {log(FV) - log(PV)} {log(1 + i)}
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, for e.g. natural log (ln)
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 rate is 8% per year, compounded quarterly. What is the equivalent effective annual rate?
The 8% is a nominal rate. It implies an effective quarterly interest rate of 8%/4 = 2%. Start with $100. At the end of one year it will have accumulated to:
$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.
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt[4]{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).
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): PV = 92.06 \,
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): FV = 1248.29 \,
-
- Failed to parse (Missing texvc executable; please see math/README to configure.): n = 36 (years) \,
-
Answer:
- Failed to parse (Missing texvc executable; please see math/README to configure.): i = \sqrt[36]{\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:
Failed to parse (Missing texvc executable; please see math/README to configure.): \frac {72} {i} \approx n , where Failed to parse (Missing texvc executable; please see math/README to configure.): \ i
is the interest rate in percentage ( i.e, Failed to parse (Missing texvc executable; please see math/README to configure.): \ 100 \ i
) and Failed to parse (Missing texvc executable; please see math/README to configure.): \ 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): n = \frac{\ln 2}{\ln(1+i)}
Periodic compounding
The amount function for compound interest is an exponential function in terms of time.
Failed to parse (Missing texvc executable; please see math/README to configure.): A(t) = A_0 \left(1 + \frac {r} {n}\right) ^ {n \cdot t}
- Failed to parse (Missing texvc executable; please see math/README to configure.): t
= Total time in years
- Failed to parse (Missing texvc executable; please see math/README to configure.): n
= Number of compounding periods per year (note that the total number of compounding periods is Failed to parse (Missing texvc executable; please see math/README to configure.): n \cdot t
)
- Failed to parse (Missing texvc executable; please see math/README to configure.): r
= Nominal annual interest rate expressed as a decimal. e.g.: 6% = 0.06
As Failed to parse (Missing texvc executable; please see math/README to configure.): n
increases, the rate approaches an upper limit of Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): a(t)=1+t r\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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, 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_{t}=\frac{a'(t)}{a(t)}\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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.
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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.
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \delta=\ln(1+r)\,
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): a(t)=P\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 of n to infinity. One should consult definitions of the exponential function for the mathematical proof of this limit.
- Failed to parse (Missing texvc executable; please see math/README to configure.): a(t)=\lim_{n\to\infty}\left(1+\frac{r}{n}\right)^{n \cdot t}
- Failed to parse (Missing texvc executable; please see math/README to configure.): a(t)=e^{r \cdot t}
The amount function is simply
- Failed to parse (Missing texvc executable; please see math/README to configure.): A(t)=A_0 e^{r \cdot t}
A common mnemonic device considers the equation in the form
- Failed to parse (Missing texvc executable; please see math/README to configure.): A = P e^{r \cdot t}
called 'PERT' where P is the principal amount, e 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 convention
To convert an interest rate from one compounding basis to another compounding basis, the following formula applies:
- Failed to parse (Missing texvc executable; please see math/README to configure.): r_2=\left[\left(1+\frac{r_1}{n_1}\right)^\frac{n_1}{n_2}-1\right]{\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 continuously compounded:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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 American tribe that accepted goods worth 60 guilders for the sale of Manhattan in 1626 had invested the money in a Dutch bank at 6.5% interest, compounded annually, then in 2005 their investment would be worth over €700 billion (around USD $1,000 billion), more than the assessed value of the real estate in all five boroughs of New York City. With a 6.0% interest however, the value of their investment today would have been €100 billion (7 times less!).
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. [2]
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.[3][4]
See also
References
- ^ 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.
- ^ This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain.
- ^ Lewin, C G (1970). "An Early Book on Compound Interest - Richard Witt's Arithmeticall Questions". Journal of the Institute of Actuaries 96 (1): 121-132.
- ^ Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries 108 (3): 423-442.
External links
et:Liitintress fr:Intérêts composés is:Vaxtavextir ja:複利 pl:Procent składany ru:Сложный процент sv:Sammansatt ränta
