Compound interest

Compound interest, is interest which is added to the original principal. New interest is then calculated, not only on the principal, but also on the interest that has been added. The more frequently interest is compounded, the faster the principal grows. Yearly compounded interest is considered the norm unless it is specified to be otherwise.

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 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. Albert Einstein is (mis)quoted as saying that "compound interest is the most powerful force in the universe." Instead, Einstein was quoted for saying compounding is "the greatest mathematical discovery of all time."

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.

Mathematics of interest rates

The Rule of 72

A principle discovered by Albert Einstein, the Rule of 72 is a very simple way of illustrating the growth potential of compound interest. The rule says simply this:

${\frac {72}{InterestRate}}$ = the number of years required for the fund to double.

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 $({\frac {72}{12}}=6)$ years. This calculation shows only the gross amount, taxes must be factored into the final amount receivable at the close of the account for taxable and tax deferred accounts (such as stocks, mutual funds, 401(k)'s, IRA's, etc.

http://en.wikipedia.org/wiki/Rule_of_72

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 $n\cdot t$)

• $r$ = 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:

$a(t)=1+tr\,$

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

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 formulas. This is called the force of interest.

The force of interest is defined as the following:

$\delta _{t}={\frac {a'(t)}{a(t)}}\,$

$a(n)=e^{\int _{0}^{n}\delta _{t}\,dt}\,$. 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 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

$A(t)=A_{0}e^{r\cdot t}$

Compounding bases

See Day count convention

To convert an interest rate from one compounding basis to another compounding basis, the following formula applies:

$r_{2}=\left[\left(1+{\frac {r_{1}}{n_{1}}}\right)^{\frac {n_{1}}{n_{2}}}-1\right]{\times }n_{2}$

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:

$R=n{\times }\ln {\left(1+{\frac {r}{n}}\right)}$

where R is the interest rate on a continuous compounding basis and r is the stated interest rate with a compounding frequency n

Another formula to calculate compound interest is $y=a\cdot b^{x}$ (y is equal to a times b to the x power) where y is the money after interest is calculated a is the initial deposit b is the interest rate in a decimal form (divided by the number of compoundings if applicable) x is the exponent of b which represents the time (multiplied by the number of compoundings if applicable)

A Simpler Understanding and Calculation

Disclosure

Interest rates must be comparable in order to be useful. Since most people think of rates as a yearly percentage, many governments require financial institutions to disclose the comparable yearly interest rate on deposits or advances. It can be called variously Annual Percentage Yield, or Annual Equivalent Rate or Effective Annual Rate.

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 also: Day count convention). The total calculation is ((100-P)/P)*((365/t)*100)
• 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).

Translating different compounding periods

Each time unpaid interest is compounded and added to the principal, the resulting principal is now P(1+i%).

A) You are told that 2% interest will be charged every quarter year. 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
You don't need any math to 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 is simpler still. Using the Future Value of a Dollar function, input

• PV = 100
• n = 4
• i = 2
• 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.
$100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104
The math to find the 0.9853% is discussion at Time value of money, but using a financial calculator 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. Input

• PV = 100,000
• n = 4
• FV = 160,000
• solve for interest = 12.47%

See also

• Rate of return on investment
• Credit card interest
• Fisher equation
• Term Structure of Interest Rates

External Links

• Compound Interest Interactive Calculator