Advanced Wealth Builders
Compound Interest
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). A loan, for example, may have its interest compounded every month: in this case, a loan with $1000 principal and 1% interest per month would have a balance of $1010 at the end of the first month.

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).

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.


    * 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.[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 formula 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: 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.
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.

The above two formula 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.
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.

    $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%

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:
is where is the interest rate in percentage ( i.e, ) and 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 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:

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 er. 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.

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:

. Note that this equation contains an ERROR given the previous equation. The below is a deemed correction.

When the above formula is written in differential equation format, the force of interest is simply the coefficient of amount of change.
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:

Continuous compounding

For interest compounded a certain number of times, n, per year, such as monthly or quarterly, the formula is:  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.  The amount function is simply

A common mnemonic device considers the equation in the form 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:
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:
where R is the interest rate on a continuous compounding basis and r is the stated interest rate with a compounding frequency n.


   1. ^ 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.

   2. ^ This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain.

   3. ^ 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.?

   4. ^ Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries 108 (3): 423-442.?   

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How Does Compound Interest Work?
If you are making any kind of investments and if your return is added to your investment, you are benefiting from compound interest. Compound interest is a way to increase your money at a faster rate.

How does compound interest work? It works like simple math. Imagine you have a bank deposit with 6% interest compounded yearly.
You’ve got 10,000 deposited. At the end of the year, you’ll still have your deposit of 10,000 plus interest of $600 (10,000 x 6%). That’s a new deposit balance of $10,600.
At the end of the second year, you’ll have your deposit balance of $10,600 plus interest of $636. That’s a new deposit balance of $11,236. You notice that the amount of interest saw an increase of $36 from the compounding effect.

Compound interest is not only compounded yearly, but it can be compounded on a quarterly basis, a monthly basis or even a daily basis.
The shorter the time period for the compounding basis, the better the terms are for you as a depositor. Compound interest helps your money grow faster.

The idea of compounding doesn’t just apply to bank deposits, it can apply to any kind of investment where the return is applied to the investment. It can apply to mutual fund investments, dividend reinvestment programs, and even real estate investments.
Now that you know how compound interest works, consider one of the more interesting peculiarities of compound interest: The rule of 72.

It works like this: When anything is compounded on a yearly basis, when the added together interest rates equal 72, then the amount is doubled. For example, if you earn 6% per year then in 12 years (6 X 12 = 72), the amount will double.
That means our mythical deposit of $10,000 at 6% interest will become $20,000 in twelve years. If the deposit is getting 8% then in 9 years it will double.

Compound interest can work for you if you allow it to. Compound interest helps your money grow faster than you think is possible. Make sure that any money you invest is compounded at the lowest time frame possible to increase your returns.

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