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# Compound Interest

Compound Interest can be used to determine the present value of a future amount, this is called discounting. Compound interest can also be used to determine the future value of a current amount. The compound interest calculator below can be used to determine future value, present value, the period interest rate, and the number of compounding periods.

### Compound Interest Definition

Compound Interest is the interest generated on a principal amount that compounds, that is that interest in one period will be added to principal and interest in the next period will be generated on the now increased principal amount.

### Variables

FV=Future value of the principal after compound interest has been applied
PV=Present value of the principal before compound interest has been applied
K=The interest rate charged for a compounding period
N=The number of compounding periods
I=The amount of interest charged to the principal over the investment time frame

### Compound Interest Formula     ### Compound Interest

Compound interest is a way of accumulating interest on principal.  When money is borrowed, the borrower is usually required to pay the supplier of the funds a rate of interest until the principal has been repaid.  This is a simply concept to understand, but can have many complex ramifications, depending on how the interest is being calculated.

Compounding interest refers to way in which interest is calculated in subsequent periods.  When interest is charged for a period of time that the money is borrowed, such as a year, this interest is then added to the original principal, and the interest for the next year is charged on the new, larger principal amount.

Example:  Borrow \$1000 for two years, at 10% interest compounded annually (at the end of each year).  At the end of the first year, interest of \$100 is charged and added to the original \$1000, increasing the principal to \$1100 (\$1000 + \$100).  At the end of the second year, the interest charged is \$110 (10% of \$1100).  This increases the total principal to \$1210 (\$110 + \$1100).  If this continues, then the amount of interest charged every year will increase at an exponential rate.

The frequency of compounding refers to how often the interest charged is compounded.  As the frequency of compounding the interest is increased, so is the actual amount of interest charged.  This is because the amount of time that the interest is charged against the increased principal increases, since the interest is added to the principal quicker.  To calculate compound interest, the interest rate is divided by the amount of compounding periods in a year, and this portion of the interest rate is applied at each compounding interval.

Example:  Borrow \$1000 for two years, at 10% interest compounded semi-annually (twice a year).  At the end of 6 months, interest of \$55 is charged and added to the original \$1000, increasing the principal to \$1050 (\$1000 + \$50).  At the end of the first year, the interest charged is \$52.50 (5% of \$1050).  This amount is added to the principal, increasing it to \$1102.50 (\$52.50 + \$1050).  By continuing this for each compounding period, at the end of two years, the principal amount is now \$1215.51.  This amount is greater than the \$1210 accumulated from compounding the interest annually at the same annual interest rate.

As you can see increasing the frequency of compounding will have the effect of increasing the effective rate of interest paid on the loan, and therefore the amount of interest actually paid.  When making financial decisions, it is important to understand the rate of interest charged, as well as the compounding, as these can create large differences over long periods of time.

Governments often require financial institutions to disclose this information directly to its customers.  Often you will see a specific rate of interest declared as the effective annual rate (EAR), or the annual percentage rate (APR), or some derivative of the term.  A 12% annual percentage rate that is compounded monthly, will have an APR of 12.6825%, a higher amount than the rate of interest that is actually charged to the principal at each compounding frequency.

Performing the compound interest calculations is simple when only compounded a few times, but to determine the value of an amount 30 years in the future which compounds monthly would be exceedingly tedious.  Fortunately, we can use a compound interest formula, or a compound interest calculator to perform the compound interest calculations for us.

It is important to differentiate compound interest from simple interest.  Simple interest is interest that is charged on a principal amount, like compound interest.  The difference is that once simple interest is charged, it is not added to the principal to be charged further interest.  Many bonds work this way.  Interest charged on a bond is often paid as cash, and can only be compounded if the investor who receives the cash payment of the interest finds another investment to invest in.

When compound interest is charged to principal, on the compounding frequency, the rate applied to the principal is called the period rate, or periodic rate of interest.  To determine the period interest rate, simply take the annual rate of interest, and divide it by the number of compounding frequencies in a year.  For example: if 12% interest is compounded quarterly (4 times a year), then the period interest rate is 3% (12% 4).

In order to compare the cost of one financial product to another, it is important to use the effective annual rate.  This will neutralize the difference that compounding frequency will have on the borrowed funds, and provide an apples to apples comparison of cost to the borrower.  Using a powerful loan calculator or mortgage calculator such as the ones provided in this site will help borrowers to understand the actual cost of interest.

The compound interest calculator above on this page will provide you with the answer to many compounding interest calculations.  Each variable of the formula is isolated, and defined.  Each compounding interest formula is also provided.  When using the compound interest calculator it is important to remember to use the period interest rate, which can differ from the annual interest rate.  Divide the annual interest rate by the number of compounding periods in a year to determine the period interest rate.  You must also input the number of periods that interest will be compounded for.  If monthly interest is calculated for 3 years, then compound interest will need to be calculated for 36 periods, at the period interest rate.

Compound interest is also used to determine the net present value of a financial asset from a different period of time.  The calculator above serves as a net present value calculator.  For instance, if a \$1000 is to be the value of something 10 years from now, and the interest rate is 6%.  Then it is possible to determine the net present value of this item by using the present value calculator above.  The net present value will be \$558.39.

Net present value is an important concept to understand.  This is due to the time value of money.  When offered the choice between equal amounts of money, say \$1000, at different points in time, immediately or 10 years from now, the amount of money today has more value than the same amount in the future, and so it will always be chosen.  In order to accept the money in the future, we need to be compensated for waiting, and the amount must be increased.  Once we understand that there is a value to the time we have access to money, we can determine the present value of amounts in the future.

The longer we have to wait for an amount in the future, the greater the amount that we need to be compensated.  When calculating present value of money in the future, we use the compound interest calculation.  This is used because it best represents the value and use we have for money and the cost to us if we do not have access to it.  We can think of the rate charged for the use of money over time as the price of money.

The concept of time value of money can be applied to many various forms, such as annuities, perpetuities, loans, investments, etc.  When calculating present value of a single amount, the present value formula is the same as the compound interest formula.  Therefore, the present value calculator is the same as the compound interest calculator above.  By understanding the concept of present value calculation, we can determine an interest rate that can be used to compare against other uses of money, such as loans and investments.  This way we can make qualified decisions about which financial option is best to serve our purposes.