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Simple interest
Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains unpaid.
The amount of simple interest is calculated according to the following formula:
where r is the period interest rate (I/m), B0 the initial balance and mt the number of time periods elapsed.
To calculate the period interest rate r, one divides the interest rate I by the number of periods mt.
For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple interest rate is 12.99% per annum. The interest added at the end of 3 months would be,
and they would have to pay $2581.19 to pay off the balance at this point.
If instead they make interest-only payments for each of those 3 months at the period rate r, the amount of interest paid would be,
Their balance at the end of 3 months would still be $2500.
In this case, the time value of money is not factored in. The steady payments have an additional cost that needs to be considered when comparing loans. For example, given a $100 principal:
Credit card debt where $1/day is charged: 1/100 = 1%/day = 7%/week = 365%/year.
Corporate bond where the first $3 are due after six months, and the second $3 are due at the year's end: (3+3)/100 = 6%/year.
Certificate of deposit (GIC) where $6 is paid at the year's end: 6/100 = 6%/year.
There are two complications involved when comparing different simple interest bearing offers.
When rates are the same but the periods are different a direct comparison is inaccurate because of the time value of money. Paying $3 every six months costs more than $6 paid at year end so, the 6% bond cannot be 'equated' to the 6% GIC.
When interest is due, but not paid, does it remain 'interest payable', like the bond's $3 payment after six months or, will it be added to the balance due? In the latter case it is no longer simple interest, but compound interest.
A bank account that offers only simple interest, that money can freely be withdrawn from is unlikely, since withdrawing money and immediately depositing it again would be advantageous.
[edit]Composition of interest rates
In economics, interest is considered the price of credit, therefore, it is also subject to distortions due to inflation. The nominal interest rate, which refers to the price before adjustment to inflation, is the one visible to the consumer (i.e., the interest tagged in a loan contract, credit card statement, etc.). Nominal interest is composed of the real interest rate plus inflation, among other factors. A simple formula for the nominal interest is:
Where i is the nominal interest, r is the real interest and is inflation.
This formula attempts to measure the value of the interest in units of stable purchasing power. However, if this statement were true, it would imply at least two misconceptions. First, that all interest rates within an area that shares the same inflation (that is, the same country) should be the same. Second, that the lenders know the inflation for the period of time that they are going to lend the money.
One reason behind the difference between the interest that yields a treasury bond and the interest that yields a mortgage loan is the risk that the lender takes from lending money to an economic agent. In this particular case, a government is more likely to pay than a private citizen. Therefore, the interest rate charged to a private citizen is larger than the rate charged to the government.
To take into account the information asymmetry aforementioned, both the value of inflation and the real price of money are changed to their expected values resulting in the following equation:
Here, is the nominal interest at the time of the loan, is the real interest expected over the period of the loan, is the inflation expected over the period of the loan and is the representative value for the risk engaged in the operation.
[edit]Cumulative interest or return
This section requires expansion. (January 2009)
The calculation for cumulative interest is (FV/PV)-1. It ignores the 'per year' convention and assumes compounding at every payment date. It is usually used to compare two long term opportunities.[citation needed]
[edit]Other conventions and uses
Exceptions:
US and Canadian T-Bills (short term Government debt) have a different calculation for interest. 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). This is equivalent to calculating the price by a process called discounting at a simple interest rate.
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).
Flat Rate Loans and the Rule of .78s: Some consumer loans have been structured as flat rate loans, with the loan outstanding determined by allocating the total interest across the term of the loan by using the "Rule of 78s" or "Sum of digits" method. Seventy-eight is the sum of the numbers 1 through 12, inclusive. The practice enabled quick calculations of interest in the pre-computer days. In a loan with interest calculated per the Rule of 78s, the total interest over the life of the loan is calculated as either simple or compound interest and amounts to the same as either of the above methods. Payments remain constant over the life of the loan; however, payments are allocated to interest in progressively smaller amounts. In a one-year loan, in the first month, 12/78 of all interest owed over the life of the loan is due; in the second month, 11/78; progressing to the twelfth month where only 1/78 of all interest is due. The practical effect of the Rule of 78s is to make early pay-offs of term loans more expensive. For a one year loan, approximately 3/4 of all interest due is collected by the sixth month, and pay-off of the principal then will cause the effective interest rate to be much higher than the APY used to calculate the payments. [9]
In 1992, the United States outlawed the use of "Rule of 78s" interest in connection with mortgage refinancing and other consumer loans over five years in term.[10] Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans.[9]
Rule of 72: The "Rule of 72" is a "quick and dirty" method for finding out how fast money doubles for a given interest rate. For example, if you have an interest rate of 6%, it will take 72/6 or 12 years for your money to double, compounding at 6%. This is an approximation that starts to break down above 10%.
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