Compound Interest

The sum of all interest payments made on a loan over a certain time period.

Cumulative interest is sometimes used as a measure of loan economics to determine which loan is most economical. However, cumulative interest alone does not account for other important factors such as initial loan costs (if those costs are paid out of pocket as opposed to being rolled into the loan’s balance) and the time value of money (different rates of interest over the life of different loans can lead to savings early in the life of one loan that can offset higher interest rates later in the life of that loan).

Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding. A bank account, for example, may have its interest compounded every year: in this case, an account with $1000 initial principal and 20% interest per year would have a balance of $1200 at the end of the first year, $1440 at the end of the second year, and so on.

In order to define an interest rate fully, and enable one to compare it with other interest rates, the interest rate and the compounding frequency must be disclosed. Since most people prefer to think of rates as a yearly percentage, many governments require financial institutions to disclose the equivalent yearly compounded interest rate on deposits or advances. For instance the yearly rate for a loan with 1% interest per month is approximately 12. 68% per annum (1.01^12). This equivalent yearly rate may be referred to as annual percentage rate (APR), annual equivalent rate (AER), annual percentage yield, 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 compare the actual costs of borrowing more easily.

For any given interest rate and compounding frequency, an “equivalent” rate for any 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 is standard 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, divided by the amount of the principal. The periodic rate is used primarily for calculations, and is rarely used for comparison. The nominal annual rate or nominal interest rate is defined as the periodic rate multiplied by the number of compounding periods per year. For example, a monthly rate of 1% is equivalent to an annual nominal interest of 12%.

Effective annual rate: this reflects the effective rate as if annual compounding were applied: in other words it is the total accumulated interest that would be payable up to the end of one year, divided by the principal.

Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may however be the one quoted instead. When quoted together 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 the nominal rate cannot be directly compared with loans that have a 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 or marketing demands, for simplicity or for other reasons.
Exceptions

* US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated as (100 ? P)/Pbnm, 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)