### End of Period, Discrete Compound Interest Model

Traditionally the markets where money has been borrowed and loaned have charged a fee based on time. The fee is usually charged on the amount borrowed, i.e. owed at the start of an accounting period. When the loan extends beyond the end of the first accounting period the amount owed includes the borrowed amount plus the fee or interest for the first period less any payments made during the period. This produces a net amount owed for the next period.

At the end of the next period the amount of the interest is calculated on the net amount owed. This process continues with each period considered as a seperate entity. As the number of periods progresses the process is repeated and a new amount owed is calculated at the end of each accounting period. Thus the interest is said to be compounded.

The usual variable names are:

• i = interest rate as fraction (or percentage) of the amount owed.
• n = number of accounting periods.
• 0 = the Present time, (arbitrary time = 0) or when the process starts.
• P = Present Value, i.e. value at time 0
• F = Future Value, i.e. value at some time n.
• A = Uniform series of paments at times 1, 2, 3,...,n

The amount owed (F) at some time n: Amount of loan (P) multiplied by (1 + i) to nth power, or:
F = P x (1 + i) n

The end of period compound interest model is a method of computing incremental change in a base quantity. It assumes no change during an interval and makes the change at the end of the interval. It is a convenient method of book keeping because it ignores positive or negative effects during the period. It makes net adjustments either at the beginning or end of the period.

A continuous exponential model ern will produce the effect of continuous compounding or discounting. This type of model can be calibrated to produce exactly the same effect as the discrete compound interest model. It however is not the traditional way the financial institutions operate.