Simple Capitalization and Financial Interest Laws
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Simple Capitalization
That type of capitalization is characterized by the fact that the interests that occur in each period are not added to the capital to produce new interests in the following period. That is, simple interest capitalizations are not productive.
Properties of Simple Capitalization
- The capital that remains invested in each time period is always the same.
- Therefore, the interests produced in each period are also equal.
Financial Laws
To verify two or more capitals invested at different points in time, one must find their value at a single moment. To keep the funds at the same time, we need to use a financial law. They can be of two types:
- Financial Law of Update: Used to calculate the value of capital at a later time and move it to an earlier time.
- Financial Law of Capitalization: Allows calculating the value of capital at an earlier time and determining its value or moving it to a later time.
Final Capital Calculation
The Final Capital (or Upright) is the sum of the initial capital plus the total interest. The formula is: Cn = Co + It. Replacing the simple capitalization value for It: Cn = Co + Co * n * i. Removing common factors of Co: Cn = Co (1 + n * i).
Calculating the Time Period
The duration of the operation is the number of periods in years. We know that: Cn = Co (1 + n * i). Therefore: Cn / Co = 1 + n * i; Cn / Co - 1 = n * i. Doing the least common multiple in the first term of equality: (Cn - Co) / Co = n * i; (Cn - Co) / (Co * i) = n; n = (Cn - Co) / (Co * i).
Calculating the Interest Rate
The interest rate of the operation is expressed as a percentage. We know that: Cn = Co (1 + n * i). Therefore: Cn / Co = 1 + n * i; Cn / Co - 1 = n * i. Doing the least common multiple in the first member of the equality: (Cn - Co) / Co = n * i; (Cn - Co) / (Co * n) = i; i = (Cn - Co) / (Co * n).
Equivalent Interest Rates
Where the interest type is expressed as a percentage and annual compounding periods relate to other time units, and where n and i are not expressed in the same unit of time, we must establish a match between them. There are two alternatives: adapt the time to the units in which the interest rate is expressed (convert n to i), or adjust the interest rate compounding periods to adapt i to n. We call i the annual interest rate and im the fractional interest rate for periods less than one year.